Handbook of Teichmüller Theory, Volume VII. 9783037192030, 3037192038

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IRMA Lectures in Mathematics and Theoretical Physics 30

This volume is the seventh in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into three parts.

The second part consists of three historico-geometrical articles on Tissot (a precursor of the theory of quasiconfomal mappings), Grötzsch and Lavrentieff, the two main founders of the modern theory of quasiconformal mappings. The third part comprises English translations of five papers by Grötzsch, a paper by Lavrentieff, and three papers by Teichmüller. These nine papers are foundational essays on the theories of conformal invariants and quasiconformal mappings, with applications to conformal geometry, to the type problem and to Nevanlinna’s theory. The papers are followed by commentaries that highlight the relations between them and between later works on the subject. These papers are not only historical documents; they constitute an invaluable source of ideas for current research in Teichmüller theory.

ISBN 978-3-03719-203-0

ems.press / ems-ph.org

Papadopoulos VII | IRMA 30 | FONT: Rotis Sans | Farben: Pantone 287, Pantone 116 | 170 x 240 mm | RB: 37 mm

Volume VII

The first part contains surveys on various topics in Teichmüller theory, including the complex structure of Teichmüller space, the Deligne–Mumford compactification of the moduli space, holomorphic quadratic differentials, Kleinian groups, hyperbolic 3-manifolds and the ending lamination theorem, the universal Teichmüller space, barycentric extensions of maps of the circle, and the theory of Higgs bundles.

Handbook of Teichmüller Theory

Volume VII Athanase Papadopoulos, Editor

Athanase Papadopoulos, Editor

Handbook of Teichmüller Theory

Handbook of Teichmüller Theory Volume VII Athanase Papadopoulos Editor

IRMA Lectures in Mathematics and Theoretical Physics 30 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. For a complete listing see our homepage at www.ems-ph.org. 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) 23 Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Lizhen Ji and Athanase Papadopoulos (Eds.) 24 Free Loop Spaces in Geometry and Topology, Janko Latschev and Alexandru Oancea (Eds.) 25 Takashi Shioya, Metric Measure Geometry. Gromov‘s Theory of Convergence and Concentration of Metrics and Measures 26 Handbook of Teichmüller Theory, Volume V, Athanase Papadopoulos (Ed.) 27 Handbook of Teichmüller Theory, Volume VI, Athanase Papadopoulos (Ed.) 28 Yann Bugeaud, Linear Forms in Logarithms and Applications 29 Eighteen Essays in Non-Euclidean Geometry, Vincent Alberge and Athanase Papadopoulos (Eds.)

Handbook of Teichmüller Theory Volume VII Athanase Papadopoulos Editor

Editor: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

2010 Mathematics Subject Classification: 30F60, 32G15, 30C20, 14H60, 30C35, 30C62, 30C70, 30C75, 37F30, 57M50 01A60, 01A55, 20F65, 20F67, 22E40, 30D30, 30D35, 30F45, 37F30, 53A30, 57M50 Key words: Riemann surface, Teichmüller space, Deligne–Mumford compactification, universal Teichmüller space, complex geodesic, holomorphic differential, quadratic differential, projective structure, Mostow rigidity, hyperbolic structure, Fuchsian group, quasi-Fuchsian group, Kleinian group, ending lamination, Higgs bundle, higher Teichmüller theory, Douady–Earle extension, quasisymmetric map, quasiconformal mapping, type problem, conformal invariant, extremal length, extremal domain, Tissot indicatrix, almost analytic function, measurable Riemann Mapping Theorem, value distribution, Modulsatz, reduced module, line complex, Speiser tree

ISBN 978-3-03719-203-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

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Typeset using the authors’ TEX files: Marco Zunino, Savona, Italy Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Foreword Eighty years have passed since Teichmüller published his first foundational papers which led to what became known as Teichmüller theory, and the field is still wide open to new concepts, ideas and problems, stirring up a continuous flow of people. The amount of work done in this area is impressive, especially in the last forty years, driven by ideas of Thurston whose work set Teichmüller space as a central object of investigation in low-dimensional geometry and topology. The variety of the ideas involved in Teichmüller theory and the relations these ideas make between different mathematical fields are a pledge for the lastingness of this subject. Our motivation for editing this handbook is the conviction that the mathematical community is in need of good surveys on important topics, especially in very active fields like Teichmüller theory. Reading the foundational texts of a theory is a major source for new ideas. Thus, together with surveys on modern research, the present Handbook contains a number of English translations of important original texts with commentaries. These texts will become in this way easily accessible to the mathematical community. I would like to take this opportunity to thank Mrs. Renate Grötzsch for correspondence regarding her father’s works, Vincent Alberge for his valuable help concerning the present volume, in particular in the choice of the papers by Grötzsch that are translated, and Marco Zunino for his excellent work in the typesetting of the final version of this book. Saying that the success of such a book depends on the energy, time and care spent by the contributing authors is stating the obvious. Each of these authors has written an invaluable chapter, accepting without hesitation my (sometimes heavy) editorial remarks, and I would like to thank them all here. Athanase Papadopoulos Strasbourg and Saint Petersburg, May 2019

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Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction to Teichmüller theory, old and new, VII by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part A. Surveys Chapter 1. The Deligne–Mumford compactification and crystallographic groups by Yukio Matsumoto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Complex geometry of Teichmüller domains by Subhojoy Gupta and Harish Seshadri . . . . . . . . . . . . . . . . . .

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Chapter 3. Holomorphic quadratic differentials in Teichmüller theory by Subhojoy Gupta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Mostow strong rigidity of locally symmetric spaces revisited by Lizhen Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Models of ends of hyperbolic 3-manifolds. A survey by Mahan Mj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Universal Teichmüller spaceas a non-trivial example of infinite-dimensional complex manifolds by Armen Sergeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7. Generalized conformal barycentric extensions of circle maps by Jun Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 8. Higgs bundles and higher Teichmüller spaces by Oscar García-Prada . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part B. Essays on the early works on quasiconformal mappings Chapter 9. A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 10. Memories of Herbert Grötzsch by Reiner Kühnau (transl. by Annette A’Campo-Neuen) . . . . . . . . . . .

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Chapter 11. A note about Mikhaïl Lavrentieff and his world of analysis in the Soviet Union by Athanase Papadopoulos (with an appendix by Galina Sinkevich) . . . .

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Part C. Sources Chapter 12. A letter by Oswald Teichmüller (transl. by Annette A’Campo-Neuen) . . . . . . . .

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Chapter 13. On some extremal problems of the conformal mapping by Herbert Grötzsch (transl. by Annette A’Campo-Neuen) . . . . . . . . . .

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Chapter 14. On some extremal problems of the conformal mapping II by Herbert Grötzsch (transl. by Melkana Brakalova-Trevithick) . . . . . . .

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Chapter 15. On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem by Herbert Grötzsch (transl. by Melkana Brakalova-Trevithick) . . . . . . .

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Chapter 16. On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions by Herbert Grötzsch (transl. by Manfred Karbe) . . . . . . . . . . . . . .

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Chapter 17. On closest-to-conformal mappings by Herbert Grötzsch (transl. by Melkana Brakalova-Trevithick) . . . . . . .

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Chapter 18. On five papers by Herbert Grötzsch by Vincent Alberge and Athanase Papadopoulos . . . . . . . . . . . . . . .

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Chapter 19. On a class of continuous representations by Oswald Teichmüller (transl. by V. Alberge and A. Papadopoulos) . . . .

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Chapter 20. A commentary on Lavrentieff’s paper “Sur une classe de représentations continues” by Vincent Alberge and Athanase Papadopoulos . . . . . . . . . . . . . . .

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Contents

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Chapter 21. An application of quasiconformal mappings to the type problem by Oswald Teichmüller (transl. by Melkana Brakalova-Trevithick) . . . . .

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Chapter 22. Investigations on conformal and quasiconformal mappings by Oswald Teichmüller (transl. by M. Brakalova-Trevithick and M. Weiss) .

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Chapter 23. Simple examples for value distribution by Oswald Teichmüller (transl. by Annette A’Campo-Neuen) . . . . . . . .

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Chapter 24. Teichmüller’s work on the type problem by Vincent Alberge, Melkana Brakalova-Trevithick, and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 25. A Commentary on Teichmüller’s paper “Untersuchungen über konforme und quasikonforme Abbildungen” by Vincent Alberge, Melkana Brakalova-Trevithick, and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 26. Value distribution theory and Teichmüller’s paper “Einfache Beispiele zur Wertverteilungslehre” by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction to Teichmüller theory, old and new, VII Athanase Papadopoulos

Teichmüller theory originates in the study of moduli of complex structures on topological surfaces, but the expression denotes now a much broader field which encompasses the study of various kinds of geometric structures on surfaces and— from a more algebraic point of view—that of representations of fundamental groups of surfaces into diverse Lie groups. This theory has wide applications and connections with other fields including complex analysis, low-dimensional topology, Kleinian groups, Finsler geometry, geometric group theory, mathematical physics and others. All these interactions make Teichmüller theory a vivid illustration of the unity of mathematics. The first volume of the Handbook of Teichmüller theory appeared twelve years ago, and since that time, the subject underwent a significant evolution. New researchers came into the field, new important results were obtained, and additional surveys of the new developments and on their connections with the classical material are needed. The present volume (the seventh in the series) is divided into three parts, whose outline is the following. Part A, consisting of chapters 1–8, is a collection of surveys covering various aspects of Teichmüller theory. The topics include a new point of view on the Deligne–Mumford compactification of the moduli space of Riemann surfaces, an exposition of the theory of biholomorphic embeddings of Teichmüller spaces into finite-dimensional complex Euclidean spaces and a review of holomorphic k differentials in relation with various subjects: the complex structure of Teichmüller space, the space of singular flat structures, the theory of harmonic maps, and projective structures on surfaces. The volume continues with surveys of Mostow’s strong rigidity, of the theory of ends of 3-manifolds, of universal Teichmüller space, of conformal barycentric extensions of maps of the circle, and of Higgs bundles in higher Teichmüller theory. We believe that this collection of surveys will be useful both for specialists and for students in the field. Parts B and C are dedicated to quasiconformal mappings, a topic which plays a fundamental role in Teichmüller theory.

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Part B (chapters 9–11) consists of three essays on the works of Tissot, Grötzsch and Lavrentieff. These three authors have contributed, each in his own way, to the development of the theory of quasiconformal mappings. The three essays will give the reader an idea of three different sources of quasiconformal mappings and their applications in the sciences, from most practical (geography, mechanics, engineering) points of view to the most theoretical one. Part C of this volume, consisting of chapters 12–26, contains English translations of papers by Grötzsch, Lavrentieff and Teichmüller which concern mainly quasiconformal mappings, together with a series of commentaries on these papers, presenting background material and highlighting the relations between these papers. The names of Grötzsch, Lavrentieff and Teichmüller are usually quoted in the modern literature on Teichmüller theory, but with no precise reference to their works: in most cases, an author who refers to them has never read any of their papers but has seen their name quoted in some other paper. This situation is contrary to the rules of scholarly writing, and, moreover, the papers contain important mathematical ideas which go beyond those that are usually quoted. This is why we think that it is important, both from the historical and mathematical points of view, to make these translations available. In the rest of this introduction, we give a more detailed overview of each chapter. The first three chapters are concerned with the complex geometry of Teichmüller space. In Chapter 1, Yukio Matsumoto describes a natural set of orbifold charts for the Deligne–Mumford compactification of the Riemann moduli space of a surface Sg ;n of genus g with n punctures and negative Euler characteristic. This compactification is obtained by adjoining to the moduli space all the (equivalence classes of ) Riemann surfaces with nodes. It is known that the compactified space is a complex orbifold of dimension 3g 3 + n . In other words, the neighborhood of each point in this space is isomorphic to the quotient of a neighborhood of a point in C3g 3+n by the action of a finite group. The charts are naturally indexed by the quotient of the curve complex Cg ;n of Sg ;n by the action of the mapping class group of this surface. Here, an element of the curve complex specifies the collection of curves on the surface that are pinched to a point; the mapping class group action is necessary because, in the moduli space, surfaces are unmarked. The empty set is admitted as an element of the curve complex, and its corresponding chart parametrizes the Riemann moduli space itself. The group associated with such an orbifold chart is a quotient of the normalizer in the mapping class group of the free abelian group generated by the curves that define the given simplex associated with the chart. This is a discrete group which the author calls the Weyl group of the chart. The group associated with a maximal simplex in Cg ;n turns out to be a Euclidean crystallographic group acting on a complex (3g 3 + n) -dimensional

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Euclidean space. In this way, the author of this chapter makes connections between the moduli space of Riemann surfaces, the theory of higher-dimensional orbifold complex structures, and discrete group actions. Chapter 2 by Subhojoy Gupta and Harish Seshadri is concerned with the complex structure of Teichmüller space induced by the Bers embedding. Incidentally, we note that by a recent holomorphic rigidity result of Daskalopoulos and Mese, any two mapping class group invariant Kähler structures on Teichmüller space are biholomorphically equivalent.1 This shows that several among the known complex structures on Teichmüller space are equivalent. The Bers embedding is an embedding of Teichmüller space in a finite-dimensional complex vector space whose image is a bounded domain. The authors in Chapter 2 are particularly interested in the convexity properties of the image of Teichmüller space by this embedding. More generally, they consider Teichmüller domains, that is, arbitrary subsets of a finite-dimensional complex vector spaces which are biholomorphically equivalent to the Teichmüller space of a surface of finite type. After reviewing the definition of the Bers embedding and discussing the structure of its boundary and that of its closure, they formulate a series of open questions concerning the biholomorphism types of Teichmüller domains and the structure of their boundaries. On the same occasion, they survey the recent result of Markovic showing the non-coincidence of the Carathéodory and Kobayashi metrics on the Teichmüller spaces of closed surfaces of genus at least two, a result that settled a question that had been open for several decades. Combined with a theorem of Lempert (1981) which says that on a convex domain of a finite-dimensional Euclidean space, the Carathéodory and the Kobayashi metrics coincide, this result implies that the Bers embedding is not convex. The authors then study the notions of convexity and pseudo-convexity of a general complex domain, reviewing a result of Bers and Ehrenpreis showing that the Bers embedding is pseudo-convex. They also present a recent result of their own saying that at any boundary point of any Teichmüller domain, the space cannot be locally strictly convex. Chapter 3 by Subhojoy Gupta is an exposition of the theory of holomorphic quadratic differentials. The author surveys the different aspects in which these objects appear in Teichmüller theory, in particular in the complex structure of Teichmüller spaces of closed surfaces of genus  2 . In this setting, holomorphic quadratic differentials parametrize the space of infinitesimal deformations of complex structures. Using Kodaira–Spencer’s theory and Serre duality, the vector space of holomorphic quadratic differentials appears naturally as the cotangent space to Teichmüller space. Among the other settings in which holomorphic quadratic differentials play an important role, the author surveys singular flat structures, measured foliations, Hopf differentials of harmonic maps, Teichmüller geodesics, and Schwarzian derivatives of projective structures on surfaces. He then reports on recent works on holomorphic quadratic differentials on punctured Riemann surfaces 1 “Rigidity of Teichmüller space,” Preprint, 2013, arXiv:1502.03367 [math.DG].

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with higher order poles at the punctures, generalizing some of the classical results that are known for closed surfaces, discussed in the preceding sections. He also mentions holomorphic cubic differentials, objects that parametrize spaces of convex projective structures on surfaces, and, more generally, holomorphic k -differentials which appear in higher Teichmüller theory. Chapter 4 by Lizhen Ji is a survey of Mostow’s strong rigidity theorem for locally symmetric spaces. Roughly speaking, the result says that a weak notion of isomorphism between spaces determines a stronger isomorphism between them. A well-known particular case of this theorem (also proved by Margulis) says that if two closed hyperbolic manifolds of the same dimension  3 have isomorphic fundamental groups, then they are isometric. This result was extended by Prasad to the case of complete manifolds of finite volume. The clearest proof of this theorem is the one contained in Chapter 5 of Thurston’s Princeton lecture notes on the geometry and topology of 3-manifolds. This rigidity result is in contrast with the case of manifolds of dimension two, where there is a large space of deformations of complex structures, namely, Teichmüller space. Even though this Handbook is mainly concerned with the case of surfaces, it seemed to us interesting to have an exposition of the deformation spaces of hyperbolic structures in higher dimensions and of the more general situations to which Mostow’s strong rigidity applies, namely, finite-volume locally symmetric spaces of noncompact type. At the level of group actions, the discussion around Mostow rigidity is reduced to the comparison between the deformation theory of isomorphisms between Fuchsian groups and that of isomorphisms between lattices in semisimple Lie groups. At the same time, Chapter 4 contains a review of some aspects of the Kodaira–Spencer deformation theory of compact complex manifolds of complex dimension  2 . Mostow’s rigidity for hyperbolic 3-manifolds of finite volume admits a vast generalization to hyperbolic 3-manifolds of infinite volume where it becomes the ending lamination theorem, a subject which is treated in the next chapter of this volume. Chapter 5, by Mahan Mj, is a survey of some important results in the theory of discrete faithful representations of surface groups into PSL(2; C) , motivated by the activity on 3-manifolds and Kleinian groups that was generated by Thurston’s ending lamination conjecture. Thurston made that conjecture in 1982, and he developed several techniques that led to its proof. It states that each end of a 3-manifold with finitely generated fundamental group is completely determined by its topology and an associated geodesic lamination, called the ending lamination. This is again in contrast with the case of dimension 2, where ends of complete hyperbolic surfaces of topological finite type are easily classified (they are either cusps or funnels). The proof of this conjecture was completed by Brock, Canary and Minsky several years later As a consequence of the proof, several other conjectures (some of them belonging to Teichmüller theory) were settled, including Bers’ density conjecture.

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The latter asserts that the space of quasi-Fuchsian groups is a dense subspace of the space of discrete faithful representations of the fundamental group of a surface into the group PSL(2; C) with respect to the algebraic topology. Some techniques introduced to prove the ending lamination conjecture were used by Mahan in his proof of the existence of Cannon–Thurston maps. Originally, this is an equivariant space-filling curve from the circle to the 2-sphere which appeared in the work of Cannon and Thurston on hyperbolic 3-manifolds fibering on the circle. The space-filling curve appears in this setting as the boundary extension of a map from the hyperbolic plane into hyperbolic 3-space. The questions raised by such boundary maps were adapted later to several contexts, including geometric group theory. As an outcome of his work on Cannon-Thurston maps, Mahan proved that connected limit sets of finitely generated Kleinian groups are locally connected, settling another conjecture of Thurston. Chapter 5 constitutes an exposition of the theory of ends of 3-manifolds and in particular of the so-called bi-Lipschitz models for ends, used by Mahan to prove the existence of Cannon–Thurston maps and the local connectivity of limit sets. Besides surveying these results, the chapter contains an introduction to the theories of Fuchsian, quasi-Fuchsian and, more generally, Kleinian groups. Chapter 6, by Armen Sergeev, is a survey of the universal Teichmüller space. This is an infinite-dimensional space that can be defined as the space of quasisymmetric homeomorphisms of the unit circle in the complex plane fixing the three points 0; 1; 1 , or, equivalently, as the space of quasiconformal mappings of the unit disc where two such mappings are identified if they coincide on the unit circle up to post-composition by a Möbius transformation. It can also be defined as the quotient of the space of Beltrami differentials on the unit disc modulo a natural equivalence relation. Alternatively, it is also the space of normalized quasiconformal mappings of the Riemann sphere that are conformal in the complement of the unit disc. The universal Teichmüller space was introduced by Bers in 1965.2 The bases of that theory were developed in several papers by Bers, Ahlfors and others. The space carries various kinds of structures: a Teichmüller-like metric, a Weil–Petersson-like metric, a complex structure, etc. It contains in a natural manner all the Teichmüller spaces of surfaces. Besides, it was realized since the beginning of the theory that there are relations between the universal Teichmüller space and physics.3 In Chapter 6, Sergeev surveys the complex-geometric properties of the universal Teichmüller space equipped with its so-called Kähler quasi-metric. This quasi-metric induces on the images of the Teichmüller spaces of compact Riemann surfaces the 2 L. Bers, “Automorphic forms and general Teichmüller spaces,” in A. Aeppli, E. Calabi, and H. Röhrl (eds.), Proceedings of the Conference on Complex Analysis (Minneapolis, 1964), Springer-Verlag Berlin, 1965, 109–113. 3 The relation between the universal Teichmüller space and physics is mentioned by Bers in his paper “Universal Teichmüller space,” in Analytic methods in Mathematical Physics, Indiana University Press, 1969, 65–83. Bers writes in this paper that J. A. Wheeler conjectured that the universal Teichmüller space can serve as a model in an attempt to quantize general relativity.

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Weil–Petersson Kähler metric of these spaces. It has a regular part and a singular part. The natural images of the Teichmüller spaces of compact Riemann surfaces do not lie in the regular part. In contrast, the regular part contains the space of normalized orientation-preserving diffeomorphisms of the circle, which is a Kähler Fréchet manifold equipped with a complex structure and a compatible symplectic structure. The author also reviews the embedding of the universal Teichmüller space into the Grassmann manifold of a Kähler Hilbert space, namely, the Sobolev 1/2 space V = H0 (S 1 ; R) of half-differentiable functions defined on the circle. He then reports on the relation between the universal Teichmüller space and string theory, explaining its Dirac quantization theory using ideas from non-commutative geometry. We note that the universal Teichmüller space is considered from other points of view in several chapters of previous volumes of the present Handbook. The interested reader is referred to the articles by Penner in Vol. I, Fletcher–Markovic in Vol. II, and Šarić in Vol. II.4 Let me also recall that Thurston wrote a paper in which he developed a theory of earthquakes on the hyperbolic disc in the setting of the universal Teichmüller space.5 The theory of the universal Teichmüller space is closely related to that of self-maps of the unit circle and their extension to maps of the disc, which plays several important roles in Teichmüller theory. Particularly interesting extensions are the Beurling–Ahlfors extension of quasisymmetric homeomorphisms and the Douady–Earle extension of homeomorphisms, based on barycentric extensions of measures defined on the unit circle which is conformally natural in the sense that it is invariant by pre- and post-composition by Möbius transformations, and the extension of a map of the circle which is the restriction of a complex-analytic map is the complex-analytic map itself. There exist other conformally natural extensions of orientation-preserving homeomorphisms of the circle, e.g. Thurston’s (generally non-continuous) earthquake maps and Markovic’s harmonic extensions. Besides their use in universal Teichmüller theory, extensions of circle homeomorphisms have applications in the theory of asymptotic Teichmüller spaces 4 R. C. Penner, “Surfaces, circles, and solenoids,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. I, IRMA Lectures in Mathematics and Theoretical Physics 11, European Mathematical Society (EMS), Zürich, 2007, 205–221; A. Fletcher and V. Markovic, “Infinitedimensional Teichmüller spaces,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. II, IRMA Lectures in Mathematics and Theoretical Physics 13. European Mathematical Society (EMS), Zürich, 2009, 65–91; D. Šarić, “The Teichmüller theory of the solenoid,” in Handbook of Teichmüller theory, Vol. II, 811–858. 5 W. P. Thurston, “Earthquakes in two-dimensional hyperbolic geometry,” in D. B. A. Epstein (ed.), Low-dimensional topology and Kleinian groups (Coventry and Durham, 1984), London Mathematical Society Lecture Note Series 112, Cambridge University Press, Cambridge, 1986, 91–112; cf. also the review in J. Hu, “Earthquakes on the hyperbolic plane,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. III, IRMA Lectures in Mathematics and Theoretical Physics 17, European Mathematical Society (EMS), Zürich, 2012, 71–122.

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and in the study of other Teichmüller spaces defined as equivalence classes of hyperbolic structures on the disc. In Chapter 7, Jun Hu provides a detailed account of a class of generalized conformal barycentric extensions of circle homeomorphisms, which are also continuous conformally natural. At the same time, he reviews the qualitative and quantitative properties of several known extensions of orientation-preserving circle homeomorphisms. His review covers the following facts and results. 1. The conformal barycentric extensions are analytic homeomorphisms of the closed disc, and the map that assigns to each homeomorphism of the disc its extension is continuous. Furthermore, if a map admits a quasiconformal extension, then its Douady–Earle extension is quasiconformal. Hu also mentions several results giving a bound on the maximal dilatation of extensions, including a result stated in terms of the cross ratio norm of the original circle homeomorphism. The results are due to Douady–Earle and to Hu–Muzician. 2. Results on the maximal dilatation of extensions of locally quasisymmetric circle homeomorphisms (work of Hu–Muzician). 3. Results on the extension of symmetric circle homeomorphisms (work of Hu–Muzician and Earle–Markovic–Sǎrić). 4. Results on the existence of angular derivatives for the extension of circle diffeomorphisms (work of Earle) and the fact that such an extension is differentiable at the boundary of the closed disc (work of Hu–Pal). In Chapter 8, the last chapter of Part A, Oscar García-Prada surveys the role of Higgs bundles in higher Teichmüller theory. We recall that there is a classical identification of the Teichmüller space of a closedorientable surface of genus g  2 with a certain connected component of the moduli space of representations of the fundamental group of the surface into PSL(2; R) , and that a higher Teichmüller space may be identified with a component of the moduli space of representations of the fundamental group of the surface into a real non-compact semisimple Lie group of higher rank. Higgs bundles, introduced by Hitchin in a paper published in 1987, turned out to be useful in realizing higher Teichmüller spaces as components of representation spaces of surface groups and studying their topology. Examples of higher Teichmüller spaces are the Hitchin components for split real groups and components of groups of Hermitian type with maximal Toledo invariant. More recently, components representing higher Teichmüller spaces were shown to exist in the case of the special orthogonal groups of signature (p; q) , in agreement with a conjecture of Guichard and Wienhard relating the existence of higher Teichmüller spaces to a notion of positivity in Lie groups and to surface group representations into these groups. This notion of positivity refines a so-called Anosov condition introduced by Labourie.

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In Chapter 8, García-Prada surveys the three situations we mentioned where higher Teichmüller spaces are known to exist: split real forms, real forms of Hermitian type, and SO(p; q) . The chapter concludes with a brief description of a conjectural general picture of positivity in higher Teichmüller theory from the point of view of Higgs bundles. The picture suggests a relation between positive representations (hence, higher Teichmüller spaces) and components of the moduli space for which there exists a so-called Cayley correspondence. The latter generalizes the Cayley correspondences that are known to exist in the cases of Hermitian groups of tube type and of special components in the moduli space of SO(p; q) -Higgs bundles. Chapter 8 complements a survey by Burger, Iozzi and Wienhard that appeared in a previous volume of this Handbook, which gives another point of view on higher Teichmüller theory.6 Part B of this volume (chapters 9–11) consists of three historical essays on Tissot, Grötzsch, and Lavrentieff respectively. These three authors contributed to the foundations of quasiconformal mappings, one of the basic tools in Teichmüller theory. Before saying a few words on these chapters, I would like to quote Ahlfors, expressing his point of view on the history of this subject. In a short essay on the present and future of classical analysis,7 he writes: I remember vividly the first time I heard about quasiconformal mappings. It was in 1930. I was visiting Carathéodory in Munich, where I also met a young Frenchman by the name of Possel.8 During a conversation he asked if we had heard of a new kind of extremal problem considered by somebody in Leipzig. He was referring to Grötzsch, a student of Koebe’s, and a problem now known as Grötzsch’s box problem. I had never heard of Grötzsch, although it turned out that our work had much in common. The problem is ridiculously simple, almost a joke, and nobody had an idea that it would mushroom to become a fruitful part of the well-established theory of conformal mapping. Even the name did not exist. Actually, the theory developed from two different seeds, one planted by Grötzsch and one by Lavrentiev in the Soviet Union who recognized its importance for elliptic differential equations. To tell the truth, the idea caught on very slowly. This was partly due to the personality of Grötzsch, but later on also to the political turmoil in Germany and finally to the total lack of communications during the 6 M. Burger, A. Iozzi, and A. Wienhard, “Higher Teichmüller spaces: from SL(2; R) to other Lie groups,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. IV, IRMA Lectures in Mathematics and Theoretical Physics 19, European Mathematical Society (EMS), Zürich, 2014, 539–618. 7 Cf. L. V. Ahlfors, “Classical analysis: present and future,” in A. Baernstein II, D. Drasin, P. Duren, and A. Marden (eds.), The Bieberbach conjecture (West Lafayette, IN, 1985), Mathematical Surveys and Monographs 21, American Mathematical Society, Providence, R.I, 1986, 1–6. 8 René de Possel (1905–1974) was one of the founding members of the Bourbaki group.

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war. Ironically, because much was published in the antisemitic Deutsche Mathematik, many papers are still difficult to find. Quasiconformal mappings enter the theory of classical function theory in three ways: (i) As a simplification of proofs. This was the least important way, although historically it was the first to demonstrate their usefulness. (ii) As a generalization of conformal mappings which was interesting in its own right. It was also important because it could be generalized to several dimensions in a way that differs completely from holomorphic functions of several variables. (iii) Through the problem of extremal quasiconformal mappings, which leads to a beautiful and unexpected connection with a class of holomorphic quadratic differentials on Riemann surfaces. The third connection is based on a brilliant discovery by Teichmüller. It forms the core of his famous paper of 1939, full of proven and unproven conjectures, ideas, and theorems, difficult to separate from each other, but all highly unconventional. Whatever feelings the name of Teichmüller evokes, the depth of his ideas cannot be disputed. It spawned the theory of Teichmüller spaces, which is still pursued by an active group of devoted mathematicians.

Chapter 9 is an essay on Nicolas-Auguste Tissot, a nineteenth-century French mathematician and cartographer who is a forerunner of quasiconformal mappings and whose work is poorly known among mathematicians. Tissot introduced a tool which became known among geographers under the name Tissot indicatrix. This is a graphical representation, on a geographical map (the image of a “projection” from a subset of the sphere, representing the surface of the Earth, onto the Euclidean plane), of a field of ellipses that are the images of an infinitesimal field of circles in the domain of the map. The ellipses contain information, at each point of the map, of the distortion of the projection, both in direction and in magnitude. Tissot studied extensively the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distortion of mappings between general surfaces. His work on this subject makes him a precursor of the theory of quasiconformal mapping, but both Grötzsch and Teichmüller mention Tissot’s name in their papers. But his work remains unknown to modern quasiconformal theorists and his name is missing from all the known historical reports on quasiconformal mappings. This is why it seemed to us appropriate to include in this handbook a short chapter on his life and work. The lives and works of Grötzsch and Lavrentieff, the two main founders of modern quasiconformal theory, are also very poorly known. The two men had very different fates. Grötzsch had his career affected by Nazism and World War II, and although he was a highly gifted mathematician, he preferred to hold a position of low visibility. He worked on quasiconformal mappings only for a few years. Lavrentieff, on the contrary, had one of the most brilliant careers in the Soviet Union, as a mathematician, physicist and science organiser. He held highly important responsibilities in his country and was awarded the most prestigious prizes. In his

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work, quasiconformal mappings found many applications in fluid mechanics, civil engineering, and other branches of science. Chapter 10, by Reiner Kühnau, is a historical survey on the academic life of Grötzsch, showing how the latter had a great respect for Koebe (who was his mentor) and a great aversion to public attention, mentioning some episodes from his life in post-World War II East Germany, and explaining his shift of interest from quasiconformal mappings to graph theory. This survey complements a memorial article on Grötzsch which the same author published in German,9 and another paper by him, titled “Some historical commentaries on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale”10 which appeared in Volume VI of the present Handbook. Chapter 10 also contains information on some mathematicians who had strong relations with Grötzsch, in particular Paul Koebe. Chapter 11, by Athanase Papadopoulos, is a survey on the life and work of Mikhaïl Lavrentieff. This chapter is also the occasion to talk about the prominent role that the latter played in the Russian science landscape during the turbulent period whose span covers the whole rule of Stalin and ended five years before the start of perestroïka. Lavrentieff was also the first to develop quasiconformal mappings in higher dimensions. At the same time, Chapter 11 contains a description of the remarkable Moscow school of topology and theory of functions during the years 1920–1930 in which Lavrentieff was trained, and of some turbulent events that this school traversed. Part C of this volume, consisting of chapters 12–26, contains translations of original texts on Teichmüller theory, as well as commentaries on these texts. It starts with a letter by Teichmüller, written in 1940, after the publication of his seminal paper “Extremale quasikonforme Abbildungen und quadratische Differentiale” (Extremal quasiconformal mappings and quadratic differentials).11 The letter was communicated to us by Reiner Kühnau. It was translated from the German by Annette A’Campo. In this letter, Teichmüller informs the recipient (whom we were unable to identify) about some directions in which he plans to continue his investigations, indicating some problems which he was not able to solve. These problems include the extension of his results to the case of what he calls “higher” 9 R. Kühnau, “Herbert Grötzsch zum Gedächtnis,” Jahresberichte d. Deutschen Math.-Verein. 99 (1997), 122–145. 10 R. Kühnau, “Some historical commentaries on Teichmüllers paper Extremale quasikonforme Abbildungen und quadratische Differentiale,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VI, IRMA Lectures in Mathematics and Theoretical Physics, 27. European Mathematical Society (EMS), Zürich, 2016. 537–546. 11 O. Teichmüller, “Extremale quasikonforme Abbildungen und quadratische Differentiale,” Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940). no. 22, 197 pp.; reprinted in Gesammelte Abhandlungen (L. V. Ahlfors and F. W. Gehring, eds.), Springer-Verlag, Berlin etc. 1982, 337–531; English translation by G. Théret, “Extremal quasiconformal mappings and quadratic differentials,” in A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. V, IRMA Lectures in Mathematics and Theoretical Physics 26, European Mathematical Society (EMS), Zürich, 2016, 321–483.

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principal domains obtained from the ordinary ones by moving the distinguished points from the interior to the boundary. Teichmüller declares that he is not capable of extending the theory of extremal quasiconformal mappings to these domains. He mentions a non-symmetric notion of orthogonality between vectors that arises in the Finsler metric on the space of marked Riemann surfaces (that is, Teichmüller space) of the ordinary principal domains which he cannot extend to the higher case, and he gives some hints to an approach to this problem. He addresses then a problem involving an extension of the method of Grötzsch and Ahlfors involving trajectories of quadratic differentials to questions related to geodesics in the Teichmüller spaces of these higher domains. Teichmüller ends this letter by mentioning the fact that the metric he defined (that is, the Teichmüller metric) is not Riemannian. The rest of Part C contains translations of fundamental papers by Grötzsch, Lavrentieff, and Teichmüller on quasiconformal mapppings and their applications, as well as commentaries on these papers, including overviews of the type problem and of Nevanlinna’s theory, two domains in which Teichmüller used these mappings. Grötzsch’s work on conformal and quasiconformal mappings was published in a series of papers that appeared in the period 1928–1934. The importance of this work was realized a few years later, and Grötzsch’s results became fundamental in Teichmüller theory. In the meanwhile, Grötzsch’s mathematical interests shifted to graph theory, although in the 1950s he published a couple of papers in which he revisited and extended some of his previous results on conformal representations of plane domains (of finite and infinite connectivity).12 Grötzsch’s papers are often quoted in the modern literature on quasiconformal mappings without having been read, one reason being that they are written in German. We have included in the present volume the English translation of the following five papers that are representative of his work: 1. “Über einige Extremalprobleme der konformen Abbildung” (On some extremal problems of the conformal mapping);13 2. “Über einige Extremalprobleme der konformen Abbildung. II.” (On some extremal problems of the conformal mapping. II);14 3. “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes” 12 H. Grötzsch, “Zur Geometrie der konformen Abbildung,” Hallische Monographien, no. 16, Max Niemeyer Verlag, Halle (Saale), 1950, 5–11; H. Grötzsch, “Zum Häufungsprinzip der analytischen Funktionen,” Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 5 (1956), 1095–1100. 13 Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig, Math.-Physische Klasse, Bd. 80 (1928), 367–376. Translated by A. A’Campo-Neuen. 14 Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig, Math.-Physische Klasse, Bd. 80 (1928), 497–502. Translated by M. Brakalova-Trevithick.

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(On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem);15 4. “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche” (On the distortion of nonconformal schlicht mappings of multiply-connected schlicht regions);16 5. “Über möglichst konforme Abbildungen von schlichten Bereichen” (On closest to conformal mappings of schlicht domains).17 The five papers are interconnected: Grötzsch starts his paper 2 by declaring that the results he presents are extensions of those he obtained in his paper 1. Moreover, the title of this second paper makes it clearly a sequel of the first one, and the section numbering there is in the continuity of that in the first one. At the beginning of the paper 3, Grötzsch writes that his paper is connected with his two earlier papers 1 and 2. In the first paragraph of the paper 4, Grötzsch refers to his paper 4. At the beginning of the paper 5, Grötzsch says that his paper is very closely related to his earlier two papers 3 and 4. The five papers together show the evolution of the questions that Grötzsch addressed and the results he obtained, from the study of conformal mappings and conformal invariants to that of quasiconformal and extremal quasiconformal mappings. Let us say a few words on the content of these papers. The first two papers are concerned with conformal representations of multiplyconnected domains. Grötzsch uses the so-called length-area method, a method which is a direct application of the Cauchy–Schwarz inequality and which turns out to be very powerful in addressing various problems on conformal and quasiconformal mappings. The method was already used by Courant and Hurwitz in their textbook Funktionentheorie,18 where they discuss the continuous extension to the boundary of the Riemann Mapping Theorem. In that passage, Courant and Hurwitz attribute the method (as well as the result they prove using it) to Carathéodory. The length-area method is in turn a precursor of the so-called method of extremal length, used later by Ahlfors and Beurling, which gives a way of computing the modulus of a quadrilateral without passing through the Riemann Mapping Theorem. In the third paper, Grözsch introduces his notion of non-conformal (“nichtkonformen”) mapping,

15 Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig, Math.-Physische Klasse, Bd. 80 (1928), 503–507. Translated by M. Brakalova-Trevithick. 16 Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig, Math.-Physische Klasse, Bd. 82 (1930), 69–80. Translated by M. Karbe. 17 Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig, Math.-Physische Klasse, Bd. 84 (1932), 114–120. Translated by M. Brakalova-Trevithick. 18 A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Julius Springer, Berlin, 1922

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which he also calls mapping of bounded infinitesimal distortion (“Abbildung von beschränkter infinitesimales Verzerrung”), a notion which is very close to that of quasiconformal mapping as we intend it today. He proves that several results that hold for conformal mappings admit a natural generalization to this new setting. These results include analogues of a distortion theorem of Koebe and of the big Picard theorem. The fourth paper is concerned with quasiconformal representations of multiply-connected domains, and, again, Grötzsch proves there generalizations of results known for conformal representations. The fifth paper is concerned with mappings that Grötzsch calls closest to conformal (“möglichst konforme”), which are the mappings that we call today “extremal quasiconformal mappings.” The translations of Götzsch’s papers, which constitute chapters 13–17 of this volume, are followed by a commentary (Chapter 18) written by Vincent Alberge and Athanase Papadopoulos, in which the interconnections between these papers are explicited. Chapter 19 is an English translation of Lavrentieff ’s paper Sur une classe de représentations continues (On a class of continuous representations), published in 1935. In this paper, Lavrentieff introduces the notion of almost analytic function, another version of quasiconformal mappings. In his later papers, Lavrentieff called these mappings quasiconformal. The quasiconformality of an almost analytic function is measured by two associated functions, which Lavrentieff calls the characteristic functions. Like the Tissot indicatrix, the properties of the characteristic functions translate into properties of the ellipses which are images of the infinitesimal circles by the map. In his paper, Lavrentieff obtained several results that show that almost analytic functions constitute natural generalizations of conformal mappings. The approach is the same that Grötzsch followed in developing his notion of “non-conformal” mappings. The results include an existence and uniqueness theorem which establishes a relation between quasiconformality and the Beltrami equation and which is an early version of the measurable Riemann Mapping Theorem. Lavrentieff ’s result is a generalization of the Riemann Mapping Theorem to the case of a complex structure defined on an open simply-connected subset of the plane by a field of ellipses (in the classical case where one has an open simply-connected subset of the plane equipped by the complex structure induced from the ambient complex structure of the plane). In fact, Lavrentieff proves a version of this theorem for the case of C 0 structures. The extension to the case of measurable structures was obtained by Morrey, Ahlfors and Ahlfors–Bers.19 Lavrentieff then proves a generalization of Picard’s theorem to the setting of almost analytic functions, and a compactness result for families of almost analytic functions with convergent distortion characteristics which is close to the Montel theorem for holomorphic 19 C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166; L. Ahlfors, “Conformality with respect to Riemannian metrics,” Ann. Acad. Sci. Fenn. Ser. A. I. 1955, no. 206, 1–22; L. Ahlfors and L. Bers, “Riemann’s mapping theorem for variable metrics,” Ann. of Math. (2) 72 (1960), 385–404.

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functions. This series of results were part of an effort to free the basic theorems on analytic functions from unnecessary hypotheses. Lavrentieff also obtains distortion results for quasiconformal mappings, and results on the type problem, that is, the problem of deciding whether a simply connected Riemann surface is conformally equivalent to the complex plane or to the unit disc. This problem was one of the major problems in function theory. In particular, it was one of the main research subjects of Ahlfors in the period 1929–1941.20 The translation of Lavrentieff ’s paper is followed by a commentary (Chapter 20), written by Vincent Alberge and Athanase Papadopoulos, which makes connections between this work of Lavrentieff and works of Teichmüller and other authors. Chapters 21–23 of this volume contain translations of the following three papers by Teichmüller: 1. “Eine Anwendung quasikonformer Abbildungen auf das Typenproblem” (An application of quasiconformal mappings to the type problem), written in 1937;21 2. “Untersuchungen über konforme und quasikonforme Abbildung” (Investigations on conformal and quasiconformal mappings), written in 1938;22 3. “Einfache Beispiele zur Wertverteilungslehre” (Simple examples in value distribution theory), written in 1944.23 The first paper (translated in Chapter 21) is concerned with the type problem. The main tools are quasiconformal mappings and line complexes. We recall that a line complex is a graph associated with a surface defined as a branched covering of the Riemann sphere which is branched over a finite number points. The graph is embedded in the total space of the covering and carries combinatorial information on the covering. Teichmüller proves in particular that if two simply connected Riemann surfaces are branched coverings of the Riemann sphere with finitely many branch values and if they have the same line complexes, then they are quasiconformally equivalent. In his proof, he introduces a technique for piecing together quasiconformal mappings. At the same time, he studies the extension of orientation-preserving diffeomorphisms of the circle of class C 1 to quasiconformal mappings of the disc which are conformal near the boundary. His motivation for this work comes from the type problem which, in the present case, is the problem of deciding whether a simply connected Riemann surface defined as a branched 20 See e.g. Ahlfors’ papers “Zur Bestimmung des Typus einer Riemannschen Fläche,” Comment. Math. Helv. 3 (1931), 173–177, “Quelques propriétés des surfaces de Riemann correspondant aux fonctions méromorphes,” Bull. Soc. Math. France 60 (1932), 197–207, and “Sur le type d’une surface de Riemann,” C. R. Acad. Sci. Paris 201 (1935), 30–32, and his comments in his Collected papers edition. 21 Deutsche Math. 2 (1937), 321–327. Translated by M. Brakalova-Trevithick. 22 Deutsche Math. 3 (1938), 621–678. Translated by M. Brakalova-Trevithick and M. Weiss. 23 Deutsche Math. 7 (1944), 360–368. Translated by A. A’Campo-Neuen.

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covering of the sphere, with some combinatorial information on the branching, is conformally equivalent to the complex plane or to the unit disc. Teichmüller writes that “one is still very far away from sufficient and necessary criteria.” He shows in particular that the type of a simply connected Riemann surface is a quasiconformal invariant. In other words, the result says that the unit disc and the complex plane are not quasiconformally equivalent. The paper “Untersuchungen über konforme und quasikonforme Abbildungen” (Investigations of conformal and quasiconformal mappings, translated in Chapter 22, is Teichmüller’s Habilitationsschrift written under the supervision of Bieberbach. It is one of his longest papers. Teichmüller introduces there a notion of quasiconformal mappings w(z) of the complex plane with dilatation quotient bounded by a function of jzj with controlled growth. Under such a condition, he proves a result on the circularity of the image of a circle near infinity; more precisely, he proves that jwj  const
jzj as z ! 1 . The paper also contains a series of results on the conformal geometry of annuli and quadrilaterals: monotonicity of modules, construction of regions that are solutions of extremal problems, introduction of new conformal invariants: conformal radius, reduced module, reduced logarithmic area, etc. An important result he obtains is a distortion lemma which is a substantial improvement of the main lemma in Ahlfors’ thesis. Another important result obtained in this paper is known as Teichmüller’s Modulsatz. It gives a sufficient condition for a doubly connected subset of the complex plane to be conformally equivalent to an annulus embedded in a homotopy-equivalent way in some annulus bounded by two concentric circles, with an a priori bound on the radii of these circles. Teichmüller used this result in his work on value distribution theory. The paper also contains a result on the type problem, more precisely, a condition for a Riemann surface to be of hyperbolic type, which Teichmüller obtains using line complexes, as in the paper “Eine Anwendung quasikonformer Abbildungen auf das Typenproblem.” At the same time, Teichmüller settles (by the negative) a question posed by Nevanlinna on the type problem. The paper “Einfache Beispiele zur Wertverteilungslehre” (Simple examples in value distribution theory), translated in Chapter 23, concerns Nevanlinna’s theory. Teichmüller gives there a set of examples of meromorphic functions f for which he gives explicit formulae for the Nevanlinna counting, characteristic and deficiency functions. He uses line complexes to construct the Riemann surface associated with the function f . An explicit formula for the generating function associated with this surface gives then the information on the counting, characteristic and deficiency functions of f . The translations of the three papers by Teichmüller are followed by three essays related to them that constitute the last three chapters (chapters 24–26) of this volume.

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Chapter 24, written by Vincent Alberge, Melkana Brakalova-Trevithick and Athanase Papadopoulos, is a commentary on Teichmüller’s work on the type problem, and it contains at the same time a review of the theory of line complexes. Chapter 25, written by the same authors, is a commentary on Teichmüller’s paper “Untersuchungen über konforme und quasikonforme Abbildungen.” Chapter 26, written by Athanase Papadopoulos, consists of notes on Nevanlinna’s theory that should help the reader placing Teichmüller’s paper “Einfache Beispiele zur Wertverteilungslehre” in its proper context. As a conclusion to this part on quasiconformal mappings, let us note that in the last section of his paper “Extremale quasikonforme Abbildungen und quadratische Differentiale” (§33), Teichmüller addresses the question: Why do we study quasiconformal mappings? His answer is that, first of all, these mappings were introduced as a generalization of conformal mappings. He adds that these mappings first appeared in the works of geographers (and he mentions Tissot), who used them to build geographical maps, an then, by others, whose aim was to generalize known properties of conformal mappings (the Picard theorem, etc.) to the setting of quasiconformal mappings. But this point of view, he says, is “outdated”: quasiconformal mappings are used in the study of conformal mappings, and this leads to the consideration of a class of quasiconformal mappings that are more general than mappings with bounded dilatation quotient, namely, their dilatation quotient is bounded by a function, and not by a constant. This is indeed the class of mappings he considers in the paper “Untersuchungen über konforme und quasikonforme Abbildungen.” He says that these mappings are useful in the solution of the type problem and in value distribution theory. Teichmüller’s papers on Riemann surfaces, conformal geometry and quasiconformal mappings are difficult to read. They are written in an informal style and many proofs are sketchy and non-rigorous. Several statements are given without proof, and for the others, the proofs given are heuristic. It is important however to note that all the results in these papers, including those stated without proofs, turned out to be correct, sometimes after a long labor by several mathematicians who tried to provide new proofs. One may recall here that Ahlfors and Bers spent several years in an effort to give satisfactory proofs of statements made by Teichmüller, often trying methods that are different from those he suggested. Bers’s penultimate published paper is titled “On Teichmüller’s proof of Teichmüller’s theorem.”24 In a postscript to that paper, Bers returned to an assertion he made in a paper he wrote 26 years before, “Quasiconformal mappings and Teichmüller’s theorem,”25 in which he gave a proof of the existence part of Teichmüller’s theorem, writing: “This note is a postscript to a paper which I published many years ago and is, 24 J. Anal. Math. 46 (1986), 58–64; reprinted in Ber’s Selected Works, Part II, 219–525. 25 In Analytic functions, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1960, 89–119; reprinted in Bers’s Selected Works, Part I, 323–354.

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like that paper, essentially expository. […] In proving a basic continuity assertion, I made use of a property of quasiconformal mappings which belongs to the theory of quasiconformal mappings with bounded measurable Beltrami coefficients (and seems not to have been known to Beltrami). Some readers concluded that the use of that theory was indispensable for the proof of Teichmüller’s theorem. This is not so, and Teichmüller’s own argument is correct.” As a matter of fact, this informal and “non-rigorous” style was also characteristic of Riemann. Concerning the question of rigor, let me recall that René Thom once wrote that rigor as “a very unnecessary quality in mathematical thinking,”26 that Thurston expressed a similar idea,27 and that Poincaré was worried about theorems that are “perfectly rigorous, but perfectly useless”.28 In this respect, let me quote Thurston again, from his introduction of Hubbard’s Teichmüller Theory and Applications:29 “[…] It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof ) n remark) n style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind.” Finally, I will quote Herbert Grötzsch, to whose work a substantial part of the present volume is dedicated. Reiner Kühnau, in his biographical article (Chapter X), recalls the following sentences by him: “We must get away from formalism in mathematics,” and “the fact that a mathematical investigation is formally logically correct does not mean automatically that it is a valuable contribution to our mathematical knowledge.” The articles that are translated and commented on in Part C of this volume, with their non-rigorous style and incomplete proofs, gave the main directions to many of the theories that constitute the bases of today’s research.

26 M. Atiyah et al., “Responses to: A. Jaffe and F. Quinn, Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics” [Bull. Amer. Math. Soc. (N.S.) 29, 1993, no. 1, 1–13)], Bull. Amer. Math. Soc. (N.S.) 30, 1994, no. 2, 178–207. 27 W. P. Thurston, “On proof and progress in mathematics,” Bull. Amer. Math. Soc. (N.S.) 30, no. 2, 1994, 161–177. 28 H. Poincaré, Science et méthode, Paris, Flammarion, 1908, §5. 29 J. H. Hubbard, Teichmüller theory and applications to geometry, topology and dynamics. Vol. 1, Matrix Editions, 2006.

Part A Surveys

Chapter 1

The Deligne–Mumford compactification and crystallographic groups Yukio Matsumoto Dedicated to Professor Yoichi Imayoshi whose pioneering work has stimulated the author

Contents 1 Introduction . . . . . . . . . . . . 2 Orbifolds and curve complexes . . 3 Augmented Teichmüller spaces . . 4 The " -thick part . . . . . . . . . 5 The compactness theorem . . . . 6 Construction of the orbifold-charts 7 Crystallographic groups . . . . . . References . . . . . . . . . . . . . . .

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21 24 26 28 35 41 54 58

1. Introduction Throughout the chapter, we will fix a topological surface Σg ;n obtained by removing distinct n points from a closed connected oriented surface Σg of genus g . A surface with no punctures Σg ;0 is nothing but Σg . We will assume that the Euler characteristic (Σg ;n ) of Σg ;n is negative. The mapping class group Γg ;n of Σg ;n is defined by Γg ;n = ¹f W Σg ;n ! Σg ;n j orientation preserving homeomorphismsº/ '

where f ' g means that the homeomorphisms f and g are isotopic. Note that in this situation f is isotopic to g if and only if f is homotopic to g (see [20]). The isotopy class of f (the mapping class) is denoted by [f ] . The group structure of Γg ;n is defined by composition of maps [f ][g] = [f ı g] . A marked Riemann surface is a pair (S; w) consisting of a Riemann surface S and an orientation preserving homeomorphism wW Σg ;n ! S called a marking.

22

Yukio Matsumoto

Since we are assuming (Σg ;n ) < 0 , a Riemann surface modeled on Σg ;n has a hyperbolic metric (the Poincaré metric). Marked Riemann surfaces (S; w) and (S 0 ; w 0 ) are equivalent and denoted by (S; w)  (S 0 ; w 0 ) if there exists a biholomorphic mapping hW S ! S 0 such that the following diagram is homotopically commutative (i.e., w 0 ' h ı w ): Σg ;n

w

! S

!

Σg ;n

w0

!

h

id :

! S0

The equivalence class of (S; w) is denoted by [S; w] . The Teichmüller space Tg ;n modeled on Σg ;n is the set of equivalence classes of marked Riemann surfaces. It is a metric space endowed with the Teichmüller metric, and by Teichmüller’s theorem, it is homeomorphic to R6g 6+2n (see Chapter 5 of [30] or Chapter 6 of [28]). In what follows, we assume 3g 3 + n = 1 . By the Ahlfors–Bers theory (see [4], [5], [6], [7], and [8]), Teichmüller space Tg ;n is a complex manifold of complex dimension 3g 3 + n . The mapping class group Γg ;n acts on Tg ;n by the rule [f ][S; w] = [S; w ı f

1

]

for all [f ] 2 Γg ;n ; [S; w] 2 Tg ;n :

(1)

Note that the action changes the marking w but not the Riemann surface S . The action is properly discontinuous. It preserves the Teichmüller metric and the complex structure. (See [30] and [28].) The moduli space Mg ;n is defined to be the quotient Mg ;n = Tg ;n /Γg ;n :

This is a normal complex variety ([17]). The moduli space Mg ;n parameterizes all the isomorphism classes of Riemann surfaces homeomorphic to Σg ;n . It is known that Mg ;n is not compact. But by adding “frontier points” which parameterize Riemann surfaces with nodes, it can be compactified. This is the Deligne–Mumford compactification of the moduli space (the DM-compactification for short), see [19]. We will denote the DMcompactification x g ;n . by M Unfortunately, the arguments of Deligne and Mumford [19] are not easy to understand. Because of its importance, several authors have tried to understand the compactification by analytic methods. L. Bers in [11] and [12] was one of the authors who started an analytic approach to the DM-compactification, but his project was not completed. An analytic construction was given only in the 21st century by J. H. Hubbard and S. Koch [29]. (See [40], p. 459, for Kra’s comment on their work.)

1 The Deligne–Mumford compactification

23

The DM-compactification of the moduli space has a structure of a complex orbifold. The purpose of this paper is to construct natural and “tautological”1 x g ;n which are indexed by simplexes of Harvey’s curve complex orbifold-charts of M Cg ;n . Note that our orbifold-charts ¹(D" (); W ())º 2Cg;n are in a generalized sense, in which the groups W () are not necessarily finite, but they act on the complex manifolds D" () properly discontinuously. See §2 for a formal definition of generalized orbifold-charts. Moreover, strictly speaking, our orbifold-charts are not indexed by the curve complex Cg ;n itself but by the quotient of Cg ;n under the action of Γg ;n . The quotient complex Cg ;n /Γg ;n is a finite simplicial complex. The index set of our orbifold-charts is a finite set of simplexes, each simplex chosen from a coset of Cg ;n /Γg ;n as a representative. For each  2 Cg ;n , the chart (D" (); W ()) is defined as follows. Let Γ() be the free abelian group generated by right-handed Dehn twists about the simple closed curves belonging to  , and let N Γ() be the normalizer of Γ() in Γg ;n . The group W ( ) is the quotient group N Γ()/Γ() , called the “Weyl group” (see §4, Definition 4.2). Note that our Weyl group W () is a discrete group, but not necessarily a finite group. The manifold D" () is a certain complex manifold of complex dimension 3g 3 + n , called the “controlled deformation space” (see §6, Definition 6.2). Topologically, D" () is homeomorphic to an open cell of (real) dimension 6g 6 + 2n (see Lemma 6.7). The group W () acts on D" () properly discontinuously. Our main theorem states the following. Theorem 1.1. The finite family ¹(D" (); W ())º 2Cg;n /Γg;n is a (“tautological”) x g ;n of moduli space. atlas of complex orbifold-charts of the DM-compactification M We admit the empty set ; to be a member of Cg ;n . The chart corresponding to ; is nothing but the pair of Teichmüller space and the mapping class group: (D" (;); W (;)) = (Tg ;n ; Γg ;n ) . Essentially the same orbifold charts appear in many places in the literature. In fact, the discussion of Hubbard and Koch [29] contains them as an important part. Precisely speaking, however, to get an orbifold chart such as (D" (); W ()) , we must be a little bit careful: the definition of orbifold chart requires that the x g ;n . (See §2, Definition 2.1.) quotient D" ()/W () should be an open subset of M This is rather a delicate condition, especially in constructing a “tautological atlas.” Our (D" (); W ()) (see §6, Definition 6.2) meets the requirement, but the similar pair (D" ( ); W ()) (see §6, Definition 6.1) does not. This is the reason why we modified (D  (); W ()) to (D" (); W ()) . See Remark 6.3. 1 In the case of manifolds, an example of a “tautological atlas” is that of the n -dimensional projective space Pn = ¹(x0 : x1 : : : : : xn )º consisting of the n -dimensional affine spaces Ai = ¹(x0 ; x1 ; : : : ; xi 1 ; 1; xi +1 ; : : : ; xn )º; i = 0; 1; : : : ; n . We do not give any technical definition of “tautological atlas,” but the meaning would be understandable.

24

Yukio Matsumoto

An interesting feature of our construction is that certain Euclidean crystallographic groups appear related to the maximal simplexes of Cg ;n . In fact, our second main theorem states the following. Theorem 1.2. If  is a maximal simplex, then the Weyl group W () is a finite group, and the group N Γ() is a crystallographic group acting on Euclidean space E3g 3+n . The author believes that this result is new. For details, see §7. Theorem 1.1 was stated in [46] with the idea of the proof. It was stated again in [47] together with an argument on the " -thick part of Teichmüller space. In this paper, we will present the full discussions. Our plan after §1 is the following. In §2, we will recall the notion of orbifold and curve complex. In §3, we will review the augmented Teichmüller spaces Tyg ;n . In §4, we will study the " -thick part Tg";n of Tg ;n and the action of the mapping class group Γg ;n on it. In §5, we will give our proof of the well-known fact that the quotient space Tyg ;n /Γg ;n is compact and identified with the DM-compactification of the moduli space. In §6, we will introduce the deformation spaces D" () and the controlled deformation spaces D" () , as refinements of Bers’ deformation spaces [12]. Theorem 1.1 will be proved in this section. In §7, we will prove Theorem 1.2.

2. Orbifolds and curve complexes In this section, we will recall the formal definition of orbifolds and curve complexes.

2.1. Orbifolds Orbifolds were first introduced by I. Satake [58] under the name of V-manifolds, and were re-discovered by W. Thurston as an essential tool in the geometry and topology of 3-manifolds ([59]). The name of orbifold is due to Thurston. “Orbifolds conveniently blend the theory of manifolds with that of finite group actions,” see [14], p. 441. The definition given here is based on the simplified version due to F. Bonahon and L. C. Siebenmann [14] (see also [49]). Let Uz be a finite dimensional smooth manifold which is acted on by a (not necessarily finite) group G . Suppose the action of G is smooth and properly discontinuous. Then the isotropy group of each point of Uz is finite, and the quotient space Uz /G is a good example of an orbifold. The following definition is a somewhat generalized one, having this example in mind. Definition 2.1. A smooth m -dimensional orbifold (briefly, an m -orbifold) is a  -compact Hausdorff space M which is locally modeled on a quotient space of a finite group action on a smooth m -dimensional manifold. More precisely,

1 The Deligne–Mumford compactification

25

i . an m -orbifold M is covered by an atlas of orbifold-charts ¹(Uzi ; Gi ; 'i ; Ui )ºi 2I , each chart consisting of a smooth m -manifold Uzi , a (not necessarily finite) group Gi acting on Uzi smoothly and properly discontinuously, an open set Ui of M and a folding map 'i W Uzi ! Ui which induces a natural homeomorphism Uzi /Gi ! Ui ; i i . (compatibility condition) for x 2 Uzi and y 2 Uzj such that 'i (x) = 'j (y) 2 Ui \ Uj , there exists a diffeomorphism W Vzx ! Vzy from an open z neighborhood of x in Ui to an open neighborhood of y in Uzj such that (x) = y and 'j = 'i . Remark 2.1. The usual definition of orbifold requires the group action of Gi on Uzi to be effective. In this paper, however, we do not assume the effectiveness, aiming at the uniform treatment to cover certain sporadic cases where the actions of the mapping class groups on the Teichmüller spaces are not effective (for example the case (g; n) = (2; 0) ). Note that Hinich and Vaintrob [27], §3, also consider non-effective orbifolds. In what follows, we will always consider complex m -orbifolds, in which the manifolds Uzi are complex manifolds of complex dimension m , the gluing diffeomorphisms are biholomorphic, and the groups Gi act on Uzi holomorphically. We will often use the simplified notation ¹(Uzi ; Gi )ºi 2I to represent the orbifoldcharts ¹(Uzi ; Gi ; 'i ; Ui )ºi 2I .

2.2. Curve complexes Given a surface Σg ;n , W. Harvey [25] introduced an abstract simplicial complex called the complex of curves (or curve complex) Cg ;n = C(Σg ;n ) . Definition 2.2. A vertex of the curve complex Cg ;n is an isotopy class of an essential simple closed curve on Σg ;n , where a closed curve is said to be essential if it is neither null-homotopic nor homotopic to a puncture. A simplex  of Cg ;n is a collection of disjoint, mutually non-isotopic essential simple closed curves on Σg ;n :  = hC1 ; : : : ; Ck i: The number k of simple closed curves contained in  will be denoted by j j . It is known that jj 5 3g 3 + n . Clearly we have dim  = jj 1 . Harvey [25] introduced the curve complex to study the “boundary structure” of Teichmüller space and the action of Γg ;n on the boundary. The mapping class group Γg ;n naturally acts on Cg ;n as automorphisms. N. V. Ivanov [33] proved the converse, assuming g = 2 . M. Korkmaz [38] and F. Luo [42] extended Ivanov’s results to cover the cases g = 0; 1 . The final form of their results is the following.

26

Yukio Matsumoto

Theorem 2.1 ([42]). (a) If 3g 3 + n = 2 and (g; n) ¤ (1; 2) , then any automorphism of Cg ;n is induced by a self-homeomorphism of the surface Σg ;n . (b) Any automorphism of C1;2 preserving the set of vertices represented by separating loops is induced by a self-homeomorphism of the surface. (c) There is an automorphism of C1;2 which is not induced by any homeomorphisms. Remark 2.2 (cf. [48], Lemma 3). Suppose 3g 3 + n = 1 and (g; n) ¤ (2; 0); (1; 2); (1; 1); (0; 4) . Then the natural homomorphism Γg ;n ! Aut(Cg ;n ) is injective. If (g; n) = (2; 0); (1; 2) or (1; 1) , the kernel of Γg ;n ! Aut(Cg ;n ) is isomorphic to Z2 . If (g; n) = (0; 4) , the kernel of Γ0;4 ! Aut(C0;4 ) is isomorphic to Z2 ˚ Z2 . Combining Remark and Theorem 2.1, we have the following result. Corollary 2.2.

If 3g

3 + n = 2 and (g; n) ¤ (1; 2) , then ´ Γg ;n for (g; n) ¤ (2; 0); Aut(Cg ;n ) =  Γ2;0 /Z2 for (g; n) = (2; 0);

where Γg ;n denotes the extended mapping class group containing orientation reversing homeomorphisms.

3. Augmented Teichmüller spaces In terms of Kleinian groups, L. Bers in [11] and W. Abikoff in [1] and [3] introduced the augmented Teichmüller space Tyg ;n by attaching to Tg ;n a special type of degenerate quasi-Fuchsian groups called regular b-groups. Regular b -groups are a disguised form of Riemann surfaces with nodes. Thus the augmented Teichmüller space is the Teichmüller space to which Riemann surfaces with nodes are attached. The most natural construction of the augmented Teichmüller space Tyg ;n would be to take the metric completion of Teichmüller space with respect to the Weil–Petersson metric. (See [66], Theorem 4.4.) As mentioned in §1, the Teichmüller metric is a natural metric on Tg ;n from the viewpoint of quasi-conformal deformations. But if we view Tg ;n as a Riemannian manifold, it is not so natural a metric, because the Teichmüller metric is not defined by ds 2 ([60], p. 384). The Weil–Petersson metric is a natural metric from this point of view (see [60] and [4], §4). S. Wolpert [61] and T. Chu [18] proved, however, that Teichmüller space equipped with the Weil–Petersson metric is not complete. In fact, there exist geodesics which terminate in a finite length. H. Masur [44], S. Wolpert [64], [65], and [66], and S. Yamada [67] studied in detail the behavior of the metric tensors on this metric completion Tyg ;n .

1 The Deligne–Mumford compactification

27

Let  = hC1 ; : : : ; Ck i be a simplex in the curve complex Cg ;n . Then a singular surface, denoted by Σg ;n ( ) , is obtained from Σg ;n by pinching each curve Ci 2  to a point. On a marked Riemann surface (S; w) , there is a disjoint union of simple closed geodesics ¹c1 ; : : : ; ck º such that ci represents the isotopy class of w(Ci ) (i = 1; : : : ; k) . If each ci shrinks to a point as [S; w] approaches the “boundary” of Tyg ;n , then in the limit we have a Riemann surface with nodes. (See [67], Proposition 1.) This nodal Riemann surface is modeled on Σg ;n () . Here we will recall from Bers’ paper [11] the formal definition of a Riemann surface with nodes. Definition 3.1 ([11]). A Riemann surface with nodes (or a nodal Riemann surface for short) S is a connected complex space such that every point p 2 S has arbitrarily small neighborhood isomorphic either to the set jzj < 1 in C or to the set jzj < 1 , jwj < 1 , zw = 0 in C2 . In the second case, p is called a node. Every component of the complement of the nodes is called a part of S . Each part is assumed to have negative Euler characteristic. (Thus each part has a hyperbolic metric.) If Σg ;n () has no complex structures, the meaning of a “part” or a “node” of Σg ;n () will be clear. A marking of a nodal Riemann surface S (modeled on Σg ;n () ) is a homeomorphism wW Σg ;n () ! S which is orientation preserving on each part. Teichmüller space of marked Riemann surfaces with nodes, T () , is defined similarly to Tg ;n . This is isomorphic to the product of Teichmüller spaces modeled on the parts of Σg ;n () ([44], p. 624). Its complex dimension is given as follows (see [44] and [67]): dimC T () = 3g

3+n

jj:

Remark 3.1. Teichmüller space Tg ;n is a bounded domain of C3g 3+n homeomorphic to an open cell. (See [30], §6.1.4.) Being a product of the Teichmüller spaces of its parts, Teichmüller space T ( ) of Σg ;n () is again a bounded domain of C3g 3+n j j , and homeomorphic to an open cell. Following [67], we will call Tyg ;n n Tg ;n the frontier set or the boundary and will denote it by @Tg ;n . T () has its own Weil–Petersson metric, and its metric completion Ty ( ) is meaningful ([67], and Wolpert [65]). Each component of the boundary Teichmüller spaces is totally geodesic (Theorem 2 of [67]), and we have [ [ Tyg ;n = T () and @Tg ;n = T (); (2)  2Cg;n

where T (;) = Tg ;n .

;¤ 2Cg;n

28

Yukio Matsumoto

The equalities (2) are essentially the same as those on p. 330 of [67], because [ Ty () = T ( );  " for any other essential simple closed curve C  Σg ;n . By the definition, F " () \ F " ( ) = ;;

We call the set

if  ¤ :

def

@Tg";n = ¹p 2 Tg ;n j L(p) = "º the boundary of Tg";n . The boundary is a disjoint union of the facets: [ @Tg";n = F " (): ;¤ 2Cg;n

In what follows, we will put m = 3g 3 + n , for simplicity. By adding suitable m k simple closed curves Ck +1 ; : : : ; Cm to the members of  , we get a maximal simplex of Cg ;n : ˜ = hC1 ; : : : ; Ck ; Ck +1 ; : : : ; Cm i: Recall that with ˜ are associated the Fenchel–Nielsen coordinates. In fact, ˜ decomposes the surface Σg ;n into a union of (generalized) pairs of pants P1 ; : : : ; P2g 2+n . Here a generalized pair of pants means a surface homeomorphic to a 2 -sphere with the interiors of three disjoint disks removed (this is an ordinary one), or a once punctured annulus, or twice punctured disk. A pair of pants has the Euler characteristic 1 . A marked Riemann surface (S; w) representing p 2 Tg ;n is decomposed into a union of hyperbolic pants by the system of m simple closed

30

Yukio Matsumoto

geodesics hc1 ; : : : ; cm i on S , where ci is homotopic to w(Ci ) , i = 1; : : : ; m . The Fenchel–Nielsen coordinate li (p) (> 0) is the hyperbolic length of ci , and i (p) is the amount of the “twist” on gluing pairs of pants along ci . Following [65] and [66] we measure the magnitude of twist not by the angle but by the hyperbolic length of the movement along ci . The Fenchel–Nielsen coordinates (l1 ; : : : ; lm ; 1 ; : : : ; m ) are real analytic global coordinates of Tg ;n (see [2] and [30]), which give a real analytic isomorphism m " Tg ;n Š Rm +  R . With respect to these coordinates, F () is written by l1 =


= lk = ";

lk +1 > "; : : : ; lm > ":

It immediately follows that dimR F " () = 2m

k;

where k = jj:

Note that the facet F " () always contains a factor of twist-coordinate space Rm = ¹(1 ; : : : ; m )º , because the coordinates (1 ; : : : ; m ) are free from any constraints. If another simplex  2 Cg ;n (with jj = l ) contains  as a face, then with suitable simple closed curves Ck +1 ; : : : ; Cl on Σg ;n , we have  = hC1 ; : : : ; Ck ; Ck +1 ; : : : ; Cl i:

Choosing further simple closed curves Cl +1 ; : : : ; Cm on Σg ;n , we have a maximal simplex ˜ 0 = hC1 ; : : : ; Ck ; : : : ; Cl ; : : : ; Cm i . With respect to the associated Fenchel– Nielsen coordinates, F " ( ) is written as l1 =


= lk = lk +1 =


= ll = ";

ll +1 > "; : : : ; lm > ":

Thus we have proved the following result. Lemma 4.3. If jj = k , the facet F " () is real analytically isomorphic to k m " " Rm >"  R . Furthermore, the closure F () contains F ( ) in its boundary if and only if  ˆ  (i.e.  is a proper face of  ). Given a simplex  with jj < m , there are infinitely many simplexes  with  <  . Thus F " () is surrounded by infinitely many facets F " ( ) . In this sense, a facet F " () itself is an infinite polyhedron (unless the simplex  is maximal). The following theorem is regarded as a toy analogue of the Royden–Earle–Kra theorem [57] and [21], or of the Masur–Wolf theorem [45]. This theorem is not quite necessary in order to prove our main theorem, but it gives a global perspective to our construction of orbifold-charts. Theorem 4.4. Suppose 3g 3 + n = 2 and (g; n) ¤ (2; 0); (1; 2) . Considering Tg";n as an infinite polyhedron, we have Aut(Tg";n ) = Γg ;n ; where Aut(Tg";n ) is the orientation preserving automorphism group of the polyhedron.

1 The Deligne–Mumford compactification

31

Proof. The totality of the closed facets on @Tg";n makes a complex (the facet complex) in the following sense. i . Two closed facets F " ( ) and F " ( ) are disjoint, or else intersect in a common closed facet F " () , where  = h;  i . i i . Given a closed facet F " ( ) , there are only a finite number of closed facets F " () such that F " ()  F " () . These properties are easily proved by Lemma 4.3. A flag in the facet complex means a finite sequence of the closed facets: F " (1 )  F " (2 ) 


 F " (u ):

This flag corresponds uniquely to a flag in Cg ;n : 1 <  2 <


<  u : Tg";n

Now an automorphism of induces an automorphism of the facet complex on the boundary @Tg";n , and that of the flag complex of the facet complex. Since the latter complex is (inversely) isomorphic to the flag complex of Cg ;n (which is in turn isomorphic to the barycentric subdivision of Cg ;n ), we get an automorphism of Cg ;n . By Theorem 2.1, this automorphism of Cg ;n is induced by a unique element of Γg ;n . Conversely an element of Γg ;n induces an automorphism of Tg ;n and that of Tg";n . The argument is closed, and the proof is complete. □ Essentially the same arguments have been done in A. Papadopoulos [53] and K. Ohshika [52]. Remark 4.1.

If (g; n) = (2; 0) , we have " Aut(T2;0 ) = Γ2;0 /Z2 :

This is proved by the same arguments as of Theorem 4.4, by using Corollary 2.2 instead of Theorem 2.1. Recall that, for a non-empty simplex  = hC1 ; : : : ; Ck i of Cg ;n , Γ() is the free abelian subgroup of Γg ;n generated by the right-handed Dehn twists  (Ci ) about Ci , i = 1; : : : ; k . Let N Γ() be the normalizer of Γ() in Γg ;n . Theorem 4.5. Suppose 3g 3 + n = 1 . A mapping class [f ] belongs to the normalizer N Γ() if and only if [f ] permutes the isotopy classes of the curves Ci , i = 1; : : : ; k; of  . Proof.

Let [f ] be an element of N Γ() , Ci any curve taken from  . Since  (f (Ci )) = [f ] (Ci )[f ]

1

2 Γ();

32

Yukio Matsumoto

the Dehn twist  (f (Ci )) commutes with every  (Cj ) ( Cj 2  ). It is known that if two simple closed curves C and C 0 cannot be separated by any isotopy of Σg ;n , then the Dehn twists (C ) and  (C 0 ) do not commute in Γg ;n (see Ishida [31]). Applying this to f (Ci ) , we may assume that f (Ci ) \ Cj = ;; for j = 1; : : : ; k . Let Σ0 denote the component of the (possibly non-connected) surface Σg ;n n  that contains f (Ci ) , where Σg ;n n  denotes the surface obtained by cutting open Σg ;n along the simple closed curves Cj , j = 1; : : : ; k . Claim. If an essential simple closed curve C0 in a compact, connected, oriented surface Σ0 is not peripheral, that is, not isotopic in Σ0 to any boundary curve, then there is a simple closed curve C 0 in Σ0 which intersects C0 and such that no isotopy of Σ0 can separate it from C0 . Proof of the claim. If C0 is a non-separating curve in Σ0 , then there exists a simple closed curve C 0 which transversely intersects C0 in a point. This curve C 0 has the required property. On the other hand, if C0 is a separating curve in Σ0 , then both the components Σ01 , Σ02 of the cut open surface Σ0 n C0 have negative Euler characteristic (because C0 is not peripheral). Take two points a; b on C0 , then we can find an embedded arc on each of Σ01 and Σ02 which has a; b as end points and is not isotopic (fixing ¹a; bº ) into C0 by any isotopy of the component. Joining the two arcs, we get a simple closed curve C 0 in Σ0 which intersects C0 in the two points ¹a; bº . From the construction, C 0 cannot be separated from C0 by any isotopy of Σ0 . This completes the proof of the claim. △ Now we return to the proof of Theorem 4.5. By the claim above, if f (Ci ) were not peripheral in Σ0 , then there would be a simple closed curve C 0 in Σ0 which intersects f (Ci ) and such that no isotopy of Σ0 can separate it from f (Ci ) . By [31], the Dehn twist  (f (Ci )) does not commute with the Dehn twist (C 0 ) . Every Dehn twist  (Cj ) , however, commutes with  (C 0 ) , because Cj is disjoint from C 0 . This contradicts the assumption that  (f (Ci )) belongs to Γ( ) . Thus f (Ci ) must be isotopic in Σ0 to a boundary curve, which is a copy of some Cj . Since we took Ci from  arbitrarily, this proves that [f ] permutes the isotopy classes of curves Cj , j = 1; : : : ; k . Conversely, if [f ] permutes the isotopy classes of curves Cj , j = 1; : : : ; k , [f ] clearly belongs to the normalizer N Γ() . The proof of Theorem 4.5 is complete. □ Corollary 4.6. When Γg ;n acts on Tg";n , the subgroup which preserves a facet F " () is precisely the normalizer N Γ( ) . Proof. Recall from (1) of §1 that a mapping class [f ] maps a point p = [S; w] to [f ](p) = [S; w ı f 1 ] , and that this action does not change the Riemann surface S but changes the marking. If p belongs to the facet F " () , the set of closed geodesics ¹c1 ; : : : ; ck º on S , each ci being homotopic to w(Ci ) , is precisely the set of closed geodesics of hyperbolic length " .

1 The Deligne–Mumford compactification

33

If as assumed the image [f ](p) belongs to the same facet F " () , then the set of closed geodesics ¹c10 ; : : : ; ck0 º on S , each ci0 being homotopic to w ı f 1 (Ci ) , is precisely the set of closed geodesics of hyperbolic length " . On the same Riemann surface S , the two sets of closed geodesics must coincide: ¹c1 ; : : : ; ck º = ¹c10 ; : : : ; ck0 º:

The set of isotopy classes of simple closed curves ¹f 1 (C1 ); : : : ; f 1 (Ck )º on Σg ;n must coincide with ¹C1 ; : : : ; Ck º , and by Theorem 4.5, [f ] belongs to the normalizer N Γ() . Let us prove the converse. By Lemma 4.1, we have a general formula [f ](F " ()) = F " (f ()):

(3)

where f () = hf (C1 ); : : : ; f (Ck )i . If [f ] belongs to N Γ() , it permutes C1 ; : : : ; Ck by Theorem 4.5, and [f ] satisfies f () =  . Thus we have [f ](F " ()) = F " (f ()) = F " ();

hence [f ] preserves F " () . The proof of Corollary 4.6 is complete.

□

For a simplex  = hC1 ; : : : ; Ck i of Cg ;n , let us introduce the following group: Definition 4.2 (“Weyl group”). We denote the quotient N Γ()/Γ() by W ( ) , and for the formal resemblance would like to call it the Weyl group associated with the simplex  . Our W () is not necessarily a finite group. Corollary 4.7. Σg ;n () .

The group W ( ) is the mapping class group of the nodal surface

Proof. Let [f ] be any element of N Γ() . By Theorem 4.5, [f ] induces a permutation  on the set of the isotopy classes of Ci , i = 1; : : : ; k: Thus we may assume f (Ci ) = C(i) ; i = 1; : : : ; k: The nodal surface Σg ;n () is obtained from Σg ;n by pinching each curve Ci to a point (§3). Thus clearly, f induces an orientation preserving self-homeomorphism f˜ of Σg ;n () . Conversely, given an orientation preserving self-homeomorphism of Σg ;n () , we can find a self-homeomorphism f of Σg ;n which permutes the isotopy classes of Ci , i = 1; : : : ; k , and such that the homeomorphism f˜ induced on Σg ;n () is isotopic to the given one. Let g be a self-homeomorphism of Σg ;n which permutes the isotopy classes of Ci , i = 1; : : : ; k , then f˜ and g˜ are isotopic on Σg ;n () , if and only if f is isotopic to g on Σg ;n up to the Dehn twists about Ci , i = 1; : : : ; k , in other words, if and only if [f ] = [g] 2 N Γ()/Γ() . The proof of Corollary 4.7 is complete. □

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Definition 4.3 (fringe). Let  = hC1 ; : : : ; Ck i be a (non-empty) simplex of Cg ;n . The fringe FR" () bounded by the facet F " () is defined by def [ ı FR" () = F (): (4) 0 : The facet F " () which is written as l1 =


= lk = ";

lk +1 > "; : : : ; lm > "

moves in FR () , and converges to T () , or at worst to Ty () , as " ! 0 . This rather intuitive explanation is also justified metrically, because near T () the Weil–Petersson metric tensor G is “almost product” of two metrics: the metric along T () and the metric along Earle–Marden’s plumbing coordinates, see [43] and [22]. The plumbing coordinates of the facet F " ( ) converge to 0 as " ! 0 . (See [44], and Proposition 3 of [67].) A sharper estimate is provided by Wolpert. Let dT () be the distance on Tyg ;n to Ty () . Wolpert proved ([65], Corollary 4.10) k k  X 1/2 X  5/2 dT ( ) = 2 li +O li : i =1

i =1

(7)

1 The Deligne–Mumford compactification

35

Thus as " ! 0 , points on F " () (for which l1 =


= lk = " ) converge to points in Ty () . To be more specific, we have to use in advance the argument which will be given in the proof of Lemma 6.4 below. The argument starts with the fact that Tyg ;n is a CAT(0) space. (For the definition of CAT(0) space, see [16], [65], [66], and [67].) By Yamada [67], given a  2 Cg ;n there exists a family of geodesics in Tyg ;n with the property that any point p 2 Tg ;n is connected by a geodesic in the family to a unique point  (p) 2 Ty () such that the length of the geodesic segment between p and  (p) is equal to the distance dT ( ) (p) . This property together with Wolpert’s formula (7) assures that the family of geodesics are transverse to F " () , and that as " ! 0 F " () converges along the geodesic family to Ty () . □ Furthermore, if we use an argument supplied by Yamada (see the proof of Lemma 6.4), we can prove the following more precise statement Addendum to Lemma 4.8.

F " () approaches T () as " ! 0 .

By Lemma 4.8 and its Addendum, FR" ()[T () is a metric space embedded in the augmented Teichmüller space Tyg ;n . We have the following partition extending (6): [ Tyg ;n = Tg";n (FR" () [ T ()): (8) ;¤ 2Cg;n

This is a disjoint union. Remark 4.2. The action of the mapping class group Γg ;n on Tyg ;n preserves the partition (8). A mapping class [f ] 2 Γg ;n sends FR" ( ) [ T () to FR" (f ()) [ T (f ()) . This follows from the rule (3). In particular, we have the following formula: [f ](T ()) = T (f ()):

5. The compactness theorem Although the following theorem is well known (cf. [23], [24], [3], [41], [56], [27], and [29]), we will give our own proof in this section: Theorem 5.1. As a topological space, the quotient Tyg ;n /Γg ;n is compact, and it x g ;n . is identical with the Deligne–Mumford compactification M Hinich and Vaintrob [27], Earle and Marden [22], and Hubbard and Koch [29] gave a complex structure to Tyg ;n /Γg ;n , and established the analytic identification x g ;n . of Tyg ;n /Γg ;n with the algebraic variety M

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Yukio Matsumoto

For the sake of exposition, we will give somewhat explicit description of the moduli spaces, including facets and fringes, and will use this to derive the result. We start with the partition (8). By Definition 4.3, the fringe FR" ( ) is foliated by the facets ¹F ı ()º0  , we have the same conclusion. The proof of Lemma 6.2 is complete. □ By Lemma 6.1, the normalizer N Γ() acts on U" () and on U" () . In particular, Γ() acts on them.

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Yukio Matsumoto

Definition 6.1 (deformation space). The quotient space def

D" () = U" ()/Γ( )

is called the deformation space associated with the simplex  . Definition 6.2 (controlled deformation space). The quotient space def

D" () = U" ()/Γ()

is called the controlled deformation space associated with the simplex  . These spaces are considered as refinements of Bers’ deformation spaces, see [11] and [12]. In our previous papers [46] and [47], we constructed controlled deformation spaces in a more complicated manner, depending on 2m numbers 0 < "1 < 1 < "2 < 2 <


< "m < m < M . This was because certain similarity between controlled deformation spaces and handle-body decompositions of manifolds was taken into account. In the present paper, we have adopted simpler construction without noticing the similarity. Lemma 6.3.

The deformation space D" () is an open 2m -cell homeomorphic to (∆" )k  T ();

where ∆" is the open 2 -disk of radius " . Proof.

In terms of the Fenchel–Nielsen coordinates (l1 ; : : : ; lm ; 1 ; : : : ; m )

associated with a maximal simplex ˜ = hC1 ; : : : ; Ck ; Ck +1 ; : : : ; Cm i

which contains  = hC1 ; : : : ; Ck i , the open set U" () \ Tg ;n is described as S1 


 Sk 

m Y

(R+  R);

j =k +1

where Si denotes the strip region Q Si = ¹(li ; i ) j 0 < li < "º . The Dehn twist  (Ci ) (with i 2 ¹1; : : : ; kº ) leaves jm=k +1 (R+  R) invariant and acts on Si by  (Ci )(li ; i ) = (li ; i + li ):

(18) ∆0"

The quotient space Si /h (Ci )i is identified with pthe punctured " -disk := ∆" n ¹0º by the correspondence (li ; i ) 7! li e 2 1i /li . Thus the quotient (U" () \ Tg ;n )/Γ() is identified with (∆0" )k

m Y  (R+  R): j =k +1

1 The Deligne–Mumford compactification

45

Filling the punctures of ∆0" ’s by the continuous extension (15), we get U" ()/Γ() = (∆" )k 

m Y

(R+  R):

j =k +1

Q The continuous extension (15) assures that the factor jm=k +1 (R+  R) gives coordinates of ¹0º  T () . Since T () is homeomorphic to an open (2m 2k) -cell, D" () is homeomorphic to an open 2m -cell. The proof of Lemma 6.3 is complete. □

Remark 6.1. Comparing lemmas 6.3 and 5.2 we see that the factor Cone" (T k ) of the quotient of (FR" () [ T ())/Γ() is embedded in the factor (∆" )k of the deformation space D" () as the cone over the “corner torus” (@∆" )k . The quotient group W ( ) = N Γ()/Γ() acts on D" () and on D" () . Lemma 6.4.

The Weyl group W ( ) acts on D" () properly discontinuously.

Proof. A natural argument would be to appeal to Yamada’s result [67] that the Weil–Petersson completion Tyg ;n is a CAT(0) space and Ty () is a complete convex subset in Tyg ;n . (For the definition of CAT(0) space, see [16] Chapter II.1, [65], §4, and [66], Chapter 5.) This implies the following (see [67], p. 342, and [16], Proposition 2.4). i . For every p , there exists a unique point  (p) 2 Ty () such that d (p;  (p)) = d (p; Ty ()) , where d (
;
) is the distance. i i . If p 0 belongs to the geodesic segment connecting p and  (p) , then  (p 0 ) =  (p) . i i i . Given p … Ty () and q 2 Ty () , if q ¤  (p) then the Alexandrov angle (as defined in [16], p. 9) † (p) (p; q) = /2 . These properties provide us with a “brush” of geodesics with the core Ty () . This brush structure is invariant under the action of N Γ() . Confining our attention to geodesic segments of short lengths, say ı > 0 , attached to T () , and passing to the quotient space by Γ() , we obtain a region Bı in D" () which has the structure of a brush with short geodesic fibers attached to T () . The region Bı is equipped with the projection  W Bı ! T () along the geodesic fibers, which is equivariant under the action of W () . Being the mapping class group of the nodal surface Σg ;n () (Corollary 4.7), the group W () acts on T () properly discontinuously. The existence of the equivariant projection  W Bı ! T ( ) assures that its action on Bı is also properly discontinuous. (In the above argument, we changed the core of the brush from Ty () to T () . The geometric justification of this passage is provided by the fact that in Tyg ;n the stratum T () “meets” another T ( 0 ) along Ty ( [  0 ) ( Ty () \ Ty ( 0 )) at right angles if  „  0 nor

46

Yukio Matsumoto

 0 „  . See [66], Chapter 6, §4. The shortest geodesics attached to the points of T () meet T () at right angles. See property ( i i i ) above. Thus the domain of the geodesic projection  W Tyg ;n /Γ() ! Ty () is “stratified” according to the stratification of the target Ty () , and we can regard D" () as contained in the domain of the projection whose target is T ( ) . The author learned this argument from Sumio Yamada.) Under the free action of Γ() , the complement (D" () n Bı/2 ) is a quotient of a sub-region of U" ( ) on which N Γ() acts properly discontinuously. Thus W () = N Γ()/Γ() acts on (D" () n Bı/2 ) properly discontinuously. Since D" () = (D" () n Bı/2 ) [ Bı , we get the assertion of Lemma 6.4 by combining the two results above. □

Theorem 6.5 (Hubbard and Koch [29]). The deformation space D" () is a complex m -manifold homeomorphic to an open 2m -cell. This theorem is essentially Theorem 10.1 of Hubbard and Koch [29], because their space QΓ (see §7 of [29]) is almost the same as our deformation space D" () . The only difference is that their QΓ has no restriction on the “size” in the directions which are normal to the core T () , while our D" () has the size restriction " . Hubbard and Koch proved that QΓ is a complex manifold. Their main concern is to define a complex structure in a neighborhood of a point p = (0; u) of ¹0º  T () (in our notation). Their complex coordinates near the point p are (like [44], §2, [63], §4, [64], §2, and [22], §13) the product of the plumbing coordinates (cf. Earle and Marden [43] and [22]) and the coordinates in an open set U  T () . More specifically, they started from a  -marked family2 of nodal surfaces Y (0; u) over ¹0º  U , and constructed a plumbed family Y = Y (t1 ; : : : ; tk ; u) over (∆ı )k  U , where (t1 ; : : : ; tk ) are plumbing coordinates, each ti belonging to an open ı -disk ∆ı (with a small ı > 0 ). Let (l1 ; : : : ; lm ; 1 ; : : : ; m ) be the Fenchel–Nielsen coordinates associated with a maximal simplex ˜ containing  2 Cg ;n . The fiber of Y ! (∆ı )k  U over a point (t1 ; : : : ; tk ; u) has its position in D" () described by the “extended” Fenchel–Nielsen coordinates (see the proof of Lemma 6.3) (l1 e 2

p

11 /l1

; : : : ; lk e 2

p

1k /lk

; lk +1 ; : : : ; lm ; k +1 ; : : : ; m ):

The map Φı ;U W (∆ı )k  U ! D" () which sends (t1 ; : : : ; tk ; u) to (l1 e 2

p

11 /l1

; : : : ; lk e 2

p

1k /lk

; lk +1 ; : : : ; lm ; k +1 ; : : : ; m )

is continuous ([29], Proposition 9.1). Hubbard and Koch ([29] §9) proved that if we take ı and U sufficiently small, Φı ;U is a topological embedding, and stratumwise it is analytic. Using Φı ;U as a local chart, they put the complex coordinates (t1 ; : : : ; tk ; u) to the open neighborhood Φı ;U ((∆ı )k  U ) of p . Note 2 A  -marking means a marking modulo Γ() . See the proof of Lemma 6.6 below.

1 The Deligne–Mumford compactification

47

that the restriction Φı ;U j¹0º  U is the identity id¹0ºU . Therefore their complex structure about p = (0; u) is a product (∆ı )k  U . The point p was arbitrarily taken from ¹0º  T () , thus as a consequence, D" () is a complex m -manifold, and T () is a complex submanifold of D" () . On the other hand, we already proved that D" () is homeomorphic to an open 2m -cell (Lemma 6.3). Remark 6.2 (Bers’ conjecture). Bers ([11], p. 47) made an announcement to the effect that, in our notation, D" () is a bounded domain in Cm . But he did not give any proof. We would like to call this statement Bers’ conjecture. When introducing the complex structure to D" () as above, if we could take U = T () and thus D" () would contain an open submanifold biholomorphic to (∆ı )k  T () , then by taking an "0 > 0 smaller than " , we would be able to show that D"0 () is a bounded domain. Hubbard and Koch, however, gave a warning that we could not take U = T () , ([29], Remark 8.1). This implies that their construction does not give a bounded domain D" () . Earle and Marden ([22], §13, Theorem II) states that D" () can be topologically embedded in (∆ı )k  T () by using the plumbing coordinates. If this is the case, D" () would again be a bounded domain. But unfortunately, any proof of their Theorem II has not yet been published (except for the case of a maximal simplex  2 Cg ;n , for which Kra [39] has proved the corresponding result: if  is maximal, D" () is a bounded domain). There is a bad news: Hinich [26], p. 152, claimed that his result contradicts a partial consequence of Earle–Marden’s theorem stated as a corollary on p. 346 of [43]. Wolpert [64] proved the following estimate: at a point p = (0; u) 2 ¹0º  T () and at a node pi on the singular fiber Xp of Y (0; u) , let ti be the plumbing coordinate with which we open up the node pi , and let li be the length of a simple closed geodesic which appears on the opened up smooth fiber X ti , then Wolpert ([64]) gave the estimate li = 2 2 ( log jti j)

1

+ O(log jti j)

2

:

If the second term O(log jti j) 2 is a uniform estimate with respect to the position of p 2 ¹0º  T () , then we would find some ı such that Φı ;T () ((∆ı )k  T ())  D" ():

Furthermore, in this case, if we take smaller "0 such that D"0 ()  Φı ;T ( ) ((∆ı )k  T ()) , D"0 () would be a bounded domain, and Bers’ conjecture would follow. But in a discussion with the author, Wolpert ascertained that the second term O(log jti j) 2 is not a uniform estimate with respect to p . Thus the above argument fails. In conclusion, it seems that at present there is no complete proof of Bers’ conjecture.

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Lemma 6.6.

The action of W () on D" ( ) is holomorphic.

Proof. This follows from the universality of the marked family of stable curves constructed by Hubbard and Koch [29]. They constructed a Γ -marked proper flat family of stable curves with sections YΓ ! QΓ ([29], §10). Their Γ is nothing but our  = hC1 ; : : : ; Ck i (see Definition 1.4 of [29]), and they defined QΓ by [ QΓ := UΓ /∆Γ ; where UΓ := SΓ0  Tyg ;n : Γ0 Γ

(See §7 of [29].) Their ∆Γ is our Γ() ([29] §0.1), and SΓ is nothing but our T () ([29], §2.1). Since discussions with two different systems of notation are inconvenient, let us follow their arguments using our own notation. Given a simplex  = hC1 ; : : : ; Ck i 2 Cg ;n , we define [ def U () = Tg ;n T ()  Tyg ;n :  0 . Let KX denote the canonical bundle of X .

2 Complex geometry of Teichmüller domains

65

A holomorphic quadratic differential on X is a holomorphic section of KX ˝KX . Let Q(X ) be the space of holomorphic quadratic differentials on X . A Beltrami differential on X is an L1 -section of KX 1 ˝ KxX . We denote the space of Beltrami differentials on X by BD(X ) . In a holomorphic chart U  X , an element of BD(X ) has the form d z¯ dz 1 where  2 L (U ) is called a Beltrami coefficient. By the Uniformization Theorem, the universal cover of the surface X is the upper half plane H , and we have 

X = H/Γ

for Γ a Fuchsian group, namely a discrete subgroup of Aut(H) = PSL2 (R) . A quasiconformal map between two Riemann surfaces X and Y is a homeomorphism f W X ! Y with weak partial derivatives (in the sense of distributions) that are locally square-integrable, such that the Beltrami coefficient =

fz¯ fz

(1)

satisfies kk1 < 1 . Such a map lifts to a quasiconformal map f˜W H ! H

that extends to a homeomorphism from H = H [ R to itself. The corresponding ˜ then lies in the open unit ball of the complex Banach space Beltrami coefficient  of bounded Beltrami coefficients ˜ 2 L1 (H)W  ˜ is Γ BDΓ (H) = ¹

invariantº:

From this perspective, one can define Teichmüller space by considering quasiconformal maps from a fixed basepoint, up to homotopy (or isotopy) of the maps relative to the punctures: Tg ;n = ¹(Y; k)W Y is a Riemann surface, kW X ! Y is quasiconformal mapº/ 

where (Y; k)  (Z; h) if and only if there is a conformal homeomorphism cW Y ! Z such that the composition f =h

1

ı c ı kW X ! X

is a quasiconformal map that lifts to a map f˜W H ! H that extends to the identity map on R . Since X is of finite analytical type, the last condition is equivalent to the map f being homotopic to the identity map. The foundation for the complex-analytic theory of Teichmüller spaces is the Measurable Riemann Mapping Theorem (see [4] for an exposition).

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y be the Riemann sphere and let Theorem 2.1 (Ahlfors, Bers, and Morrey). Let C 1 y  2 L (C) satisfy kk1 < 1 . Then there is a quasiconformal map y !C y f W C

with Beltrami coefficient  , that is, satisfying the Beltrami equation (1). Such a solution is unique up to a post-composition by a Möbius transformation y). A 2 Aut(C y ) , namely, Moreover, the solutions vary analytically in the parameter  2 L1 (C y for any fixed z 2 C , the assignment z 7 ! f  (z)

is analytic in  .

3. The Bers embedding 3.1. Construction of the embedding As earlier, we fix a Riemann surface X = H/Γ . In this section we describe the Bers embedding BX W Tg ;n ! C3g 3+n : (2) We refer to the image as the Bers domain. The fact that the complex structure on Tg ;n acquired via this embedding does not depend on the choice of a basepoint X is due to the analytical dependence of parameters in Theorem 2.1. The starting point is the simultaneous uniformization theorem of Bers ([7], see Theorem 3.3) which associates to a point (Y; g) 2 T (X ) a quasiconformal map y !C y FWC that fixes ¹0; 1; 1º , and conjugates the action of Γ to the action of a Kleinian y ) = PSL2 (C) , such that group ΓY < Aut(C • F is conformal on H , where H˙ denotes the upper/lower half-plane respectively, and • ΓY leaves invariant the images of H and H+ with quotients Xy and Y respectively. The idea of the proof of the existence of such a quasiconformal map F is simple: lift the Beltrami coefficient (g) to the upper half plane H+ , and extend y by defining it to be identically zero on H . ˆ on C to a Beltrami coefficient  The above map F is then the normalized solution to the Beltrami equation with ˆ . Since  ˆ is Γ -invariant, we have, by the uniqueness of Beltrami coefficient  solutions (see Theorem 2.1) that F ı = ( ) ı F

for each 2 ΓY ;

where W 1 (X ) ! PSL2 (C) has an image ΓY , a Kleinian group.

(3)

2 Complex geometry of Teichmüller domains

67

ˆ H  0 , and is moreover univalent as it is Note that F jH is conformal as j the restriction of a homeomorphism. Recall that the Schwarzian derivative of a univalent function f W H ! C is the meromorphic quadratic differential  f 000 3  f 00 2  S (f )(z) = dz 2 (4) f0 2 f0

that measures the deviation of f from being Möbius. In particular, S (f ) vanishes y). identically if and only if f 2 Aut(C By the equivariance (3), and the property that S(A ı f ) = S (f )

y ); for any A 2 Aut(C

the Schwarzian derivative of F jH thus defines a ΓY -invariant quadratic differential that descends to a meromorphic quadratic differential Y on the Riemann surface Xy . The Schwarzian derivative has poles of order at most one, and by Riemann–Roch, y ) Š C3g 3+n . The we know that the space of such quadratic differentials Q(X assignment BX (Y ) = Y defines the map in (2). The fact that this is well defined relies on the observation that if g and h are homotopic quasiconformal maps, then the corresponding maps F1 jH+ and F2 jH+ determine the same maps on R that is the common boundary of both half-planes H and H+ . The conformal maps F1 jH and F2 jH consequently determine the same Schwarzian derivatives. A similar argument proves that BX is an embedding: if the Schwarzian derivatives Y1 and Y2 agree on H , then F1 jH and F2 jH must differ by an automorphism of H that fixes ¹0; 1; 1º on the boundary R ; consequently F1 jR = F2 jR and considering the restrictions F1 jH+ and F2 jH+ , we see that the surfaces Y1 and Y2 are equivalent in Tg ;n . The Bers embedding we just described has the following two key properties, that we shall now briefly discuss. BX is bounded. We equip the vector space Q(H) of holomorphic quadratic differentials on H with the norm kq(z)dz 2 k = sup 4=(z)2 jq(z)j: z 2H

The invariance of the 1 -form

dz =(z)

under the Möbius group implies that the norm y ). is invariant under the Fuchsian group Γ and determines a finite norm on Q(X With respect to this norm, Theorem 3.1 (Nehari). y ). Q(X

The image of BX is contained in the ball B(0; 3/2) 

68

Subhojoy Gupta and Harish Seshadri

The proof of this is a short computation involving (4) and an application of the classical Area Theorem; the reader is referred to [19] or [4] for the details. BX is open. For a quadratic differential q(z)dz 2 2 Q(H) , define the Beltrami coefficient q on C by q (z) = 2y 2 q(¯ z) (5)

on H+ and identically zero on H . Let f q be the solution of the Beltrami equation as in Theorem 2.1. Then, Theorem 3.2 (Ahlfors and Weill [3]). If kqk < 1/2 , the Schwarzian derivative of the restriction of f q to the upper half-plane is q . The idea is to consider the two linearly independent solutions 1 ; 2 of the Schwarzian equation: 1 00 + q = 0 (6) 2 on the lower half-plane, whose ratio f = 1 /2 has Schwarzian derivative q . It can be checked that f is injective, and the map F (z) =

1 (z) + (¯ z 2 (z) + (¯ z

z)01 (z) z)02 (z)

defines an extension to the upper half-plane which is quasiconformal (this uses the bound on the norm of q ) having Beltrami coefficient (5). See Chapter VI.C of [4] for the details. An equivariant version of this result defines an inverse to the Bers embedding y ) . This construction works for (smaller) BX in an open ball B(0; 1/2)  Q(X neighborhoods of an arbitrary point in the image, showing that the image of the embedding is open.

3.2. Structure of the Bers boundary In what follows, a Bers domain shall be the image of any Bers embedding BX ; the geometry of this domain in complex space can be studied by analyzing the construction sketched in the previous section. First, the images of the upper and lower half-planes H˙ by the quasiconformal map F in (3) are quasidisks. The image group ΓY < PSL2 (C) of the representation  leaves these invariant, and the quotient of each yields the Riemann surfaces Xy and Y respectively. Such a representation is called quasi-Fuchsian and thus Bers proved what is known as the Simultaneous uniformization theorem. Theorem 3.3 (Bers [7]). The space of quasi-Fuchsian representations QF(1 (X )) is homeomorphic to Tg ;n  Tg ;n .

2 Complex geometry of Teichmüller domains

69

The Bers embedding BX as in (2) is thus a slice of the subset QF(1 (X )) of the PSL2 (C) representation variety (when the conformal structure on one of the factors is kept fixed at X ). The image of a quasi-Fuchsian representation ΓY 2 QF(1 (X )) is a Kleinian group, a discrete subgroup of PSL2 (C) that acts freely and properly-discontinuously on H3 with quotient a hyperbolic 3 manifold M . Topologically, M is the interior of the product S I of a surface and an interval, where S is the underlying topological surface of X , and the boundary components S  ¹0; 1º acquire a conformal structure from the action of Γ at the sphere at infinity @H3 . It is well known that discrete faithful representations form a closed subset, denoted by AH(S ) , of the PSL2 (C) representation variety; in particular, the representations that arise in the boundary of a Bers slice correspond to hyperbolic 3 -manifolds with cusps, when the conformal structure on either boundary degenerates by pinching a curve, or degenerate “ends”, when the conformal structure degenerates by pinching an “ending lamination” (see [35] for a survey). Thus points in the Bers boundary can be studied using tools from hyperbolic geometry; see §3.2 for some results for the case when S is a punctured torus. As an example, Sullivan’s theorem on rigidity of the “totally degenerate” hyperbolic 3-manifolds yield the fact that such boundary points are peak convex, that is, Theorem 3.4 (Theorem 2 in [45]). Let f W ∆ ! @BX be a holomorphic map such that f (0) is a totally degenerate boundary point. Then f is constant. In contrast, at any non-maximal cusp, the remaining moduli of the conformal boundary parametrize an open set in @BX . The simplest Teichmüller spaces are those of the punctured torus and the four-punctured sphere, as they have complex dimension 1 . Much is known about the Bers embedding in these cases. See [25] and [24] for visualizations, computational aspects, which rely on solving the Schwarzian equation numerically and then checking for discreteness of the resulting holonomy. We highlight here some results for the case when X is a punctured torus. • The Bers domain BX is a Jordan domain; this is a consequence of the deep work of Y. Minsky in [35] that proves the Ending Lamination Conjecture for punctured-torus groups. The boundary circle can be identified with projective classes of measured laminations on the punctured torus. • H. Miyachi proved a folklore conjecture in [36] showing that the rational points on the Bers boundary of BX are 2/3 cusps; in particular, though a Jordan domain, it is not a quasi-disk. • Boundary points exhibit spiralling behavior, which is described in work of D. Goodman in [17]. The last two papers crucially use Minsky’s Pivot Theorem in [35] and study the behaviour of rational pleating rays introduced by L. Keen and C. Series in [21]. The following question is still open.

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Question 3.5 (Canary in [11]). What is the Hausdorff dimension of the boundary of BX when X is a once-punctured torus? For Bers domains of higher dimensions, very little is known. In particular, it is not known whether the closure is homeomorphic to a closed ball. In fact, the following is still open. Conjecture 3.6 (Bromberg in [10]). The closure of a Bers domain of dimension greater than one in QF(S ) is not locally connected. One can also raise the following question. Question 3.7. What does the tiling of the Bers domain by fundamental domains for the mapping class group action look like? In particular, for a fixed fundamental domain F , does the Euclidean diameter of i
F tend to zero for a diverging sequence i ! 1 (for i ! 1 ) of mapping classes in MCG(S ) ?

4. Invariant metrics on Teichmüller domains 4.1. The Kobayashi and Carathéodory metrics The Bers domain is often studied in terms of metrics invariant under the automorphism group of the domain. We will focus on the Kobayashi and Carathéodory metrics, which are complete Finsler metrics intimately related to issues of Euclidean convexity of the domain. On any bounded domain Ω in CN , the infinitesimal Kobayashi metric is defined by the following norm for a tangent vector v at a point X 2 Ω : KΩ (X; v) =

inf

hW∆!Ω

jvj ; jh0 (0)j

(7)

where the infimum is over all holomorphic maps hW ∆ ! Ω such that h(0) = X and h0 (0) is a multiple of v . The Kobayashi metric dΩK on Ω is then the distance defined in the usual way: one first defines lengths of piecewise C 1 curves in Ω using the above norm and then takes the infimum of lengths of curves joining two given points to get the distance between them. The Carathéodory metric dΩC on Ω is defined by dΩC (p; q) = sup¹dH (f (p); f (q))º;

where dH denotes the Poincaré metric on ∆ and the supremum is over all holomorphic maps from Ω to ∆ . Note that dH is the distance function associated ˝dz to the Kähler metric 4 (1dz jzj 2 )2 on ∆ .

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The following is immediate from the definitions. For any p; q 2 Ω , dΩC (p; q)  dΩK (p; q)

(8)

and we have the contracting property dΩK2 (f (p); f (q))  dΩK1 (p; q)

and dΩC2 (f (p); f (q))  dΩC1 (p; q)

(9)

for any holomorphic map f W Ω1 ! Ω2 between the domains. Remark 4.1. The Kobayashi and the Carathéodory metrics coincide on the unit ball Bn  Cn . Moreover, they are equal to the norm of the complex hyperbolic K metric on Bn . In particular, d∆ is the Poincaré (hyperbolic) metric on the ball.

4.2. Complex geodesics Let Ω  Cn be a bounded domain. A complex geodesic in Ω is a holomorphic and isometric embedding  W (H; dHK ) ! (Ω; dΩK ):

As noted above, the Kobayashi dHK on H is the distance function of the hyperbolic metric, and is isometric to the Poincaré disk (∆; dH2 ) . It is a remarkable fact, due to Lempert [29] and generalized by Royden and Wong [43], that complex geodesics exist in abundance when Ω is a bounded convex domain. Theorem 4.1 (Lempert [29] and Royden and Wong [43]). Let Ω  Cn be a bounded convex domain. i . For any p; q 2 Ω there exists a complex geodesic  W H ! Ω with  (0) = p and (z) = q for some z 2 H . i i . For any p 2 Ω and v 2 Cn there exists a complex geodesic W H ! Ω with  (0) = p and  0 (0) = tv for some t > 0 . This was proved for domains with C2 -smooth strongly convex domains by Lempert and generalized to arbitrary convex domains by Royden and Wong. Apart from convex domains there are very few domains for which the existence of complex geodesics with prescribed data is known. Interestingly, this is known to be true for Teichmüller domains. Theorem 4.2 (Earle, Kra, and Krushkal [12]). Let Ω  Cn be a Teichmüller domain. Then the conclusions of Theorem 4.1 hold. We will describe complex geodesics in Teichmüller space in more detail in the next subsection. Returning to convex domains, more is known about complex geodesics, in addition to their existence. In fact, Lempert and Royden–Wong–Krantz proved the following theorem.

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Theorem 4.3 (Lempert [29] and Royden, Wong, and Krantz [43]). Let Ω  C n be a bounded convex domain. For every complex geodesic  W H ! Ω there is a holomorphic retract ΦW Ω ! Ω for  , i.e., there is a holomorphic map ΦW Ω ! Ω such that Φ(Ω)   (H) and Φ(z) = z for all z 2  (H) . By composing Φ with  1 we get a map, which we continue to denote by Φ , from Ω to H . It is not difficult to see that Φ is an extremal map for the Carathéodory metric and, in fact, we have Corollary 4.4.

If Ω  Cn is a bounded convex domain, then dΩK = dΩC .

Theorems 4.1 and 4.2 prompt a natural question: can a Teichmüller domain be bounded and convex? On the other hand, an analogue of Theorem 4.3 and Corollary 4.4 was not known to hold for Teichmüller space. In fact, Markovic proved recently that Corollary 4.4 fails for Teichmüller space, thereby answering the question in the negative. We will discuss his work and related issues in later sections.

4.3. The Teichmüller metric It turns out that the Kobayashi metric on Tg ;n can be described in terms of extremal quasiconformal maps. In fact, the following definition of the Teichmüller metric predates that of the Kobayashi metric. The Teichmüller distance between two marked surfaces X and Y is defined by 1 dT (X; Y ) = inf ln K(f ) (10) 2 f where the infimum is taken over quasiconformal homeomorphisms preserving the marking and fixing the punctures, and 1 + kk1 K(f ) = 1 kk1 where  , as in (1), is the quasiconformal dilatation of f . The infimum is, in fact, attained by a Teichmüller map ΨW X ! Y . It turns out that such a map is an affine stretch in local Euclidean charts on X associated to a holomorphic quadratic differential q on X . Indeed, if the quadratic differential q can be expressed as d  2 in a local chart given by the complex coordinate  = x + iy , then a Teichmüller map can be written as Ψ() = K 1/2 x + iK

1/2

y;

where K is the minimum dilation in (10). A remarkable feature of the Teichmüller metric is that it admits a geometric description of geodesic paths: namely, the one real-parameter worth of such affine stretches Ψ t () = e t x + ie t y (11)

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yields a family of surfaces Y t that lie along a (parametrized) Teichmüller geodesic ray. These, in fact, extend to complex geodesics, as defined earlier, by allowing a complex parameter  = t + i 2 H , in which case Ψ () = e t cos x + ie

t

sin y

(12)

in each coordinate chart. The locus of Riemann surfaces obtained by varying  2 H is then a totally geodesic disk in Teichmüller space called a Teichmüller disk. The induced metric on the disk is then the Poincaré (hyperbolic) metric on H . Alternatively, a holomorphic quadratic differential q 2 Q(X ) gives rise to a holomorphic embedding D(q)W H ! BD(X ) defined by 1  jqj Dq () = : 1+ q On the other hand, as in §2, we have a natural holomorphic map W BD(X ) ! Tg ;n which associates to  the quasiconformal map [] with complex dilatation  . It can be checked that the composition  ı D(q)W H ! Tg ;n coincides with the Teichmüller disk above. Moreover, such a disk is a complex geodesic as defined in the previous section (§4) (See for example, §9.5 of [28], for more on their construction and properties). As a side note, we point out a recent result of S. Antonakoudis [5] that any such isometric embedding of (D; d ) in (Tg ;n (X ); dT ) is necessarily holomorphic or anti-holomorphic. By the work of Royden, we have the following result. Theorem 4.5 (Royden [42]). The Kobayashi and Teichmüller metrics coincide. Remark 4.2. The above construction in fact gives a Teichmüller disk D through any point and any (complex) direction; using the fact that it is an isometric embedding, the contracting property of the Kobayashi metric (see (9)) implies that d K  dT ; the main work is to prove the reverse inequality, see [12] or Chapter 7 of [16] for a proof using the Slodkowski extension theorem. In what follows, let S be a closed orientable surface of genus g  2 ; the discussion extends to the case of a hyperbolic surface with punctures which we elide. The mapping class group MCG(S) is the group of self-homeomorphisms of S up to homotopy. It is immediate from the definitions that each element of MCG(S ) is a holomorphic isometry of (Tg ; dT ) . It is a fact, due to Royden again, that these are essentially all the isometries. Theorem 4.6 (Royden [42]). Isom(Tg ; dT ) = MCG(S ) when g  3 . For g = 2 , the natural homomorphism MCG(S ) ! Isom(Tg ; dT ) has a kernel of order two, generated by the hyperelliptic involution (that acts trivially on Tg ). Since biholomorphic mappings are isometries for the Kobayashi metric, this implies the followig result.

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Corollary 4.7. Aut(Tg ) equals MCG(S ) when g  3 , and when g = 2 , equals its quotient by the order-two subgroup generated by the hyperelliptic involution. Note that by (12), Teichmüller disks arise as orbits of the linear group SL2 (R) which act linearly on the local  = x + iy coordinates. A full account of the dynamics of this action is beyond the scope of this article (see [53] for a readable survey), however we mention the following fundamental ergodicity result which we will use later in the paper. We note first that the identification of the cotangent space TX Tg with the vector space Q(X ) of holomorphic quadratic differentials on X , combined with (11) gives rise to the Teichmüller geodesic flow Ψ t on the unit cotangent bundle T1 Tg . Theorem 4.8 (Masur [32] and Veech [50]). There is a MCG(S ) -invariant measure on the unit cotangent bundle T1 Tg such that 1. the Teichmüller geodesic flow  t on T1 Tg is ergodic, and 2. the induced measure on Tg is equivalent to the top-dimensional Hausdorff measure of dT . We end this section by remarking that there are other invariant metrics on Teichmüller which are important in other settings and have been extensively studied, notably the Weil–Petersson metric, which is an incomplete Kähler metric on Tg ;n (for a survey see [52]). For comparisons between various metrics, see [55] or the series of papers of Liu, Sun, and Yau (for example [30]).

5. Comparison with products and bounded symmetric domains It is known that Teichmüller space endowed with the Teichmüller metric dT is not negatively curved, even in a “coarse” sense. More precisely, it is not Gromov hyperbolic (see [33]). On the other hand, quite remarkably, Tg ;n exhibits some features of complex structures of strictly negatively curved Kähler manifolds. In particular, one has the following “rigidity” result. Theorem 5.1 (Corollary 1 of [48]). Let f W D  D ! Tg ;n be a holomorphic map such that f jD ¹wº W D ! Tg ;n is proper for some w 2 D . Then f (z; w1 ) = f (z; w2 ) for all z; w1 ; w2 2 D . An immediate result is the following corollary. Corollary 5.2 ([48] and [37]). manifolds.

Tg ;n is not biholomorphic to a product of complex

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The proof of Theorem 5.1 uses the interpretation of the Bers domain as a subset of quasi-Fuchsian characters; here we provide a sketch of the proof. Recall from §3 that the “totally degenerate” points on the Bers boundary are of full measure; this remains true for the boundary limits of any holomorphic disk as in (10) with respect to the Lebesgue measure on the boundary circle (such limits exist in almost every radial direction by Fatou’s theorem). However, as a consequence of Sullivan rigidity, totally degenerate boundary points do not admit holomorphic deformations (cf. Theorem 3.4). The proof of the rigidity result can then be derived from the abundance of such rigid boundary points, and an application of F. and M. Riesz’s theorem that says that if a holomorphic map on the disk extends to a map that is constant almost everywhere on the boundary, then it is constant. When g = 1 , the Teichmüller space T1 is biholomorphic to the upper half-plane H2 , so its geometry is that of this symmetric space. Indeed, the factor 12 in the definition of the Teichmüller metric in (10) is to ensure that it coincides with the hyperbolic metric. When g  2 , the discreteness of the automorphism group (Corollary 4.7) immediately implies that Tg is never biholomorphic to a bounded symmetric domain. Even stronger still is the result of Farb–Weinberger who showed that the isometry group of any MCG(S) -invariant complete finite co-volume smooth Finsler metric on Tg has to be a finite extension of MCG(S) if g  3 (see [13]). In this connection we also note the result of S. Antonakoudis who proved in [6] that, except for the torus case, there are no holomorphic isometric embeddings of Teichmüller space into a bounded symmetric domain (or vice versa). However, the analogy with symmetric spaces continues to shed light on the geometry of Teichmüller space and its compactifications (see, for example, [20] for an account of the latter).

6. Convexity and pseudoconvexity Pseudoconvexity of Teichmüller domains. The notion of “pseudoconvexity” of a domain (or, more generally, a complex manifold) is fundamental in complex geometry, as it is closely related to the notion of a “domain of holomorphy” or a “Stein manifold” (see, for example, the insightful survey of Siu in [46]). We provide a brief discussion here. A domain Ω  Cn is said to be strictly (or strongly) pseudoconvex if there is 2 a C -smooth function 'W Cn ! R such that 1. Ω = ¹z 2 Cn W '(z) < 0º , 2. '

1

(0) = @Ω and r' ¤ 0 on @Ω ,

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3. the following convexity property is satisfied at any p 2 @Ω : the Hermitian quadratic form (the Levi form) defined by L' (p; v) =

n X i; j =1

@2 ' (p)vi v¯j @zi @¯ zj

is positive for all v 2 Cn n¹0º such that vN := hrz¯ '(p); vi = 0 . Here
rz¯ '(p) = @' ; : : : ; @' . @¯ z @¯ z Note that the condition vN = 0 is equivalent to the requirement that v lies in the maximal complex subspace of Tp (@Ω) . It can be shown that the above condition can be restated as follows: D is strictly pseudoconvex at p 2 @D if there exists a neighborhood U of p in Cn such that @D \ U is C2 -smooth and a biholomorphism ΦW U ! Φ(U )  Cn such x is strictly convex at p , that is, there is a local defining function that Φ(U \ D) for the boundary defined near p that has a positive-definite Hessian. n More generally, S1a bounded domain D  C is pseudoconvex if it can be written as a union D = i =1 Di where each Di is strictly pseudoconvex and Di  Di +1 . Note that we do not require @D to be C2 -smooth to define pseudoconvexity. It is clear that pseudoconvexity is invariant under biholomorphisms between domains. Examples of pseudoconvex domains include convex domains: this can be seen, for instance, by constructing a C1 -smooth strongly convex exhaustion function on such a domain (see Theorem 2.40 of [27]). The pseudoconvexity of Bers domains was established by Bers and Ehrenpreis in [9] (see also [44] for the stronger property of “polynomial convexity”). In [51], Wolpert proved that the geodesic length function of a “filling” system of curves (i.e. one whose complement is a disk) is a smooth proper, strictly plurisubharmonic exhaustion function on Tg ;n (plurisubharmonic means that the function has a positive definite Levi form as in condition (3) at every point). Another such exhaustion was provided by A. J. Tromba in [49], where he showed that the Dirichlet energy of a harmonic diffeomorphism hW Y ! X between hyperbolic surfaces, as Y varies in Tg ;n , is strictly plurisubharmonic. In [54] S.-K. Yeung constructed bounded smooth strictly plurisubharmonic exhaustion functions. In [38] Miyachi constructed other plurisubharmonic exhaustions by considering extremal lengths of filling pairs of measured foliations. Convexity of Teichmüller domains. Unlike pseudoconvexity, Euclidean convexity of domains is not natural in the setting of complex geometry, as it is not preserved under biholomorphisms. Nevertheless, there are several important results about the complex geometry of such domains. In particular, one has the striking results of Lempert, which we touched upon in §4.2. A basic question regarding the geometry of Teichmüller domains is the following folklore conjecture, recently proved by Markovic.

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Conjecture 6.1. The Teichmüller space of a closed surface of genus g  2 is not biholomorphic to a bounded convex domain Ω  C3g 3 . As described in S.-T. Siu’s survey [47], Conjecture 6.1 fits in the context of the following theorem of S. Frankel. Theorem 6.2 (Frankel [15]). Let Ω  Cn a bounded convex domain. If there is a discrete subgroup Γ  Aut(Ω) acting freely on Ω such that Ω/Γ is compact, then Ω is biholomorphic to a bounded symmetric domain. As for Tg , it is known that there is a discrete subgroup Γ  Aut(Tg ) acting freely on Tg such that the quotient Tg /Γ has finite Kobayashi volume. However Tg /Γ is noncompact and a generalization of Frankel’s result to the case of convex domains with finite-volume quotients is not known. By Corollary 4.4, the existence of a convex embedding of Tg would imply that the Kobayashi and Carathéodory metrics coincide, that is, we have dTK = dTC . In fact, it was proved by I. Kra [26] that these two metrics agree on Teichmüller disks associated to Abelian differentials. The recent result of Markovic alluded to above is the stronger fact that: Theorem 6.3 (Markovic [31]).

If g  2 , we have dTK ¤ dTC on Tg .

In particular, Tg cannot be biholomorphic to a bounded convex domain, answering Conjecture 6.1. Around the same time, the authors of the current article proved the following local property of the boundary of a Teichmüller domain. We say that a domain Ω  Cn is locally convex at p 2 @Ω if Ω \ B(p; r) is convex for some r > 0 , where B(p; r) denotes the Euclidean ball with center p and radius r . Moreover, it is locally strictly convex if Ω \ B(p; r) is strictly convex. Theorem 6.4 (Gupta and Seshadri [18]). For g  2 , any Teichmüller domain for Tg cannot have any locally strictly convex point on its boundary. In fact, minor modifications of the proof yield a more general statement: a Teichmüller domain cannot have a locally strictly convexifiable boundary point. By definition, if p 2 @Ω is such a point, then there exists r > 0 and a holomorphic embedding F W Ω \ B(p; r) ! C3g 3 such that F (Ω \ B(p; r)) is strictly convex. We remark that this local convexity is significantly weaker than that of global convexity of the domain; indeed, it is an elementary and well-known fact that any bounded domain with C 2 -smooth boundary has a locally strictly convex boundary

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point (see §4.1 of [18]). If we consider local convexifiability, we have a more concrete example: it is shown in [40] that the domain p D = ¹(z1 ; z2 ) 2 C2 W jz1 z1 z2 j2 +  + jz2 j2 < 1º is, for small enough  , strongly pseudoconvex, diffeomorphic to B2 and not biholomorphic to a convex domain. The strong pseudoconvexity property implies that every boundary point is locally strictly convexifiable in the above sense. One motivation for Theorem 6.4 is the technique of localization in complex analysis, that we shall elaborate on in §8. A version of that was used by Frankel in his proof of Theorem 6.2, and he observes in the paper that one can replace the assumption of global convexity of the domain with a suitable local convexity at certain orbit accumulation points. The next two sections are devoted to discussions of the above results of Markovic and Gupta and Seshadri.

7. Kobayashi ¤ Carathéodory on Teichmüller domains This section provides an extended discussion of some of the ideas involved in Markovic’s proof of Theorem 6.3 proving that the Carathéodory and Kobayashi metrics are not equal on Teichmüller domains. His first observation is that T0;5 , the Teichmüller space of the five-punctured sphere, admits a holomorphic embedding into Tg for any g  2 that is also an isometric embedding with respect to the Teichmüller metrics. Hence, by the distance-decreasing properties of the two metrics and the domination of the Carathéodory metric by the Kobayashi metric (see (9) and (8) in §4), it suffices to prove that dTK0;5 ¤ dTC0;5 . This involves several steps, and crucially relies on the following observation regarding the restrictions of these metrics to Teichmüller disks. Let q 2 Q(X) be a holomorphic quadratic differential and Dq the Teichmüller disk associated to q as in (12). We say that Dq is extremal for a holomorphic function ΦW Tg ;n ! H if Φ ı Dq 2 Aut(H) . Properties (8) and (9) of the Kobayashi and Carathéodory metrics then imply that the two metrics agree on Dq if and only if Dq is extremal for some holomorphic map ΦW Tg ;n ! H . Step 1: a criterion for dTK = dTC on certain Teichmüller disks in T0;5 . We say that q 2 Q(X ) is a Jenkins–Strebel differential (J-S differential, in short) if q induces a decomposition of X into a finite number of annuli Πj , j = 1; : : : ; k , foliated by closed horizontal trajectories of q . (Recall that a horizontal trajectory is an integral curve of directions v satisfying q(v; v) > 0 .) Let mj denote the

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conformal modulus of Πj . If the mj have rational ratios then q is said to be a rational J-S differential. Assume that q is rational. Let 1 ; : : : ; k be a collection of disjoint simple closed curves on X , with j homotopic to Πj . Markovic observes that the Teichmüller disk Dq determined by q is stabilized by the infinite cyclic subgroup of MCG(S) generated by a product T of certain powers of Dehn twists about j . An averaging procedure then yields the following statement: if Dq is extremal y Tg ;n ! H , then there is a holomorphic function for some holomorphic function ΦW ΦW Tg ;n ! H such that i . (Φ ı Dq )() =  for all  2 H , i i . (Φ ı T )(Y ) = Φ(Y ) + t for every Y 2 Tg ;n for some t > 0 . The next construction is a holomorphic map from a poly-plane Hk to Tg ;n associated to Teichmüller disks arising from rational J-S differentials. Fix a marked Riemann surface S 2 Tg ;n and let  be a rational J-S differential on S . Let h1 ; : : : hk denote the heights, with respect to the singular flat metric jj , of the corresponding annuli Π1 ; : : : ; Πj . We assume that each hj > 0 . Let Hk denote the k -fold Cartesian product of H and define FW Hk ! BD1 (S ) by  i   jj j F() = i + j  p on each Πj , where  = (1 ; : : : ; k ) and i = 1. This gives rise to the poly-plane map EW Hk ! Tg ;n

(13)

defined by E() = [F()]: Note that E is holomorphic and its restriction to the diagonal is the Teichmüller disk Dq . It turns out that the quasiconformal map with dilatation F() is an affine map in suitable holomorphic charts (cf. (11)) and one can use this to prove the following crucial property of E : E( + (t; : : : ; t )) = (T ı E)();

where t > 0 and T 2 MCG(S) is as above. If we let f = Φ ı E , then the holomorphic map f W Hk ! H satisfies certain properties, arising from the defining properties of Φ and the equation above. Moreover it turns out that one can classify all holomorphic maps from Hk to H satisfying these properties. Indeed, for k = 2 (this suffices for the main result), one concludes that there exist ˛1 ; ˛2 > 0 such that f (1 ; 2 ) = ˛1 1 + ˛2 2 2

for all (1 ; 2 ) 2 H . To summarize, one has the following criterion for the equality of the Kobayashi and Carathéodory metrics.

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Theorem 7.1. Let X be biholomorphic to a Riemann sphere with five punctures. Let q be a rational J-S differential on X that decomposes the surface into exactly two annuli swept out by the closed horizontal trajectories of q . Then dTK and dTC agree on the Teichmüller disk Dq if and only if the function ΦW E(H2 ) ! H given by Φ(E(1 ; 2 )) = ˛1 1 + ˛2 2 ; for all (1 ; 2 ) 2 H2 can be extended to a holomorphic function ΦW T0;5 ! H . Step 2: Special Teichmüller disks in T0;5 . The goal here is to construct certain Teichmüller disks in T0;5 for which the criterion in Theorem 7.1 fails, to conclude that dTK0;5 ¤ dTC0;5 on these disks. These special examples arise from L-shaped pillowcases: let a > 0; b  0 and 0 < q < 1 and let L(a; b; q) denote the L -shaped polygon in Figure 1. Let S = S (a; b; q) denote the double of L , as a Riemann surface. The form dz 2 descends to a J-S differential = (a; b; q) on S and S decomposes into two non-degenerate annuli Π1 and Π2 swept out by the closed horizontal trajectories of . The vertices of the polygon with interior angles /2 give rise to poles of order one on the doubled surface S , and these determine five marked points; the remaining vertex of angle 3/2 determines a zero of order one for the resulting differential.

Figure 1. This L-shaped table determines a rational J-S differential with two cylinders

Fix (a0 ; b0 ; q0 ) 2 Q3 . The J-S differential 0 = 0 (a0 ; b0 ; q0 ) is then rational. Let E0 W H2 ! T0;5 denote the poly-plane map corresponding to 0 as in (13). It can be seen that b a E0 ; = S(a; b; q0 ) b0 a0 for all a; b > 0 . The above equation combined with Theorem 7.1 and a continuity argument shows the existence of a holomorphic function ΨW T0;5 ! H such that Ψ(a; b; q0 ) = (a + bq0 )i

for all a > 0; b  0 .

(14)

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Now fix q0 2 (0; 1) and suppose that dTK0;5 = dTC0;5 on the Teichmüller disk D (a0 ;b0 ;q0 ) for some a0 ; b0 > 0 . The final and perhaps the most delicate part of the argument involves the smooth ( C 1 ) path S (t ) = S (a0 ; 0; q0 t ) in T0;5 with t > 0 sufficiently small. The contradiction arises from the following claim. Claim. The composition Ψ ı S is not smooth at t = 0 . The proof of this claim is based on the expansion t log t 1  t2  t2 + ˇ2 (1 + o(1)) + o log t 1 log t 1 for some ˇ1 and ˇ2 ¤ 0 . This is obtained by an application of (14) and some involved calculations of Schwarz–Christoffel maps for L -shaped polygons. Ψ(S(a; 0; q

t )) = Ψ(S(a; 0; t )) + ˇ1 (1 + o(1))

8. Local convexity of Teichmüller domains While Markovic’s result settles the question of convexity of Teichmüller domains, it turns out that one can extract finer information about such domains. In particular, one can show that no boundary point can be locally strictly convex, which we stated as Theorem 6.4. In this section we shall outline the proof of this result. The strategy of the proof is inspired by K.-T. Kim’s proof of the following result. Theorem 8.1 (Kim [22]).

For g  2 the Bers embedding of Tg is not convex.

As in Kim’s proof, our proof involves two distinct components. Let Ω  C3g be a Teichmüller domain with a locally strictly convex boundary point p 2 @Ω .

3

Step 1. This is the main part of [18]. The goal is to show that any point p as above is an orbit accumulation point for Aut(Ω) = MCG(S ) . In Kim’s work, one has the corresponding result for the Bers domain: every point of the Bers boundary is an orbit accumulation point. This follows from McMullen’s result that cusps are dense in the Bers boundary ([34]), since powers of a Dehn twist converge to such cusps. Note that the orbit accumulation point property may not be preserved under biholomorphisms of domains as biholomorphisms may not extend in an absolutely continuous manner (or indeed, even as a well-defined function!) to the closures of the domains.

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Step 2. This step consists of rescaling at a “smooth” orbit accumulation point to obtain an unbounded convex domain Ω1 in the limit with the following properties: i . Ω1 is biholomorphic to Ω and i i . Aut(Ω1 ) contains a one-parameter subgroup. These lead to a contradiction: Royden’s theorem (Corollary 4.7) and ( i ) together imply that Aut(Ω1 ) is the discrete group MCG(S). Localization. The second step requires the technique of “localization” or complex rescaling, which is a simple and powerful method in complex analysis pioneered by S. Pinchuk ([41]), typically used to analyze the boundary behaviour of holomorphic mappings. We briefly discuss the notion of smoothness needed. If Ω is a domain in CN , we say that a point q 2 @Ω is Alexandrov smooth if @Ω is the graph, near q , of a function which has a second-order Taylor expansion at q . It is a theorem of A. D. Alexandrov that almost every point of @Ω is smooth in this sense if Ω is convex. In particular, if Ω is locally convex at q then we can assume that q is Alexandrov smooth, without loss of generality. The original rescaling argument of Pinchuk requires the C 2 -smoothness of the boundary of the domain under consideration. However, in Frankel’s setting (for his proof of Theorem 6.2) or the case of the Bers domain, the boundary is far from being smooth. In his work Frankel introduced a rescaling technique along an orbit of the automorphism group accumulating at a boundary point which dispenses with the C 2 -smoothness assumption. Subsequently, Kim and Krantz [23] developed a variant of Pinchuk’s method in the convex case which does not require a C 2 -smooth boundary and proved that, in fact, this recovers Frankel’s rescaling. In brief, they show the following. Suppose Ω is a domain, locally convex at an Alexandrov smooth point q 2 @Ω . Suppose that pj = j (p) ! q for j 2 Aut(Ω) and p 2 Ω . Then there exist invertible complex affine transformations Aj W CN ! CN such that the biholomorphic embeddings Aj ı j W Ω ! CN converge, on compact sets, to an embedding W Ω ! CN . Moreover, the convex domains Aj ı j (Ω) converge in the local Hausdorff sense (The local convexity at a boundary point is crucial to obtain a subconvergent sequence of domains.) The limiting image (Ω) is convex and contains an affine copy of R in its boundary. This immediately implies R  Aut(Ω) where the elements act by translations. Orbit accumulation points. The proof of the orbit accumulation property in the first step is based on two basic results in Teichmüller theory. First, the abundance of Teichmüller disks, that are complex geodesics (as defined in §4.2) in Teichmüller space. More precisely, by Theorem 4.2, one knows that through every point p 2 Ω and every direction v 2 Tp Tg , there exists a Teichmüller disk  W ∆ ! Tg with  (0) = p and  0 (0) = tv for some t > 0 .

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Second, Theorem 4.8 about the ergodicity of the Teichmüller geodesic flow due to Masur [32] and Veech [50]. Their work implies, in particular, that for any Teichmüller disk, almost every radial ray gives rise to a geodesic ray in Tg that projects to a dense set in moduli space Mg := Tg / MCG(S) . In particular, there is a sequence of points along such rays that recur to any fixed compact set in Mg . Given these facts, the proof involves an elementary but delicate analysis of the boundary behaviour of holomorphic functions on the unit disk in C . We choose a point p 0 2 Ω which is close to the strictly convex point p 2 @Ω and take a complex geodesic  W ∆ ! Ω with  (0) = p 0 . Since the boundary point p is locally strictly convex, there is a pluriharmonic “barrier” function h , namely the height from a supporting hyperplane at p , whose sub-level sets nest down to the single point p . This allows us to “trap” the complex geodesic  , by proving the existence of a positive measure set of radial directions in ∆ which under the holomorphic map  limit to boundary points arbitrarily close to p . One can then apply the Masur–Veech ergodicity result to infer the existence of an orbit points shadowing such a radial ray, which accumulate arbitrarily close to p . This concludes our discussion of the proof of Theorem 6.4. Further questions. A natural question is whether one can relax the strict convexity assumption above to convexity. Conjecture 8.2. The Teichmüller space of a closed surface of genus g  2 cannot be biholomorphic to a bounded domain Ω  C3g 3 that is locally convex at some boundary point. It would be interesting to see if the analysis in Step 1 producing the orbit accumulation points can be extended to understand the statistics of the distribution of accumulation points on the boundary; even for the Bers domain, this would be a step towards a better understanding of the structure of the boundary. Acknowledgments. Subhojoy Gupta acknowledges support of the center of excellence grant “Center for Quantum Geometry of Moduli Spaces” from the Danish National Research Foundation (DNRF95)—this work was started during a stay at QGM at Aarhus, and he is grateful for its hospitality. His visit there was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7 th European Union Framework Programme (FP7/2007–2013) under grant agreement № 612534, project MODULI – Indo European Collaboration on Moduli Spaces. Subhojoy Gupta also acknowledges the SERB, DST (Grant № MT/2017/000706) and the Infosys Foundation for its support.

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Chapter 3

Holomorphic quadratic differentials in Teichmüller theory Subhojoy Gupta

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 The co-tangent space of Teichmüller space . . 3 Singular-flat geometry . . . . . . . . . . . . . 4 Extremal maps and Teichmüller geodesics . . . 5 Hopf differentials of harmonic maps . . . . . . 6 The Hubbard–Masur theorem . . . . . . . . . . 7 Schwarzian derivative and projective structures 8 Meromorphic quadratic differentials . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Let S be a closed oriented smooth surface of genus g  2 . The Teichmüller space Tg of S is the space of marked complex structures on S , as well as marked hyperbolic structures on S . These two equivalent viewpoints give rise to a rich interaction between the complex-analytic and geometric tools in Teichmüller theory. This survey aims to highlight some of this interplay, in the context of holomorphic quadratic differentials, which have been a vital part of Teichmüller theory since its inception (see, for example, [119] and the commentary [2]). The selection of topics here is guided by our own interests, and does not purport to be comprehensive; the purpose is to provide a glimpse into several different aspects of Teichmüller theory that involve quadratic differentials in an essential way. We have aimed to keep the exposition light, to make this survey accessible to a broad audience; in particular, we shall often refer to other sources for proofs, or a more detailed treatment. Indeed, there is a vast literature on these topics, and we provide an extensive, but necessarily incomplete, bibliography at the end of this article. For some standard books on the subject, see [115], [40], or [66], just to name a few.

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In the final section, we focus on some recent work concerning holomorphic quadratic differentials on punctured Riemann surfaces, with higher order poles at the punctures. There, we describe generalizations of some of the results for a closed surface discussed in the preceding sections. One of the novel features of this non-compact setting is that one needs to define structures on the non-compact ends that depend on some additional parameters at each pole. In a broader context, holomorphic k -differentials where k  3 arise in higher Teichmüller theory—see [18], and the conclusion of §5. In particular, holomorphic cubic differentials are known to parametrize the space of convex projective structures on a surface (see [79] and [85]) and there has been recent work that develops the correspondence for meromorphic cubic differentials as well (see [84], [26], and [103]). It is an active field of research to develop tools and theorems that generalize the results in the quadratic ( k = 2 ) case, that this survey discusses, to the case of higher order differentials.

2. The co-tangent space of Teichmüller space Let X be a Riemann surface of genus g  2 . We shall assume that X is marked, where a marking is a choice of a homotopy class of a diffeomorphism from a fixed smooth surface S to X . Throughout, two such marked surfaces equipped with a structure (e.g. a complex structure or hyperbolic metric) are equivalent if they are isomorphic via a map that preserves the markings. We formally introduce the central objects of this survey. A holomorphic quadratic differential on X is a holomorphic section of K ˝ K , where K is the canonical line bundle on X . Locally, such a holomorphic tensor has the form q(z) dz ˝ dz (often written q(z) dz 2 ) where q(z) is a holomorphic function. Throughout this article, let Q(X ) be the complex vector space of holomorphic quadratic differentials on X . Since the canonical line bundle K has degree 2g 2 , the Riemann–Roch formula implies dim Q(X ) = deg(K 2 ) g + 1 = 3g 3 . For details of this computation, refer to [33] or [70], or any standard reference for Riemann surfaces. The holomorphic quadratic differentials defined above immediately arise in Teichmüller theory as objects that parametrize the space of infinitesimal deformations of hyperbolic or complex structures on a fixed surface S . Theorem 2.1. Let X 2 Tg . Then the cotangent space TX Tg can be identified with the space Q(X ) of holomorphic quadratic differentials on X . We sketch three proofs that arise from different ways of defining Teichmüller space.

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• A pair of conformal structures in Tg are related by a quasiconformal map f between them that determines a Beltrami differential  = fz¯ /fz determining its dilatation—see the beginning of §4. Thus, Beltrami differential is a ( 1; 1) -differential: it is is locally expressed as (z) ddzz¯ where (z) is a measurable function having sup-norm less than 1 . Any variation of conformal structures is then a derivative ˇ d t ˇˇ = ˙ (1) dt ˇ t =0 that lies in the Banach space L1 ( 1;1) (X ) . Teichmüller’s Lemma (see §3.1 of [102]) then asserts that the subspace of Beltrami differentials that determine a trivial variation is given by ² ˇ ³ Z ˇ ˇ N =  ˇ h; i :=  = 0 for every  2 Q(X ) (2) X

and thus the tangent space TX Tg can be identified with the quotient space L1 ( 1;1) (X )/N . The non-degenerate pairing h
;
i in the definition above then identifies the cotangent space TX Tg with Q(X ) . For more details, see §7 of [27], and see [1] for a more comprehensive introduction to Teichmüller theory from this point of view. Remark 2.1. The Teichmüller space Tg can be seen to be homeomorphic to R6g 6 via the Fenchel–Nielsen parametrization that considers length and twist parameters for the 3g 3 pants curves of a choice of a pants decomposition on the genus- g surface (see [32]). One way to find a basis of TX Tg is to consider the quadratic differentials that are dual to length and twist deformations—see [134], [92], or [46] for some recent related work. On hyper-elliptic surfaces, it is often possible to also give a fairly explicit set of bases (cf. [122]). • From a differential geometric standpoint, the infinite-dimensional space H of hyperbolic metrics on the (smooth) surface S has a tangent space at g 2 H that has an orthogonal decomposition Tg H = S(0;2) (g) ˚ ¹Lie derivatives of g along smooth vector fields on Sº; where S(0;2) (g) is the space of traceless and divergence-free symmetric (0; 2) -tensors on the hyperbolic surface (S; g) . By quotienting by the action of the self-diffeomorphisms of S , we see that the tangent space to Teichmüller space Tg at (S; g) is S(0;2) (g) . A brief computation (see p. 46, in §2.4 of [121]) then shows that any such symmetric (0; 2) -tensor is in fact the real part of a holomorphic quadratic differential (and vice versa), and we again obtain an identification with Q(X) .

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• Yet another approach comes from algebraic geometry: Teichmüller space is the universal orbifold cover of the moduli space of algebraic curves Mg . By the Kodaira–Spencer deformation theory, the tangent space TX Tg is canonically identified with H 1 (X; K 1 ) where the dual of the canonical line bundle K 1 is the sheaf of holomorphic vector fields on X . By Serre duality, this space is dual to H 0 (X; K 2 ) , and so the co-tangent space is identified with Q(X ) . For more details, see the concluding remarks of Appendix A10 in [66]. We now mention a couple of other aspects of the infinitesimal theory of Teichmüller space, that will crop up in other parts of this survey.

Variation of extremal length The extremal length of a homotopy class of a curve on a Riemann surface X is defined to be L ( )2 ExtX ( ) = sup (3) A  where  varies over all conformal metrics on X of finite  -area A , and L ( ) is the infimum of the  -lengths of all curves homotopic to . It is also the reciprocal of the modulus of , denoted Mod( ) . For a path ¹X t º 1 0 such that • if horiz is of length L , then it is mapped to an arc on Y of geodesic curvature less than C e ˛R and of hyperbolic length 2L ˙ C e ˛R , and • if vert is of length L , then it is mapped to an arc of hyperbolic length less than L
C e ˛R , where R > 0 is the radius of an embedded disk (in the q -metric) containing either segment, that does not contain any singularity. We shall see some examples in §8.3, in the context of harmonic maps from C to the Poincaré disk. As a consequence of this estimate, Wolf showed that his parametrization extends to the Thurston compactification by projectivized measured foliations, that we mentioned in §3.

Non-abelian Hodge correspondence Wolf’s parametrization of Teichmüller space is equivalent to that in Hitchin’s seminal work ([63]—see §11 of that paper) which was subsequently subsumed in a much more general correspondence, proved by Simpson (see [114]). In particular, there is a Kobayashi–Hitchin correspondence between the variety of surface-group representations in SLn (R) up to conjugation, and the “Hitchin section” of the moduli space of stable rank- n Higgs bundles on a Riemann surface X . A key intermediate step is the existence of equivariant harmonic maps from the universal cover of X , to the symmetric space SLn (R)/ SO(n) , which for n = 2 is precisely the hyperbolic plane, that was discussed in this section. For a survey focusing on this interaction of harmonic maps and Teichmüller theory, see [19].

6. The Hubbard–Masur theorem It is a consequence of classical Hodge theory that the space of holomorphic differentials ( 1 -forms) on a Riemann surface X can be identified with the first

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cohomology of the underlying smooth surface with real coefficients: H 0 (X; K) Š H1 (S; R) = H 1 (S; R) R where the identification is via the real parts of periods, namely, ! 7! i Re(!) where i varies over a basis of homology. The analogue of this identification for holomorphic quadratic differentials, is the following theorem of Hubbard and Masur ([65]), where the objects on the topological side are measured foliations, that we introduced in §3.

Theorem 6.1 (Hubbard–Masur). Let X 2 Tg where g  2 . The map ΦHM W Q(X ) ! MF that assigns to a holomorphic quadratic differential its vertical measured foliation, is a homeomorphism. Remark 6.1. The same holds, of course, for the horizontal measured foliation, but the sketch of the proof at the end of the section is most natural for the vertical one. As discussed in §3, two measured foliations are equivalent if the induced transverse measures are theR same. Moreover, for the vertical foliation, the measure p of a transverse loop is jRe( q)j , and thus these transverse measures can thus be thought of as encoding “periods” of the quadratic differential. The original proof involved a construction of a continuous section F of the cotangent bundle T  Tg , for each measured foliation F 2 MF , such that (X ) 2 Q(X ) has an induced vertical foliation F . The case when the foliation F had prong-type singularities of higher order involved a delicate analysis of “perturbations” into a generic singularity set. A shorter proof was given by Kerckhoff ([73], see also [40] for an exposition) where he used the existence (and uniqueness) of Jenkins–Strebel differentials having one cylinder of prescribed height. Both these proofs used the fact that such special Jenkins–Strebel differentials are dense in T  Tg , as proved by Douady and Hubbard ([21]). Wolf’s proof uses harmonic maps to R -trees (see [132] and [131]); it is this strategy, that we briefly sketch below, that was employed in the generalization of the Hubbard–Masur theorem to meromorphic quadratic differentials, that we shall discuss in §8. An equivalent approach was developed by Gardiner and others (see [39] and [41]) in which the holomorphic quadratic differential on X emerges as the solution to an extremal problem, similar to the case of Jenkins–Strebel differentials. We begin with two definitions. First, an R -tree is a geodesic metric space, such that any two points has a unique embedded path between them that is isometric to a real interval; this generalizes the notion of a simplicial tree (and in particular, may have locally infinite branching). In our context, the leaf-space TF of the lift of a measured foliation F to the universal cover is an R -tree (see Chapter 11 of [72] or [104] for an exposition).

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Second, an equivariant map from the universal cover of a Riemann surface to an R -tree is harmonic if convex functions pull back to subharmonic functions (see [19]).

In particular, there is a well-defined Hopf differential, which is holomorphic, as in the case of smooth targets. This forms a special case of a deeper theory of harmonic maps to such non-positively curved metric spaces, developed by Korevaar and Schoen ([75] and [76]).

Strategy of Wolf’s proof Fix a compact Riemann surface X on which we aim to realize a measured foliation F . Passing to the universal cover, one can show that there is a 1 (X ) -equivariant harmonic map from Xy to TF , the R -tree dual to the lift of F . The proof of this existence theorem proceeds by showing that an energy-minimizing sequence is equicontinuous; the uniform boundedness, in particular, is a consequence of a finite energy bound that we have because of compactness of X . The preimage of a point on TF is a vertical leaf of the Hopf differential q˜ , since it is clearly along directions of maximal collapse. Recall that each point of TF also corresponds to a leaf of Fy ; a topological argument shows that in fact the vertical foliation of q˜ is precisely the lift of F . By the equivariance of the map, q˜ descends to the desired holomorphic quadratic differential on X .

7. Schwarzian derivative and projective structures A (complex) projective structure on a surface S is an atlas of charts to C P1 such that the transition maps on the overlaps are restrictions of Möbius transformations. Piecing together the charts by analytic continuation in the universal cover Xy , one can define a developing map f W Xy ! C P1 that is equivariant with respect to the holonomy representation W 1 (S) ! PSL2 (C) . The space Pg of marked projective structures on a genus g  2 surface S (up to isotopy) admits a forgetful projection map pW Pg ! Tg that records the underlying Riemann surface structure. Such structures arise in connection with the uniformization theorem—indeed, hyperbolic surfaces provide examples of projective structures which have Fuchsian holonomy, and a developing image that is a round disk D  C P1 . Other examples of projective structures include a quasi-Fuchsian structure in which the developing image is a quasidisk Ω , and the holonomy is a discrete subgroup of PSL2 (C) . See [77] or [112] for more on the relation of projective structures to Kleinian groups. The fibers of the projection map p can be parametrized by holomorphic quadratic differentials; indeed, we have:

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Proposition 7.1. The set of projective structures on a fixed marked Riemann surface X 2 Tg is parametrized by Q(X) . As we shall see below, this identification with Q(X ) is not a canonical one, but depends on a choice of a base projective structure X0 (for example, the uniformizing Fuchsian structure) with holonomy Γ0 , that shall correspond to the zero quadratic differential. Sketch of the proof of Proposition 7.1. In one direction, consider the Schwarzian derivative of the developing map f , defined as  f 00 0 1  f 00 2 S (f ) = : f0 2 f0 This measures the deviation of any univalent holomorphic map f from being a Möbius map. In particular, one can verify that whenever A is a Möbius map, that is, an element of PSL2 (C) , we have S (A)  0;

S (A ı f ) = S(f );

S (f ı A)(z) = S (f ) ı A(z)A0 (z)2 :

This implies, in our setting, by the  -equivariance of the developing map f , that the holomorphic quadratic differential S (f ) dz 2 on Xy is invariant under the action of Γ0 and descends to the Riemann surface X . In the other direction, given a quadratic differential q 2 Q(X ) , we can define a corresponding projective structure by considering the Schwarzian equation on Xy given by 1 u00 + q˜u = 0 (15) 2 where q˜ is the lift to the universal cover. The ratio f := u1 /u2 of two linearly independent solutions u1 and u2 then defines the developing map to C P1 , and the monodromy along loops on X yields the holonomy representation  . By (15) the Wronskian u1 u02 u01 u2 is constant, and it is an exercise to show that the Schwarzian derivative S (f ) recovers q , hence establishing that the constructions in the two directions are in fact inverses of each other. □ Corollary 7.2. The space of projective structures Pg is an affine bundle over Tg modelled on the vector bundle Qg ! Tg . Projective structures in fact give rise to a global realization of Tg as a bounded complex domain in Q(X ) . This relies on the following deep observation of Bers (see [13]) concerning the quasi-Fuchsian projective structures that we defined above: Theorem 7.3 (simultaneous uniformization). For any pair (X; Y ) 2 Tg  Tg , there is a unique quasi-Fuchsian structure QF (X; Y ) determined by a discrete subgroup y with complementary Γ < PSL2 (C) that leaves invariant a quasi-circle ΛΓ in C components Ω˙ , such that Ω+ /Γ = X and Ω /Γ = Y .

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Fixing an X 2 Tg , and varying Y over Tg , then defines the Bers embedding BX W Tg ! Q(X ) Š C3g

3

by taking the holomorphic quadratic differential corresponding to the projective structure QF (X; Y ) . For more details, and various open questions about this embedding, see the survey [54]. For a more comprehensive treatment of this approach to Teichmüller theory, see the books [102] and [68].

Grafting To conclude this section, we mention that Thurston showed that any projective structure is obtained from a Fuchsian one by a geometric operation of grafting along a measured lamination  on a hyperbolic surface X . We describe this operation only in the case that  is a simple closed geodesic with weight t : here, a projective annulus A t is grafted into X along the curve, where A t is the quotient of the wedge ¹0  Arg(z)  t º  C by the hyperbolic element that is the holonomy of the curve. More generally, a measured lamination can be approximated by weighted multicurves, and a grafted surface can be defined in the limit. The resulting surface gr(X; ) is a new complex projective surface; in fact, Thurston showed that the map grW Tg  ML ! Pg is a homeomorphism. See [71] and [118] for an idea of the proof, and [78] for a more general context. By scaling the measure on the lamination  by t  1 , one obtains a oneparameter family of projective structures that project to grafting rays in Tg , that are in fact asymptotic to Teichmüller geodesic rays that we saw in §4 (see [50]). Furthermore, the work of Dumas ([24]) compares the Schwarzian derivatives of the corresponding projective structures to the holomorphic quadratic differentials corresponding to t via the Hubbard–Masur theorem. For more on this, and the topics mentioned in this section, see the extensive survey on projective structures by Dumas [25].

8. Meromorphic quadratic differentials The preceding sections were under the assumption that the underlying Riemann surface was compact; in this section we shall consider the case of a Riemann surface X of finite type, that is, of finite genus and finitely many punctures, such that the Euler characteristic is negative. Throughout, we shall assume that the quadratic differentials on X are meromorphic with finite order poles at the punctures. Here, a pole of order n  1 means that the expression of the quadratic differential q is q = (an z

n

+ an

1z

1 n

+


+ a1 z

1

+ a0 + g(z)) dz 2

(16)

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in a choice of a coordinate disk D around the pole, where ai 2 C for 0  i  n and g(z) is a holomorphic function that vanishes at the origin (where the pole is).

8.1. Simple poles A pole of order one is also called a “simple” pole; in this R case the induced singular-flat metric is still of finite area, since Areaq (X ) = X jqj = kqkL1 (X) and kz 1 dz 2 kL1 (D) < 1 . Indeed, the double cover w 7! z = w 2 branched at the simple pole pulls back q as in (16) to the holomorphic quadratic differential 4(a1 + a0 w 2 + w 2 g(w 2 )) dw 2 , and much of the analysis of this case reduces to that for holomorphic quadratic differentials (on a compact surface) via this branched cover. The space of such meromorphic quadratic differentials arises as the tangent space of the Teichmüller space Tg ;n of a surface of finite type, having genus g and n punctures, and the discussion in the previous sections can be carried forth in this case. This was in fact, the original setting of the work of Teichmüller (see the commentary in [2]). In particular, a measured foliation induced by such a meromorphic quadratic differential has a “fold” at each simple pole—see Figure 3(a); the corresponding space of such measured foliations can be parametrized by train-tracks “with stops” (see Chapter 1.8 of [106]) and the analogue of the Hubbard–Masur theorem (Theorem 6.1) holds. Considering Tg ;n to be the space of marked hyperbolic surfaces of genus g and n cusps, the analogue of Wolf’s parametrization (Theorem 5.2) is an early work of Lohkamp (see [86]).

8.2. Poles of order two In this case, the induced singular-flat metric around the puncture has the form dr 2 +rd 2 in polar coordinates, and is hence a semi-infinite Euclidean cylinder, and r2 in particular, of infinite area. One way such a pole arises is in a limit of a sequence of Jenkins–Strebel differentials where the length of one of the Euclidean cylinders in the induced metric diverges to infinity. This happens, for example, along Strebel rays, that are Teichmüller rays determined by a Jenkins–Strebel differential. The corresponding limiting quadratic differential is called a Strebel differential (though sometimes it is also called a Jenkins–Strebel differential in the literature), which is meromorphic quadratic differential with poles of order two, with the additional property that each non-critical horizontal leaf is closed. Indeed, we have the following existence (and uniqueness) theorem: Theorem 8.1 (Strebel). Let X be a compact Riemann surface of genus g  1 and let P = ¹p1 ; p2 ; : : : pk º be a non-empty set of points. Given positive real numbers a1 ; a2 ; : : : ak , there is a unique Strebel differential with poles of order two at the points of P , and such that for each 1  i  k , the semi-infinite Euclidean cylinder at pi has circumference ai .

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(a)

(b)

(c)

Figure 3. Structure of the induced horizontal foliation near (a) a simple pole, (b) a pole of order two, and (c) pole of order 4

The proof involves showing that a Strebel differential arises as the solution of an extremal problem, similar to that of Jenkins–Strebel differentials—see (8)—but involving, instead of the extremal lengths of the curves, the (reciprocal of the) reduced modulus of the punctured disks centered at the points of P . See §4.1.2 of [99] for a sketch of the proof, and Chapter VI, §23, of [115] or [6] for a longer exposition. The critical trajectories of a Strebel differential form a metric graph G that is a retract of the punctured surface, as can be seen by collapsing each semi-infinite cylinder in the vertical direction. The orientation of the surface also induces a ribbon graph structure on the embedded graph G , namely, an orientation on the half-edges emanating from each vertex. By Strebel’s theorem, there is a unique such metric ribbon graph associated to any Riemann surface with punctures (X; P ) with a set of positive real parameters at each puncture; this gives a combinatorial description of decorated Teichmüller space Tg ;n  (R+ )n that has been useful in various contexts (see for example [61] and [74], and see [105], [87], and [101] for related discussions). Strebel differentials are not the only ones with a pole of order two, though; indeed, the local structure of the horizontal foliation depends on the coefficient of the z 2 -term. In the generic case, such a foliation comprises leaves that are not closed, but spiral into the puncture (see the bottom of Figure 3(b)). A generalization of Strebel’s theorem to cover such foliations was proved in [56]; see also the remark following Theorem 8.2. For a generalization of Wolf’s theorem (Theorem 5.2) to the case of poles of order two, see [109].

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Harmonic maps Meromorphic quadratic differentials with poles of order two also arise in the context of harmonic maps; in [130] Wolf showed that such a differential arises as the Hopf differential of a harmonic map of a noded Riemann surface to a hyperbolic surface obtained by “opening up” the node. Indeed, a class of such differentials, together with a Fenchel–Nielsen twist parameter, parametrize a neighborhoods of yg (that descends to a point in the boundary of augmented Teichmüller space T the Deligne–Mumford compactification of Mg under the quotient by the action of the mapping class group). For some other instances where such differentials with double poles appear in this context, see [92] or [67].

An application As indicated in the beginning of this subsection, Strebel differentials arise as limits of Teichmüller rays determined by Jenkins–Strebel differentials. Thus, they are in fact useful in the study of the asymptotic behaviour of such Teichmüller rays—see [5] and [4]. As we shall mention in the next subsection, higher order poles arise in limits of more general Teichmüller rays, and we expect such meromorphic quadratic differentials to play a role in analogous results.

8.3. Higher order poles For simplicity, we shall assume, throughout this section, that the Riemann surface X has a single puncture where the quadratic differential has a pole of order greater than two.

Polynomial quadratic differentials The first case of interest is when X is the complex plane; an elementary computation shows that a quadratic differential of the form (z d + ad 2 z d 2 +


+ a0 ) dz 2 has a pole of order (d + 2) at infinity. The coefficients a0 ; : : : ad 2 are complex numbers, and any such polynomial of degree d can be taken to be monic and centered, as above, by composing with a suitable automorphism of C . The simplest example is the constant differential dz 2 on C ; its induced metric is just the standard Euclidean metric on the plane, and its horizontal (resp. vertical) foliation are the horizontal (resp. vertical) lines on C . In general, a polynomial quadratic differential of degree d induces a singular-flat metric on C comprising (d + 2) Euclidean half-planes, and some number of infinite Euclidean strips. The induced foliation is the standard foliation on these domains, and its leaf space is a planar metric tree. This tree has (d + 2) infinite rays dual to the foliated half-planes, and the remaining edges dual to the foliated strips, having lengths equal to the transverse measures across them. In fact, this description allows one to show that the space of such measured foliations MF(C; d + 2) is homeomorphic to Rd 1 (see, for example, [9]).

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Such a polynomial quadratic differential also arises as the Hopf differential of a harmonic map from C to the hyperbolic plane (for the existence of such a map, see [126], and for a discussion of the uniqueness, the recent work [82]). The structure of the horizontal and vertical foliations discussed above, together with the estimates mentioned in Proposition 5.3, implies that the image of such a map is an ideal (d + 2) -gon in H2 (see Figure 4), and in fact all such ideal polygons arise this way (see [60]).

Figure 4. The induced horizontal foliation of z 4 dz 2 on C (left) and the ideal polygon that is the image of the corresponding harmonic map (right). Proposition 5.3 implies that horizontal leaves in each sector that are far from the zero, map close to the geodesic sides.

Measured foliation with a pole singularity More generally, on a once-punctured Riemann surface X of higher genus, the local structure of the induced singular-flat geometry around any pole of order n  3 comprises (n 2) foliated half-planes. This can also be seen in the example of the constant quadratic differential dz 2 on C : any neighborhood U of the pole at infinity would contain exactly two half-planes that are foliated by horizontal lines. Throughout this section, we shall consider the induced horizontal foliation, with an understanding that the same results hold if considers the vertical foliation as well. By a result of Strebel (see Theorem 7.4 of [115]), there is such a neighborhood U of the pole such that any horizontal leaf entering it terminates at the pole. The induced measured foliation F of a meromorphic quadratic differential on X has additional real parameters at the pole, that records the structure of its restriction F jU . These parameters can be thought of as lengths of edges in the metric graph that is the leaf-space of the lift of the restricted foliation F jU (see Figure 5).

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1 2

3

Figure 5. The foliation around a pole singularity (shown as the black dot) comprises foliated half-planes that appear as the “petals” in the figure (left). Its leaf-space is a metric graph with an infinite ray corresponding to each half-plane (right).

Note that this metric graph has (n 2) infinite rays towards the pole, corresponding to each of the foliated half-planes. In the universal cover, the leaf-space of the lift Fy of the entire foliation is then an “augmented” R -tree, which has an equivariant collection of such infinite rays. The endpoints of these rays determine an equivariant collection of additional vertices, and consequently finite-length edges, in the augmented R -tree. Analyzing the structure of these R -trees, and in particular taking into account the length parameters of these additional edges, one can show that the space of such measured foliations MF(X; n) is homeomorphic to R6g 6+n+1 (see [57]). The generalization of the Hubbard–Masur theorem then relies on fixing the principal part P at the pole of order n , with respect p to a coordinate disk U Š D around it. This is defined to be the polar part of qjU comprising the terms for p i z where i  2 . Note that the difference qjU P has at most a simple pole at p the origin. In order to make sense of qjU when n is odd, the principal part P can be thought of as a meromorphic 1 -form defined on the branched double cover of U . Example 8.1. We illustrate the last sentence with the following computation. Consider the polynomial quadratic differential q = (z 3 + az + b) dz 2 where a; b 2 C . This quadratic differential has a pole of order 7 at 1 , where it is of the form (w 7 + aw 5 + bw 4 )dw 2 on a disk U equipped with the coordinate w obtained by y inversion. Substituting 2 = pw to pass to the branched double cover U , and taking a square root, we obtain qjUy = 2( 6 + 12 a 2 +


)d and thus the principal part P of the original quadratic differential q involves the complex coefficient a , and not b .

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When the order of the pole n is even, the real part of the residue of the principal part P is determined by the transverse measures of the induced horizontal foliation around the pole (see Lemma 5 of [57] for details). If the residue satisfies that, we say that the principal part P is compatible with the foliation; we say this compatibility always holds in the case that n is odd. Theorem 8.2 ([57]). Let X be a once-punctured Riemann surface of genus g  1 and fix a coordinate disk U around the puncture p . Let n  3 and let F 2 MF(X; n) be a measured foliation on X with a pole singularity at p . Then for any choice of principal part P that is compatible with the foliation F , there exists a unique meromorphic quadratic differential with a pole of order n at p with that principal part on U , whose horizontal foliation is F . Remarks 8.1. (1) Just like Theorem 6.1, the above holds for the vertical foliation as well; the compatibility condition then involves the imaginary part of the residue. Moreover, the assumption on the genus g is an artifact of the fact that we have assumed there is exactly one puncture; more generally, we only need the Euler characteristic (X) < 0 . (2) The same statement as above holds when n = 2 (see [56]); in this case, the residue at the pole is determined by the coefficient of z 2 at the order two pole, and determines the “spiralling” nature of the horizontal foliation. A special case of the above theorem is when the foliation F has the property that all the non-critical leaves lie in foliated half-planes. The critical graph then forms a metric spine for the punctured surface X n p , and a meromorphic quadratic differential with such a horizontal foliation is the analogue of Strebel differentials that we saw earlier. The induced singular-flat metric is what we call a half-plane structure comprising Euclidean half-planes attached to the critical graph. The choice of the principal part P determines the singular-flat geometry of the resulting “planar end” around the puncture (see [55]). Like for Strebel differentials, such half-plane structures arise as limits of singular-flat surfaces along a Teichmüller rays; see §2 of [49] for details, and [51] or [49] for some applications. A generic meromorphic quadratic differential induces a (horizontal or vertical) measured foliation such that each leaf has at least one end terminating at the pole; as for polynomial quadratic differentials on C , the induced singular-flat metric then comprises half-planes and infinite strips. This trajectory structure is related to the theory of wall-crossing, Donaldson–Thomas invariants and stability conditions in the sense of Bridgeland (see [17] and [58]). For further work investigating the trajectory structure on various strata, that can be defined for the total space of the bundle of meromorphic quadratic differentials over Tg ;1 , see [116]. For an investigation of SL2 (R) -dynamics on such strata, see [16].

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Crowned hyperbolic surfaces The generalization of Wolf’s parametrization (Theorem 5.2) to the case of meromorphic quadratic differentials involves considering hyperbolic surfaces with certain non-compact ends corresponding to the higher order poles. Such an end is a hyperbolic crown bordered by a cyclic collection of bi-infinite geodesics, each adjacent pair of which encloses a “boundary cusp.” See Figure 6. The space of such crowned surfaces of genus g  1 , such that the crown end has m  1 boundary cusps, is the “wild” Teichmüller space Tg (m) that can be shown to be homeomorphic to R6g 6+m+3 (see §3 of [48]).

Figure 6. A crowned hyperbolic surface

When m is even, the metric residue of the crown end is calculated by considering a truncation of the cusps, and taking an alternating sum of the lengths of the resulting geodesic sides (this is independent of the choice of truncation). In [48] we proved: Theorem 8.3. Let X be a once-punctured Riemann surface of genus g  1 and fix a coordinate disk U around the puncture p . Let • Q(X; p; P ) be the space of meromorphic quadratic differentials with a pole of order n at p that has principal part P on U , • Tg (P ) be the subspace of Tg (m) such that the metric residue of the crown end equals twice the real part of the residue of P , if m is even and • Tg (P ) be equal to Tg (m) if m is odd. Then for any Y 2 Tg (P ) , there is a unique harmonic map hW X n p ! Y such that the Hopf differential Ψ(Y ) lies in Q(X; p; P ) . This defines a homeomorphism ΨW Tg (P ) ! Q(X; p; P ) .

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Remarks 8.2. (1) The constraint imposed by the residue in the case m is even is a consequence of Proposition 5.3, since horizontal leaves in the foliated half-planes around the pole map close to the geodesic sides of the crown end, with twice the length. (2) The theorem above holds in the case that X = C as well; in this case Q(X; p; P ) is the space of polynomial quadratic differentials of degree (n 2) that we saw earlier, and Tg (P ) is the space of ideal n -gons. In the case of n = 4 (degree two polynomials) the correspondence is explicit, since the metric residue of an ideal quadrilateral determines its cross ratio; this correspondence was observed in a different way in [8].

Strategy of the proof In Theorem 8.3, even the existence of such a harmonic map from a non-compact domain to a crowned hyperbolic surface requires work, the additional difficulty being that such a map necessarily has infinite energy since the Hopf differential has a higher order pole. This existence proof involves the following steps. • St e p 1 . Determine a space of model harmonic maps defined on a punctured disk. A crucial observation here is that the principal part of the Hopf differential determines the asymptotic behavior at the puncture. In particular, our desired principal part P picks out a model map  that our final harmonic map should be asymptotic to, on the coordinate disk U . • St e p 2 . Define a sequence of harmonic maps ¹hi ºi 0 defined on a compact exhaustion X0  X1 


 Xi 


of the punctured surface X n p , such that each solves a Dirichlet problem with the boundary condition determined by the desired model map  . The key analytic work is to then show that there is a uniform bound on the energy of these maps when restricted to a fixed compact subsurface. This relies on the analysis of a partial boundary value problem (see [64]) on an annular region that is the difference Xi +1 n X0 , and proving that it is a uniformly bounded distance from the model map  . • St e p 3 . Finally, the convergent subsequence of harmonic maps obtained in the preceding step is shown to converge to a harmonic map on the punctured Riemann surface that has principal part P . Remark 8.3. The same strategy is used in the proof of Theorem 8.2; in that case the model harmonic maps have a target metric tree that is dual to the lift of the desired foliation on U to the universal cover.

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An alternative approach for the proof of the existence, using the method of suband super-solutions, is implicit in the work of [103] (see also §2 of [117]). In the more general context of the non-abelian Hodge correspondence mentioned in §5, the case when the Higgs field has irregular or higher order poles was considered by Biquard and Boalch in [14], where they also proved an existence theorem for the corresponding harmonic maps. In [49] we show how harmonic maps from a punctured Riemann surface to a crowned hyperbolic surface also arise in limits of harmonic maps between compact surfaces (as in Wolf’s parametrization), when the target hyperbolic surface is fixed, and the domain Riemann surfaces diverges along a Teichmüller ray which has a half-plane structure as a limit.

8.4. Meromorphic projective structures Finally, we mention some ongoing work generalizing the relation of quadratic differentials with projective structures that we saw in §7. Meromorphic quadratic differentials with poles of order two arise as Schwarzian derivatives of branched projective structures (see, for example, [88] or §1.4 of [36]). In the case of poles of higher order, the corresponding solutions of the Schwarzian equation define a space of meromorphic projective structures, and the holonomy map defined on the space of such structures was studied in the recent works [3], [52], and [53]. The Schwarzian equation is part of a broader study of linear differential systems on the complex plane, and in this context, the case of polynomial quadratic differentials on C is part of the classical literature (see, for example, [113]). It would be interesting to study other aspects of the geometry of projective structures on punctured Riemann surfaces. Meromorphic quadratic differentials are also known to arise in compactifications of various strata of the total space Qg of quadratic differentials—see [11], [10], [43], or Theorem 10 of [29]. It would also be interesting to explore the resulting compactification of the space of projective structures Pg , and the degenerations that lead to the meromorphic structures on the boundary. Acknowledgments. I wish to thank Athanase Papadopoulos for his kind invitation to write this article. I am grateful to the SERB, DST (Grant № MT/2017/000706) and the Infosys Foundation for their support. It is a pleasure to thank Michael Wolf, for sharing his insight, and the many hours of conversation that led to some of the results described in §8. Last but not the least, I am grateful to Fred Gardiner, and the anonymous referees, for their suggestions that helped improve this article.

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Chapter 4

Mostow strong rigidity of locally symmetric spaces revisited Lizhen Ji

Contents 1 2 3 4 5 6 7 8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and rigidity of Riemann surfaces . . . . . . . . . . . . . Deformation and three types of rigidity of complex manifolds . . . . . Deformation and local rigidity of locally symmetric spaces . . . . . . Strong rigidity of lattices and locally symmetric spaces . . . . . . . . Rigidity and arithmeticity . . . . . . . . . . . . . . . . . . . . . . . . Proofs of Mostow strong rigidity . . . . . . . . . . . . . . . . . . . . . Quasiconformal mappings and Mostow strong rigidity for hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The negative part of Mostow strong rigidity . . . . . . . . . . . . . . 10 Generalizations of the Mostow strong rigidity . . . . . . . . . . . . . . 11 Other major works of Mostow and another strong rigidity . . . . . . . 12 Some comments on the Mostow strong rigidity . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 127 128 132 135 137 141 142 147 153 155 157 159

1. Introduction After the notion of Riemann surfaces was introduced by Riemann in his thesis in 1851, the problem of classifying compact Riemann surfaces naturally leads to the notion of moduli spaces in a very influential paper titled Theory of Abelian functions by Riemann in 1857. More specifically, Riemann showed that every compact Riemann surface corresponds to a birational equivalence class of plane algebraic curves and raised the problem of classifying algebraic function fields of one variable, which is equivalent to the problem of classifying compact Riemann surfaces up to biholomorphic equivalence. By viewing compact Riemann surfaces as ramified coverings of CP 1 , he counted the number of moduli, or effective complex parameters, to describe them by deforming the branching points. In his counting, the idea of deformation of Riemann surfaces is essential. Riemann’s

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count suggested that the moduli space of compact Riemann surfaces is a complex analytic space of the complex dimension equal to the number of moduli. Once the uniformization theorem for Riemann surfaces was established together with the Poincaré upper half-plane model of the hyperbolic plane, the moduli space of compact Riemann surfaces can be identified with the moduli space of compact oriented hyperbolic surfaces. One natural generalization of Riemann surfaces consists of higher dimensional complex manifolds, and another natural generalization, through hyperbolic surfaces, consists of locally symmetric spaces of noncompact type. Consequently, the deformation of compact Riemann surfaces admits two generalizations: 1. deformation and rigidity of compact complex manifolds; 2. deformation and rigidity of compact locally symmetric spaces. It turns out that compact hyperbolic surfaces are essentially the only compact irreducible locally symmetric spaces of noncompact type which admit nontrivial deformations, and the Mostow strong rigidity is even a stronger rigidity property, which has the striking implication that the geometry of these compact locally symmetric spaces satisfying the Mostow strong rigidity is determined by its topology. Remark 1.1. In contrast with locally symmetric spaces, compact complex manifolds of higher dimensions corresponding to compact Riemann surfaces of genus greater than 1 are not necessarily rigid and may admit large deformation families. For example, compact complex manifolds of general type of complex dimension at least 2 usually have positive dimensional moduli spaces (see [18], [85], [35], and [83]). If a locally symmetric space is reducible and contains a compact hyperbolic surface as a factor, then it is not rigid and deformed locally symmetric spaces also contain a compact hyperbolic surface as a factor. If a compact complex manifold is the product of a compact Riemann surface of genus greater than 1 and another compact complex manifold, then it clearly admits nontrivial deformation families. In general, deformable (or non-rigid) complex manifolds do not admit such products, and the product structure is not preserved under deformation. As we shall see later, compact quotients of bounded symmetric domains of dimension greater than 1 by irreducible lattices are both locally symmetric spaces and complex manifolds, and they are rigid both as complex manifolds and locally symmetric spaces. Several other rigidity results only apply to compact complex manifolds, though they are suggested by locally symmetric spaces. Maybe one explanation for the above difference in deformation properties of the two classes of spaces is that there are, in some sense, more compact complex manifolds or ways to deform them than locally symmetric spaces. Due to the close connection between symmetric spaces and Lie groups, the Mostow strong rigidity can also be formulated as a statement on the rigidity of

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lattices of semisimple Lie groups, i.e., extension of homomorphisms between lattices to ambient Lie groups. In this chapter, we review the history of the notions of deformation and rigidity of complex manifolds, rigidity properties of locally symmetric spaces, especially two equivalent formulations of the Mostow strong rigidity. We also discuss the motivations, impact and some of less known aspects of the Mostow strong rigidity of locally symmetric spaces: (1) the implication of the rigidity on fields of definition of algebraic varieties and, (2) cases when the assumption in the Mostow strong rigidity does not hold.

2. Deformation and rigidity of Riemann surfaces When Riemann first introduced Riemann surfaces in his thesis (see [69]), he described them as ramified coverings over domains in C or CP 1 . When a compact Riemann surface Σg of genus g is realized as a ramified covering of CP 1 , the Riemann–Hurwitz formula shows that the genus g of the Riemann surface Σg is determined by the number of branch points in CP 1 and ramification multiplicities of points above them. Consequently, when the branch points move while the ramification multiplicities stay the same, we get families of compact Riemann surfaces of the same genus. As mentioned before, this was the way that Riemann first counted the number of effective complex parameters needed to describe points in the moduli space Mg of compact Riemann surfaces, and the answer is equal to 3g 3 when g  2 . Riemann called these complex parameters moduli. Since every compact Riemann surface Σ1 of genus 1 admits a double covering over CP 1 with 4 branch points, three of which can be normalized to be 0; 1; 1 , it follows that the moduli space M1 of compact Riemann surfaces of genus 1 needs one complex parameter, i.e., is of complex dimension 1. On the other hand, it can be shown that every compact Riemann surface of genus g = 0 is biholomorphic to CP 1 , and hence M0 consists of only one point.1 Therefore, in contrast to compact Riemann surfaces of positive genus, CP 1 is rigid. Right after Riemann’s papers, a lot of work has been devoted to put the above intuitive results of Riemann about the moduli space Mg , g  2 , on a firm foundation. For example, the works of Teichmüller, Ahlfors, Bers et al showed that Mg is a complex orbifold of complex dimension 3g 3 when g  2 . To achieve this, Teichmüller introduced the notion of marked Riemann surfaces, which consist of pairs of a compact Riemann surface and a choice of a set of generators of its fundamental group, and the moduli space of marked Riemann surfaces, which is called the Teichmüller space and denoted by Tg . Teichmüller also developed the 1 This is not obvious and was first proved by Alfred Clebsch and Hermann Schwarz. See [20], p. x x v i .

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theory of deformation of compact Riemann surfaces through quasi-conformal maps. One consequence of his description of extremal quasi-conformal maps through holomorphic quadratic forms implies that the cotangent space of the Teichmüller space Tg at a marked Riemann surface can be identified with the space of holomorphic quadratic forms on the Riemann surface, which also gives a concrete description of the space of infinitesimal deformations of compact Riemann surfaces. Teichmüller’s theory of quasi-conformal maps also gives a description of the global deformations of compact Riemann surfaces. He showed that the Teichmüller space is diffeomorphic to R6g 6 , defined the Teichmüller metric on Tg , and described (or outlined) construction of a natural complex structure on Tg and the universal deformation family of marked compact Riemann surfaces over Tg (i.e., Tg is a fine moduli space of marked compact Riemann surfaces). Consequently, he obtained Mg as the quotient of Tg under the proper and holomorphic action of the mapping class group of a compact Riemann surface of genus g , which gives Mg the structure of a complex orbifold. This justified Riemann’s count of complex parameters as the complex dimension of Mg . The history of the moduli space Mg and the Teichmüller space Tg is long and complicated. There are several different approaches to Mg as a complex space: complex analysis through the Teichmüller space and algebraic geometry through geometric invariant theory.2 Teichmüller made deep contributions in both approaches, but after him, there was not much interaction between the two communities working on these approaches. See the papers [1] and [30] for some detailed discussions.

3. Deformation and three types of rigidity of complex manifolds We note that by the common modern definition, Riemann surfaces are complex manifolds of dimension 1. Motivated by results of Riemann and Teichmüller on deformation of compact Riemann surfaces mentioned in the previous section, deformation theories of complex structures on a fixed compact smooth manifold of higher dimensions were initiated by Frölicher and Nijenhuis in [22] in 1957, exactly 100 years after Riemann’s paper.3 In this paper, they showed that for a compact complex manifold X , if the first cohomology H 1 (X; Θ) = 0 , where Θ is the sheaf of germs of holomorphic vector fields on X (i.e., the sheaf of holomorphic sections 2 The approach using the uniformization of Riemann surfaces and the hyperbolic geometry of Riemann surfaces does not lead to a complex structure on Tg and Mg . 3 It is perhaps helpful to note that Ahlfors and Bers’ works to redo and make rigorous some of the major results of Teichmüller started at around the same time period. Ahlfors’ major paper [3] on quasi-conformal maps appeared in 1954, and his paper with Bers [5] on solutions of Beltrami equations appeared in 1960, and Ahlfors’ paper [4] on a complex structure on Tg appeared in 1960. Bers summarized his results on the complex structure and embedding of Tg around 1960 in his talk at ICM 1960 [9].

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of the holomorphic tangent bundle TX of X , and H 1 (X; Θ) = H 1 (X; TX ) ), then every smooth family X t , t 2 ( 1; 1) , of compact complex manifolds containing X as the fiber X0 is locally trivial at t = 0 , i.e., when t is small, X t is biholomorphic to X0 . Motivated by the work [22] and the deformation of compact Riemann surfaces, Kodaira and Spencer developed the theory of deformation of compact complex manifolds in [36] and [37]. At the beginning of their paper [37], they wrote: The deformation of higher-dimensional complex manifolds, or of algebraic surfaces at least, seems to have been considered first by Max Noether in 1888. However, in sharp contrast with the case of complex dimension 1 (Riemann surfaces), the deformation of higher-dimensional complex manifolds has been curiously neglected. Last year Frölicher and Nijenhuis, as an outgrowth of their earlier work on vector-valued differential forms, obtained an important theorem which is the starting point of this study. The purpose of the present paper is to develop a more or less systematic theory of deformations of complex structures of higher dimensional manifolds.

One result about deformation of compact Riemann surfaces played a crucial role, or rather was the starting point, in the work of Kodaira and Spencer. When X = Σg is a compact Riemann surface of genus g  2 , the Serre duality implies that H 1 (X; Θ) can be identified with the dual of the complex vector space of holomorphic quadratic forms on X . As mentioned before, the space of holomorphic quadratic forms on a Riemann surface X was crucial to the theory of Teichmüller space, and this identification with, or rather interpretation by, H 1 (X; Θ) is crucial to the deformation theory of Kodaira-Spencer. According to [34], p. 21: Deformation theory had been on Spencer’s mind for some time prior to his work with Kodaira […]. As he explained it to me, the issue was that they did not know what should play the role in higher dimension of quadratic differentials in the Teichmüller theory on Riemann surfaces. The breakthrough came with (2) [The identification of H 1 (X; Θ) with the dual of the space of holomorphic quadratic forms on X ]. With that major “hint” everything began to fall into place, leading to the papers […], which brought deformation theory into the core of complex algebraic geometry.

For a general compact complex manifold X , the deformation theory of Kodaira and Spencer interprets H 1 (X; Θ) as the space of infinitesimal deformations of X , or the tangent space of deformation spaces. To put this result of Frölicher and Nijenhuis into the general framework to be used below, we introduce the following notions.

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Definition 3.1. A compact complex manifold X is called infinitesimally rigid if H 1 (X; Θ) = 0 . A compact complex manifold X is called locally rigid if every smooth deformation X t of X = X0 , t 2 ( 1; 1) , is locally trivial near t = 0 , i.e., for t sufficiently small, X t is biholomorphic to X0 . A compact complex manifold X is called globally (or strongly) rigid if every compact complex manifold Y which is homotopy equivalent to X is biholomorphic to X . In terms of the above definitions, the result in [22] says Proposition 3.1. If a compact complex manifold X is infinitesimally rigid, then it is locally rigid. As an application, they proved the following result. Proposition 3.2. rigid.

For every n  1 , the complex projective space CP n is locally

On the other hand, the strong rigidity of CP n was more difficult to prove and depends on the solution of the Calabi conjecture by Yau in [89] (Theorem 5 and Remark), after some earlier work Hirzebruch and Kodaira in [26]. Proposition 3.3. Every compact complex surface homotopy equivalent to CP 2 is biholomorphic to CP 2 . For n > 2 , every compact Kähler manifold which is homeomorphic to CP n is biholomorphic to CP n . The first part of Proposition 3.3 was improved in [67] to that any compact complex surface with the same homology group with integral coefficient as CP 2 is biholomorphic to it. Note that in high dimensions, there are differences between classification by homotopy equivalence and homeomorphism. It seems not clear if in the second statement in the above proposition that homotopy equivalence is sufficient. The above result on the strong rigidity of CP 2 was conjectured by Severi. This is probably the earliest strong rigidity result on higher dimensional compact complex manifolds, after the strong rigidity of CP 1 , which was mentioned in the previous section and is one case of the uniformization theorem for Riemann surfaces. For compact Riemann surfaces, CP 1 is strongly rigid, while those of positive genera are far from being rigid and admit positively dimensional deformation families. By the uniformization theorem, every compact Riemann surface Σg , g  2 , is a quotient of the unit disk D in C . Since the unit disk D is a bounded symmetric domain, one question is to examine the deformation or rigidity property of compact quotients of bounded symmetric domains in Cn of higher dimensions. Motivated by the results in [22] and [37], Calabi and Vesentini proved in [16] the infinitesimal rigidity of compact

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quotients of irreducible bounded symmetric domains, and hence local rigidity of such quotients by Proposition 3.2. In some sense, they worked out the general deformation theories of [22] and [37] in the concrete examples of Hermitian locally symmetric spaces. Proposition 3.4. Any compact complex manifold X which is a quotient of an irreducible bounded symmetric domain of dimension at least 2 is infinitesimally rigid and hence locally rigid. The corresponding strong rigidity was proved by Siu in [74] and [75]. Proposition 3.5. Let X be a compact quotient of an irreducible bounded symmetric domain of dimension at least 2. If Y is a compact Kähler manifold which is homotopy equivalent to X , then Y is biholomorphic to X . Note that in the above proposition, Y is required to be a compact Kähler manifold, and the most general strong rigidity requires only compact complex manifolds. It is interesting to quote from [74], p. 73, about the motivation of the problem and results: In 1960 Calabi and Vesentini proved that compact quotients of bounded symmetric domains are rigid in the sense that they do not admit any nontrivial infinitesimal holomorphic deformation. In 1970 Mostow discovered the phenomenon of strong rigidity. He proved that the fundamental group of a compact locally symmetric Riemannian manifold of nonpositive curvature determines the manifold up to an isometry and a choice of normalizing constants if the manifold admits no closed one or two dimensional geodesic submanifolds which are locally direct factors. In particular, two compact quotients of the ball of complex dimension  2 with isomorphic fundamental groups are either biholomorphic or conjugate biholomorphic. Yau conjectured that this phenomenon of strong rigidity should hold also for compact Kähler manifolds of complex dimension  2 with negative sectional curvature. That is, two compact Kähler manifolds of complex dimension  2 with negative sectional curvature are biholomorphic or conjugate biholomorphic if they are of the same homotopy type. In this paper we prove that Yau’s conjecture is true when the curvature tensor of one of the two compact Kähler manifolds is strongly negative in the sense defined in Section 2 with no curvature assumption on the other manifold.

Remark 3.1. The papers [74] and [75] made crucial use of a fundamental technique in geometric analysis, the Bochner technique, in proving the rigidity result mentioned above and other related ones. For a systematic and accessible exposition of the Bochner technique in differential geometry, see the book [88].

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As mentioned before, the question and results on the strong rigidity of the complex projective space CP n was considered before 1957 in [26]. On the other hand, the strong rigidity of quotients of bounded symmetric domains and compact Kähler manifolds with negative curvature was motivated by the Mostow strong rigidity for locally symmetric spaces of noncompact type, which have nonpositive sectional curvature. We next explain the other generalization of deformation and rigidity of Riemann surfaces in terms of locally symmetric spaces, especially the Mostow strong rigidity.

4. Deformation and local rigidity of locally symmetric spaces By the uniformization theorem, every compact Riemann surface Σg of genus g  2 admits a canonical hyperbolic metric which is conformal to the complex structure. Equivalently, there exists a faithful representation W 1 (Σg ) ! PSL(2; R)

whose image (1 (Σg )) is a discrete subgroup, denoted by Γ , such that Σg Š ΓnH2 :

This identification allows one to reduce the problem on deformation of Riemann surfaces Σg to deformation of discrete subgroups Γ of the Lie group PSL(2; R) , or equivalently to deformation of hyperbolic surfaces. In particular, the Teichmüller space Tg can be identified with the Fricke space of discrete and faithful representations of the surface group 1 (Σg ) in PSL(2; R) . Motivated by his work on the Selberg trace formula and the deformation of lattices in PSL(2; R) , Selberg initiated the study of deformation and rigidity of lattices of semisimple Lie groups in [70]. In a paper published in 1959 [70], p. 1, Selberg wrote: The theory of uniformization then easily is shown to provide us with equivalence classes of groups that have compact fundamental domains and depend continuously on as many continuous parameters as might be desired. If one looks at the similar problem in a symmetric space of higher dimension, the theory of uniformization has no counterpart, and one has so far only been able to construct groups that have some kind of arithmetical definition (if we exclude the case of a reducible symmetric space that contains the hyperbolic plane as a factor.) One is led to the question whether not all groups in these cases are equivalent to such groups with an arithmetic definition, if we make the assumption that the fundamental domain has finite volume (measured with the invariant volume-element) or the stronger assumption that the fundamental domain is compact.

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In [70], Selberg proved that every cocompact lattice in PSL(n; R) with n  3 is equivalent to an algebraic group Γ in the sense that every matrix entry of every element of Γ is an algebraic number in some real algebraic field. There are two steps in the proof in [70]. 1. There exists an one-parameter deformation Γ t , t 2 ( 1; 1) , with Γ0 = Γ such that for some small t , Γ t is an algebraic group in the sense above. 2. Under the condition n  3 , the deformation Γ t has to be trivial, i.e., there exists a family of elements g t 2 PSL(n; R) such that Γ t = g t Γ0 g t 1 . It is worthwhile to emphasize that both deformation and rigidity were used to prove the algebraicity of lattices in [70]. More details were given in a later paper [71], §4. It seems that these results of Selberg were followed immediately by the work of Calabi [15], and the closely related paper [16]; and the work of Weil [86] and [87]. Specifically, the following result was proved in the paper [15]. Proposition 4.1. Avery compact hyperbolic manifold M of dimension at least 3 does not admit any nontrivial deformation, i.e., it is locally rigid. More precisely, write M as ΓnHn , where Γ is a cocompact discrete subgroup of isometries of Hn , n  3 . Then for every family of cocompact discrete subgroups Γ t , t 2 ( 1; 1) , of isometries of Hn such that Γ0 = Γ , and Γ t is isomorphic to Γ0 , there exists a family g t of isometries of Hn such that Γ t = g t Γ0 g t 1 . In analogy with rigidity properties of compact complex manifolds, we now introduce the corresponding notions for locally symmetric spaces or discrete subgroups of Lie groups. See [71], p. 150. Let G be a noncompact semisimple Lie group, K  G a maximal compact subgroup. Then the homogeneous space X = G/K endowed with a G -invariant Riemannian metric is a symmetric space of noncompact type: it is simply connected with nonnegative sectional curvature but does not contain any Euclidean space as an isometric factor. Let Γ  G a discrete subgroup. Then Γ acts isometrically and properly on X , and the quotient space ΓnX is a locally symmetric space. Definition 4.1. With the above notation, a locally symmetric space ΓnX is called locally rigid if every deformation family of ΓnX is trivial. In other words, for every continuous family of locally symmetric spaces Γ t nX , t 2 ( 1; 1) , if Γ0 nX is isometric to ΓnX , then for all t 2 ( 1; 1) , Γ t nX is isometric to ΓnX . Equivalently, if Γ t , t 2 ( 1; 1) , is a continuous family of discrete subgroups of G , with Γ0 = Γ , then there exists a continuous family of elements g t 2 G such that Γ t = g t Γ0 g t 1 . In this case, the discrete subgroup Γ is also said to be locally rigid. Similar to deformation of complex manifolds, we can also define infinitesimal rigidity of locally symmetric spaces by the vanishing of the cohomology group H 1 (Γ; Ad) = 0 , where Ad is the adjoint representation of Γ on the Lie algebra

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g of G . Since the tangent space of the character variety (or the deformation space) of Γ is described by H 1 (Γ; Ad) , the local rigidity of Γ is implied by the infinitesimal rigidity of Γ . This is similar to deformation spaces of compact complex manifolds X , whose tangent spaces are described by the cohomology group H 1 (X; Θ) .

Definition 4.2. A locally symmetric space ΓnX is called strongly (or globally) rigid if every locally symmetric space Γ1 nX1 with Γ1 Š Γ2 is isometric to ΓnX after suitable scaling of the metrics of the irreducible factors of X1 and X2 . In this case, the discrete subgroup Γ is also called strongly rigid. Remark 4.1. In this definition, the symmetric space X1 is a priori not necessarily isometric to X . In some formulation, especially when X1 is the real hyperbolic space Hn , X2 is assumed to be equal to X1 , and the problem is to compare two discrete subgroups acting on the same symmetric space. It is interesting to compare local rigidity of complex manifolds and of locally symmetric spaces. In the former case, we consider deformation through all possible complex manifolds, and in the latter case, we consider deformation through only locally symmetric spaces. The same comment applies to strong rigidity of these two classes of spaces. It is a basic result that any bounded symmetric domain in Cn with the Bergman metric is a Hermitian symmetric space of noncompact type, i.e., a symmetric space X = G/K of noncompact type which admits a complex structure invariant under G , and every Hermitian symmetric space of noncompact type can be realized as a bounded symmetric domain. When a compact complex manifold is a quotient of a bounded symmetric domain in Cn , its local rigidity can be considered from either one of the above two points of view. In this case, the local rigidity in the category of compact complex manifolds is stronger than that in the category of locally symmetric spaces, since every locally symmetric space which is a deformation of a quotient of a bounded symmetric domain is also a quotient of a bounded symmetric domain and hence is a complex manifold. One can see this through Lie groups, or from the tangent spaces and hence Lie algebras, or the curvature properties of the symmetric spaces. On the other hand, there could be deformations of complex manifolds which are not locally symmetric spaces. Therefore, the result in [16] (see Proposition 3.4) implies the following result. Proposition 4.2. Let M be a compact quotient of an irreducible bounded symmetric domain X  Cn , n  2 , and write M = ΓnX , where Γ is a discrete subgroup of Aut(X ) . Then Γ is infinitesimal rigid and hence locally rigid. Motivated by this result in [16], Weil removed the assumption of X being a Hermitian symmetric space and proved in [87] (Theorem 1, p. 588).

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Proposition 4.3. Let G be a noncompact semisimple Lie group whose associated symmetric space X = G/K does not contain the hyperbolic plane H2 as an isometric factor. Then every cocompact discrete subgroup Γ  G is infinitesimal rigid and hence locally rigid. Among discrete subgroups of Lie groups G , the most natural class consists of lattices Γ , i.e., discrete subgroups such that the quotient ΓnG has finite volume with respect to the Haar measure on G . The work of Weil [87] was supplemented by an unpublished work of Borel on the local rigidity of arithmetic subgroups of noncompact semisimple Lie groups of Q -rank at least 2, and by an unpublished work of Garland on the local rigidity of arithmetic subgroups of linear semisimple algebraic groups which are split over Q . (See [55], p. 5). Finally Raghunathan [68] extended this local rigidity result to all arithmetic subgroups of semisimple Lie groups. Proposition 4.4. Let G be a noncompact semisimple Lie group whose associated symmetric space is not isometric to the hyperbolic plane H2 . Then every irreducible arithmetic subgroup Γ  G is infinitesimal rigid and hence locally rigid. Therefore, up to that point, 1867, the local rigidity of non-arithmetic noncocompact lattices of semisimple Lie groups was not proved.

5. Strong rigidity of lattices and locally symmetric spaces After the local rigidity of some lattices of noncompact semisimple Lie groups, for example, cocompact lattices and arithmetic subgroups, were proved, people started to try to prove the strong rigidity for them. Both Mostow and Furstenberg raised questions about global or strong rigidity of lattices, and they both had their first publications on this topic around 1966-1967, i.e., [23], [52], and [53]. But the actual dates were earlier for both of them. As Mostow write in [55], p. 6, “The phenomenon of strong rigidity for arbitrary lattices first turned up in 1965 in my search for a geometric explanation of deformation-rigidity.” He wanted to give a geometric proof of the results recalled in the previous section. On the other hand, Furstenberg was motivated by applications of his work on the Poisson boundary and random walks on groups. There are several interesting contrasts between them. Furstenberg was interested in the negative direction, i.e., proving that lattices of two nonisomorphic semisimple Lie groups are not isomorphic, while Mostow was interested in the positive direction that an isomorphism between two lattices of one common semisimple Lie group extends to an automorphism of the Lie group.

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More specifically, Furstenberg proved in [23] that lattices acting on the real hyperbolic space of any dimension are not isomorphic to lattices acting on the symmetric space SL(n; R)/ SO(n) , n  3 . Mostow proved in [53] and [54] that every two isomorphic lattices Γ1 ; Γ2 acting on the hyperbolic space Hn of dimension at least 3, with a quasi-conformal homeomorphism between the associated hyperbolic manifolds Γ1 nHn and Γ2 nHn , are conjugate. Furstenberg introduced the notion of an envelope of a discrete group and asked whether the envelope is unique, and consequently he raised the question on strong rigidity. Specifically, a Lie group G is an envelope of a discrete group of Γ if Γ is isomorphic to a lattice of G , and he tried to prove that two nonisomorphic Lie groups should not envelop the same discrete group. The motivation of Furstenberg was how Poisson boundaries of groups influence properties or structures of the groups.4 On the other hand, Furstenberg and Mostow had the same fundamental theme: how a lattice Γ of a semisimple Lie group G determines the ambient Lie group G . Furstenberg used measure theory and hence the condition of being a lattice is natural and important, but whether the lattice is cocompact or not is not essential to him. On the other hand, Mostow uses quasi-isometries, or the framework of coarse geometry. In his book [55], the condition on the existence of a quasi-conformal map between hyperbolic manifolds imposed in his paper [54] was removed if the lattices are cocompact. This completes the proof of the result that cocompact lattices acting on the hyperbolic space Hn , n  3 , are strongly rigid. This strong rigidity for compact hyperbolic manifolds was also proved by Margulis slightly earlier in [42] using similar methods. See [28], p. 16, for more detail and discussion. But both Furstenberg and Mostow used action at the maximal Furstenberg boundary of the symmetric spaces. The most general strong rigidity of compact locally symmetric spaces is usually called the Mostow strong rigidity and stated in the book [55], p. 3, in two equivalent forms.

Theorem 5.1. Let X = G/K be a symmetric space of noncompact type, and Γ  G be a cocompact torsion-free lattice. Then the compact locally symmetric space ΓnX is determined by its fundamental group Γ up to isometry and a choice of normalizing constants on the isometric factors of X , provided ΓnX does not have two dimensional geodesic subspaces which are direct factors locally. 4 Furstenberg told the author of this chapter that he realized later that the rigidity result he proved follows easily from Kazhdan property T in [32]. Specifically, for lattices Γ in SL(3; R) , H 1 (Γ; R) = 0 , while for torsion-free lattices Γ0 in SL(2; R) , H 1 (Γ0 ; R) ¤ 0 .

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Theorem 5.2. Assume G has no center and no nontrivial compact normal subgroup. Let Γ  G be a cocompact lattice. If PSL(2; R) is not a direct factor of G which is closed modulo Γ , then the pair (G; Γ) is uniquely determined by Γ . That is, given two such pairs (G; Γ) and (G 0 ; Γ0 ) and an isomorphism W Γ ! Γ0 , ¯ G ! G 0 such that  is the restriction of there exists an analytic isomorphism W ¯ to Γ , provided there is no factor Gi of G isomorphic to PSL(2; R) such that ΓGi is a closed subgroup of G . The assumption of cocompactness on lattices in the above theorem was needed to guarantee an equivariant pseudo-isometry (or quasi-isometry) between the corresponding symmetric spaces. The strong rigidity was proved by Margulis [43] for irreducible non-cocompact lattices acting on symmetric spaces of rank at least 2 , and by Prasad [65] for Q -rank 1 irreducible lattices when the symmetric space X is not isometric to the hyperbolic plane, which includes all noncocompact lattices acting on a symmetric space of rank 1 of dimension at least 3. For lattices acting on the three-dimensional hyperbolic space H3 , the strong rigidity was also proved by Marden [41] as a consequence of his study of deformations of three dimensional hyperbolic manifolds.

6. Rigidity and arithmeticity Since the classification of spaces and other structures is a basic problem in mathematics, moduli spaces are fundamental. In defining and understanding moduli spaces of certain spaces, families of the spaces under classification, or deformations of them, are fundamental objects. Consequently, deformation and rigidity are natural and important properties of the spaces. In particular, if the spaces under classification are strongly rigid, then their moduli spaces are trivial, and if they are locally rigid, then the moduli spaces are zero dimensional. In this section, we consider some other applications of deformation and rigidity to show how rigidity properties imply special features of algebraic varieties, in particular of Hermitian locally symmetric spaces of finite volume, and lattices of semisimple Lie groups.

6.1. Local rigidity of Hermitian locally symmetric spaces and fields of definition When a symmetric space X is a Hermitian symmetric space of noncompact type (i.e., a bounded symmetric domain in Cn ) and Γ is a torsion-free cocompact lattice of Aut(X ) , the quotient space ΓnX is a compact complex manifold. In [33], Kodaira proved that ΓnX can be embedded into some CP n as a projective algebraic variety. By [16] (see Proposition 3.4), when Γ is an irreducible lattice and dim ΓnX > 1 , ΓnX is locally rigid.

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The following result seems to be known to experts, but is not written down explicitly anywhere as far as we know. (See Remark 6.3 below for more information and some references.) Proposition 6.1. Let ΓnX be a compact irreducible Hermitian locally symmetric space of dimension strictly greater than 1 . Then it is a projective variety defined over a number field, i.e., a finite extension of Q . Remark 6.1. When X is the unit ball of C2 , or equivalently X is the complex hyperbolic plane H2C , then every compact quotient ΓnH2C is an algebraic variety defined over a number field. On the other hand, when X = H2 , compact Riemann surfaces are projective curves over C , but most of them are not defined over number fields. The reason is that the moduli space of compact Riemann surfaces of genus g  2 is of positive dimension and hence uncountable, but there are only countably many algebraic curves which are defined over number fields. The above proposition follows from the next folklore result.5 Proposition 6.2. Let V be a projective or a quasi-projective variety which is locally rigid, then V is defined over a number field. We could not find any proof of this result in literature and will provide several arguments or rather explanations for it. The first explanation is probably most intrinsic, but difficult to justify. Consider all algebraic varieties of the same “type” as V . Suppose there exists a moduli space M of these varieties which is an algebraic variety defined over Q . Then x /Q) acts on M . Since V is locally rigid, M is the absolute Galois group Gal(Q a zero dimensional variety and hence consists of finitely many points. Therefore, x /Q) fixes the point in the moduli space M a subgroup H of finite index of Gal(Q which represents the equivalence class [V ] of V . This implies that V is defined x fixed by the over the number field which is the subfield of the algebraic closure Q subgroup H . One difficulty with the above argument is that it is highly nontrivial, or even may not be true, that there exists such a moduli space M of varieties like V which is an algebraic variety defined over Q . The second method of proof is similar to the deformation argument in [70]. First, we assume that V is a hypersurface in CP n defined by a single homogeneous equation, X aI z I = 0; (1) I

5 One related result is stated in [48], p. 648: “It is a standard fact that rigidity implies (V; f ) is defined over a number field; otherwise the transcendental elements in its field of definition would give deformations.”

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where I = (i1 ; : : : ; in ) , and z I = z1i1


znin . Take one coefficient aI0 ¤ 0 . Multiply all coefficients by aI01 so that the new coefficient aI0 = 1 . If all other thus normalized coefficients are algebraic numbers, then V is defined over a number field already. Assume next that some coefficients aI are not algebraic numbers. We now follow the argument of Selberg [71], p. 154, and [70], p. 470. Let F be the algebraic number field which is obtained by adding only algebraic coefficients aI to Q , and K be the field obtained by adding all coefficients aI . Then K is obtained from F by adding transcendental coefficients, denoted by aI1 ; : : : ; aIr . Let Pi (x1 ; : : : ; xr ) = 0 , i = 1; : : : ; s , be a complete set of independent algebraic relations between the transcendental aI1 ; : : : ; aIr with coefficients in F , i.e., none is redundant, and other algebraic relations can be expressed in terms of them. Now these polynomial equations Pi = 0 define an algebraic variety W over F , and the transcendental coefficients aI1 ; : : : ; aIr form a generic point (aI1 ; : : : ; aIr ) of the variety W . In this variety W , when we deform this generic point and use the coordinates of a deformed point to replace the coefficients of equation 1, we get a deformation V t of the hypersurface V . There are two cases to consider. If V t is not isomorphic to V , then we have obtained a nontrivial deformation family of V . This contradicts the assumption. Otherwise, all such deformations V t of V are isomorphic to V . Since W is defined over the number field F , we can pick a point in W which has algebraic numbers as coordinates. This implies that we have obtained an algebraic variety which is defined over a number field and isomorphic to V . Consequently, V can be defined over a number field. In general, assume that V is not defined by a single equation. If V is not defined over a number field, consider a finite set of generators of the ideal which defines V . For each generator, we normalize the coefficients as above and separate out algebraic ones and transcendental ones. Then we deform the transcendental coefficients inside the variety which contains a generic point coming from the transcendental coefficients as above to obtain deformation families of the variety V and to prove that V is defined over a number field. In this case, deforming in the algebraic variety which contain the generic point formed from the transcendental coefficients is crucial since we want to preserve all relations between these generators. The third explanation is a proof in a special case and was communicated by Deligne [19]. Proposition 6.3. Let X is a smooth projective variety. If X is infinitesimally rigid, i.e., the cohomology H 1 (X; TX ) = 0 , where TX is the tangent bundle of X , then X is defined over a number field. Proof. We first can let the coefficients of the equations of the variety X move, to get f W Y ! S projective of finite type, with S of finite type over Q , so that X is the fiber of f at a complex point s of S . Cutting S in locally closed pieces,

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one can get Y to be flat over each piece, and taking the piece containing s , we may assume that Y is projective and smooth over S . Consider now over S  S the scheme Isom with fiber over (a; b) the isomorphisms between Xa and Xb . It exists by the Hilbert scheme story, and can be analyzed by deformation theory. The rigidity assumption will imply that it maps onto an open subset U which after extension of scalars from Q to C contains (s; s) . We take (s; a) in U , with a x , and we get the existence of an isomorphism between X and Xa , defined over Q x. getting X defined over Q □ Remark 6.2. When Γ is a congruence subgroup, ΓnX is a Shimura variety and defined over a canonical number field. This is much stronger than the above result of merely defining over a number field. See [49] for details about Shimura varieties and their fields of definition. Remark 6.3. After a preliminary version of this paper was finished, we found the statement of Proposition 6.3 and a different proof of it in a paper of Shimura [73]. In view of the previous remark, it is natural. We also learnt that in the book [84], p. 83, Proposition 6.1 seems to be attributed to Faltings [21]. Actually, Faltings refereed the case of compact smooth Hermitian locally symmetric spaces to [73] and proved that toroidal compactifications of quotients of bounded symmetric domains by neat arithmetic groups are defined over number fields, and hence his result complements the result of Shimura.

6.2. Superrigidity and arithmeticity of lattices A major application of rigidity of lattices is concerned with arithmeticity of lattices. As mentioned earlier, in the first paper [70], Selberg showed two results. 1. Any cocompact lattice of SL(n; R) can be deformed to a lattice whose elements have matrix coefficients contained in real algebraic number fields. 2. When n  3 , such lattices of SL(n; R) are locally rigid, and hence are conjugate to lattices whose elements have real algebraic coefficients. Selberg’s result falls short of proving that lattices are arithmetic subgroups, since matrix coefficients of a lattice may not be algebraic integers. As it is well known, this was proved by Margulis by using superrigidity (see [44]). Briefly, he used the super-rigidity of lattices over Archimedean place R to show that in suitable coordinates, the matrix coefficients of elements of an irreducible lattice of a semisimple Lie group of rank at least 2 are rational numbers, and use the superrigidity over the p -adic number Qp to show that the matrix coefficients have bounded powers of p in the denominators. Consequently, super-rigidity of lattices at all places implies the arithmeticity of the lattices.

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Remark 6.4. One result of [72] seems mysterious. On page [72], p. 104, Selberg claimed arithmeticity of irreducible lattices in the group SL(2; R)  SL(2; R); or in more general semisimple Lie groups SL(2; R) 


 SL(2; R): On the other hand, later in the paper [72], pp. 111–113, he tried to prove this result, but he could only prove it under an assumption of zeroes of certain L -functions. Therefore, it seems that he did not have a proof of the arithmeticity result at that time. The usual history on the arithmeticity of lattices is that Selberg conjectured arithmeticity of non-cocompact lattices in [72], and Piatetski-Shapiro conjectured arithmeticity of cocompact lattices (see [79] and [64], p. 3). In [58], p. 169, Mostow wrote: Prior to 1960, it had been observed that the only general constructions of lattices applicable in all dimensions were arithmetic. We own to Selberg the intuition that what cannot be constructed does not exist and the judgement to act on that intuition.

In a more recent paper [60] which contains a discussion of the work of Margulis, Mostow attributed this conjecture on arithmeticity of lattices to Siegel and wrote the following when he commented on reading a paper of Margulis: I read the title: “On discrete groups of motions on spaces of nonpositive curvature,” which was not very informative. As I read further, I realized that this was a landmark paper! This paper settled a question of C. L. Siegel suggested by a construction of G. Giroux: Are all lattices constructible arithmetically?

7. Proofs of Mostow strong rigidity The Mostow strong rigidity for locally symmetric spaces was proved in [55] using two different methods which depend on whether the associated symmetric spaces are of rank 1 or of higher rank. When the rank of a symmetric space is equal to 1, it has strictly negative sectional curvature. After the original proof of Mostow using quasi-conformal maps in [55], there has been several different proofs. For example, there is a proof for hyperbolic manifolds due to Gromov and Thurston using the simplicial volume of Gromov (see [61] and [8] for expositions of this proof), and a proof for all rank 1 locally symmetric spaces using entropy rigidity in [10] and [11]. When the rank of symmetric spaces is greater than 1 , the Mostow strong rigidity is derived from the rigidity of spherical Tits buildings (see §9 below for a brief summary). Though the Mostow strong rigidity for hyperbolic manifolds is proved in many places, for example the books [31] and [46] and the survey [24], it seems that the Mostow strong rigidity for higher rank locally symmetric spaces is not discussed or proved in books or papers besides the original book of Mostow [55].

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The papers [28] and [29] contain a summary of the proof in this case. Since the Margulis superrigidity theorem contains the Mostow strong rigidity as a special case, Margulis’s proof of his superrigidity theorem gives another proof of Mostow strong rigidity. In the next section, we will summarize the proof of the Mostow strong rigidity for hyperbolic manifolds in order to prepare for the discussion in §9 about the negative part of the Mostow strong rigidity. Before that, let us take a look at some comments of the originators of the subject of rigidity of lattices. As mentioned earlier, the problem on rigidity of locally symmetric spaces was motivated by the deformation of compact Riemann surfaces of genus at least 2. The proof of Mostow strong rigidity for the rank 1 case was motivated by Riemann surfaces. The following quote from Mostow makes the connection with the theory of Riemann surfaces even closer. In [56], p. 201, Mostow wrote: My method of proving strong rigidity evolved from an effort to understand the failure of rigidity in PL(2; R) from a geometric viewpoint. If X denotes the simply connected covering space of S and S 0 , then we may regard X as the interior of the unit disc in the complex plane. As differentiable transformation groups on X , Γ , and Γ0 are equivalent. Why then are they inequivalent in the complex analytic sense? or to rephrase the question, why are they not conjugate in PL(2; R) ? The natural conjecture is: because they are not differentiably equivalent on the boundary of X .

As we shall see, quasi-conformal maps on the boundary of hyperbolic spaces are crucial to the proof of the Mostow strong rigidity of hyperbolic manifolds in [54] and [55], and is also influential to later development. It is also interesting to quote from Selberg [72], p. 114, on Mostow’s proof of using quasi-conformal maps and boundary maps: Quite recently G. D. Mostow in a note gave a new proof of rigidity for the case of the n -dimensional hyperbolic space when n > 2 . His proof is quite simple, but it is not entirely clear how well the method which uses quasi-conformal mappings of S onto itself, and looks at the extension of this mapping to the “boundary” of S , will carry over to the general case.

Though quasi-conformal maps are not used in the proof of Mostow strong rigidity of locally symmetric spaces of the higher rank, boundary maps are a crucial ingredient.

8. Quasiconformal mappings and Mostow strong rigidity for hyperbolic manifolds In many discussions and expositions of Mostow strong rigidity, the following special case is often emphasized:

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Proposition 8.1. If two compact hyperbolic manifolds of dimension at least 3 are homotopy equivalent, then they are isometric. The generalizations in [2], [10], [11], [13], [24], [27], [31], [46], [81], and [82] all refer to this. This is natural since it is the simplest and most concrete class of locally symmetric spaces, and the proof is also the simplest among rank 1 locally symmetric spaces. In this section, we will outline a proof of the Mostow strong rigidity for hyperbolic manifolds using quasi-conformal maps, and the discussion here will motivate results on conformal structures on Gromov hyperbolic spaces and the proof of the negative part of the rigidity in the next section. To prove this rigidity property of real hyperbolic manifolds, we need quasiconformal maps on the Euclidean space Rn . When n = 2 , quasi-conformal maps on R2 are classical. The theory of quasi-conformal maps in higher dimensions was developed simultaneously by Y. G. Reshetnyak, F. W. Gehring and J. Väisälä in the period 1960–62, and was applied by Mostow in [54], after some suitable generalizations, to prove the first significant case of Mostow strong rigidity. For the convenience of the reader, we recall the definition of quasi-conformal maps on Rn , n  2 . The definition below also works for metric spaces, in particular to the Gromov boundary discussed in the next section. Let Ω  Rn be an open subset, and 'W Ω  Rn ! Rn

be a continuous map such that ' is a homeomorphism from Ω to its image '(Ω) . Let d (
;
) be the distance on Rn with respect to the standard Euclidean metric. For every point p 2 Rn , and r > 0 , define L' (p; r) =

sup

d ('(q); '(p));

qWd (q;p)=r

l' (p; r) =

inf

qWd (q;p)=r

d ('(q); '(p));

and the dilatation L' (p; r) : l' (p; r) r !0 As in the two-dimensional case, the map ' is called a K -quasi-conformal map, where K is a positive constant, if for every point p 2 Ω , H' (p) = lim sup

H' (p)  K:

Remark 8.1. In some sense, the Mostow strong rigidity for real hyperbolic manifolds is the most important application of the theory of quasi-conformal maps on the Euclidean spaces. In comparison, the most important application of quasiconformal maps on Riemann surfaces is the Teichmüller theory. There is a marked difference between compact hyperbolic manifolds of higher dimension and compact

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hyperbolic surfaces, due to the Mostow strong rigidity for the former case and the existence of deformation in the latter case. Therefore, it is interesting that quasiconformal maps are used essentially in both cases. Remark 8.2. It might be important to explain the natural role quasi-conformal maps take in the Teichmüller theory of Riemann surfaces. When one consider the category of complex manifolds, the natural morphisms between the spaces are holomorphic maps. Now we consider the category of compact Riemann surfaces of genus g  2 and take holomorphic maps between them as morphisms. Unfortunately, there is no nonconstant holomorphic map between two nonisomorphic compact Riemann surfaces Σg and Σ0g . This can be proved easily by the Riemann–Hurwitz formula. In view of this, this category is not adequate. A major theorem of Teichmüller tells us that there exists a unique extremal quasi-conformal map between two marked compact Riemann surfaces. This is the starting point in the detailed deformation theory of compact Riemann surfaces, especially in the global study of Teichmüller spaces. For example, the Teichmüller metric depends on this, and the Teichmüller dynamics on the Teichmüller space also depends on this. On the other hand, for compact complex manifolds of higher dimensions, there are no analogues of extremal quasi-conformal maps between nonbiholomorphic compact complex manifolds. Therefore, the Kodaira-Spencer deformation theory for compact complex manifolds is only local or infinitesimal, and there is no analogue of Teichmüller metric. To study and understand global properties of moduli spaces of compact complex manifolds, some maps beyond diffeomorphisms connecting nonisomorphic compact complex manifolds will be very helpful. One important result beyond the local rigidity results of locally symmetric spaces discussed in Section 4 was the following one on hyperbolic manifolds by Mostow [54] in 1968. We quote the original statement of [54], Corollary 12.2. Proposition 8.2. Let Y and Y 0 be complete Riemannian manifolds having constant negative curvature and finite volume. Assume that there is a quasi-conformal homeomorphism from Y to Y 0 . Then there is a unique conformal mapping from Y onto Y 0 inducing the same isomorphism between the fundamental groups, provided the dimension of Y and Y 0 is greater than two. The conformal mapping may be taken to be an isometry if Y and Y 0 have the same curvature. We note that in the above proposition, the negative constant curvatures of the manifolds Y and Y 0 might take different values, and a scaling is needed. If they are both assumed to be hyperbolic, i.e., of constant curvature 1 , then no scaling is needed, and they are isometric. This was the first time that a strong rigidity phenomenon was proved for this class of locally symmetric spaces under a reasonable assumption.

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Remark 8.3. In an earlier paper [52], p. 418, on rigidity of locally symmetric spaces of finite volume, Mostow made a strong assumption that there exists an equivariant homeomorphism between symmetric spaces which induces a diffeomorphism on the maximal Furstenberg boundaries in the Satake compactification of the symmetric spaces. This condition seems difficult to check. The assumption on the existence of a quasi-conformal homeomorphism between Y and Y 0 is crucial for the proof and it does not follow obviously from the assumption that Y and Y 0 have isomorphic fundamental group. Later Mostow [55] and Margulis [42] proved the now standard strong rigidity for compact hyperbolic manifolds as stated in Proposition 8.1. The idea of the proof of Proposition 8.1 is to obtain an appropriate quasiconformal map on the boundary @Hn of the hyperbolic space Hn from an quasi-conformal map in the interior Hn . Let W Γ ! Γ0 be an isomorphism between the fundamental groups of compact hyperbolic manifolds ΓnHn and Γ0 nHn . Then there exists a coarse isometry (or quasi-isometry) 'W Hn ! Hn

which is equivariant with respect to W Γ Š Γ0 . The crucial step is to show that ' extends to a quasi-conformal map '0 on the boundary @Hn . Then the method of [54] can be applied to prove that the boundary map '0 is conformal if n  3 and hence ΓnHn and Γ0 nHn are isomorphic. The brief discussion in the next section explains why the boundary map '0 is quasi-conformal in the general setup of Gromov hyperbolic spaces and Gromov boundary. Now we briefly explain how quasi-conformal maps were used in [54] to prove the strong rigidity result. Let 'W ΓnHn ! Γ0 nHn

be a quasi-conformal map. Then ' lifts to an equivariant quasi-conformal map 'W Hn ! Hn with respect to an isomorphism W Γ ! Γ0 : It is known that the boundary @Hn can be canonically identified with the unit sphere S n 1 . The following are two basic results of [54], theorems 10.1 and 10.2. Proposition 8.3. A quasi-conformal mapping of Hn onto itself induces a homeomorphism on the boundary @Hn . Proposition 8.4. An equivariant quasi-conformal surjective map 'W Hn ! Hn extends continuously to the boundary @Hn = S n 1 , and the boundary map '0 W S n

1

! Sn

1

(2)

is also a quasi-conformal map with respect to the standard metric on the unit sphere S n 1  Rn .

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As Mostow stated on [54], p. 54, he was directly motivated by a basic result in the theory of quasi-conformal mappings of A. Mori [50], Theorem 4, on the extension of quasi-homeomorphisms of the open unit disc in C to the boundary S 1 . It follows from the general theory of quasi-conformal mappings [54], Theorem 9.1, that the quasiconformal boundary mapping (2) is differentiable almost everywhere for all n  2 . When n  3 , the sphere S n 1 has dimension at least 2, and the differential of '0 is invertible at almost every point [54], Theorem 9.4, and p. 99. This is one crucial point of the proof. Remark 8.4. Though Mostow gave a self-contained exposition of quasi-conformal maps in high dimensions, there was a lot of earlier work on quasi-conformal maps both in dimension 2 and above by many people. For a survey and extensive references, see [45]. Remark 8.5. When n = 2 , if '0 W S n 1 ! S n 1 does not come from a Möbius transformation of H2 , then 'o is completely singular, i.e., the derivative of '0 exists almost everywhere and is zero whenever it exists. Therefore, when n = 2 . the boundary map '0 is either a Möbius transformation or completely singular. The second case can happen, and completely singular boundary maps '0 correspond to nontrivial deformations of the compact hyperbolic surface ΓnH2 . This was Mostow’s explanation for the non-rigidity of compact hyperbolic surfaces. See [57], p. 117. Another crucial ingredient in the proof of Mostow strong rigidity for hyperbolic manifolds is that the actions of Γ and Γ0 on @Hn are ergodic. More specifically, the boundary at infinity @Hn can be canonically identified with the unit sphere S n 1 in the tangent space Tp Hn at any base point b 2 Hn , and the Riemannian inner product on Tp Hn induces the Lebesgue measure on S n 1 , which is transferred to a measure p on @Hn . It is reasonable to expect and indeed it is true that this measure p depends on the choice of the base point p . On the other hand, for a different base point p 0 , the measures p and p0 are absolutely continuous with respect to each other. This implies that even though the action of Γ on @Hn does not leave the measure p invariant, it preserves the equivalence classes of the measures p , and hence the notion of ergodic action is well defined. The ergodicity of the action of Γ on @Hn follows from the ergodicity of the geodesic flow of a strictly negatively curved locally symmetric space of finite volume [47]. Mostow also gave a self-contained proof of a slightly weaker result in [54], Theorem 11.1: the topological transitivity of a related action, which was sufficient for the proof in [54]. See [76], §2.2, for a summary on the ergodicity of the actions of lattices on the boundary of symmetric spaces and of geodesic (or more general homogeneous) flows of locally homogeneous spaces, and for an outline of a proof of the Mostow strong rigidity for hyperbolic manifolds. Since the boundary map '0 W @Hn ! @Hn is equivariant with respect to the isomorphism W Γ ! Γ0 and the actions of Γ and Γ0 are ergodic, it implies:

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Proposition 8.5. The boundary map '0 is conformal. Consequently, the two cocompact lattices are conjugate by a Möbius transformation. This result was proved in two steps. Under the assumption n  3 , the Jacobian of '0 is positive almost everywhere. For every point x 2 @Hn such that '0 is differential at x and its differential d'0 (p) is invertible, the ergodicity of the Γ -action implies that d'0 (p)W Tp @Hn ! T'0 (p) @Hn is a conformal map. In fact, the ergodicity of the Γ -action implies that the distortion is the same in every direction in Tp @Hn . This finishes the first step. Second, it follows from a general fact that any 1 -quasi-conformal map W S n 1 ! S n 1 is conformal if n  3 [54], Lemma 12.2, which is a version of the Liouville’s theorem for conformal mappings. The assumption that n  3 is needed because of the use of moduli of spherical shells to identify conformal mappings [54], Remark, p. 102. This completes the proof of Proposition 8.1. Remark 8.6. The fact that an equivariant quasi-conformal mapping of S n 1 , n  3 , with respect to ergodic actions of discrete groups having an invertible differential at some point is conformal is not surprising. In some sense, the ergodicity of the action swears out the invertibility of the differential to other points and forces invariance at each point as well, which implies that the map is conformal. Remark 8.7. In the above outline of a proof of the Mostow strong rigidity for compact hyperbolic manifolds of dimension greater than 2, we did not completely follow the proof in [54]. Instead of the ergodic action of lattices Γ on the boundary @Hn of the hyperbolic spaces, Mostow used the topological transitivity of a semigroup isomorphic to the semi-group (0; 1)  (R>0 ; ) in [54], Theorem 11.1. This was used crucially to prove that at a differentiable point, the boundary map '0 induces a conformal map on the tangent spaces in [54], pp. 99–101. In later expositions of proofs of the Mostow strong rigidity following [54], people usually use the steps we outlined above, for example in [57], p. 116, [55], p. 6, and [76], §2.2. One reason is that the proof using the ergodicity of lattice actions on the boundary @Hn is more conceptional and is also related to Mostow’s proof of strong rigidity for other rank-1 symmetric spaces in [55].

9. The negative part of Mostow strong rigidity The best known case of the Mostow strong rigidity theorem deals with compact hyperbolic manifolds and was discussed in Proposition 8.1 in the previous section. It is a subcase of the following special case of the general Mostow strong rigidity. Proposition 9.1. Let G be a semisimple noncompact Lie group G without center and nontrivial normal compact Lie subgroup. If G ¤ PSL(2; R) , then any two isomorphic lattices Γ1 ; Γ2 of G are conjugate by an element of G .

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This special result in Proposition 9.1 is also the crucial step in the proof of the general case. Specifically, we can formulate the Mostow strong rigidity for general locally symmetric spaces in the following two statements. 1. If two lattices Γ1 ; Γ2 of two semisimple Lie groups G1 ; G2 are isomorphic, then G1 and G2 are isomorphic. (Though at this stage, the isomorphism Γ1 ! Γ2 may not be the restriction of an isomorphism G1 ! G2 yet.) 2. For a fixed semisimple Lie group G and two lattices Γ1 ; Γ2 in G , any isomorphism between Γ1 and Γ2 comes from the conjugation by an element of G . Statement 2 is Proposition 9.1. In this section, we explain a proof of the Statement 1 through a particular case of the following result. Proposition 9.2. Let G1 ; G2 be two noncompact semisimple Lie groups with trivial center and no compact factors, and let Γ1  G1 , Γ2  G2 be two lattices. If G1 is not isomorphic to G2 , then Γ1 cannot be isomorphic to Γ2 . Equivalently, if the associated symmetric spaces X1 = G1 /K1 and X2 = G2 /K2 are not isometric up to scaling, then compact locally symmetric spaces Γ1 nX1 and Γ2 nX2 associated with them are not homotopy equivalent. When the reader sees this statement, one natural reaction might be to deduce it from the conclusion of the Mostow strong rigidity. As explained above, this is one step in the Mostow strong rigidity. The purpose of this section is to give a direct proof of this through a basic example. Proposition 9.2 is more in the spirit of Furstenberg [23]. Since this is a negative version of Step 1, we call it the negative part of Mostow strong rigidity as in the heading of this section. Remark 9.1. An equivalent formulation of the Mostow strong rigidity for rank-1 locally symmetric spaces is the following. Let X1 and X2 are two symmetric spaces of rank 1. If two compact locally symmetric spaces Γ1 nX1 and Γ2 nX2 are not isometric up to scaling, then they are not homotopy equivalent. The assumption here is clearly weaker than the assumption in Proposition 9.2 where it deals with symmetric spaces of all possible ranks. In the book [55], the proof of the general Mostow strong rigidity is reduced to the following two cases. 1. When the ranks of symmetric spaces X; X 0 are at least 2, the strong rigidity follows from the rigidity of the spherical Tits buildings of X and X 0 , which are infinite simplicial complexes whose underlying geometric spaces can be identified with the spheres at infinity of X and X 0 respectively. More

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specifically, since ΓnX and Γ0 nX 0 are compact, a homotopy equivalence between them 'W ΓnX ! Γ0 nX 0 lifts to a (Γ; Γ0 ) -equivariant pseudo-isometry (now usually called quasiisometry) ˜ X ! X 0: 'W This map '˜ induces an isomorphism between the spherical Tits buildings of X and X 0 , and hence the rigidity of spherical Tits buildings (see [80]) implies that the Lie groups G1 ; G2 are isomorphic, or equivalently the symmetric spaces X and X 0 are isometric after a suitable scaling of the metrics. 2. When X and X 0 are of rank one, i.e, if they have strictly negative curvature, their boundaries @X , @X 0 are one-point compactifications of nilpotent Lie groups. In fact, for every point z 2 @X , the stabilizer Gz of z in G is a parabolic subgroup, and the unipotent radical Nz of Gz is a nilpotent Lie group and acts simply transitively on the complement @X ¹zº . When X is the real hyperbolic space, Nz is abelian, and the boundary of X is a sphere S dim X 1 as we saw in the previous section. When X is the complex hyperbolic space, Nz is the (generalized) Heisenberg group. For other rank-1 symmetric spaces, Nz is a two-step nilpotent Lie group. As in the previous case, an equivariant quasi-isometry ˜ X ! X0 'W

induces a map '˜0 between the boundaries @X and @X 0 . A theory of quasi-conformal structures and quasi-conformal maps on the boundary of symmetric spaces of rank one, or on the nilpotent Lie groups Nz , was developed in [55], §20, and it was applied in [55], §21. One crucial point is that the quasi-conformal structures on the boundaries @X and @X 0 are defined with respect to sub-Riemannian metrics, and the boundary map '˜0 is quasi-conformal with respect to these sub-Riemannian metrics on the boundaries [55], Lemma 21.2. A certain differentiability property of the map '˜0 , see Proposition 21.3 of [55] (and the correction in [59]), implies that X and X 0 are isometric [55], Corollary 21.5. Combining with the ergodic action of Γ on the boundary @X again, one can show that '˜0 is induced from an isometry between X and X 0 . In reducing the general Mostow strong rigidity to the two cases above, the following result was proved in [55], Lemma 11.3 and the summary on p. 79. Proposition 9.3. Let ΓnX and Γ0 nX 0 be two locally symmetric spaces of finite volume. If there exists a homotopy equivalence 'W ΓnX ! Γ0 nX 0 , then the rank of X is equal to the rank of X 0 . In particular, a rank- 1 locally symmetric space cannot be homotopic equivalent to a higher rank locally symmetric space.

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The basic idea is that the rank of X is equal to the maximal rank of abelian subgroups of Γ consisting of semisimple elements. Besides the original exposition of Case 1 in [55], a fairly detailed summary together with discussion on buildings is given in [28]. For Case 2 on rank- 1 locally symmetric spaces, most expositions concentrate on the case of real hyperbolic spaces as we saw in the previous section. On the other hand, there are some exceptions. Besides the approach of using the entropy rigidity in [11], a summary of the proof using quasi-conformal maps for the case of X being a complex hyperbolic plane H2C is given in [17], pp. 135–138. In the rest of this section, we explain through a particular example a proof of the following special case of Proposition 9.1, which is also natural and basic in itself. It is known [55], §19, that there are four types of symmetric spaces of rank 1: the real hyperbolic spaces Hn , the complex hyperbolic spaces Hm C , the quaternionic hyperbolic spaces, and the hyperbolic Cayley plane. Proposition 9.4. Suppose that the rank of both symmetric spaces X and X 0 is equal to 1 , but X and X 0 are not of the same type and hence not isometric, for example, X is a real hyperbolic space HnR , and X 0 is a complex hyperbolic 0 0 space Hm C . Then locally symmetric spaces associated with them, ΓnX and Γ nX , are not homotopy equivalent. This proposition was the first step in the proof of the rank 1 case of Mostow strong rigidity in [55] and was proved in [55], Corollary 21.5. To prove this proposition, we need to show that if X and X 0 are not of the same type, then there is no homotopy equivalence ' between ΓnX and Γ0 nX 0 . Lemma 9.5. If X and X 0 have compact quotients ΓnX and Γ0 nX 0 which are homotopy equivalent, then X and X 0 have the same dimension. In particular, the boundaries @X and @X 0 are topological spheres of the same dimension. Proof. We only need to note that when Γ; Γ0 are torsion-free, the cohomology dimensions of Γ and Γ0 are equal to dim X and dim X 0 respectively. □ To distinguish between X and X 0 , we need to show that different internal geometric structures on the symmetric spaces X; X 0 induce different additional structures at the boundary at infinity @X; @X 0 . To distinguish their spheres at infinity, the notions of quasi-conformal structures on the boundaries and quasi-conformal maps between them turn out to be sufficient and convenient. We will explain briefly how it works in the case X = H4R , the real hyperbolic space, and X 0 = H2C , the complex hyperbolic space. We will explain that the conformal structure on the sphere at infinity @H2C is related to the conformal structure of the Heisenberg group H , which is a two-step nilpotent Lie group of dimension 3.

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In view of applications of Mostow strong rigidity and quasi-conformal maps to geometric group theory mentioned below, we introduce the notions of quasiconformal structures and quasi-conformal maps for Gromov ı -hyperbolic geodesic spaces. The basic references are [14], [12], and [63], which were motivated by [55] and [42]. Definition 9.1. On a topological space Y , a family of metrics which are quasiisometric to each other defines a quasi-conformal structure on Y . If the metrics in the family are conformal (i.e., 1 -quasi-conformal) to each other, then they define a conformal structure. Let (X; d ) be a proper ı -hyperbolic geodesic metrics space, and let @X be its Gromov boundary. For any basepoint p 2 X and x; y 2 X , let (x
y)p be the Gromov product with respect to the basepoint p : 1 (x
y)p = (d (x; p) + d (y; p) d (x; y)): 2 Then it is known that the Gromov product extends to the boundary @X , see [14], p. 432. Definition 9.2. A metric  on the Gromov boundary @X of a Gromov hyperbolic space is called a visual metric if there exist positive constants " , k1 ; k2 such that for all x; y 2 @X , k1 e "(x
y)p  (x; y)  k2 e "(x
y)p : Every visual metric induces the topology on @X , which is often called the visual topology. Hence it is called a visual metric. See [14], p. 434, for more detail. It is clear from definition that every two visual metrics are quasi-isometric to each other. It is known that for every Gromov hyperbolic space X , its boundary @X admits visual metrics. The visual metrics are quasi-isometric to each other, and hence they define the canonical quasi-conformal structure on the boundary @X . Proposition 9.6. Let X and Y be two proper geodesic metric spaces which are Gromov hyperbolic space. Then any quasi-isometry X ! Y induces a quasiconformal homeomorphism @X ! @Y with respect to their canonical quasi-conformal structures. For proof, see [12] and also [14], p. 436. When X is a CAT(-1)-space and " is sufficiently small, the function " (x; y) = e

"(x
y)p

defines a metric on @X and hence is a visual metric by definition. It was also shown in [12] that when X is a CAT(-1)-space, by using the Gromov product with

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respect to basepoints on the boundary @X , we can obtain a canonical conformal structure on @X . On the other hand, for a general Gromov ı -hyperbolic space, the function e "(x
y)p does not define a metric on @X , but we can produce an associated metric out of this. See [14], p. 434f. Now we are ready to explain the idea of the proof of Proposition 9.4, i.e., the Mostow strong rigidity when X = H4R and X 0 = H2C : 1. The conformal structure on the boundary H4R is the canonical conformal structure on S 3 , which is equal to the conformal structure induced from the standard conformal structure on R3 when S 3 is identified with R3 [ ¹1º . More specifically, this follows from the fact that when H4R is realized as the unit ball B 4 (1) in R4 and its boundary @H4R is identified with the unit sphere S 3 , then the visual metric e (x
y)0 on B 4 (1) with respect to the origin (as a basepoint) is equal to a multiple of the restriction of the Euclidean metric on R4 [17], p. 50. 2. The boundary @H2C of H2C is also equal to S 3 and can be identified with the one point compactification of the Heisenberg group H : @H2C Š H [ ¹1º:

Recall [17], p. 11, that the Heisenberg group H is defined by ˇ 8 9 0 1 ˇ 1 x1 x3 < = ˇ H = x = @0 1 x2 A 2 GL(3; R) ˇˇ x1 ; x2 ; x3 2 R : : ; ˇ 0 0 1 Under this identification, the conformal structure on H2C induces the conformal structure on H defined by a sub-Riemannian metric (or the Koranyi metric) on H . This follows from the fact that when H2C is realized as the unit ball in C2 , the visual metric with respect to the origin (as a basepoint) is compatible with the Koranyi metric, and when the basepoint in the Gromov product is moved to a boundary point, the associated visual metric converges to the Koranyi metric [17], pp. 54f. Specifically (see [17], p. 18), the Koranyi metric on H is defined by: for x; y 2 H , dK (x; y) = jjy 1 xjjK ; (3) where the Koranyi gauge jjxjjK is defined by 4 jjxjjK = (x12 + x22 )2 + 16x32 :

(4)

Lemma 9.7. The conformal structures on @H2C and @H4R are not quasi-conformal to each other.

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Proof. It can be seen from the fact (equations 3 and 4) that the Koranyi metric or the Koranyi gauge jjxjjK on H hasinhomoheneous dilation in the coordinates 1 x1 x3 x1 ; x2 ; x3 of an element x = 0 1 x2 in H , while the Euclidean metric on R3 0 0

1

is clearly homogeneous in the coordinates of (x1 ; x2 ; x3 ) 2 R3 ,

□

This lemma and Proposition 9.6 imply the following result. Corollary 9.8. There is no homotopy equivalence between compact locally symmetric spaces ΓnH4R and Γ0 nH2C . Proof. Any homotopy equivalence between ΓnH4R and Γ0 nH2C induces a quasiisometry between H4R and H2C . By Proposition 9.6, it induces a quasi-conformal homeomorphism between @H2C and @H4R , which contradicts Lemma 9.7. □ Remark 9.2. This important application to the Mostow strong rigidity to complex hyperbolic manifolds motivated the theory of quasiconformal mappings of the Heisenberg group, in particular the papers [38], [39], and [62]. These works also motivated quasiconformal mappings in other spaces such as Gromov hyperbolic spaces and CAT( 1) -spaces as mentioned above. Remark 9.3. One important application of quasi-conformal maps on the boundary of the other two types of rank 1 symmetric spaces is a stronger rigidity result than the Mostow strong rigidity. In [62], Pansu obtained the following rigidity result: every quasi-isometry of quaternionic or octonionic spaces has bounded distance from an isometry, which does not hold for real and complex hyperbolic spaces. We note that in the Mostow strong rigidity, the quasi-isometry between symmetric spaces is equivariant with respect to lattices. There is no equivariance here.

10. Generalizations of the Mostow strong rigidity There have been several generalizations of the Mostow strong rigidity. In this section, we briefly discuss several of them.

10.1. Strong rigidity for lattices of p -adic semisimple Lie groups As it is known, R is the completion of Q with respect the Archimedean norm, and for each prime number p , there is a p -adic norm on Q , and the completion of which gives the field Qp of p -adic numbers. For many problems in number theory and representation theory of Lie groups, it is important and fruitful to treat

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all these norms on Q on equal footing. One example is the introduction of adeles and ideles. The above discussion suggests that lattices in the real Lie group SL(n; R) and SL(n; Qp ) should be considered together. Therefore, it is not surprising that the problem on the strong rigidity of lattices of p -adic semisimple Lie groups was raised already. We note that in this case, every lattice is cocompact. In [66], Prasad proved the Mostow strong rigidity for them when the rank of the building associated with the p -adic semisimple Lie group is greater than or equal to 2 . Instead of considering real and p -adic semisimple Lie groups separately, it is important to consider products of real and p -adic semisimple Lie groups and lattices in them. Such discrete groups arise naturally in the theory of automorphic forms. For example, for each prime number p , let Z p1 be the ring obtained from Z by  
inverting 2 . Then SL n; Z p1 is a lattice of the product SL(n; R)  SL(n; Qp ) . Margulis [44] considered and proved the superrigidity for irreducible lattices in such products. As a corollary, these lattices satisfy the Mostow strong rigidity.

10.2. Generalizations to infinite volume locally symmetric spaces In the above discussion on rigidity of discrete subgroups of Lie groups, we always assume that they are lattices, or even cocompact lattices. On the other hand, discrete subgroups which are not lattices arise naturally in the theory of Fuchsian groups acting on the real hyperbolic plane H2 and Kleinian groups acting on the real hyperbolic space Hn , n  3 . We mention some generalizations of the Mostow strong rigidity for them. We note that PSL(2; C) acts isometrically on the three-dimensional real hyperbolic space H3 . If Γ  PSL(2; R) is a cocompact lattice, then it can be embedded into PSL(2; C) as a discrete subgroup by composing with the map PSL(2; R) ! PSL(2; C) . In this case, it is known that the noncompact hyperbolic manifold ΓnH3 is of infinite volume and not locally rigid. See [41] for example. For a discrete subgroup Γ acting on Hn which is not necessarily a lattice, one correct formulation of the Mostow strong rigidity is as follows. Consider the action of Γ on the sphere at infinity S n 1 = @Hn . Suppose there is a homeomorphism f W S n 1 ! S n 1 which conjugates the action of Γ on S n 1 to the action of another discrete subgroup Γ0 . If f is a Möbius transformation, we say that Γ is strongly rigid. As we explained above in §8, when Γ is a lattice, this is a crucial step in the proof of the Mostow strong rigidity, after the boundary map '0 W S n 1 ! S n 1 is constructed from an isomorphism W Γ ! Γ0 . We emphasize that the assumption that Γ is a lattice is used crucially in constructing the boundary map. In [2], [77], and [78], the strong rigidity was proved for discrete subgroups of divergence type when the boundary map f is quasi-conformal. In [81], the strong rigidity was proved when f is differentiable with a non-vanishing Jacobian

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at a radial limit point of Γ . Some related results and improvements were obtained in [27]. More generally, we can also replace the whole sphere S n 1 by a subset A  S n 1 which is invariant under Γ . Usually A is taking to be a suitable limit set of Γ . Several results of this type were obtained in [82]. Briefly, one result says that, if f is measurable and injective, then either f is the restriction of a Möbius transformation or there is a subset of full measure whose image under f has zero measure.

10.3. Rigidity results in differential geometry As explained above, the Mostow strong rigidity can be viewed either as a rigidity property of special Riemannian manifolds of nonpositive curvature or as a rigidity property of lattices of semisimple Lie groups. In the book [6], a strengthening of the Mostow strong rigidity was proved by replacing one of the two compact locally symmetric spaces with isomorphic fundamental groups by a compact Riemannian manifold of nonpositive sectional curvature of the same volume and the same fundamental group. As mentioned before, the strong rigidity for Hermitian locally symmetric spaces in [74] and [75] were also motivated by the Mostow strong rigidity and can be considered as a generalization or strengthening of it.

11. Other major works of Mostow and another strong rigidity To understand better the Mostow strong rigidity, it will be helpful to take a look at some other major results of Mostow. For this purpose, the description of Mostow’s work by the Wolf Prize committee, when Mostow was awarded the Wolf prize in 2013, is a concise and comprehensive summary: George D. Mostow made a fundamental and pioneering contribution to geometry and Lie group theory. His most celebrated accomplishment in this fields is the discovery of the completely new rigidity phenomenon in geometry, the Strong Rigidity Theorems. These theorems are some of the greatest achievements in mathematics in the second half of the 20 th century. This established a deep connection between continuous and discrete groups, or equivalently, a remarkable connection between topology and geometry. Mostow’s rigidity methods and techniques opened a floodgate of investigations and results in many related areas of mathematics. Mostow’s emphasis on the “action at infinity” has been developed by many mathematicians in a variety of directions. It had a huge impact in geometric group theory, in the study of Kleinian groups and of low dimensional topology, in work connecting ergodic theory and Lie groups. Mostow’s contribution to mathematics is not limited to strong

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We want to explain another strong rigidity result of Mostow which was not explicitly mentioned in the above citation, but is closely related to the Mostow strong rigidity and has had a huge impact on geometric topology through the Borel conjecture. It is well known that a discrete subgroup Λ  Rn is a lattice if and only if the quotient ΛnRn is compact, and any two lattices Λ1 ; Λ2  Rn are isomorphic as groups, and their associated quotient spaces Λ1 nRn and Λ2 nRn are diffeomorphic smooth manifolds. One natural generalization is to replace Rn by a simply connected nilpotent Lie group N . Then every lattice Λ  N is cocompact, and the quotient ΛnN is called a nilmanifold. In this case, the fundamental group of ΛnN is isomorphic to Λ . One result of Malcev in [40] says that if two nilmanifolds are homotopy equivalent, i.e., their corresponding lattices in nilpotent Lie groups are isomorphic, then they are homeomorphic. This is a strong rigidity property of nilmanifolds. Mostow generalized this rigidity to solvmanifolds in [51]. Specifically, a manifold is called a solvmanifold if a solvable Lie group acts transitively on it. Equivalently, a solvmanifold can be written in the form H nG where G is a solvable Lie group, and H  G is a closed subgroup such that the quotient H nG is compact. One result of Mostow in [51] says that two compact solvmanifolds with isomorphic fundamental groups are hommeomorphic. After reading this paper, Borel made a conjecture in a letter to Serre in 1954 which states: If two closed (i.e., compact without boundary) aspherical manifolds are homotopy equivalent, then they are homeomorphic. z By definition, a manifold M is aspherical if its universal covering space M is contractible. It is known that nilmanifolds and solvmanifolds are aspherical, and locally symmetric spaces of noncompact type are also aspherical. Consequently, the Borel conjecture is a natural generalization of the strong rigidity of solvmanifolds in [51]. Remark 11.1. Since two compact hyperbolic surfaces with isomorphic fundamental groups are diffeomorphic, the Mostow strong rigidity for locally symmetric spaces implies that the Borel conjecture holds when both aspherical manifolds are locally symmetric spaces of noncompact type. Because of this, some people thought that the Borel conjecture was motivated by the Mostow strong rigidity for locally symmetric spaces.

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Due to existence of exotic spheres, the conclusion of the existence of homeomorphisms in the Borel conjecture cannot be improved to the existence of diffeomorphisms. Naturally, for aspherical Riemannian manifolds, it cannot be improved to an isometry as in the case of the Mostow strong rigidity. The Borel conjecture has been proved for large classes of manifolds and is still open in general. It is closely related to the Novikov conjectures in topology, and is one of the most important conjectures in geometric topology. See the paper [7] and references there.

12. Some comments on the Mostow strong rigidity Though the Mostow strong rigidity is a well known result to many people, especially to geometers, topologists and Lie group theorists, our discussions above showed that there might be some aspects which are not so well known to some people. In the final section, we will take a look at some general aspects of the Mostow strong rigidity and its impact, and use this as an example of deep result in mathematics. In particular, we are interested in the following questions. 1. Is the statement or result simple to be stated and easy to be understood, while the proof being highly nontrivial? 2. Does it shed a new light on the object under study? 3. Does it establish new connections with other subjects? 4. Are there other problems and results motivated by it? 5. Are the methods introduced in the proof useful for other problems? 6. Does it open a few topic or even a subject? Does it provide a new perspective on mathematics? The answer seems to be positive to all the above questions. We will comment on them briefly. 1. The Mostow strong rigidity for compact hyperbolic manifolds of dimension at least 3 is simple and clean: the geometry of such hyperbolic manifolds is completely determined by the fundamental group. This is in sharp contrast with compact hyperbolic surfaces. The Mostow strong rigidity for general locally symmetric spaces can also stated simply, and it can also be stated simply in terms of lattices of semisimple Lie groups: a lattice in a semisimple Lie group uniquely determines the Lie group besides the obvious exception. On the other hand, its proof is sophisticated and innovative. It makes use of multiple existing complex methods and develop new ones. 2. It expresses a striking relation between two aspects of a natural class of Riemannian manifolds in a clean statement. In particular, the volume provides a homotopy invariant of compact hyperbolic manifolds.

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Locally symmetric spaces were introduced by E. Cartan as a natural generalization of Riemann manifolds of constant curvature, i.e., those Riemannian manifolds whose curvature tensor is parallel. Locally symmetric spaces of finite volume have been extensively studied in the theory of automorphic forms since Klein and Poincaré. For many problems, there is no substantial difference between spaces of dimension 2 or above. Hence the Mostow strong rigidity brings out a new aspect of the geometry of locally symmetric spaces. When we view the Mostow strong rigidity in terms of Lie groups and discrete subgroups, it establishes a strong and unexpected connection between discrete groups and Lie groups which are continuous objects by definition. To answer the question “Was there a prevailing opinion beforehand that the theorem should be true?”, Mostow said in an interview after receiving the Wolf Prize in 2013: “Quite the contrary. The theorem attracted attention because it was unexpected.” As Helgason write in his review [25], p. 510, of the book [55]: His strong rigidity theorem not only proves that isomorphic subgroups Γ and Γ0 are necessarily conjugate under an automorphism of G , but also that all automorphisms of Γ are induced by automorphisms of G ; the latter being mostly inner automorphisms, this implies considerable limitations on Γ as an abstract group for it to be imbeddable in G as a discrete cocompact subgroup.

3. The Mostow strong rigidity brought together several subjects such as analysis (quasi-conformal maps), dynamics, geometry, and group theory. Mostow’s use of quasi-conformal maps is probably the most important application in the theory of quasi-conformal analysis in the higher dimensional Euclidean spaces and nilpotent Lie groups, and has stimulated its development. 4. The Mostow strong rigidity amounts to the rigidity of actions of lattices of semisimple Lie groups on symmetric spaces. There are other infinite discrete groups and their actions, and rigidity problems are important for them. One particular example is the mapping class group of surfaces and its action on the Teichmüller space. As we explained earlier, the Mostow strong rigidity also motivated several results about compact complex manifolds. 5. Mostow [55] introduced the notion of coarse geometry and made effective use of the idea of pushing things to infinity. Similar ideas were also introduced by Margulis [42] in his proof of the Mostow strong rigidity of hyperbolic manifolds. These ideas were developed systematically by Gromov into the subject of large scale geometry of spaces and groups, which has changed the subjects of geometric group theory and geometric topology.

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6. Rigidity and deformation are two contrasting properties in mathematics. There are many results related to them or their interactions. The Mostow strong rigidity provides a new way to think about the rigidity phenomenon and has stimulated a lot of work. The references in the survey papers [28] and [29] provide ample evidence to support this. Acknowledgments. I would like to thank P. Deligne for the very helpful correspondence [19] which provides a proof of Proposition 6.3, and N. A’Campo, I. Dolgachev, R. Spatzier, S. K. Yeung, and an anonymous referee for helpful comments and constructive suggestions. This work is partially supported by Simons grant #353785.

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Chapter 5

Models of ends of hyperbolic 3-manifolds. A survey Mahan Mj

Contents 1 Introduction: Fuchsian surface groups . . . . . 2 Kleinian surface groups . . . . . . . . . . . . . 3 Building blocks and model geometries . . . . . 4 Hierarchiesand the ending lamination theorem 5 Split geometry . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction: Fuchsian surface groups The aim of this survey is to describe models for degenerate ends of hyperbolic 3 -manifolds. The main problem in the area was Thurston’s Ending Lamination Conjecture [27], now a theorem due to Brock, Canary and Minsky [17] and [2]. The ending lamination theorem says roughly that the asymptotic topology of an end, encoded by the ending lamination, captures the geometry of the end completely. As a direct consequence of the techniques developed in [17] and [2], a number of other major conjectures in the area were directly resolved. Arguably, the most important of these was the Bers density conjecture asserting that quasi-Fuchsian groups are dense in the space AH(S ) of discrete faithful representations of the fundamental group 1 (S ) of a surface S into PSL2 (C) (up to conjugacy) equipped with the algebraic topology. The main tool developed in [17] and [2] is a combinatorial model for a degenerate end. This model was used by the author in [23] to prove that Cannon–Thurston maps exist and hence connected limit sets of finitely generated Kleinian groups are locally connected, thus settling another problem in [27]. We should also mention here work of Bowditch [5] giving a different proof of the ending lamination theorem. We shall survey some of these developments here. But to set the context, we shall first give a quick introduction to Fuchsian and Kleinian surface groups. A curious commentary on mathematical terminology and attribution is the fact

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that Fuchs had little to do with Fuchsian groups and Klein had little to do with Kleinian groups (though he did have something to do with Fuchsian groups); while Poincaré had everything to do with both. The terms “Fuchsian groups” and “Kleinian groups” are both due to Poincaré. The theme of this survey is the study of discrete faithful representations of surface groups into PSL2 (C) and so the story begins with Poincaré’s study of discrete faithful representations of surface groups into PSL2 (R) . The Lie group PSL2 (R) can be equivalently described as the group of Mobius transformations Mob(∆) of the unit disk, and hence as the group of Mobius transformations Mob(H) of the (conformally equivalent) upper half plane. The natural metric of constant negative curvature on the upper half plane is given by 2 2 ds 2 = dx y+dy the hyperbolic metric. The resulting space is denoted as H2 . The 2 orientation preserving isometries of H2 are exactly the conformal automorphisms of H2 . The boundary circle S 1 compactifies H2 , or equivalently, ∆ . This has a geometric interpretation. It encodes the “ideal” boundary of H2 , consisting of asymptote classes of geodesics. The topology on S 1 is induced by a metric which is defined as the angle subtended at 0 2 ∆ . The geodesics turn out to be semicircles meeting the boundary S 1 at right angles. To give the reader an idea of where we are heading let us proceed to construct an explicit example of a discrete subgroup of Isom(H2 ) . The genus two orientable surface Σ2 can be described as a quotient space of an octagon with edges labeled a1 ; b1 ; a1 1 ; b1 1 ; a2 ; b2 ; a2 1 ; b2 1 , where the boundary has the identification induced by this labeling. In order to construct a metric of constant negative curvature on it, we have to ensure that each point has a small neighborhood isometric to a small ball in H2 . To ensure this it is enough to do the above identification on a regular hyperbolic octagon (all sides and all angles equal) such that the sum of the interior angles is 2 . To ensure this, we have to make each interior angle equal 2 . The 8 infinitesimal regular octagon at the tangent space to the origin has interior angles equal to 3 . Also the ideal regular octagon in H2 has all interior angles zero. See 4 figure 1. Hence by the intermediate value theorem, as we increase the size of the octagon from an infinitesimal one to an ideal one, we shall hit interior angles all equal to 4 at some stage. The group G that is generated by side-pairing transformations corresponds to a Fuchsian group, or equivalently, a discrete faithful representation of the fundamental group of a genus 2 surface into Isom(H2 ) . We let W Σ2 ! PSL2 (R) denote the associated representation. A quick look at the construction above shows that it can be generalized considerably. All we required are: 1. sides identified are isometric; 2. total internal angle for the octagon is 2 . By modifying lengths and angles subject to these constraints, this allows us to obtain the entire space of hyperbolic structures on Σ2 , also called the Teichmüller space Teich(Σ2 ) .

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Figure 1

2. Kleinian surface groups We now turn to PSL2 (C) . As above, this Lie group appears in three incarnations: y ) = Isom+ (H3 ) . A Kleinian group G is a discrete subgroups PSL2 (C) = Mob(C of PSL2 (C) . This gives us three closely intertwined perspectives on the field: 1. studying discrete subgroups G of the group of Mobius transformations y ) emphasizes the complex analytical/dynamical aspect; Mob(C 2. studying discrete subgroups G of PSL2 (C) emphasizes the Lie group/matrix group theoretic aspect; 3. studying discrete subgroups G of Isom+ (H3 ) emphasizes the hyperbolic geometry aspect. We shall largely emphasize the third perspective. Since G is discrete, we can pass to the quotient M 3 = H3 /G . Thus, we are studying hyperbolic structures on 2 2 +dz 2 3 -manifolds. The hyperbolic metric is given by ds 2 = dx +dy on the upper z2 half space. Note that the metric blows up as one approaches z = 0 . Equivalently y consists of ideal we could consider the ball model, where the boundary S 2 = C y end-points of geodesic rays as before. The metric on C is given by the angle subtended at 0 2 H3 . Since Isom(H2 )  Isom(H3 ) , we can look upon the discrete group G constructed in §1 above also as a discrete subgroup of Isom(H3 ) . In figure 2, the x = R [ ¹1º . equatorial circle is identified with R

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Figure 2

In the above picture two things need to be observed. 1. The orbit G:o accumulates on the equatorial circle. This is called the limit set ΛG . 2. The complement of ΛG consists of two round open discs. On each of these disks, G acts freely (i.e. without fixed points) properly discontinuously, by conformal automorphisms. Hence the quotient is two copies of the “same” Riemann surface (i.e. a one dimensional complex analytic manifold). The y n ΛG = ΩG is called the domain of discontinuity of G . complement C y with the We proceed with slightly more formal definitions identifying C 2 sphere S . Definition 2.1. If x 2 H3 is any point, and G is a discrete group of isometries, the limit set ΛG  S 2 is defined to be the set of accumulation points of the orbit G:x of x . The domain of discontinuity for a discrete group G is defined to be ΩG = S 2 n Λ G . Proposition 2.1 ([26], Proposition 8.1.2). If G is not elementary, then every non-empty closed subset of S 2 invariant by G contains the limit set ΛG .

2.1. Quasi-Fuchsian groups Suppose that G is abstractly isomorphic to the fundamental group of a finite-area hyperbolic surface S , and W 1 (S ) ! PSL2 (C) is a discrete faithful representation with image G . Suppose further that  is strictly type-preserving, i.e. g 2 1 (S ) represents an element in a peripheral (cusp) subgroup if and only (g) is parabolic. In this situation, we shall refer to G as a Kleinian surface group. A recurring theme in the context of finitely generated, infinite covolume Kleinian groups is that the general theory can be reduced to the study of Kleinian surface groups. Thus, we study the discrete faithful elements of the representation space Rep(1 (S); PSL2 (C)) .

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y ) , the dynamics of the action of G Regarding G as a subgroup of Mob(C y on C) emerges. Recall that the limit set ΛG of G is the set of accumulation y for some (any) o 2 H3 . The limit set may be thought points of the orbit G:o in C of as the locus of chaotic dynamics of the action of G on C . The complement y n ΛG = ΩG is called the domain of discontinuity of G . C On the other hand regarding G as a subgroup of Isom(H3 ) , we obtain a quotient hyperbolic 3 -manifold M = H3 /G with fundamental group G . A substantial part of the theory of Kleinian groups tries to understand the relationship between the dynamic and the hyperbolic geometric descriptions of G . The Bers simultaneous uniformization theorem states that given any two y) conformal structures 1 ; 2 on a surface, there is a discrete subgroup G of Mob(C whose limit set is topologically a circle, and whose domain of discontinuity quotients to the two Riemann surfaces 1 ; 2 . See figure 3. The limit set is a quasiconformal map of the round circle.

Figure 3

Definition 2.2. A Kleinian surface group G is quasi-Fuchsian if its limit set ΛG is homeomorphic to a circle. These (quasi Fuchsian) groups can be thought of as deformations of Fuchsian groups (Lie group theoretically) or quasiconformal deformations (analytically). Ahlfors and Bers proved that these are precisely all quasiconvex Kleinian surface groups. Let QF(S ) consist of quasi Fuchsian representations W 1 (S) ! PSL2 (C) , where two such representations are regarded as equivalent if they are conjugate in PSL2 (C) . Thus the Ahlfors–Bers theorem can be summarized as:

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QF(S ) = Teich(S )  Teich(S ) .

The convex hull CHG of ΛG is the smallest non-empty closed convex subset of H3 invariant under G . It can be constructed by joining all pairs of points on limit set by bi-infinite geodesics, iterating this construction, and finally taking the closure. The quotient of CHG by G , which is homeomorphic to S  [0; 1] , is called the convex core CC(M ) of M = H3 /G . A slight generalization of quasi-Fuchsian groups is given by the following: Definition 2.3. A Kleinian group is geometrically finite group if for some  > 0 , the volume of the  -neighborhood N (CC(M )) is finite. A Kleinian group is geometrically infinite if it is not geometrically finite. The “thickness” of CC(M ) for a quasi-Fuchsian surface group, measured by the distance between S  ¹0º and S  ¹1º is a geometric measure of the complexity of the quasi-Fuchsian group G . If we assume that S has no parabolics, then QF(S ) coincides precisely with geometrically finite representations without parabolics. For any such G , M n CC(M ) consists of two hyperbolically flaring pieces, called geometrically finite ends, each homeomorphic to S  [0; 1) . The metric on such a geometrically finite end E is of the form dt 2 + sinh(t ) ds 2 . On the other hand, geometrically infinite Kleinian surface groups G without parabolics are those for which the convex core CC(M ) is homeomorphic to S  [0; 1) or S  ( 1; 1) . The geometry of such ends is far more complicated and we shall return to this topic later.

2.2. Laminations and pleated surfaces An essential technical tool in the study of Kleinian surface groups is the theory of laminations and pleated surfaces [26]. Definition 2.4. A geodesic lamination on a hyperbolic surface is a foliation of a closed subset with geodesics. A geodesic lamination equipped with a transverse measure is a measured lamination. The space of measured laminations on S is denoted as ML(S) . It is equipped with the weak topology on transverse measures. Projectivizing (by removing the zero element and identifying elements that differ by a scaling), we obtain the projectivized measured lamination space PML(S ) . A fundamental theorem of Thurston shows that PML(S ) is homeomorphic to S 6g 7 (assuming that S is closed of genus g ). Further, the Thurston compactification of Teich(S ) adjoins PML(S ) to the Teichmüller space (homeomorphic to R6g 6 ) in a natural way, so that the action of the mapping class group Mod(S ) on Teich(S) extends naturally and continuously to the compactification.

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Geodesic laminations arise naturally in a number of contexts in the study of hyperbolic 2- and 3-manifolds: 1. as stable and unstable laminations corresponding to a pseudo-Anosov diffeomorphism of a hyperbolic surface; 2. as the pleating locus of a component of the convex core boundary @ CC(M ) of a hyperbolic 3 -manifold M ; 3. as the ending lamination corresponding to a geometrically infinite end of a hyperbolic 3 -manifold. In this section, we shall discuss briefly how each of these examples arises.

2.2.1. Stable and unstable laminations We consider the torus T 2 equipped with a diffeomorphism  , whose p action pon homology is given by a 2  2 matrix with irrational eigenvalues, e.g. 3+2 5 ; 3 2 5 . Then the eigendirections give rise to two sets of foliations by dense copies of R : the stable and unstable foliation. Such a diffeomorphism is called a Anosov diffeomorphism. Anosov diffeomorphisms of the torus may be characterized in terms of their action on 1 (T ) as not having periodic conjugacy classes. Now consider the stable (or unstable) foliation (minus a point  ) on S = (T 2 n ¹º) . Equip S with a complete hyperbolic structure of finite volume and straighten every leaf of the foliation to a complete geodesic. The leaf through ¹º is simply removed. The resulting union of leaves is called the stable (resp. unstable) lamination of the diffeomorphism  on the hyperbolic surface S . One of the fundamental pieces of Thurston’s work [8] shows that the existence of such a stable and unstable lamination generalizes to all hyperbolic surfaces. A diffeomorphism  of a hyperbolic surface S preserving punctures (or boundary components, according to taste) is called pseudo-Anosov if the action of  on 1 (S ) has no periodic conjugacy classes. Thurston proved the existence of a unique stable and unstable lamination without any closed leaves for any pseudo Anosov homeomorphism  acting on a hyperbolic surface S .

2.2.2. Pleating locus We quote a picturesque passage from [26]. Consider a closed curve  in Euclidean space, and its convex hull H ( ) . The boundary of a convex body always has non-negative Gaussian curvature. On the other hand, each point p in @H ( ) n  lies in the interior of some line segment or triangle with vertices on  . Thus, there is some line segment on @H () through p , so that @H ( ) has non-positive curvature at p . It follows that @H () n  has zero curvature, i.e., it is developable. If you are not familiar with this idea, you can see it by bending a curve out of a piece of stiff wire (like a coat-hanger). Now roll the wire around on a big piece of paper, tracing out a curve where

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This forces @H (K) equipped with its intrinsic metric to be a hyperbolic surface. However, there are complete geodesics along which it is bent (but not crumpled). Thus each boundary component S , and hence its universal cover Sx , carries a metric that is intrinsically hyperbolic. However, as a (possibly immersed) surface in H3 , the universal cover Sx is bent along a geodesic lamination. S is an example of a pleated surface: Definition 2.5 ([26], Definition 8.8.1). A pleated surface in a hyperbolic threemanifold N is a complete hyperbolic surface S of finite area, together with a map f W S ! N such that with respect to the metric induced from N , f is an isometry. Further, every x 2 S is in the interior of some straight line segment which is mapped by f to a straight line segment. Also, f must take every cusp of S to a cusp of N . The pleating locus of the pleated surface f W S ! M is the set  S consisting of those points in the pleated surface which are in the interior of unique line segments mapped to line segments. Proposition 2.3 ([26], Proposition 8.8.2). The pleating locus is a geodesic lamination on S . The map f is totally geodesic in the complement of .

2.2.3. Ending laminations The notion of an ending lamination comes up in the context of a geometrically infinite group. We shall deal with these groups in greater detail in the next section. Thurston defined the notion of a geometrically tame end E of a manifold M as follows.

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Definition 2.6. Let E be an end of a hyperbolic manifold M such that @E = S . The end E is said geometrically tame if there exists a sequence of simple closed curves ¹ i º on S such that the corresponding sequence ¹i º of geodesic realizations in M exit E . From the sequence ¹i º of geodesic realizations in M exiting E , it is in fact possible to construct a sequence ¹Σi º of pleated surfaces exiting E in the following sense. Fix a neighborhood E0 of E homotopy equivalent to S  (0; 1) . We say that ¹Σi º exits E , if, for any sequence of compact sets Kn exhausting E , and any n 2 N , there exists N 2 N such that Σi \ Kn = ; for all i  N . For such an end E , and for a sequence of simple closed curves ¹i º exiting E , the limit of ¹i º in the projectivized measured lamination space PML(S ) is a lamination  (the reader will not be much mistaken if (s)he thinks of the Hausdorff limit on the bounding surface S of E ). It turns out that  is independent of the sequence ¹n º and is called the ending lamination of the end E .

2.3. Degenerate groups The most intractable examples of Kleinian surface groups are obtained as limits of quasi Fuchsian groups. In fact, it has been recently established by Minsky et al. [17] and [2] that the set of all Kleinian surface groups (or equivalently all discrete faithful representations of a surface group in PSL2 (C) ) are given by quasi-Fuchsian groups and their limits. This is known as the Bers density conjecture. To construct limits of quasi Fuchsian groups, one allows the thickness of the convex core CC(M ) to tend to infinity. There are two possibilities: a. let only 1 degenerate. i.e. I ! [0; 1)

(simply degenerate case); b. let both 1 ; 2 degenerate, i.e. I ! ( 1; 1)

(doubly degenerate case). Thurston’s double limit theorem [28] says that these limits exist. In the doubly y. degenerate case the limit set is all of C

3. Building blocks and model geometries Let N be the convex core of a hyperbolic 3 -manifold without parabolics. The proof of the tameness conjecture (see [1] and [7]) shows that any end E of N has a neighborhood homeomorphic to S  [0; 1) , where S is a closed (or more generally finite volume hyperbolic) surface; in other words ends of hyperbolic

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3 -manifolds are topologically tame. Further, Thurston and Bonahon [26] and [3] and Canary [6] establish that geometrically infinite ends are geometrically tame i.e. there exists a sequence of pleated surfaces exiting them. However, the geometry of such ends can be quite complicated. We shall now proceed to describe model geometries of ends of hyperbolic 3 -manifolds following [16], [17], [2], [21], [22], [24], and [23] and leading to the most general case. In what follows in this section, we shall describe different kinds of models for building blocks of E : thick, thin, amalgamated. Each building block is homeomorphic to S  [0; 1] , where S is a closed surface of genus greater than one. What is common to all these three model building blocks is that the top and bottom boundary components are uniformly bi-Lipschitz equivalent to a fixed hyperbolic S . In the next section, a more general model geometry will be described.

Definition 3.1. A model Em is said built up of blocks of some prescribed geometries glued end to end if 1. Em is homeomorphic to S  [0; 1) ; 2. there exists L  1 such that S  [i; i + 1] is L -bi-Lipschitz to a block of one of the prescribed geometries S  [i; i + 1] will be called the (i + 1) -th block of the model Em . The thickness of the (i + 1) -th block is the length of the shortest path between S  ¹i º and S  ¹i + 1º in S  [i; i + 1]( Em ) .

3.1. Bounded geometry Definition 3.2 ([16] and [14]). An end E of a hyperbolic M has bounded geometry if there is a (uniform) positive lower bound 20 for lengths of closed geodesics in E . If M has finitely many ends and all the ends of M have bounded geometry, we say that M itself has bounded geometry. We briefly sketch the steps of the proof of the ending lamination theorem by Minsky [14] in the special case of bounded geometry ends. Let E be a simply degenerate end of N . Then E is homeomorphic to S [0; 1) for some closed surface S of genus greater than one. Thurston [26] established the density of pleated surfaces: Lemma 3.1. There exists D1 > 0 such that for all x 2 N , there exists a pleated surface gW (S;  ) ! N with g(S)\BD1 (x) ¤ ; . The following Lemma now follows easily from the fact that the injectivity radius i nj N (x) > 0 (where 0 is as in Definition 3.2): Lemma 3.2 ([3] and [26]). There exists D2 > 0 such that if the map gW (S;  ) ! N is a pleated surface, then dia(g(S )) < D2 .

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The following theorem due to Minsky [13] follows from compactness of pleated surfaces and a geometric limit argument. Theorem 3.3 ([13]). Fix S and  > 0 . Given a > 0 there exists b > 0 such that if gW (S;  ) ! N and hW (S; ) ! N are homotopic pleated surfaces which are isomorphisms on 1 and injN (x) >  for all x 2 N , then

where dTeich

dN (g(S ); h(S ))  a H) dTeich (; )  b , denotes Teichmüller distance.

The universal curve over X Teich(S ) is a bundle whose fiber over x 2 X is x itself. One can now show: Lemma 3.4. There exist K;  and a homeomorphism h from E to the universal curve S over a semi-infinite Lipschitz path in Teichmüller space, such that h˜ from Ex to the universal cover of S is a (K; ) -quasi-isometry. Proof. We can assume that S ¹0º is mapped to a pleated surface S0  N under the homeomorphism from S [0; 1) to E . We shall construct inductively a sequence of “equispaced” pleated surfaces Si  E exiting the end. Assume that S0 ; : : :; Sn have been constructed such that 1. if Ei be the non-compact component of EnSi , then Si +1  Ei ; 2. the Hausdorff distance between Si and Si +1 is bounded above by 3(D1 + D2 ) ; 3. dN (Si ; Si +1 )  D1 + D2 ; 4. from Lemma 3.3 and condition (2) above there exists D3 depending on D1 , D2 and S such that dTeich (Si ; Si +1 )  D3 . Next choose x 2 En , such that dN (x; Sn ) = 2(D1 + D2 ) . Then by Lemma 3.1, there exists a pleated surface gW (S;  ) ! N such that dN (x; g(S))  D1 . Let Sn+1 = g(S) . Then by the triangle inequality and Lemma 3.2, if p 2 Sn and q 2 Sn+1 , D1 + D2  dN (p; q)  3(D1 + D2 ) . This allows us to continue inductively. The lemma follows. □ Finally, Minsky in [13] and [14] establishes that the Lipschitz path in Lemma 3.4 is a quasi-geodesic tracking a unique Teichmüller geodesic ray. As a consequence he shows that any two ends of bounded geometry with the same ending lamination are bi-Lipschitz homeomorphic to each other. The ending lamination theorem in this case now follows in a straightforward fashion by using a theorem due to Sullivan [25]. Lemma 3.4 also furnishes a bi-Lipschitz model Em for E by gluing a sequence of thick blocks end-to-end.

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We describe below the construction of a thick block, as this will be used in all the model geometries that follow. Let S be a closed hyperbolic surface with a fixed but arbitrary hyperbolic structure.

Thick block. Fix a constant L and a hyperbolic surface S . Let B0 = S  [0; 1] be given the product metric. If B is L -bi-Lipschitz homeomorphic to B0 , it is called an L -thick block. The following statement is now a consequence of Lemma 3.4 (see [14] and also [18] and [21]). Remark 3.1. For any bounded geometry end, there exists L such that E is bi-Lipschitz homeomorphic to a model manifold Em consisting of gluing infinitely many L -thick blocks end-to-end.

3.1.1. Cannon–Thurston maps for bounded geometry The bounded geometry model of Remark 3.1 above was used to prove the existence of Cannon–Thurston maps in this case (see [14] for the case of surface groups and [18], [9], [4], [19], and [20] for other Kleinian groups). The main tool in [18] is the construction of a hyperbolic ladder L  Ex for any geodesic in Sx . Then Ex can be thought of as (is quasi-isometric to) a tree T of spaces, where 1. the vertices of T are given by ¹0º [ N and the edges of T are given by [i; i + 1] , where i = 0; 1; 2; : : : , making T isometric to R+ ; 2. all the vertex and edge spaces are (intrinsically) isometric to Sx ; 3. the edge space to vertex space inclusions are uniform quasi-isometries. Given a geodesic  = 0  Sx , we briefly outline the construction of a ladder L  Ex containing  . Index the vertices by n 2 ¹0º [ N . Since the edge-to-vertex space inclusions are quasi-isometries, this induces a quasi-isometry n from Sx  ¹nº to Sx  ¹n + 1º for n  0 . These quasi-isometries n induce maps Φn from geodesic segments in Sx  ¹nº to geodesic segments in Sx  ¹n + 1º for n  0 by sending a geodesic in Sx  ¹nº joining a; b to a geodesic in Sx  ¹n + 1º joining n (a); n (b) . Inductively define: • j +1 = Φj (j ) for j  0 , S • L = j j . The set L is called a hyperbolic ladder. The ladder L turns out to be quasiconvex in X , see [18]. This suffices to prove the existence of Cannon–Thurston maps.

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3.2. i-bounded geometry The next model geometry is satisfied by degenerate Kleinian punctured-torus groups as shown by Minsky in [15]. Definition 3.3. An end E of a hyperbolic M has i-bounded geometry [22] if the boundary torus of every Margulis tube in E has bounded diameter. An alternate description of i-bounded geometry can be given as follows. We start with a closed hyperbolic surface S . Fix a finite collection C of disjoint simple closed geodesics on S and let N (i ) denote the  -neighborhood of i ( i 2 C ). Here  is chosen small enough so that no two lifts of N (i ) to the universal cover Sz intersect. Thin block. LetS I = [0; 3] . Equip S  I with the product metric. Let
Bc = S  I N ( )  [1; 2] . Equip B c with the induced path-metric.  i i Note that any geodesic realization of the meridian on @N (i )  [1; 2] has length (4 + 2) . For each resultant torus component of the boundary of B c , perform Dehn filling on some (1; ni ) curve to obtain B homeomorphic to S  I (the ni ’s may vary from block to block but we do not add on the suffix for B to avoid cluttering notation), which goes ni times around the meridian and once round the longitude. ni will be called the twist coefficient. Foliate the torus boundary of B c by translates of (1; ni ) curves and arrange so that the solid torus Θi thus glued in is hyperbolic and foliated by totally geodesic disks bounding the (1; ni ) curves. Θi equipped with this metric will be called a Margulis tube. The resulting copy of S  I obtained, equipped with the metric just described, is called a thin block. The following statement is a consequence of [22]. Proposition 3.5. An end E of a hyperbolic 3 -manifold M has i-bounded geometry if and only if it is bi-Lipschitz homeomorphic to a model manifold Em consisting of gluing thick and thin blocks end-to-end. Figure 4 below illustrates a model Em , where the black squares denote Margulis tubes and the (long) rectangles without black squares represent thick blocks. Minsky proved the ending lamination theorem for punctured torus groups in [15] by constructing a bi-Lipschitz model for ends. This is done by looking at the sequence of short curves exiting the end and tracking the sequence on the curve complex C(S1;1 ) of a punctured torus S1;1 . The curve complex in this case is the Farey graph dual to the Farey triangulation in figure 5. An ending lamination corresponds to an irrational number z on the boundary circle. Given two such ending laminations z ; z+ corresponding to irrational numbers, Minsky ([15], Section 4) extracts a pivot sequence as follows. There is a well-ordered set E of bi-infinite geodesics in the Farey triangulation separating

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z ; z+ . Let P be the set of vertices of the Farey triangulation which belong to at least two edges in E . The vertices in P admit a natural order. Suppose that there are exactly ni consecutive triangles with vertex vi separating z ; z+ .

Figure 4. Model of i-bounded geometry (schematic)

Figure 5

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The Minsky model in this case has the i -th block Bi with short curves (vi ; vi +1 ) on its two boundary components. Further, ni agrees precisely with the Dehn filling coefficient described above. The main theorem of [15] then proves that a doubly degenerate manifold with ending laminations coded by z˙ is bi-Lipschitz homeomorphic to the model thus constructed. The ending lamination theorem for Kleinian punctured torus groups immediately follows using Sullivan’s theorem [25].

3.2.1. Cannon–Thurston maps for i-bounded geometry The existence of Cannon–Thurston maps for punctured torus groups was first proven by McMullen [12]. Following [22], we give a quick sketch of the construction of the ladder L and the proof of the existence of Cannon–Thurston maps in this case. First electrify all the Margulis tubes in the model. This ensures that in the resulting electric geometry, each block is of bounded geometry. More precisely, there is a (metric) product structure on S  [0; 3] such that each ¹xº  [0; 3] has uniformly bounded length in the electric metric. Further, since the curves in C are electrified in a block, Dehn twists are isometries from S  ¹1º to S  ¹2º in a thin block. This allows the construction of the ladder L to go through as before and ensures that it is quasiconvex in the resulting electric metric. Finally given an electric geodesic lying outside large balls modulo Margulis tubes one can recover a genuine hyperbolic geodesic tracking it outside Margulis tubes. The existence of Cannon–Thurston maps follows in this case.

3.3. Amalgamation geometry The next model geometry was used for proving the existence of Cannon–Thurston maps in a special case [24]. As before, we start with a closed hyperbolic surface S . An amalgamated geometry block is similar to a thinSblock, except that
we impose very mild control on the geometry of S  [1; 2] n N ( )  [1; 2] .  ij j Definition 3.4 (amalgamated block). As before I = [0; 3] . We will describe a geometry on S  I . There exist ; L (these constants will be uniform over blocks of the model Em ) satisfying the following conditions. 1. B = S  I . Let K = S  [1; 2] under the identification of B with S  I . 2. We call K the geometric core. In its intrinsic path metric, it is L -biLipschitz to a convex hyperbolic manifold with boundary consisting of surfaces L-bi-Lipschitz to a fixed hyperbolic surface. It follows that there exists D > 0 such that the diameter of S  ¹iº is bounded above by D (for i = 1; 2 ). 3. There exists a multicurve on S each component of which has a geodesic realization on S  ¹i º for i = ¹1; 2º with (total) length at most  . Let denote its geodesic realization in K .

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4. There exists a regular neighborhood Nk ( )  K of which is homeomorphic to a union of disjoint solid tori, such that Nk ( ) \ S  ¹iº is homeomorphic to a union of disjoint open annuli for i = 1; 2 . Denote Nk ( ) by T and call it the Margulis tube(s) corresponding to . 5. S  [0; 1] and S  [2; 3] are given the product structures corresponding to the bounded geometry structures on S  ¹iº , for i = 1; 2 respectively. 6. The lift to the universal cover of each component of K n Nk ( ) is quasiconvex.

Definition 3.5. An end E of a hyperbolic 3 -manifold M has amalgamated geometry if it is bi-Lipschitz homeomorphic to a model manifold Em consisting of gluing thick and amalgamated blocks end-to-end.

Remark 3.2. It turns out that i-bounded geometry is a special case of amalgamated geometry. The difference is that amalgamated geometry imposes relatively mild conditions on the geometry of the complement K T . The components of K T shall be called amalgamation components of K .

Figure 6 illustrates schematically what the model looks like. Filled squares correspond to solid tori along which amalgamation occurs. The adjoining piece(s) denote amalgamation blocks of K . The blocks which have no filled squares are the thick blocks and those with filled squares are the amalgamated blocks. The model geometry of a doubly degenerate 5-holed sphere group satisfies a slightly weaker version of amalgamation geometry. The amalgamation components are homeomorphic to S0;4  I or S0;3  I . The S0;3  I amalgamation components are uniformly quasiconvex, while the S0;4  I amalgamation components satisfy a weak form of uniformly quasiconvexity, called uniform graph quasiconvexity (see Proposition 5.6 for instance). To construct the ladder L we electrify amalgamation components as well as Margulis tubes. This ensures that in the electric metric 1. each amalgamation block has bounded geometry; 2. the mapping class element taking S  ¹1º to S  ¹2º induces an isometry of the electrified metrics. Quasiconvexity of L in the electric metric now follows as before. To recover the data of hyperbolic geodesics from quasigeodesics lying close to L , we use (uniform) quasiconvexity of amalgamation components and existence of Cannon–Thurston maps follows.

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Figure 6. Model of amalgamated geometry (schematic)

4. Hierarchies and the ending lamination theorem We recapitulate the essential aspects of hierarchies and split geometry from [10], [11], and [17]. The exposition here follows [23] in part.

4.1. Hierarchies We fix some notation first. • (Sg ;b ) = 3g + b is the complexity of a compact surface S = Sg ;b of genus g and b boundary components. • For an essential subsurface Y of S , C(Y ) will be its curve complex and P(Y ) its pants complex. • ˛ will be a collection of disjoint simple closed curves on S corresponding to a simplex ˛ 2 C(Y ) . • ˛; ˇ in C(Y ) fill an essential subsurface Y of S if all non-trivial nonperipheral curves in Y have essential intersection with at least one of ˛ or ˇ , where ˛ and ˇ are chosen to intersect minimally. • Given ˛; ˇ in C(S ) , form a regular neighborhood of ˛ [ ˇ , and fill in all disks and one-holed disks to obtain Y which is said filled by ˛; ˇ . • For an essential subsurface X  Z let @Z (X ) denote the relative boundary of X in Z , i.e. those boundary components of X that are non-peripheral in Z .

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• A pants decomposition of a compact surface S , possibly with boundary, is a disjointScollection of 3-holed spheres P1 ; : : : ; Pn embedded in S such that S n i Pi is a disjoint collection of non-peripheral annuli in S , no two of which are homotopic. • A tube in an end E  N is a regular R -neighborhood N ( ; R) of an unknotted geodesic in E . Definition 4.1 (tight geodesics and component domains). Let Y be an essential subsurface in S . If (Y ) > 4 , a tight sequence of simplices ¹vi ºi 2I  C(Y ) (where I is a finite or infinite interval in Z ) satisfies the following conditions: 1. for any vertices wi of vi and wj of vj where i ¤ j , dC(Y ) (wi ; wj ) = ji

2. for ¹i

1; i; i + 1º  I , vi equals @Y F (vi

j j; 1 ; vi +1 ) .

If (Y ) = 4 then a tight sequence is the vertex sequence of a geodesic in C(Y ) . A tight geodesic g in C(Y ) consists of a tight sequence v0 ; : : : ; vn , and two simplices in C(Y ) , I = I(g) and T = T (g) , called its initial and terminal markings, such that v0 (resp. vn ) is a vertex of I (resp. T ). The length of g is n . vi is called a simplex of g . Y is called the domain or support of g and is denoted as Y = D(g) . g is said supported in D(g) . For a surface W with (W )  4 and v a simplex of C(W ) we say that Y is a component domain of (W; v) if Y is a component of W n collar(v) , where collar(v) is a small tubular neighborhood of the simple closed curves. If g is a tight geodesic with domain D(g) , we call Y  S a component domain of g if for some simplex vj of g , Y is a component domain of (D(g); vj ) . The notion of a hierarchy below is a special case of hierarchies without annuli in [11] and is the notion used in [23], §3. Definition 4.2 (hierarchies). A hierarchy path in P(S ) joining pants decompositions P1 and P2 is a path W [0; n] ! P (S ) joining (0) = P1 to (n) = P2 such that 1. there is a collection ¹Y º of essential, non-annular subsurfaces of S , called component domains for  , such that for each component domain Y there is a connected interval JY  [0; n] with @Y  (j ) for each j 2 JY ; 2. for a component domain Y , there exists a tight geodesic gY supported in Y such that for each j 2 JY , there is an ˛ 2 gY with ˛ 2 (j ) . A hierarchy path in P(S ) is a sequence ¹Pn ºn of pants decompositions of S such that for any Pi ; Pj 2 ¹Pn ºn , i  j , the finite sequence Pi ; Pi +1 ; : : : ; Pj 1 ; Pj is a hierarchy path joining pants decompositions Pi and Pj .

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The collection H of tight geodesics gY supported in component domains Y of  will be called the hierarchy of tight geodesics associated to  . The tight geodesic in the hierarchy H whose component domain is all of S is called the bottom geodesic of the hierarchy. Definition 4.3. A slice of a hierarchy H associated to a hierarchy path  is a set  of pairs (h; v) , where h 2 H and v is a simplex of h , satisfying the following properties: 1. a geodesic h appears in at most one pair in  ; 2. there is a distinguished pair (h ; v ) in  , called the bottom pair of  , where h is the bottom geodesic. 3. for every (k; w) 2  other than the bottom pair, D(k) is a component domain of (D(h); v) for some (h; v) 2  . A resolution of a hierarchy H associated to a hierarchy path W I ! P(S ) is a sequence of slices i = ¹(hi 1 ; vi 1 ); (hi 2 ; vi 2 ); : : : ; (hi ni ; vi ni )º

(for i 2 I , the same indexing set) such that the set of vertices of the simplices ¹vi1 ; vi2 ; : : : ; vi ni º is the same as the set of the non-peripheral boundary curves of the pairs of pants in (i ) 2 P(S) . In [17], Minsky constructs a model manifold M associated to end-invariants  . M [0] denotes M minus the collection of Margulis tubes and horoball neighborhoods of cusps. M [0] is built up as a union of standard “blocks” of a finite number of topological types. Definition 4.4 (Minsky blocks [17], §8.1). A tight geodesic in H supported in a component domain of complexity 4 is called a 4 -geodesic and an edge of a 4 -geodesic in H is called a 4 -edge. Given a 4-edge e in H , let g be the 4 -geodesic containing it, and let D(e) be the domain D(g) . Let e and e + denote the initial and terminal vertices of e . Also collar(v) denotes a small tubular neighborhood of v in D(e) . To each e a Minsky block B(e) is assigned as as follows: B(e) = (D(e)  [ 1; 1]) n (collar(e )  [ 1; 1/2) [ collar(e + )  (1/2; 1]):

That is, B(e) is the product D(e)  [ 1; 1] , with solid-torus trenches dug out of its top and bottom boundaries, corresponding to the two vertices e and e + of e . The horizontal boundary of B(e) is @˙ B(e)  (D(e) n collar(e ˙ ))  ¹˙1º:

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The horizontal boundary is a union of three-holed spheres. The rest of the boundary (including the walls of the trenches) is a union of annuli and is called the vertical boundary. The top (resp. bottom) horizontal boundaries of B(e) are (D(e) n collar(e + ))  ¹1º (resp. (D(e) n collar(e ))  ¹ 1º . The blocks constructed this way are the internal blocks of [17]. A related set of blocks are also constructed to form boundary blocks which figure in the construction of geometrically finite ends. One of the main aims of [17] is to construct a model manifold using these building blocks. A model M [0] is constructed in [17] by taking the disjoint union of all the Minsky blocks and identifying them along three-holed spheres in their gluing boundaries. The rule is that whenever two blocks B and B 0 have the same three-holed sphere Y appearing in both @+ B and @ B 0 , these boundaries are identified using the identity on Y . The hierarchy serves to organize these gluings and insure that they are consistent. The following theorem is then shown in [17]. Theorem 4.1 (Minsky [17], Theorem 8.1). serving embedding ΨW M [0] ! S  R .

M [0] admits a proper orientation-pre-

The tori boundary components of M [0] will be referred to as Margulis tori.

4.1.1. Tori and meridinal coefficients Let T be a Margulis torus in M [0] . Then T has the structure of a Euclidean torus and gives a unique point !T in the upper half plane, the Teichmüller space of the torus. The real and imaginary components of !T have a geometric interpretation. Suppose that the Margulis tube T corresponds to a vertex v 2 C(S ) . Let twT be the signed length of the annulus geodesic corresponding to v , i.e. it counts with sign the number of Dehn twists about the curve represented by v . Next, note that by the construction of the Minsky model, the vertical boundary of T consists of two sides—the left vertical boundary and right vertical boundary. Each is attached to vertical boundaries of Minsky blocks. Let the total number of blocks whose vertical boundaries, are glued to the vertical boundary of T be nT . Similarly, let the total number of blocks whose vertical boundaries, are glued to the left (resp. right) vertical boundary of T be nT l (resp. nT r ) so that nT = nT l + nT r . In §9 of [17], Minsky shows the following result Theorem 4.2 ([17]). There exists C0  0 , such that the following holds: !T

(twT +i nT )  C0 :

If twT   and nT  ! , then T is said (; !) -thin.

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Definition 4.5. The union of M [0] as in Theorem 4.1 and model Margulis tubes as above is the Minsky model. In [2], Brock, Canary, and Minsky showed that the Minsky model manifold built in [17] is in fact bi-Lipschitz homeomorphic to the hyperbolic manifold with the same end-invariants. Theorem 4.3 (bi-Lipschitz model theorem). Given a surface S of genus g , there exists L (depending only on g ) such that for any doubly degenerate hyperbolic 3 -manifold M without accidental parabolics homotopy equivalent to S , there exists an L -bi-Lipschitz map from M to the Minsky Model for M . The pre-image of M [0] to M under the L -bi-Lipschitz map of Theorem 4.3 will be denoted as M (0) and its boundary consisting of tori by @M (0) . Once the bi-Lipschitz model theorem 4.3 is in place, the ending lamination theorem is an immediate consequence of Sullivan’s theorem [25] as before. Theorem 4.4 (ending lamination theorem, [2]). Let Mi ( i = 1; 2 ) be two doubly degenerate hyperbolic 3 -manifolds without accidental parabolics homeomorphic to S  R with the same ending laminations. Then M1 and M2 are isometric.

5. Split geometry We come now to the last model geometry of this paper and indicate how it is used to apply the bi-Lipschitz model Theorem 4.3 to prove the existence of Cannon–Thurston maps in general.

5.1. Split level surfaces Let E be a degenerate end of a hyperbolic 3 -manifold N . Let T denote a collection of disjoint, uniformly separated tubes in ends of N such that 1. all Margulis tubes in E belong to T ; S 2. there exists 0 > 0 such that injradx (E) > 0 for all x 2 E n T 2T Int(T ) . (Here injrad denotes the injectivity radius.) Let (Q; @Q) be the unique hyperbolic pair of pants such that each component of @Q has length one. Q will be called the standard pair of pants. An isometrically embedded copy of (Q; @Q) in (M (0); @M (0)) will be said flat. Definition 5.1. A split level surface associated to a pants decomposition ¹Q1 ; : : : ; Qn º of aScompact surface S (possibly with boundary) in M (0)  M is an embedding f W i (Qi ; @Qi ) ! (M (0); @M (0)) such that

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1. each f (Qi ; @Qi ) is flat; 2. f extends to an embedding (also denoted f ) ofSS into
M such that the interior of each annulus component of f S n Q lies entirely in i i
S F Int (T ) . T 2T Let Sis denote the union of the collection of flat pairs of pants in the image of the embedding Si . Note that Si n Sis consists of annuli properly embedded in Margulis tubes. The class of all topological embeddings from S to M that agree with a split level surface f associated to a pants decomposition ¹Q1 ; : : : ; Qn º on Q1 [


[ Qn will be denoted by [f ] . We define a partial order E on the collection of split level surfaces in an end E of M as follows: f1 E f2 if there exist gi 2 [fi ] , i = 1; 2 , such that g2 (S ) lies in the unbounded component of E n g1 (S) . A sequence Si of split level surfaces exits an end E if i < j implies Si E Sj and further for all compact subsets B  E , there exists L > 0 such that Si \ B = ; for all i  L . Definition 5.2. A curve v in S  E is l -thin if the core curve of the Margulis tube Tv ( E  N ) has length less than or equal to l . A tube T 2 T is l -thin if its core curve is l -thin. A tube T 2 T is l -thick if it is not l -thin. A curve v splits a pair of split level surfaces Si and Sj ( i < j ) if v occurs as a boundary curve of both Si and Sj . A pair of split level surfaces Si and Sj ( i < j ) is a l -thin pair if there exists an l -thin curve v splitting both Si and Sj . The collection of all l -thin tubes is denoted as Tl . The union of all l -thick tubes along with M (0) is denoted as M (l) . Definition 5.3. A pair of split level surfaces Si and Sj ( i < j ) is k -separated if a. for all x 2 Sis , dM (x; Sjs )  k ; b. similarly, for all x 2 Sjs , dM (x; Sis )  k . As a consequence of the bi-Lipschitz model theorem 4.3 we have the following: Theorem 5.1 ([17] and [2]). Let N be the convex core of a simply or doubly degenerate hyperbolic 3 -manifold minus an open neighborhood of the cusp(s). Further assume that N has no accidental parabolics. Let S be a compact surface, possibly with boundary, such that N is homeomorphic to S  [0; 1) or S  R according as N is simply or doubly degenerate. There exist L  1 , ; !; ; 1 > 0 , a collection T of (; !) -thin tubes containing all Margulis tubes in N , a 3 -manifold M , and an L -bi-Lipschitz homeomorphism F W N ! M such that the following holds.

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S Let M (0) = F N n T 2T Int(T ) and let F (T) denote the image of the collection T under F . Let E denote the partial order on the collection of split level surfaces in an end E of M . Then there exists a sequence Si of split level surfaces associated to pants decompositions Pi exiting E satisfying the following properties. 1. Si E Sj if i  j . 2. The sequence ¹Pi º is a hierarchy path in P(S) . 3. If Pi \ Pj = ¹Q1 ; : : : ; Ql º , then fi (Qk ) = fj (Qk ) for k = 1; : : : ; l , where fi ; fj are the embeddings defining the split level surfaces Si ; Sj respectively. 4. For all i , Pi \ Pi +1 = ¹Qi;1 ; : : : ; Qi;l º consists of a collection of l pairs of pants, such that S n (Qi;1 [


[ Qi;l ) has a single non-annular component of complexity 4 . Further, there exists a Minsky block Wi and an isometric map Gi of Wi into M (0) such that fi (S n (Qi;1 [


[ Qi;l ) (resp. fi +1 (S n (Qi;1 [


[ Qi;l ) ) is contained in the image under Gi of the bottom (resp. top) gluing boundary of Wi . 5. For each flat pair of pants Q in a split level surface Si there exists an isometric embedding of Q  [ ; ] into M (0) such that the embedding restricted to Q  ¹0º agrees with fi restricted to Q . 6. For each T 2 T , there exists a split level surface Si associated to pants decompositions Pi such that the core curve of T is isotopic to a nonperipheral boundary curve of Pi . The boundary F (@T ) of F (T ) with the induced metric dT from M (0) is a Euclidean torus equipped with a product structure S 1 Sv1 , where any circle of the form S 1 ¹t º  S 1 Sv1 is a round circle of unit length and is called a horizontal circle, and any circle of the form ¹t º  Sv1 is a round circle of length lv and is called a vertical circle. 7. Let g be a tight geodesic other than the bottom geodesic in the hierarchy H associated to the hierarchy path ¹Pi º , let D(g) be the support of g and let v be a boundary curve of D(g) . Let Tv be the tube in T such that the core curve of Tv is isotopic to v . If a vertical circle of (F (@Tv ); dTv ) has length lv less than n1 , then the length of g is less than n .

Two consequences of Theorem 5.1 that are needed in [23] are given below. Lemma 5.2 ([23], Lemma 3.6). Assume the setup of Theorem 5.1. Given l > 0 there exists n 2 N such that the following holds. Let v be a vertex in the hierarchy H such that the length of the core curve of the Margulis tube Tv corresponding to v is greater than l . Next suppose (h; v) 2 i for some slice i of the hierarchy H such that h is supported on Y , and D is a component of Y n collar v . Also suppose that h1 2 H such that D is the support of h1 . Then the length of h1 is at most n .

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The idea behind Lemma 5.2 is that the existence of a lower bound l is equivalent to the existence of upper bounds 0 and !0 on the real and imaginary coefficients of the Margulis torus @Tv . These in turn translate to the upper bounds on the lengths of hierarchy paths in Lemma 5.2 via Theorem 5.1. A further refinement of this idea furnishes the following. Lemma 5.3 ([23], Lemma 3.7). Assume the setup of Theorem 5.1. Given l > 0 and n 2 N , there exists L2  1 such that the following holds. Let Si ; Sj ( i < j ) be split level surfaces associated to pants decompositions Pi ; Pj such that a. (j i)  n ; b. Pi \ Pj is a (possibly empty) pants decomposition of S n W , where W is an essential (possibly disconnected) subsurface of S such that each component Wk of W has complexity (Wk )  4 ; c. for any k with i < k < j , and (gD ; v) 2 k for D  Wi for some i , no curve in v has a geodesic realization in N of length less than l . Then there exists an L2 -bi-Lipschitz embedding GW W  [ 1; 1] ! M , such that 1. W admits a hyperbolic metric given by W = Q1 [


[ Qm where each Qi is a flat pair of pants; 2. W  [ 1; 1] is given the product metric; 3. fi (Pi n Pi \ Pj )  W  ¹ 1º and fj (Pj n Pi \ Pi )  W  ¹1º .

5.2. Split surfaces and weak split geometry The union of M (0) along with Margulis tubes with core curve of length at least l is denoted M (l) . Definition 5.4. An L -bi-Lipschitz split surface in M (l) associated to a pants decomposition ¹Q1 ; : : : ; Qn º of S and a collection ¹A1 ; : : : S ; Am º of S complementary S annuli (not necessarily all of them) in S is an embedding f W i Qi i Ai ! M (l) such that S 1. the restriction f W i (Qi ; @Qi ) ! (M (0); @M (0)) is a split level surface; 2. the restriction f W Ai ! M (l) is an L -bi-Lipschitz embedding; 3. f extends to an embedding (also denoted f ) of S S into M

that S such the interior of each annulus
component of f S n Q [ A lies i i i i S entirely in F Int (T ) . T 2Tl A split level surface differs from a split surface in that the latter may contain bi-Lipschitz annuli in addition to flat pairs of pants. We denote split surfaces by Σi . Let Σsi denote the union of the collection of flat pairs of pants and bi-Lipschitz annuli in the image of the split surface (embedding) Σi . The next theorem is one of the technical tools from [23] and follows from combining the bi-Lipschitz model Theorem 5.1 with lemmas 5.2 and 5.3.

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Theorem 5.4 ([23], Theorem 4.8). Let N; M; M (0); S; F be as in Theorem 5.1 and E an end of M . For any l less than the Margulis constant, let M (l) = ¹F (x)W injradx (N )  lº . Fix a hyperbolic metric on S such that each component of @S is totally geodesic of length one (this is a normalization condition). There exist L1  1 , 1 > 0 , n 2 N , and a sequence Σi of L1 -bi-Lipschitz, 1 separated split surfaces contained in M (l) and exiting the end E of M such that for all i , one of the following occurs: 1. an l -thin curve splits the pair (Σi ; Σi +1 ) , i.e. the associated split level surfaces form an l -thin pair; 2. there exists an L1 -bi-Lipschitz embedding Gi W (S  [0; 1]; (@S )  [0; 1]) ! (M; @M ) Σsi

such that = Gi (S  ¹0º) and Σsi+1 = Gi (S  ¹1º) . Finally, each l -thin curve in S splits at most n split level surfaces in the sequence ¹Σi º . Pairs of split surfaces satisfying alternative (1) of Theorem 5.4 will be called an l -thin pair of split surfaces (or simply a thin pair if l is understood). Similarly, pairs of split surfaces satisfying alternative (2) of Theorem 5.4 will be called an l -thick pair (or simply a thick pair) of split surfaces. We refer back to the diagram of a model of i-bounded geometry after Proposition 3.5 for a schematic description of thin and thick pairs. The blocks without black squares may be thought of as thick blocks and their boundary surfaces as thick pairs. Blocks with black squares, on the other hand, may be thought of as thin blocks and their boundary surfaces as thin pairs. Definition 5.5. A model manifold satisfying the following conditions is said to have weak split geometry. 1. A sequence of split surfaces Sis exiting the end(s) of M , where M is marked with a homeomorphism to S  J ( J is R or [0; 1) according as M is totally or simply degenerate). Sis  S  ¹i º . 2. A collection of Margulis tubes T in N with image F (T) in M (under the bi-Lipschitz homeomorphism between N and M ). We refer to the elements of F (T) also as Margulis tubes. 3. For each complementary annulus of Sis with core  , there is a Margulis tube T 2 T whose core is freely homotopic to  such that F (T ) intersects Sis at the boundary. (What this roughly means is that there is an F (T ) that contains the complementary annulus.) We say that F (T ) splits Sis . 4. There exist constants 0 > 0 , K0 > 1 such that for all i , either there exists a Margulis tube splitting both Sis and Sis+1 , or else Si (= Sis ) and Si +1 (= Sis+1 ) have injectivity radius bounded below by 0 and bound a block Bi that is K0 -bi-Lipschitz homeomorphic image of S  I .

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5. F (T ) \ Sis is either empty or consists of a pair of boundary components of Sis that are parallel in Si . 6. There is a uniform upper bound n = n(M ) on the number of surfaces that F (T ) splits. Theorem 5.4 then gives the following result. Theorem 5.5 ([23]). Any degenerate end of a hyperbolic 3 -manifold is bi-Lipschitz homeomorphic to a Minsky model and hence to a model of weak split geometry.

5.2.1. Split blocks and hanging tubes We shall now define split geometry. Definition 5.6. Let (Σsi ; Σsi+1 ) be a thick pair of split surfaces in M . The closure of the bounded component of M n (Σsi [ Σsi+1 ) between Σsi ; Σsi+1 will be called a thick block. Note that a thick block is uniformly bi-Lipschitz to the product S  [0; 1] and that its boundary components are Σsi ; Σsi+1 . Definition 5.7. Let (Σsi ; Σsi+1 ) be an l -thin pair of split surfaces in M and F (Ti ) be the collection of l -thin Margulis tubes that split both Σsi ; Σsi+1 . The closure
S of the union of the bounded components of M n (Σsi [ Σsi+1 ) F (T )2F (Ti ) F (T ) between Σsi ; Σsi+1 will be called a split block. The closure of any bounded component is called a split component. Each split component may contain Margulis tubes, which we shall call hanging tubes (see below) that do not split both Σsi ; Σsi+1 . Topologically, a split block B s  B = S  I is a topological product S s  I for some connected S s . However, the upper and lower boundaries of B s need only be be split subsurfaces of S s . This is to allow for Margulis tubes starting (or ending) within the split block. Such tubes would split one of the horizontal boundaries but not both. We shall call such tubes hanging tubes. See figure 7. The vertical lengths of hanging tubes are further required to be uniformly bounded below by some 0 > 0 . The existence of such an 0 > 0 is ensured by the choice of the sequence of split surfaces ¹Σsi ºi . Further, each such annulus has cross section a round circle of length 0 . Definition 5.8. Hanging tubes intersecting the upper (resp. lower) boundaries of a split block are called upper (resp. lower) hanging tubes. Electrifying split components, we obtain a new electric metric called the graph metric dG on E .

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Figure 7. Split block with hanging tubes

Definition 5.9. A model of weak split geometry is a model of split geometry if the convex hull of each split component has uniformly bounded dG -diameter. Equivalently we say that split components are uniformly graph quasiconvex. A key technical proposition of [23] asserts what follows. Proposition 5.6. quasiconvex.

For E a degenerate end, split components are uniformly graph

As an immediate consequence of Theorem 5.5 and Proposition 5.6 we have the following. Theorem 5.7 ([23]). Any degenerate end of a hyperbolic 3 -manifold is bi-Lipschitz homeomorphic to a Minsky model and hence to a model of split geometry. Once it is established that M has split geometry, the proof of the existence of Cannon–Thurston maps proceeds as for amalgamation geometry by electrifying split components, constructing a hyperbolic ladder L and finally recovering a hyperbolic geodesic from an electric one. We summarize this as follows. Theorem 5.8 ([23]). Let W 1 (S) ! PSL2 (C) be a simply or doubly degenerate Kleinian surface group. Then a Cannon–Thurston map exists. Acknowledgments. The author thanks the anonymous referee for a careful reading and detailed comments. The author is partly supported by a DST JC Bose Fellowship, CEFIPRA project № 5801-1 and a SERB grant MTR/2017/000513.

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References [1] I. Agol, Tameness of hyperbolic 3 -manifolds. Preprint, 2004. arXiv:math/0405568 [math.GT] R 173 [2] J. F. Brock, R. D. Canary, and Y. N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture. Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381 Zbl 1253.57009 R 165, 173, 174, 185, 186 [3] F. Bonahon, Bouts des variétés hyperboliques de dimension 3. Ann. of Math. (2) 124 (1986), no. 1, 71–158. MR 0847953 Zbl 0671.57008 R 174 [4] B. H. Bowditch, The Cannon–Thurston map for punctured surface groups. Math. Z. 255 (2007), no. 1, 35–76. MR 2262721 Zbl 1138.57020 R 176 [5] B. H. Bowditch, The ending lamination theorem. Preprint, 2015. https://homepages.warwick.ac.uk/~masgak/papers/elt.pdf R 165 [6] R. D. Canary, Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc. 6 (1993), no. 1, 1–35. MR 1166330 Zbl 0810.57006 R 174 [7] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3 -manifolds. J. Amer. Math. Soc. 19 (2006), no. 2, 385–446. MR 2188131 Zbl 1090.57010 R 173 [8] A. Fathi, M. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces. Séminaire Orsay. With an English summary. Astérisque, 66-67. Société Mathématique de France, Paris, 1979. MR 0568308 Zbl 0406.00016 R 171 [9] E. Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere. Amer. J. Math. 121 (1999), no. 5, 1031–1078. MR 1713300 Zbl 1011.30035 R 176 [10] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338 Zbl 0941.32012 R 181 [11] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal. 10 (2000), no. 4, 902–974. MR 1791145 Zbl 0972.32011 R 181, 182 [12] C. T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus. Invent. Math. 146 (2001), no. 1, 35–91. MR 1859018 Zbl 1061.37025 R 179 [13] Y. N. Minsky, Teichmüller geodesics and ends of hyperbolic 3 -manifolds. Topology 32 (1993), no. 3, 625–647. MR 1231968 Zbl 0793.58010 R 175 [14] Y. N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic 3 -manifolds. J. Amer. Math. Soc. 7 (1994), no. 3, 539–588. MR 1257060 Zbl 0808.30027 R 174, 175, 176 [15] Y. N. Minsky, The classification of punctured-torus groups. Ann. of Math. (2) 149 (1999), no. 2, 559–626. MR 1689341 Zbl 0939.30034 R 177, 179 [16] Y. N. Minsky, Bounded geometry for Kleinian groups. Invent. Math.‘146 (2001), no. 1, 143–192. MR 1859020 Zbl 1061.37026 R 174 [17] Y. N. Minsky, The Classification of Kleinian surface groups. I. Models and bounds. Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036 Zbl 1193.30063 R 165, 173, 174, 181, 183, 184, 185, 186 [18] M. Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces. J. Differential Geom. 48 (1998), no. 1, 135–164. MR 1622603 Zbl 0906.20023 R 176 [19] H. Miyachi, Semiconjugacies between actions of topologically tame Kleinian groups. Preprint, 2002. R 176 [20] M. Mj, Cannon–Thurston maps for pared manifolds of bounded geometry. Geom. Topol. 13 (2009), no. 1, 189–245. MR 2469517 Zbl 1166.57009 R 176

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Chapter 6

Universal Teichmüller space as a non-trivial example of infinite-dimensional complex manifolds Armen Sergeev

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Definition of the universal Teichmüller space . 3 Subspaces of the universal Teichmüller space . 4 Grassmann realization of universal Teichmüller 5 Universal Teichmüller space and string theory References . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction There is an opinion that it is impossible to construct any general theory of infinitedimensional complex manifolds. The reason is that such manifolds differ a lot from each other so that many of them deserve a separate theory and these separate theories cannot be united into one general theory. It is hard to say at the moment whether this point of view is true or not but at least it justifies an extensive study of non-trivial examples of infinite-dimensional complex manifolds. One such example is provided by the universal Teichmüller space and we shall present in this survey the main complex geometric features of this remarkable infinite-dimensional manifold. The chapter is organized as follows. In §2 we introduce the universal Teichmüller space. We start with some necessary properties of quasiconformal maps of domains x (identified with the Riemann sphere). The universal in the extended complex plane C Teichmüller space T is, by definition, the space of normalized quasisymmetric homeomorphisms of the unit circle S 1 , i.e. orientation-preserving homeomorphisms of S 1 , extending to quasiconformal maps of the unit disk ∆ and fixing three points on S 1 . We also give two other definitions of T . According to the first of them,

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which may be called real-analytic, T is identified with the quotient of the space of Beltrami differentials in ∆ modulo a natural equivalence relation. In the second definition, which may be called complex analytic, T is treated as the space of x . The x , conformal in the complement of ∆ normalized quasiconformal maps of C universal Teichmüller space can be provided with a complex structure induced by the Bers embedding of T into the complex Banach space of holomorphic quadratic differentials in a disk. It can also be provided with the Teichmüller metric, a metric which is not Kähler. It turns out that a Kähler metric on T , compatible with the Teichmüller topology, is only defined on a dense subset. (However, as it is shown in [19], it is possible to introduce a Kähler metric on T compatible with a different topology.) More information on universal Teichmüller space can be found in [9], [10], [15], and [20]. In §3, we study subspaces of T . First of all we show that all classical Teichmüller spaces T (G) , associated with compact Riemann surfaces, are embedded into T as complex subspaces. This explains the origin of the name “universal Teichmüller space” for T . The Kähler quasimetric on T under this embedding is reduced to the Weil–Petersson Kähler metric on T (G) . The image of T (G) n ¹0º in T lies outside the regular part of T . This is in sharp contrast with the space S of normalized orientation-preserving diffeomorphisms of S 1 which is placed inside the regular part of T . The space S is a Kähler Fréchet manifold provided with a symplectic structure compatible with the complex structure of S . The §3 is devoted to the Grassmann realization of the universal Teichmüller space T . Namely, we construct an embedding of T into the Grassmann manifold 1/2 of a Hilbert space which coincides with the Sobolev space V = H0 (S 1 ; R) of half-differentiable functions on the circle. This is a Kähler Hilbert space. Its complexification V C can be decomposed into the direct sum of subspaces W˙ of functions holomorphic inside (respectively, outside) the unit disc ∆ . The group QS(S 1 ) of quasisymmetric homeomorphisms of S 1 acts on the Sobolev space V by reparametrizations. This action preserves the symplectic form of V but, in general, changes its complex structure. More precisely, by the Nag–Sullivan theorem, this action defines an embedding of T into the space J(V ) = Sp(V )/ U(W+ ) of complex structures on V compatible with the symplectic structure. This space can be realized as an infinite-dimensional Siegel disk. One more remarkable property of T is its relation to string theory considered in §5.4. We start with the smooth string theory having its phase space identified with the space Ωd of smooth loops with values in the d -dimensional Minkowski space R1;d 1 . To construct the Dirac quantization of this phase space one has to use a complex structure J 0 on Ωd . However, this complex structure is changed under the action of the group Diff+ (S 1 ) of reparametrizations of loops. The set of admissible complex structures on Ωd , obtained from the reference complex structure J 0 by this action, is parameterized by points of the Kähler Fréchet manifold S = Diff+ (S 1 )/Möb(S 1 ) . The main step in the construction of the quantization is the Segal theorem asserting that the action of the group Diff+ (S 1 ) on S by

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left translations can be pulled up to a projective unitary action of this group on the Fock spaces FJ . These Fock spaces FJ are defined over the Sobolev space V using admissible complex structures J . Having quantized the phase space of smooth string theory, one may ask why we deal only with smooth strings. The only “physical parameter” in our theory is the symplectic form ! on Ωd . But this 1/2 form may be extended to the Sobolev completion Vd = H0 (S 1 ; R1;d 1 ) of the 1 space Ωd consisting of half-differentiable functions on S with values in R1;d 1 . Moreover, this space is maximal among the Sobolev spaces H0s (S 1 ; R1;d 1 ) , s > 0 , on which this form is correctly defined. So we take Vd as the phase space of non-smooth string theory. However, with this choice of phase space, we are not able to construct the corresponding classical system associated with Vd . Instead, we define directly a quantum system, corresponding to this phase space, using the ideas from the noncommutative geometry. This construction is presented in the last subsection of the chapter (cf. also [15], [20], and [18]).

2. Definition of the universal Teichmüller space 2.1. Quasiconformal maps We list below the main properties and definitions related to the theory of quasiconformal mappings. A detailed presentation of this theory may be found in Ahlfors’ book [1]. Let wW D ! D 0 be an orientation-preserving homeomorphism mapping a domain x , identified with the Riemann sphere, onto D in the extended complex plane C 0 x another domain D  C . Suppose that w has locally integrable first derivatives in 1 D in the sense of distributions, i.e. w belongs to the Sobolev space Hloc (D) . Definition 2.1. The orientation-preserving homeomorphism w mapping D onto D 0 is called quasiconformal if there exists a bounded measurable function  2 L1 (D; C) with norm kk1 =: k < 1 such that the following Beltrami equation @w @w = @¯ z @z holds for almost all z 2 D . The function  is called the Beltrami differential. In particular, if k = 0 , i.e.   0 , then the map w is conformal. If w is smooth, then the above definition means that the tangent map of w at any point z 2 D sends the circles centered at the origin of Tz D to ellipses on Tw(z) D 0 centered at the origin and having the ratio of the big half-axis to the small one bounded by a common constant K for all z 2 D . One can set the quasiconformality constant K > 1 equal to K = 1+k . 1 k

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The term “Beltrami differential” used for  reflects the behavior of  under conformal maps. In order to preserve the form of the Beltrami equation,  should transform under such maps as a differential of type ( 1; 1) . Solutions of the Beltrami equation are uniquely determined up to conformal maps. More precisely, if there are two solutions w1 , w2 of this equation with the same Beltrami differential  then the maps w1 ı w2 1 and w2 ı w1 1 are conformal. Quasiconformal self-homeomorphisms wW ∆ ! ∆ of the unit disk ∆ form a group with respect to composition. All such maps extend continuously (and even Hölder-continuously) to the boundary S 1 = @∆ to orientation-preserving homeomorphisms S 1 ! S 1 . One can ask when the converse is true, i.e. when a given homeomorphism f W S 1 ! S 1 , preserving the orientation, extends to a quasiconformal map wW ∆ ! ∆ . The answer to this question is given in terms of cross ratios. Recall that the cross ratio of four pairwise-distinct points z1 ; z2 ; z3 ; z4 in the complex plane is defined as the double ratio z4 z1 z3 z1 (z1 ; z2 ; z3 ; z4 ) := W : z4 z2 z3 z2 The equality of two cross ratios (z1 ; z2 ; z3 ; z4 ) = (1 ; 2 ; 3 ; 4 ) is necessary and sufficient for the existence of a fractional-linear map sending (z1 ; z2 ; z3 ; z4 ) to (1 ; 2 ; 3 ; 4 ) . The quasiconformal maps can change the cross ratios but in a controlled manner as we can see from the following Theorem 2.1 (Beurling–Ahlfors theorem, cf. [1]). Let f be an orientation-preserving homeomorphism f W S 1 ! S 1 . Suppose that f satisfies the following condition: there exists " , 0 < " < 1 , such that the following inequality 1 " 1+"  (f (z1 ); f (z2 ); f (z3 ); f (z4 ))  (1) 2 2 holds for any four-tuple of pairwise-distinct points z1 ; z2 ; z3 ; z4 on S 1 with cross ratio (z1 ; z2 ; z3 ; z4 ) = 1/2 . Then f can be extended to a quasiconformal map wW ∆ ! ∆ . Another important result from the theory of quasiconformal maps, which we shall use in the sequel, is the following Theorem 2.2 (Ahlfors existence theorem, cf. [1]). For any function  2 L1 (D) in x , satisfying the condition kk1 < 1 , there exists a quasiconformal a domain D  C map w which is a solution of the Beltrami equation in D with Beltrami coefficient equal to  .

2.2. Universal Teichmüller space Definition 2.2. An orientation-preserving homeomorphism f mapping S 1 onto S 1 is called quasisymmetric if it extends to a quasiconformal homeomorphism wW ∆ ! ∆ of the unit disc ∆ .

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Grace to the Beurling–Ahlfors theorem, this definition is equivalent to the following: an orientation-preserving homeomorphism f W S 1 ! S 1 is quasisymmetric if it satisfies the estimate (1). Since the quasiconformal maps ∆ ! ∆ form a group with respect to composition, the same is true for quasisymmetric homeomorphisms S 1 ! S 1 . The group of such homeomorphisms of S 1 is denoted by QS(S 1 ) and plays a crucial role in the sequel. It can be included into the following chain of embeddings Möb(S 1 )  Diff+ (S 1 )  QS(S 1 )  Homeo+ (S 1 ) where Homeo+ (S 1 ) denotes the group of all orientation-preserving homeomorphisms of S 1 , Diff+ (S 1 ) consists of orientation-preserving diffeomorphisms of S 1 and Möb(S 1 ) is the Möbius group of fractional-linear automorphisms of the unit disc ∆ , restricted to S 1 . Definition 2.3.

The quotient T := QS(S 1 )/Möb(S 1 )

is called the universal Teichmüller space. This space can be identified with the space of normalized quasisymmetric homeomorphisms of the circle fixing the points ˙1; i on the circle S 1 . It follows from the above chain of embeddings that T contains the space S of normalized diffeomorphisms of S 1 defined as S = Diff+ (S 1 )/Möb(S 1 ):

Using Ahlfors’ existence theorem, we can give two other definitions of T directly in terms of Beltrami differentials. Let us start from the definition which may be called “real-analytic.” Given a Beltrami differential  in the unit disk ∆ , we can extend it by symmetry with x defined by the formula ˆ in C respect to S 1 to the Beltrami differential  1 z2 ˆ  := (z) 2 for z 2 ∆: z z¯ By Ahlfors’ existence theorem there exists a unique normalized quasiconformal x having  ˆ as its Beltrami differential. Since w should homeomorphism w in C preserve S 1 we can associate with it a quasisymmetric homeomorphism w jS 1 of S 1 . We introduce an equivalence relation of Beltrami differentials in ∆ by setting:    if and only if w  w on S 1 . Denote by B(∆) the space of Beltrami differentials in ∆ which can be identified with the unit ball in the complex Banach space L1 (∆; C) . Then we have T = B(∆)/  :

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Another definition of T may be obtained by extending a given Beltrami x to a Beltrami differential  x . Denote ˇ in C differential  2 B(∆) by zero outside ∆  x with by w a unique quasiconformal solution of the Beltrami equation in C ˇ Beltrami differential  which we normalize by the condition that it fixes the points x of the closed x n∆ 0; 1; 1 . Note that w  is conformal in the complement ∆ := C x unit disk ∆ . Introduce a new equivalence relation of Beltrami differentials in ∆ :    if and only if w   w  in ∆ . It turns out that this equivalence relation coincides with the previous one (cf. [1] and [9]). Now we can give another, “complex-analytic,” definition of T as the space of x !C x which are conformal in ∆ . normalized quasiconformal maps C

2.3. The complex structure of the universal Teichmüller space The universal Teichmüller space T has a natural metric, called the Teichmüller distance, which is defined as follows. Consider points of T as classes [f ] of quasisymmetric homeomorphisms f W S 1 ! S 1 . Then the distance between two points [f ]; [g] 2 T is equal to 1 inf¹log Kg ıf 1 W f 2 [f ]; g 2 [g]º 2 where Kg ıf 1 is the quasiconformality constant of the quasiconformal map wW ∆ ! ∆ obtained by extension of the quasisymmetric map g ı f 1 . The space T , provided with this distance, becomes a complete metric space (cf. [9]). We shall construct an embedding of T into the space of holomorphic quadratic differentials in a disk which will allow us to introduce global complex coordinates on T . Consider a point [] 2 T as the normalized quasiconformal homeomorphism x !C x , conformal in ∆ . Since w  is conformal in ∆ we can take its wW C Schwarzian derivative S (w  j∆ ) which is a holomorphic function in ∆ which does not depend on the choice of  2 [] and transforms as a quadratic differential under conformal changes of variable. (The definition and properties of Schwarzian derivative can be found in [9].) The map [] 7 ! S(w  j∆ )

dist([f ]; [g]) =

is an embedding since the equality S(w  ) = S (w  ) on ∆ w  j∆ = w  j∆ , i.e.    . So we obtain an embedding

implies that

ΨW T ! B2 (∆ )

of T into the space B2 (∆ ) of holomorphic quadratic differentials in ∆ is called the Bers embedding.

which

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The space B2 (∆ ) is a complex Banach space provided with the hyperbolic norm: B2 (∆ ) = ¹

=

(z)dz 2 W k kB2 := sup (1 z 2∆

jzj2 )2 j (z)j < 1º:

It may be proved (cf. [9]) that Ψ is a homeomorphism of T onto an open contractible subset of B2 (∆ ) such that B(0; 2)  Ψ(T)  B(0; 6)

where we denote by B(0; r) the ball of radius r in B2 (∆ ) centered at the origin. We now introduce a complex structure on T induced from B2 (∆ ) via Bers embedding. Since we have proved that the complex-analytic equivalence relation  coincides with the real-analytic one  , we can also introduce a complex structure on T which is induced from B(∆)  L1 (∆; C) by the natural projection B(∆) ! T = B(∆)/  :

It turns out that both complex structures are equivalent to each other in the sense that the composite map B(∆) ! T = B(∆)/  ! B2 (∆ )

is a holomorphic map of complex Banach spaces (cf. [10]). After we have introduced the complex structure on T , it is natural to ask whether it is possible to provide T with a Kähler metric (note that the Teichmüller metric is not Kähler). In order to answer this question, consider the Ahlfors map ΦW L1 (∆) ! B2 (∆)

given by the formula (which can be obtained by composing the Bers embedding with the map 1 7! 1/z) : Z () Φ[](z)  '(z) = ¯ 4 d d;  =  + i 2 ∆: (1 z ) ∆

It is reasonable to try to introduce a Kähler metric on T via this Ahlfors map. Namely, we can try to define it first at the origin o of T and then extend to the other points of T using left translations by elements of QS(S 1 ) . So let us define an inner product on To T by setting it equal on tangent vectors []; [] 2 To T to Z Z (z)() (; )  h; Φ[]i = d d dxdy ; z = x + iy 2 ∆: (1 z)4 ∆ ∆

However, such product is correctly defined only on a dense subset of To T . The reason is that for a general vector  2 L1 (∆) its image Φ[] 2 B2 (∆) may be non integrable. In fact, the inner product introduced is correctly defined only for sufficiently smooth ;  (cf. [10]).

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3. Subspaces of the universal Teichmüller space 3.1. Classical Teichmüller spaces We explain now the origin of the name “universal Teichmüller space.” It is motivated by the fact that all the classical Teichmüller spaces T (G) can be realized as complex subspaces of T . Recall that we can associate with any Fuchsian group G a compact Riemann surface X = ∆/G uniformized by the unit disk ∆ (cf. [5]). The corresponding Teichmüller space T (G) consists of quasisymmetric homeomorphisms f 2 QS(S 1 ) of the circle which are G -invariant in the following sense: for any g 2 G there exists a Möbius transformation g such that f ı g = g ı f: 1

In other words, the group f Gf is again Fuchsian. In terms of Beltrami differentials, this means that the Beltrami differential  , associated with f , is invariant under G as a ( 1; 1) -differential. If we denote by QS(S 1 )G the subgroup of QS(S 1 ) consisting of G -invariant quasisymmetric homeomorphisms then T (G) is defined as T (G) = QS(S 1 )G /Möb(S 1 ):

Different interpretations of T , given above, remain meaningful also for T (G) if we add everywhere the term “ G -invariant.” For example, T (G) = B(∆)G / ;

where B(∆)G is the subspace of G -invariant Beltrami differentials in ∆ . We can associate with any G -invariant Beltrami differential  in ∆ the Fuchsian group G = w Gw 1 . Then we have a natural quasiconformal map between Riemann surfaces X = ∆/G ! X := ∆/G :

This map is a homeomorphism which is holomorphic if and only if the quasisymmetric homeomorphism, associated with  , is Möbius. Hence, the Teichmüller space T (G) parameterizes via the map  7! X various complex structures on X obtained from the original one by quasiconformal deformations. This is related to the original Teichmüller definition of T (G) . The Kähler quasimetric introduced on T can be reduced to the Weil–Petersson Kähler metric on T (G) (cf. [10]). What can be said about the image of T (G) in the universal Teichmüller space T ? There is an interesting result of Bowen [2] asserting that this image cannot not belong to the regular part of T . More precisely, let us call a point [] 2 T regular if it corresponds to a smooth normalized quasisymmetric homeomorphism w W S 1 ! S 1 or, which is the same, to a quasiconformal map w  W ∆ ! C with smooth boundary w  (S 1 ) . Bowen has proved that any point [] of T (G) , apart from the origin, corresponds to the quasiconformal map w  W ∆ ! C with fractal

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boundary of Hausdorff dimension dH which can take any value between 1 and 2, i.e. 1 < dH < 2 .

3.2. The space of normalized diffeomorphisms We have already introduced above the space S = Diff+ (S 1 )/Möb(S 1 )

of normalized diffeomorphisms of S 1 . In contrast with classical Teichmüller spaces T (G) , this space is contained in the regular part of T and intersects T (G) only at the origin. It is a Fréchet manifold with the tangent space at the origin given by the Fréchet vector space Vect(S 1 )/sl(2; R) where Vect(S 1 ) is the Lie algebra of Diff+ (S 1 ) . The elements of Vect(S 1 ) are given by vector fields v = v()d /d on S 1 where v() is a real-valued smooth 2 -periodic function of variable  2 R . They have Fourier decompositions of the form 1 X

v=

vn e i n

n= 1

where v¯n = v with

n.

The subalgebra sl(2; R) consists of the vector fields v = v()d /d v() =

X

vn e i n :

n=0;˙1

It follows that the elements v = v()d /d of the tangent space To S at the origin, identified with Vect(S 1 )/sl(2; R) , have Fourier decompositions of the form X v= vn en with v¯n = v n : n¤0;˙1

There is a unique complex structure I on S which is invariant under the action of the group Diff+ (S 1 ) on S by left translations (cf. [11]). Its restriction to the space To S is given by the formula X X Iv( ) = i vn e i n + i vn e i n : n>1

n< 1

It also has a Diff+ (S ) -invariant symplectic form w whose restriction to To S is given in terms of Fourier decomposition by the formula 1

w(u; v) = 2 Im

1 X

(n3

n)un v¯n

n=2

for any two vectors u; v 2 To S . The form w is compatible with the complex structure I in the sense that they generate together a Diff+ (S 1 ) -invariant Kähler metric g on S , given by g(u; v) := w(u; Iv) for u; v 2 To S . Thus, S is a Kähler Fréchet manifold.

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We can also introduce a complex structure on S induced from T by the embedding S  T . This complex structure is equivalent to I in the sense that the embedding S  T is a holomorphic map (cf. [11]).

4. Grassmann realization of universal Teichmüller space 4.1. Sobolev space of half-differentiable functions The Sobolev space of half-differentiable functions 1/2

V = H0 (S 1 ; R)

is the Hilbert space consisting of real-valued functions f 2 L2 (S 1 ; R) with Fourier decompositions of the form X f (z) = fk z k with f¯k = f k ; z = e i ; k ¤0

having finite Sobolev norm of order 1/2 kf k21/2 :=

X

jkjjfk j2 = 2

k ¤0

1 X

kjfk j2 < 1:

k =1

We can consider V as a natural Hilbert completion of the loop space ΩR := C01 (S 1 ; R) , consisting of smooth real-valued functions on S 1 with zero average, with respect to the introduced Sobolev norm. We provide ΩR with complex and symplectic structures which are compatible with each other. In terms of Fourier series, a symplectic form ! on ΩR is given by the formula X !(; ) = 2 Im kk ¯k : k >0

on any two vectors  = k ¤0 k z ,  = k ¤0 k z k , belonging to ΩR . A complex structure J 0 on ΩR is defined by X X J 0 () = i k z k + i k z k : P

k

P

k >0

k 0

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The structures !; J 0 and g 0 we introduced extend to the completion V of ΩR . Moreover, V is the largest space in the Sobolev scale of Hilbert spaces H0s (S 1 ; R) , s > 0 , on which ! and g 0 are correctly defined. This fact plays a crucial role in string theory. The complexification 1/2 V C = H0 (S 1 ; C) of the Sobolev space V consists of complex-valued functions f 2 L2 (S 1 ; C) with zero average and finite Sobolev norm of order 1/2. We extend the symplectic form ! and the complex structure J 0 to V C complex-linearly while g 0 is extended to V C as a Hermitian form. The complexified Sobolev space V C admits a decomposition V C = W+ ˚ W

into the direct sum of () -eigenspaces of the operator J 0 2 End V C . These subspaces consist of functions f 2 V C which extend holomorphically to the disk x n ∆+ ). ∆+  ∆ (respectively, ∆ := C

4.2. Grassmann realization of T 1

The group QS(S ) of quasisymmetric homeomorphisms of S 1 acts on the Sobolev space V by change of variable. More precisely, we associate with a homeomorphism f 2 QS(S 1 ) the operator Tf , acting by the formula (Tf h) = h ı f

1 2

Z2

h(f (e i ))d;

h 2 V:

0

Then we have the following Theorem 4.1 (Nag–Sullivan theorem). The operator Tf acts from V to V if and only if f 2 QS(S 1 ) . If f 2 QS(S 1 ) then !(Tf h; Tf g) = !(h; g) for any h; g 2 V . Moreover, the complex-linear extension of Tf to the complexified Sobolev space V C preserves the subspaces W˙ if and only if f 2 Möb(S 1 ) and in this case it acts on W˙ as a unitary operator. This theorem implies that there is an embedding T = QS(S 1 )/Möb(S 1 ) ! Sp(V )/ U(W+ )

where Sp(V ) is the group of linear bounded symplectic operators on V and U(W+ ) is its subgroup, consisting of unitary operators on W+ . In terms of the decomposition V C = W+ ˚ W , the operators A 2 Sp(V ) have block representations of the form   a b A= ¯ b a¯

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with entries aW W+ ! W+ , bW W ! W+ satisfying the reality conditions: a¯t a b t b¯ = 1 , a¯t b = b t a¯ . Here, at , b t denote the transposed operators at W W+0 ! W+0 , b t W W+0 ! W 0 . The unitary group U(W+ ) is embedded into Sp(V ) as the subgroup of block-diagonal matrices of the form   a 0 A= 0 a¯ with a 2 U(W+ ) . The space J(V ) = Sp(V )/ U(W+ )

may be considered as a space of complex structures on V compatible with the symplectic form ! . Indeed, any complex structure J of this type generates a decomposition x VC =W ˚W (2) into a direct sum of (i ) -eigenspaces of the operator J which are isotropic with respect to ! . Conversely, any such decomposition determines a complex structure x . So J on V C , compatible with ! and equal to iI on W and +iI on W the group Sp(V ) acts transitively on the space of complex structures J of the above type. On the other hand, the isotropy subgroup of Sp(V ) at the point J 0 , corresponding to the original complex structure on V , coincides with the group of unitary operators preserving the subspaces W˙ , i.e. with U(W+ ) . Hence, J(V ) is indeed the space of complex structures on V compatible with ! . The space J(V ) may be realized as an infinite-dimensional Siegel disk. By definition, this disk is the set D = ¹ZWW+ ! W is a bounded linear symmetric operator x < I º: with ZZ x < I means that the symmetric operator I ZZ x Here, the inequality ZZ is positively defined. In order to identify J(V ) with D , consider the action of the group Sp(V ) on D by operator fractional-linear transformations   a b ¯ Sp(V ) 3 A = ¯ W Z 7 ! (¯ aZ + b)(bZ + a) 1 : b a¯

This map establishes a one-to-one correspondence J(V ) = Sp(V )/ U(W+ ) ! D:

The constructed embedding T ,! J(V ) also induces an embedding S = Diff+ (S 1 )/Möb(S 1 )  J(V ):

But in the case of S , we can assert that the image of this map is contained in the space of Hilbert–Schmidt complex structures JHS (V ) := SpHS (V )/ U(W+ )

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where SpHS (V ) is the subgroup of Sp(V ) consisting of operators   a b A= ¯ 2 Sp(V ) b a¯ with b being a Hilbert–Schmidt operator. The space JHS (V ) of Hilbert–Schmidt complex structures on V , compatible with ! , can also be realized as the Hilbert–Schmidt Siegel disk DHS = ¹ZW W+ ! W is a symmetric Hilbert–Schmidt operator x < I º: with ZZ

5. Universal Teichmüller space and string theory 5.1. Classical system associated with smooth string theory It was pointed out in physics papers dealing with string theory that the phase space of the theory of smooth strings, taking values in the d -dimensional vector space R1;d 1 (provided with Lorentz metric), may be identified with the space Ωd := C01 (S 1 ; R1;d 1;d

1

)

1

of smooth loops in R with zero average (cf. [3]). The space Ωd has a natural symplectic form ! given by the formula 1 !(; ) = 2

Z2 h(); 0 ()i d 0

where  = (e i ) ,  = (e i ) are smooth maps S 1 ! R1;d 1 . (An expression of ! in terms of Fourier series was given earlier while discussing the properties of the Sobolev space V .) The classical system, associated with this string theory, is given by the pair (phase space, algebra of observables) where the phase space coincides with Ωd while the algebra of observables is the Lie algebra given by the semi-direct product Ad := heis(Ωd ) Ì Vect(S 1 )

where heis(Ωd ) is the Heisenberg algebra of Ωd generated by the coordinate functions on Ωd , and Vect(S 1 ) is the Lie algebra of tangent vector fields on S 1 . This algebra may be considered as an infinite-dimensional analogue of the Poincaré algebra in Minkowski space. In our case, the role of the translation subalgebra is played by the Heisenberg algebra and that of the Lorentz rotation subalgebra by the algebra Vect(S 1 ) . The quantization of a classical system, given by the pair (Ωd Ad ) , is an irreducible linear representation rW Ad ! End (H )

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of observables f 2 Ad by linear selfadjoint operators r(f ) acting in a complex Hilbert space H , called the quantization space. This representation should send the Poisson brackets of observables to commutators of the corresponding operators, i.e. 1 (r(f )r(g) r(g)r(f )) i for any f; g 2 Ad . We also impose the normalization condition: r(1) = id. In the case of the infinite-dimensional algebra Ad , it is more natural to look for projective representations of Ad . Having found such a representation of Ad , f f we can construct the quantization of the extended system (Ωd ; A d ) where Ad is a suitable central extension of Ad determined by the cocycle of the projective representation. The loop space Ωd has a complex structure J 0 defined, as in the case of the Sobolev space, by the formula X X (J 0 )() = i k e i k + i k e i k r(¹f; gº) =

k >0

k 0 , on which this form ! is correctly defined. Then why not take this Sobolev space, already “chosen by the symplectic form ! itself,” for the phase space of string theory? So we choose Vd as the phase space of non-smooth string theory. Next we should fix a natural algebra of observables Ad on this phase space. For the first component of Ad we take, as before, the Heisenberg algebra heis(Vd ) , describing the kinematics. Turning to the second component of Ad , recall that in the case of the algebra Ad we have taken the Lie algebra Vect(S 1 ) of the Lie group Diff+ (S 1 ) , acting on the phase space Ωd by reparameterizations. In the same way, in the case of the algebra Ad , we can ask: what is the maximal group acting on Vd by reparameterizations? The answer to this question is given by Nag–Sullivan theorem—it is the group QS(S 1 ) of quasisymmetric homeomorphisms of the circle. We would like to take for the second component of Ad the Lie algebra of this group and quantize the classical system (Vd ; Ad ) in the same way, as in the case of smooth loops. However, the action of QS(S 1 ) on Vd is not smooth. It implies that we cannot define the classical system, corresponding to the phase space Vd provided with the action of QS(S 1 ) . So what to do in such a situation? The idea is to construct a quantum system, associated with Vd , directly, passing by the stage of construction of a classical system associated with Vd .

5.4. Quantization of the Sobolev space Vd We start with substituting the Dirac definition of quantization, which we have been using up to this point, by the definition due to Connes [4]. In Connes’ definition, the algebra of observables A is an associative algebra provided with an involution and exterior derivative d . The quantization of such a system is given by an irreducible linear representation  of observables from A by closed linear operators which are densely defined in the quantization space H . Under this representation, the involution in A should transform into Hermitian conjugation while the derivation operator d is sent to the commutator with some symmetry operator S being a selfadjoint operator in H with square S 2 = I . In other words, W df 7 ! d q f := [S; (f )];

f 2 A:

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The Connes approach may also be formulated in the language of Lie algebras. For that, consider the Lie algebra Der(A) of differentiations of A , i.e. linear maps A ! A , satisfying the Leibniz rule. In terms of the algebra Der(A) , the Connes quantization is an irreducible representation of the Lie algebra Der(A) in the Lie algebra End H , provided with commutator as a Lie bracket. In the case when all observables from A are given by smooth functions on the phase space, the Connes definition is essentially equivalent to that of Dirac. But if we allow A to contain non-smooth functions, the differential df of a non-smooth observable f 2 A has no sense. However, it may happen that its quantum analogue d q f is correctly defined. Consider as an example the algebra of observables coinciding with the algebra A = L1 (R; C) of bounded functions on the real line R . Any function f 2 A generates a bounded multiplication operator Mf in the Hilbert space H = L2 (R; C) acting by the formula Mf W h 2 H 7 ! f h 2 H

for h 2 H:

The symmetry operator S in this case is given by the Hilbert transform: Z h(t ) (Sh)(x) = dt; h 2 H; x t R

where the integral is taken in the principal value sense. The differential of a general observable f 2 A is not defined in the classical sense but its quantum analogue d q f := [S; Mf ] is correctly defined as a bounded linear operator in H (even for functions f 2 BMO(R) ). Namely, this operator is given by Z f (x + t ) f (x) (d q f )(h)(x) = h(t ) dt; h 2 H: t R

The quasiclassical limit of this operator, established by restricting it to smooth functions and taking the residue at t = 0 , coincides with the multiplication operator h 7! f 0
h . In other words, the quantization procedure in this example essentially reduces to the replacement of the derivative by its finite-difference analogue. Such a quantization, given by the correspondence A 3 f 7 ! d q f W H ! H;

is called the “quantum calculus” by analogy with the finite-difference calculus (cf. [4] and [16]). Returning to the quantization of the Sobolev space Vd , it would be more convenient to switch from S 1 to the real line R . Then Vd will be replaced by the Sobolev space H 1/2 (R) of half-differentiable vector-functions on the real line (which we continue to denote by Vd ) and QS(S 1 ) will be substituted by the group QS(R) of quasisymmetric homeomorphisms of R , extending to quasiconformal

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homeomorphisms of the upper half-plane. Then for any h 2 Vd we introduce the operator d q hW VdC ! VdC by the formula Z h(x + t ) h(t ) q (d h)(v)(x) = v(t ) dt; v 2 VdC : t R

According to [12] (cf. also [6]), the tangent space to QS(R) at the origin coincides with the Zygmund space Λ(R) of functions f (x) satisfying the condition jf (x + t ) + f (x

t)

2f (x)j  C jt j

uniformly for x 2 R , t > 0 , and growing not faster than const
x 2 for jxj ! 1 . This motivates the introduction of the following operator d q g for g 2 QS(R) Z g(x + t ) + g(x t ) 2g(x) d q g(v) = v(t ) dt; v 2 VdC : t R

We define now the quantized infinitesimal action of QS(R) on VdC as the composition Tgq h := d q h(g) ı d q g of the introduced operators. The quasiclassical limit of this operator is equal to the operator of multiplication by h0 (g)g 0 . This action extends to the whole Fock space F0 in the following way. We define it first on the elements of the orthonormal basis of F0 , given by monomials PK (z) , by the Leibniz rule. Then we extend this operator to the whole algebra S(W+ ) of symmetric polynomials on W+ by linearity. The closure of the obtained operator yields an operator Tgq h in the Fock space F0 . In the same way, the operator d q h is extended to a closed operator d q h in F0 . The desired quantum algebra of observables Aqd is the Lie algebra generated by the constructed operators d q h and Tgq h in F0 with g 2 QS(R) , h 2 Vd . We remark in conclusion that it is also possible to construct the quantization of Vd in the frame of the Dirac definition of quantization as it is done in [17]. Acknowledgments. I am grateful to the referee for the careful reading of the chapter and making valuable remarks. While preparing this survey the author was partially supported by RFBR grants 18-51-41011, 19-01-00474, and Presidium of RAS program “Nonlinear dynamics.”

References [1] L. Ahlfors, Lectures on quasiconformal mappings. Manuscript prepared with the assistance of C. J. Earle, Jr. Van Nostrand Mathematical Studies, 10. D. Van Nostrand Co., Toronto etc., 1966. MR 0200442 Zbl 0138.06002 R 197, 198, 200 [2] R. Bowen, Hausdorff dimension of quasicircles. Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 329–333. MR 0951982 Zbl 0606.30023 R 202

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[3] M. J .Bowick and S. G. Rajeev, The holomorphic geometry of closed bosonic string theory and Diff S 1 /S 1 . Nuclear Phys. B 293 (1987), no. 2, 348–384. MR 0908048 R 207, 209 [4] A. Connes, Géométrie non commutative. InterEditions, Paris, 1990. MR 1079062 Zbl 0745.46067 R 210, 211 [5] H. M. Farkas and I. Kra, Riemann surfaces. Second edition. Graduate Texts in Mathematics, 71. Springer-Verlag, New York etc., 1992. MR 1139765 Zbl 0764.30001 R 202 [6] F. P. Gardiner and D. P. Sullivan, Symmetric structures on a closed curve. Amer. J. Math. 114 (1992), no. 4, 683–736. MR 1175689 Zbl 0778.30045 R 212 [7] R. Goodman and N. R. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347 (1984), 69–133. MR 0733047 Zbl 0537.22015 Zbl 0514.22012 R 209 [8] V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinitedimensional Lie algebras. Advanced Series in Mathematical Physics, 2. World Scientific Publishing Co., Teaneck, N.J., 1987. MR 1021978 Zbl 0668.17012 R 209 [9] O. Lehto, Univalent functions and Teichmüller spaces. Graduate Texts in Mathematics, 109. Springer-Verlag, New York, 1987. MR 0867407 Zbl 0606.30001 R 196, 200, 201 [10] S. Nag, The Complex Analytic Theory of Teichmüller Spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, New York, 1988. MR 0927291 Zbl 0667.30040 R 196, 201, 202 [11] S. Nag and A.Verjovsky, Diff(S 1 ) and the Teichmüller spaces. Comm. Math. Phys. 130 (1990), no. 1, 123–138. MR 1055689 Zbl 0705.32013 R 203, 204 [12] M. Reimann, Ordinary differential equations and quasiconformal mappings. Invent. Math. 33 (1976), no. 3, 247–270. MR 0409804 Zbl 0328.30019 R 212 [13] G. Segal, Unitary representations of some infinite-dimensional groups. Comm. Math. Phys. 80 (1981), no. 3, 301–342. MR 0626704 Zbl 0495.22017 R 209 [14] A. Sergeev, Kähler geometry of loop spaces. MSJ Memoirs, 23. Mathematical Society of Japan, Tokyo, 2010. MR 2654515 Zbl 1191.58001 R 209 [15] A. Sergeev, Lectures on universal Teichmüller space. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2014. MR 3243752 Zbl 1297.30003 R 196, 197 [16] A. Sergeev, Quantum calculus and quasiconformal mappings. Mat. Zametki 100 (2016), no. 1, 144–154 In Russian. English translation, Math. Notes 100 (2016), no. 1-2, 123–131. MR 3588834 Zbl 1359.30037 R 211 [17] A. Sergeev, Quantization of Sobolev space of half-differentiable functions. Mat. Sb. 207 (2016), no. 10, 96–104. In Russian. English translation, Sb. Math. 207 (2016), no. 9-10, 1450–1457. MR 3588973 Zbl 1360.30019 R 212 [18] A. Sergeev, The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms. Georgian Math. J. 23 (2016), no. 4, 615–622. MR 3565990 Zbl 1350.81018 R 197 [19] L. A. Takhtajan and L-P. Teo, Weil–Petersson metric on the universal Teichmüller space. Mem. Amer. Math. Soc. 183 (2006), no. 861. MR 2251887 Zbl 1243.32010 R 196 [20] A. Yu. Vasiliev and A. G. Sergeev, Classical and quantum Teichmüller spaces. Uspekhi Mat. Nauk 68 (2013), no. 3(411), 39–110. In Russian. English translation, Russian Math. Surveys 68 (2013), no. 3, 435–502. MR 3113857 Zbl 1285.30026 R 196, 197

Chapter 7

Generalized conformal barycentric extensions of circle maps Jun Hu

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized conformal barycentric extensions of continuous circle maps Qualitative and quantitative properties of the extensions of circle homeomorphisms in different classes . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 217 227 236

1. Introduction x be Let S1 be the unit circle in the complex plane C centered at the origin and D the closed disk bounded by S1 . Denote by F the collection of all continuous maps from S1 to itself. A procedure to extend the maps f in F to maps Φ(f ) from x to itself is said to be conformally natural if it satisfies the following properties. D i. The extensions are compatible under precomposition and postcomposition by the conformal homeomorphisms of the unit disk D ; that is, for any f 2 F and any two Möbius transformations A and B preserving D , Φ(A ı f ı B) = A ı Φ(f ) ı B:

(1)

i i. If a map f of F is equal to the restriction to S of a complex analytic map F that preserves D , then Φ(f ) = F . These extensions are called conformally natural extensions. The first property is called the conformal naturality of the construction procedure or the extensions with respect to the class F . Let  be a probability measure supported on S1 and f be a homeomorphism of S1 . Douady and Earle [7] proved that  has a unique conformal barycenter if  has no atom. Then using the conformal barycenter of the pushforward measure under f of the harmonic measure on S1 viewed from each point z 2 D , they constructed a conformally natural extension Φ(f ) of f . It is also pointed out in [7] that if  has atoms and the measure of each atom is strictly less than 12 , then  1

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has a conformal barycenter B() in D . Under this assumption, B() is the unique fixed point of the so-called MAY iterator defined for  (see [1]). Furthermore, if  has only one atom of measure greater than or equal to 12 (called a strong atom of  ), then the MAY iterator defined by  has a unique fixed point on the boundary S1 , which is equal to that atom [2]. So replacing conformal barycenters by unique fixed points of MAY iterators, conformally natural extensions are defined for monotone degree ˙1 continuous maps from S1 to itself [1]. Again using conformal barycenters, conformally natural extensions are introduced for continuous maps f from S1 to itself assuming that f is not constant on any set of positive Lebesgue measure [23]. Finally, using conformal barycenters and strong atoms of probability measures, conformally natural extensions are constructed for arbitrarily continuous maps from S1 to itself (see [24]; some parts of this work were first reported on in [33]). Conformal barycentric extensions of homeomorphisms are real analytic homeomorphisms of D (see [7]). Such extensions of quasisymmetric homeomorphisms of S1 have found many applications in the study of Teichmüller spaces and asymptotic Teichmüller spaces. For example, one can find 1. the importance of the conformal naturality of the extensions in [42]; 2. an application in the study of the rigidity of the groups of homeomorphisms of S1 with uniformly bounded quasisymmetry in [29]; 3. an application in the proof of the coincidence of the Teichmüller and Kobayashi metrics on the Teichmüller spaces of Riemann surfaces in [10]; 4. applications to prove the contraction property of various subspaces of the universal Teichmüller space in [5], [6], [11], [14], [31], and [38]; 5. applications in the study of the topological characterizations of Teichmüller spaces and asymptotic Teichmüller spaces in terms of Thurston’s earthquake measures in [32] and [13], and in terms of shears in [37] and [12]. Investigations on regularities of the conformal barycentric extensions of circle homeomorphisms in different classes are given in [29], [20], [21], [22], and [25]. Note that Thurston’s earthquake maps of orientation-preserving homeomorphisms of S1 are conformally natural extensions, but the extensions are not necessarily continuous on the disk D (see [41]). For a self-contained introduction and a study of Thurston’s earthquake maps from different aspects, we refer to [16], [17], [18], [36], or [19]. Lately, Markovic proved that every quasisymmetric homeomorphism of S1 admits a (unique) harmonic extension to D (see [30]). Harmonic extensions are conformally natural. It is proved in [26] that there are quasisymmetric homeomorphisms f of S1 whose barycentric extensions are not harmonic. Therefore, for the class of quasisymmetric homeomorphisms of S1 , there are three different ways to construct conformally natural extensions. These are the conformal barycentric extensions, Thurston’s earthquake maps and the harmonic extensions. As pointed out in the previous paragraph, Thurston’s earthquake maps are not necessarily continuous on D , and in the meantime there is not much known on the existence of a harmonic

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extension for an arbitrary orientation-preserving homeomorphism of S1 . In this survey, we emphasize that generalized conformal barycentric extensions are well defined for arbitrarily continuous maps from S1 to itself. The survey is comprised of two sections. In one section, we first recall how generalized conformal barycentric extensions are defined for arbitrarily continuous maps f from D to itself and why they are conformally natural; then we provide a shortened proof for the continuity of the extensions; thirdly we give a summary of the properties of the extensions developed in [24] and [23]. In another section, we provide a brief account of qualitative and quantitative properties of such extensions for circle homeomorphisms, quasisymmetric circle homeomorphisms, symmetric circle homeomorphisms, and circle diffeomorphisms, which are respectively developed in [7], [29], [20], [21], [22], and [25].

2. Generalized conformal barycentric extensions of continuous circle maps 2.1. Definition Let us first give some background on the existence of a conformal barycenter. Let  be a probability measure supported on S1 . A point w of D is called a conformal barycenter of  and denoted by B() if Z  w d() = 0: ¯ 1 w S1

It is proved in [7] that if  has no atom, then B() exists uniquely. Douady and Earle [7] also pointed out that if  has no atom of measure  12 (called an admissible measure), then B() exists uniquely as well. To see the existence and uniqueness of B() , the following smooth vector field  on D is considered in [7]: Z  w  (w) = (1 jwj2 ) d(); w 2 D: (2) ¯ 1 w S1

In fact, the vector field  (w) is relatively easy to handle if the scalar factor 1 jwj2 is dropped (see [23]). So let us set 1 ˜ (w) =  (w): (3) 1 jwj2 Then the work to reach the existence and uniqueness of B() is comprised of two steps. One can see first that if  is admissible, then  (w) (and hence ˜ (w) ) points inside the circle centered at the origin and passing through w when w is sufficiently close to the boundary S1 . (In fact, if  has no atom, then ˜ (w) x and points inside at every extends to a continuous vector field on the closed disk D

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point on S1 .) This implies the existence of a singular point of the vector field, which is a conformal barycenter of  . Secondly, in order to have the uniqueness of B() , it suffices to know that the Jacobian of the vector field is positive at every point of D . Now one can find that it is quite easy to achieve this property by using ˜ (w) since it satisfies ˜ (w) = ˜(gw ) () (0);

(4)

w where gw () = 1 w and (gw ) () is the pushforward measure of  under the ¯ map gw (see 8), and hence the values of ˜ at two points z and w in D are related by ˜ (z) = ˜(gw ) () (gw (z)): (5) Then the Jacobian of ˜ at w is the product of the Jacobian of ˜(gw ) () at 0 and the Jacobian of gw at w . Therefore, it is sufficient to show that the Jacobian of ˜ is positive at 0 , which is “ 1 Jac(˜ )(0) = j2  2 j2 d()d() > 0: (6) 2

S1  S1

Now by the Poincaré-Hopf Index Theorem, we know that the singular point of  in D is unique. The details for the first step can be found in [24]. Correspondingly, a probability measure  on S1 is said to be non-admissible if it has one atom of measure  12 . Then any atom with measure  12 is called a strong atom. Clearly,  can have at most two strong atoms. Given a point z 2 D , let z be the normalized harmonic measure on S1 as viewed from z ; that is, for any Borel set E  S1 , Z Z 1 1 1 jzj2 z (E) = jdgz ()j = jd j: (7) 2 2 jz j2 E

E

1

If I is an arc on S and z (I ) is the radian of the angle of I as viewed from z (in the hyperbolic metric on D ), then z (I ) is the ratio of z (I ) to 2 . For each element f of F , the pushforward measure f (z ) is defined as f (z )(E) = z (f

1

(E))

(8)

for any Borel set E  S1 . Then one can easily see that if f is a constant map, then f (z ) has one atom of measure 1 ; if f is not constant, then f (z ) can have at most one strong atom by using the connectivity for a continuous map. Therefore, whether or not f is constant, f (z ) has at most one strong atom, which we denote by z if it exists. This observation is the key to define generalized conformal barycentric extensions for the maps f 2 F .

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Definition 2.1 ([33]). Given each f 2 F , we divide the points of D into two subsets: ıf = ¹z 2 DW f (z ) has a strong atomº

and ˇf = D n ıf :

(9)

We denote by z the strong atom of f (z ) if it exists. Then the generalized conformal barycentric extension Φ(f ) of f is defined as 8 ˆ if z 2 S1 ; 0 , there exists ı > 0 such that for any z 2 B(z0 ; ı) = ¹zW jz z0 j < ıº , jf (z )(E)

f (z0 )(E)j <  Leb(f

1

(E))

for any measurable subset E of S1 , where we denote the harmonic measure 0 by Leb . Corollary 2.3. Let ¹zn º1 n=1 be a sequence of points in D converging to a point z0 2 D , and let n = f (zn ) and 0 = f (z0 ) . Then for any  > 0 there exists a positive integer N such that for any n > N and any measurable subset E of S1 , jn (E)

0 (E)j <  Leb(f

1

(E)):

Lemma 2.2 implies the following result. Proposition 2.4. subset of D .

For each f 2 F , ˇf is an open subset of D and ıf is a closed

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For each point a 2 S1 , define ıf (a) to be the collection of all points z 2 D such that f (z ) has a strong atom at a . Clearly, ıf (a) and ıf (a0 ) are disjoint for any two distinct points a and a0 on S1 since f (z ) can not have two strong atoms for each z 2 D . Corollary 2.3 implies Proposition 2.5. of D .

For each f 2 F and each a 2 S1 , ıf (a) is a closed subset

The next property of ıf (a) is not trivial. So we recapitulate the proof given in [24] or [33], which is obtained from the following lemma. Lemma 2.6. If ¹zn º1 n=1 is a sequence of points in ıf converging to a point z0 2 ıf (a) for some point a 2 S1 , then there exists a positive integer N such that for any n > N , zn 2 ıf (a) . Proof. Suppose that the conclusion is not true. Then there is a subsequence ¹znk º1 nk =1 such that znk 2 ıf (bnk ) with bnk ¤ a for any k . If ¹bnk W k 2 Nº is finite, then ¹bnk º has a subsequence with all elements equal to a constant b . For brevity of notation, we continue to denote this subsequence by ¹bnk º . Applying Corollary 2.3, we know 1 f (z0 )(¹bº) = lim f (znk )(¹bº) = lim f (znk )(¹bnk º)  : 2 k !1 k !1 This means that b is also a strong atom of f (z0 ) . By the uniqueness of strong atom, b = a . This is a contradiction. If ¹bnk W k 2 Nº is infinite, then it has an accumulation point b . By passing to a subsequence, we may assume that ¹bnk º converges to b . By Corollary 2.3, for any  > 0 , there exists m such that for any k > m and any measurable subset E jf (z0 )(E)

f (znk )(¹bº)(E)j < :

Then for each k > m , by letting E = ¹bnk º , 1 ; 2 which contradicts that f (z0 ) is a probability measure. f (z0 )(¹bnk º) >

□

The previous Lemma 2.6 implies Proposition 2.7. If a point z0 of ıf (a) is an interior point of ıf , then z0 is also an interior point of ıf (a) . In the remaining part of this subsection, we recapitulate the proof of the following theorem (see [24] or [33]). Note that the second property of the following theorem is also given in [34].

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Theorem 2.8 (conformal naturality). The generalized conformal barycentric extension has the following two properties. 1. For any element f 2 F and any two conformal homeomorphisms A and B of D , Φ(A ı f ı B) = A ı Φ(f ) ı B:

2. If f is the restriction to S1 of a complex analytic map Ψ defined on x , then Φ(f ) = Ψ . a neighborhood of D Proof.

(1) We prove that for any z 2 D (11)

Φ(A ı f ı B)(z) = A ı Φ(f ) ı B(z):

By Proposition 2.1, ˇAıf ıB = B

1

(ˇf )

and

ıAıf ıB = B

1

(ıf ):

The proof of (11) is divided into two cases by considering z 2 B z 2 B 1 (ıf ) . Recall that given a point a 2 D , ga (z) = 1z a¯az .

1

(ˇf ) or

Case 1. Assume that z 2 B 1 (ˇf ) . Let b = B(z) , w = Φ(f )(b) and w  = A(w) . Then b 2 ˇf , and b and w satisfy Z gw ı f ı gb 1 ()jd j = 0: S1

Note that there exist 1 and 2 such that (B ı gz 1 )

1

= gB

1 (b)

1

ıB

= e i1 gb

and gw  ı A = gA(w) ı A = e i2 gb :

Then Z

gw  ı A ı f ı B ı gz 1 ()jd j

S1

Z =

(e i2 gw ) ı f ı (gb )

1

(e

i1

)jd j

S1

= e i1

Z

S1

= 0:

(gw ) ı f ı (gb )

1

()jd j

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This means Φ(A ı f ı B)(z) = w  = A(w) = A(Φ(f )(b)) = A ı Φ(f ) ı B(z):

Case 2. Assume that z 2 B Then

1

(ıf ) . Let b = B(z) , Φ(f )(b) = w , and w  = A(w) .

(A ı f ı B) (z )(¹w  º) = (f ı B) (z )(¹wº) = Leb(gz 1 ı B 1

= Leb(gb ı f

1 1

ıf

1

(¹wº))

(¹wº)

= f (b )(¹wº): 

Therefore, w is a strong atom of (A ı f ı B) (z ) if and only if w is a strong atom of f (b ) . Furthermore, Φ(A ı f ı B)(z) = w  = A ı Φ(f ) ı B(z):

(2) We show that if f is the restriction to S1 of a complex analytic map Ψ x , then Φ(f ) = Ψ . Such a map Ψ is a Blaschke defined on a neighborhood of D product preserving D . Let f = ΨjS1 . Clearly, ıf is empty. It is proved in [23] and [34] that Φ(f ) = Ψ . The proof is very short and goes as follows. Given a point a 2 D , let b = Ψ(a) . Let f˜ = gb ıf ıga 1 and Fx = gb ıΨıga 1 . By the first part of this theorem, it suffices to show that Φ(f˜)(0) = 0 ; that is, R ˜ S1 f (z)jdzj = 0 . By Cauchy’s Integral Formula, we obtain Z Z Z x 1 F (z) ˜ x f (z)jdzj = F (z)jdzj = dz = 2 Fx(0) = 0: □ i z S1

S1

S1

The following theorem is proved in [24]. Theorem 2.9 (anti-conformal naturality). The generalized conformal barycentric extension is anti-conformally natural in the sense that it has the following two properties. 1. For any element f 2 F and any two anti-conformal homeomorphisms A and B of D , Φ(A ı f ı B) = A ı Φ(f ) ı B: 2. If f is the restriction to S1 of a complex anti-analytic map Ψ defined on x , then Φ(f ) = Ψ . a neighborhood of D

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2.3. Continuity of the extension In this subsection, we give a shortened proof for the continuity of the extension Φ(f ) obtained in [24], which was first reported on in [33]. Theorem 2.10 (continuity). For each f 2 F , the generalized conformal barycentric x to itself. extension Φ(f ) of f is a continuous map from D This theorem is proved in [24] by considering four cases for the points z 2 D : 1: z 2 ˇf ; 2: z 2 int(ıf );

3: z 2 S1 ; 4: z 2 @ıf \ D:

The proof for either case (1) or case (2) is straightforward; the proof for case (3) follows the same idea used in [7] to prove the continuity of Φ(f ) at a boundary point of D when f is a homeomorphism of S1 ; the proof for case (4) presented in [24] shows how close to z an input z 0 needs to be in order to have the corresponding output Φ(f )(z 0 ) to be close to Φ(f )(z) . The proof for case (4) given in [24] is neat, but it is quite long. Using a proof by contradiction, we present in this subsection a different and slightly shorter proof for case (4). In order to present a complete proof of Theorem 2.10 to a reader, we include the proof for the first three cases. Proof. We may assume that f is not a constant map. We consider the following four cases. Case 1. Assume that z 2 ˇf . Let w = B(f (z )) . Then z and w satisfy F (z; w) = 0;

where F (z; w) =

1 2

Z S1

f () w 1 jzj2
jd j: ¯ () jz j2 1 wf

(12)

By using the conformal naturality of the extension and through precomposition and postcomposition by conformal homeomorphisms of D , we may assume z = 0 and w = 0 ; that is, Φ(f )(0) = 0 . Then the partial derivatives of w to z and z¯ at z = 0 are expressed in [20] as follows: @w (0) = @¯ z

@F @F (0; 0) (0; 0) @w @¯ z ˇ ˇ2 ˇ @F ˇ ˇ ˇ ˇ @w (0; 0)ˇ

@F @F (0; 0) (0; 0) ¯ @w @z ˇ ˇ2 ˇ @F ˇ ˇ (0; 0)ˇˇ ˇ @w ¯

@w (0) = @z

@F @F (0; 0) (0; 0) @w @z ˇ ˇ2 ˇ @F ˇ ˇ ˇ ˇ @w (0; 0)ˇ

@F @F (0; 0) (0; 0) ¯ @w @¯ z : ˇ ˇ2 ˇ @F ˇ ˇ ˇ (0; 0)ˇ ˇ @w ¯

and

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The denominator of these two expressions can be written as ˇ ˇ2 ˇ @F ˇ ˇ ˇ ˇ @w (0; 0)ˇ

ˇ ˇ2  1 2 Z2Z2 ˇ @F ˇ ˇ ˇ (0; 0)ˇ = 2 sin2 (h(s) ˇ @w ¯ 2 0

h(t ))dsdt > 0;

0

where hW R ! R is a lift of f to the real line R ; that is, f (e i u ) = e ih(u) for any u 2 R and h(u + 2) = h(u) + 2 . §3.1 presents the detail on how to derive this expression when f is a homeomorphism of S1 , which can be copied to obtain the same expression in this case for an arbitrarily non-constant continuous map f from S1 into itself. It follows that if f is a nonconstant continuous map from S1 to itself, then @w (0) and @w (0) exist. Then w = Φ(f )(z) is differentiable at z = 0 . Hence @z @¯ z Φ(f ) is continuous at every point z 2 ˇf . Case 2. Assume that z 2 int(ıf ) . Proposition 2.7 implies that Φ(f ) is continuous at each point z 2 int(ıf ) . Hence it is constant on each connected component of int(ıf ) . Case 3. Assume that z 2 S1 . The essential work to handle this case is to prove the following claim. Let I  S1 be an arc and J = f (I ) . Define ° ° 2± 1± U (I ) =  2 DW  (I )  and V (J ) =  2 DW  (J )  [ J: 3 4

Claim.

Then Φ(f )(U (I ))  V (J ) . Let z 0 2 U (I ) and  = f (z 0 ) . Then (J )  2/3 . Thus, if J consists of a point then z 0 2 ıf and V (J ) = J . From the definition, it follows that Φ(f )(U (I )) = J = V (J ) . Therefore, it remains to prove the claim for the general situation that J = f (I ) contains at least two points. The proof is achieved by considering two situations: z 0 2 ˇf or z 0 2 ıf . Situation i . Assume that z 0 2 ˇf . Then J is an arc and V (J ) has an interior in D . Let w 2 Γ = @V (J ) \ D . There exists a conformal homeomorphism g of D such that

7 ;e )

g(J ) = (e

i /4

i /4

and

g(w) = 0:

Let  = g () . Then (g(J ))  2/3 . By Lemma 1 in [7], Re( (0)) > 0;

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where the vector field  is defined by (2). This means that  points to the right half coordinate plane. Using the conformal naturality, we know that  points to the interior of V (J ) at every point w 2 Γ = @V (J ) \ D . Since  is a continuous vector field on D and points into D at every point on a circle contained in D and sufficiently close to S1 , it follows that B() 2 V (J ) and hence Φ(f )(U (I ) \ ˇf )  V (J ): Situation i i . Assume that z 0 2 ıf . Then there is an atom ˛ 2 S1 such that (¹˛º)  1/2 . By definition, Φ(f )(z 0 ) = ˛ . In the meantime, (J c ) = f (z 0 )(J c ) = z 0 (f

1

(J c ))  z 0 (I c ) = 1

z 0 (I )  1/3;

where J c and I c are the complements of J and I in S1 respectively. Thus, ˛ 2 J  V (J ):

Therefore, Φ(f )(z 0 ) 2 J and hence Φ(f )(U (I ) \ ıf )  V (J ): Therefore, we have obtained Φ(f )(U (I ))  V (J ): 1

Now let z 2 S and w = f (z) . Let J be a neighborhood of w on S1 . Since f is continuous, there is a neighborhood I of z on S1 such that f (I )  J . Let J 0 = f (I ) . Using the claim and the fact that J 0  J , we obtain Φ(f )(U (I ))  V (J 0 )  V (J ):

Therefore, Φ(f ) is continuous at every point z 2 S1 . Case 4. Assume that z 2 @ıf \ D . Let ¹zn º1 n=1 be a sequence of points in D converging to z . It suffices to show that Φ(f )(zn ) converges to Φ(f )(z) as n ! 1 . The proof is divided into the following two steps. Step 1. For any subsequence ¹znk º1 of ¹zn º1 n=1 contained in ıf , we show that k =1 Φ(f )(znk ) is eventually equal to Φ(f )(z) . By Lemma 2.6, there is a large enough N such that for all nk > N , znk belongs to ıf (a) for some a 2 S1 as z does. From the definition, it follows Φ(f )(znk ) = Φ(f )(z0 ) for all nk > N . Thus, Φ(f )(znk ) converges to Φ(f )(z) as soon as ¹znk ºnk =1 is a sequence of points in ıf approaching z . Step 2. For any subsequence ¹znk º1 of ¹zn º1 n=1 contained in ˇf , we show that k =1 Φ(f )(znk ) converges to Φ(f )(z) as znk converges to z . For brevity of notation, we continue to use ¹zn º1 n=1 to denote the subsequence; that is, we may assume that ¹zn º1  ˇ . Now let b be a limiting point of ¹Φ(f )(zn )º1 f n=1 n=1 . By passing to a subsequence again, we may assume further that Φ(f )(zn ) converges to b as n ! 1 . Let w = Φ(f )(z) . By Proposition 2.4, w 2 S1 . It remains to prove that b = w.

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Let n = f (zn ) and  = f (z ) , and let bn = B(n ) . Then by the assumption, bn = Φ(f )(zn ) converges to b as n ! 1 . We first show that b … D . Suppose that b 2 D . Then the conformal homeomorphism An () =



bn

1

bn 

converges to A() =

 b ¯ 1 b

uniformly on S1 as n ! 1 . Therefore, given any  > 0 , ˇZ ˇ ˇ  b ˇ  b  ˇ ˇ n d () ˇ ˇ 12 . Let (I ) = 12 + 2ı , where ı > 0 . By Corollary 2.3, there exists N > 0 such that n (I )  12 + ı for any n > N .

7 Generalized conformal barycentric extensions

227

Now let gn be the conformal homeomorphism of D that maps bn to 0 and w to 1 , and let In = gn (I ) and n = (gn ) n . Then n = (gn ı f ) (zn ) and n (In ) = n (gn 1 (In )) = n (I )  12 + ı . Since bn approaches b 2 S1 and b … I , it follows that the arc length of In converges to 0 and In shrinks to the point 1 . By the conformal naturality, gn ı Φ(f )(zn ) = 0 . This means that the origin is a conformal barycenter of n for each n . On the other hand, if n is sufficiently large then In is a sufficiently short arc containing 1 and having n (In )  12 + ı . Using the definition of conformal barycenter, we can see that this property implies that the origin can not be the conformal barycenter of n when n is sufficiently large. A contradiction is reached. Therefore, the assumption that b ¤ w can not hold. Thus, b = w . Now we can conclude that Φ(f ) is continuous at every point z 2 @(ıf )\ D . □

2.4. Criterion for surjectivity x with the same constant value. If f is a constant map, then so is Φ(f ) on D Clearly, this is an example of f 2 F such that Φ(f ) is not a surjective map from x to itself. A criterion is obtained in [24] to determine when the extension Φ(f ) D is surjective.

Theorem 2.11 ([24]). x to D x. from D

If f 2 F0 and deg(f ) ¤ 0 , then Φ(f ) is a surjection

For continuous circle maps f of degree 0 , it is obvious that if f is not x to itself; if f is surjective surjective from S1 to itself then neither is Φ(f ) from D from S1 to itself then Φ(f ) can be either surjective or not surjective. Theorem 2.12 ([24]). There exists a degree- 0 continuous surjection f W S1 ! S1 such that Φ(f )W D ! D is not surjective; there also exists a degree- 0 continuous surjection f W S1 ! S1 such that Φ(f )W D ! D is surjective.

3. Qualitative and quantitative properties of the extensions of circle homeomorphisms in different classes In this section, we give a summary of qualitative and quantitative properties of the conformal barycentric extensions for circle homeomorphisms, quasisymmetric circle homeomorphisms, symmetric circle homeomorphisms, and circle diffeomorphisms, which are respectively developed in [7], [29], [20], [21], [22], and [25].

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3.1. Extensions of homeomorphisms In this subsection, we prove the following theorem of Douady and Earle [7]. Theorem 3.1 ([7]). If f is an orientation-preserving homeomorphism of S1 , then x. the conformal barycentric extension Φ(f ) is a homeomorphism of D Proof. Given a point z 2 D , f (z ) has no atom and hence it has a unique conformal barycenter B(f (z )) in D . Thus, w = Φ(f )(z) = B(f (z )) , where z and w satisfy F (z; w) = 0; with F (z; w) is defined by (12); that is, Z 1 f () w 1 jzj2 F (z; w) =
jd j: ¯ () jz j2 2 1 wf

(13)

S1

x , it suffices to prove In order to conclude that Φ(f ) is a homeomorphism of D that the Jacobian of w to z is positive at any point z 2 D . Using the conformal naturality of Φ , we may assume that z = 0 and w = 0 by precomposition and postcomposition by conformal homeomorphisms of D . Under this assumption, the Jacobian of w to z at the origin can be explicitly expressed in terms of the partial derivatives of F at the origin (0; 0) . If we let Z @F 1 ¯ ()jd j; c1 = (0; 0) = f (14a) @z 2 S1

@F 1 c 1= (0; 0) = @¯ z 2

Z

(14b)

f ()jd j; S1

and @F d1 = (0; 0) = @w

1;

d

1

@F 1 = (0; 0) = ¯ @w 2

Z

f ()2 jd j;

(15)

S1

then @w (0) = @¯ z

@F @F (0; 0) (0; 0) @w @¯ z ˇ ˇ2 ˇ @F ˇ ˇ ˇ (0; 0) ˇ @w ˇ

@F @F (0; 0) (0; 0) ¯ @w @z ˇ ˇ2 ˇ @F ˇ ˇ ˇ (0; 0) ˇ @w ˇ ¯

(16)

@w (0) = @z

@F @F (0; 0) (0; 0) @w @z ˇ ˇ2 ˇ @F ˇ ˇ ˇ ˇ @w (0; 0)ˇ

@F @F (0; 0) (0; 0) ¯ @w @¯ z : ˇ ˇ2 ˇ @F ˇ ˇ ˇ (0; 0)ˇ ˇ @w ¯

(17)

and

7 Generalized conformal barycentric extensions

229

Furthermore, the Jacobian of Φ(f ) at 0 is equal to ˇ ˇ2 ˇ ˇ2 ˇ @F ˇ ˇ @F ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ (0; 0)ˇˇ ˇ @z (0; 0)ˇ ˇ @¯ ˇ @w ˇ2 ˇ @w ˇ2 jc1 j2 z ˇ ˇ ˇ ˇ =ˇ (0) (0) = ˇ ˇ ˇ ˇ @z ˇ ˇ @¯ ˇ ˇ @F ˇ2 ˇ @F ˇ2 z jd1 j2 ˇ ˇ ˇ ˇ (0; 0) (0; 0) ˇ @w ˇ ˇ @w ˇ ¯

jc 1 j2 : jd 1 j2

(18)

Now let hW R ! R be a lift of f to the real line R ; that is, f (e i u ) = e ih(u) for any u 2 R , where h(u + 2) = h(u) + 2 . Then  1 2 Z Z 2 2 jd 1 j = f (z)2 f () jdzjjd j 2

=

 1 2 2

S1 S1 Z2Z2 0

=

 1 2

2ih(t)

dsdt

0

2 Z2Z2 0

e 2ih(s) e

cos 2(h(s)

h(t ))dsdt;

0

and hence one can rewrite  1 2 Z2Z2 jd 1 j = 2 sin2 (h(s) 2

2

2

jd1 j

0 2

h(t ))dsdt:

(19)

0

2

It follows that jd1 j jd 1 j > 0 . In [7], Douady and Earle expressed jc1 j2 jc1 j2

jc

2 1j =

 1 2 2

Z

u=0

jc

sin u

2 1j

as

Z2

(20)

H (t; u)dtdu

t =0

with H (t; u) = sin(h(t + u)

h(t )) + sin(h(t + 2)

+ sin(h(t +  + u)

h(t + u + ))

h(t + )) + sin(h(t + )

h(t + u)):

Let ˛j , j = 1; 2; 3; 4 , be the differences input inPthe sine functions in H (t; u) . Clearly, all ˛j are nonnegative and their sum ˛j = 2 . By applying the summation formula in trigonometry, it is obtained in [2] that 4 X j =1

sin ˛j = 4 sin

˛1 + ˛2 ˛1 + ˛3 ˛2 + ˛3 sin sin : 2 2 2

(21)

Now we can easily see that jc1 j2 jc 1 j2 > 0 since H (t; u)  0 for all t and u and is not identically equal to 0 . It follows that the Jacobian of Φ at the origin is positive. By the conformal naturality, the Jacobian of Φ is positive at every point

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Jun Hu

z 2 D , which implies that Φ is an orientation-preserving diffeomorphism of D and x. a homeomorphism of D □

Furthermore, the following two theorems are also proved in [7] by Douady and Earle. Theorem 3.2 ([7]). If f is an orientation-preserving homeomorphism of S1 , then the conformal barycentric extension Φ(f ) is real analytic on D . x ) ) be the space of all orientationTheorem 3.3 ([7]). Let H (S1 ) (respectively, H (D x ), equipped with the preserving circle homeomorphisms of S1 (respectively, D C 0 -topology, and Diff(D) the space of all diffeomorphisms of the open unit disk x ) is continuous. with C 1 -topology. Then the map ΦW H (S1 ) ! Diff(D) \ H (D

3.2. Extensions of quasisymmetric homeomorphisms Let f be a quasisymmetric homeomorphism of S1 and Φ(f ) the conformal x. barycentric extension of f to the closed unit disk D x , then Φ is Theorem 3.4 ([7]). If f admits a quasiconformal extension to D quasiconformal.

In [7], Douady and Earle showed that Φ is quasiconformal if f admits x . In fact, the following quantitative result is given a quasiconformal extension to D in the proof of Proposition 7 of [7]. p p Theorem 3.5 ([7]). Let K  1 and ˛(K) = 4(1 + 2)(16 3)K . If f admits x , then the maximal dilatation K(Φ(f )) of a K -quasiconformal extension to D Φ(f ) of f satisfies K(Φ(f ))  64227K ˛(K)7 /3:99:

(22)

Using the Beurling–Ahlfors extension [4], we know that if f is quasisymmetric x . Therefore, the Douady–Earle then f admits a quasiconformal extension to D extension Φ of f is quasiconformal if f is quasisymmetric. Recently, a new method was developed in [20] to derive the quasiconformality of Φ directly from the quasisymmetry of f on S1 . In fact, the method of [20] provides an upper bound for the maximal dilatation K(Φ) of Φ on D in terms of the cross-ratio distortion norm kf kcr of f on S1 . In this subsection, we give a summary of the main results obtained in [20]. Given a quadruple Q = ¹a; b; c; d º consisting of four points a; b; c; d on the unit circle S1 arranged in counterclockwise order, we denote by cr(Q) the following cross ratio of Q (b a)(d c) cr(Q) = : (23) (c b)(d a)

7 Generalized conformal barycentric extensions

231

It is easy to check that a quadruple Q has cr(Q) = 1 if and only if the geodesic ac from a to c is perpendicular to the geodesic bd from b to d . Given an orientation-preserving homeomorphism f of S1 , we define the cross-ratio distortion norm of f as kf kcr = sup j ln cr(f (Q))j; (24) cr(Q)=1

where cr(f (Q)) =

(f (b) (f (c)

f (a))(f (d ) f (b))(f (d )

f (c)) : f (a))

(25)

We say that f is quasisymmetric if kf kcr is finite. This definition is equivalent to the one requiring bounded ratio distortion under f for all symmetric triples on the unit circle S1 . Let f be a quasisymmetric circle homeomorphism and F be the Beurling– x . By a result of Lehtinen in [27], Ahlfors extension of f to the closed unit disk D the maximal dilatation K(f ) of F is bounded from above by 2e kf kcr . (For background on quasisymmetric and quasiconformal homeomorphisms, we refer to [3] or [28], and for similar applications of the cross-ratio distortion norm in the quantitative study of earthquake maps we refer to [16], [17] and [18].) By combining this inequality with the previous theorem, we obtain the following upper bound for K(Φ(f )) in terms of kf kcr . Theorem 3.6 (Douady, Earle, and Lehtinen). There exist two universal positive constants C1 and C2 such that for any quasisymmetric circle homeomorphism f , the maximal dilatation K(Φ(f )) of Φ(f ) satisfies ln K(Φ(f ))  C1 e kf kcr + C2 :

(26)

By developing a new method to estimate the maximal dilatation K(Φ(f )) directly from the cross-ratio distortion norm of f , it is proved in [20] that ln K(Φ(f )) has an upper bound depending on kf kcr linearly rather than exponentially. Theorem 3.7 ([20]). There exist two universal positive constants C1 and C2 such that for any quasisymmetric circle homeomorphism f , the maximal dilatation K(Φ(f )) of the Douady–Earle extension Φ(f ) of f satisfies ln K(Φ(f ))  C1 kf kcr + C2 :

(27)

Note that the method of [20] also provides a new and direct proof of the result that Φ(f ) is quasiconformal if f is quasisymmetric. When the cross-ratio distortion norm kf kcr is small enough, the constant C2 in the inequality (27) can be dropped. More precisely, the following is shown in Corollary 2 of [7].

232

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Theorem 3.8 ([7]). For any  > 0 , there is ı > 0 such that, for any quasisymmetric homeomorphism f , if f admits a K -quasiconformal extension with K  1 + ı , then K(Φ(f ))  K 3+ : (28) It is proved by Beurling and Ahlfors ([4]) that K(f )  (e kf kcr )2 = e 2kf kcr :

Therefore, the following theorem follows. Theorem 3.9 (Ahlfors, Beurling, Douady, and Earle). There exist two positive constants C and ı such that, for any quasisymmetric circle homeomorphism f , if kf kcr  ı , we have ln K(Φ(f ))  C kf kcr : Then theorems 3.7 and 3.9 together imply the following result. Theorem 3.10 ([20]). There exists a universal constant C > 0 such that, for any quasisymmetric circle homeomorphism f , ln K(Φ(f ))  C kf kcr :

(29)

In the remaining part of this subsection, we summarize the relationship between the maximal dilatations of Φ(f ) and an extremal extension of f . Let f be a quasisymmetric homeomorphism of S1 and let Ke (f ) be the infimum of the maximal dilatations of the quasiconformal extensions of f . Clearly, Ke (f )  1 . Theorem 3.5 can be briefly expressed as follows. Corollary 3.11. There exist two positive constants C1 and C2 such that, for any quasisymmetric circle homeomorphism f , ln K(Φ(f ))  C1 Ke (f ) + C2  (C1 + C2 )Ke (f ): On the other hand, Theorem 3.10 shows that ln K(Φ(f )) has a similar upper bound in terms of kf kcr . Therefore, it is a natural problem to compare these two upper bounds. In other words, one may ask: what is the quantitative relationship between Ke (f ) and kf kcr ? This relationship is investigated in [20] by exploring these two quantities for a one-parameter family of quasisymmetric homeomorphisms defined by the earthquake maps along one leaf. Let f be a left earthquake map with only one leaf at the geodesic connecting 1 to 1 and the amount of shearing to the left on the upper half disk equal to 2 and the same amount of shearing to the left on the lower half disk, where  > 0 and f0 is the identity map. We call f a simple left earthquake. Clearly, each f is symmetric with respect to the origin. Thus, Φ(f )(0) = 0 .

7 Generalized conformal barycentric extensions

Theorem 3.12 ([20]).

233

We have ln K(Φ(f ))(0) 1  :  4 !1 lim

(30)

Now note that kf kcr =  . Then Theorems 3.10 and 3.12 imply that ln K(Φ(f )) grows to 1 at the same rate as kf kcr when  ! 1 . It follows from Strebel’s work in [39] and [40] that f has a unique extremal extension Fe with a constant maximal dilatation at all points, that can be explicitly expressed in terms of  as p 2 + 4 2 2 + 4 K(Fe ) = 1 + = O(2 ); 2 2 that is, Ke (f ) = O(kf k2cr ) when  ! 1: This result implies the following result. Theorem 3.13 ([20]). For the map f , when  is sufficiently large, the upper bound in Theorem 3.10 on K(Φ(f )) is substantially smaller than the upper bound in Corollary 3.11. Theorems 3.12 and 3.13 together imply the following theorem. Theorem 3.14 ([20]). The extension Φ(f ) stays exponentially far away from being extremal as  ! 1 . Finally, by ignoring the exact values of the constants C1 and C2 in Corollary 3.11 and Theorem 3.10, it is shown in [20] that the upper bound on ln K(Φ(f )) in Theorem 3.10 implies the upper bound in Corollary 3.11 asymptotically by proving the following. Theorem 3.15 ([20]). Let f be a quasisymmetric homeomorphism and Ke (f ) be the infimum of the maximal dilatations of quasiconformal extensions of f . Then kf kcr lim  : kf kcr !1 Ke (f )

3.3. Extensions of locally quasisymmetric homeomorphisms The method of [20] is generalized in [21] to show that the quasiconformality of Φ(f ) and the upper bound for ln K(Φ(f )) obtained in [20] are local properties. It emphasizes that the local conformality can be obtained before concluding the global quasiconfromality of Φ(f ) when f is a global quasisymmetric homeomorphism of S1 . Furthermore, it is shown in [21] that the asymptotic conformality of Φ(f ) also depends locally on the symmetry of f .

234

Jun Hu

Quasisymmetric homeomorphisms of S1 are defined in the previous subsection. Similarly, a homeomorphism f of S1 is said to be quasisymmetric on an open arc I on S1 if kf jI kcr is finite by considering all quadruples Q  I with cr(Q) = 1 . A homeomorphism f of S1 is said to be symmetric if ln cr(f (Q)) converges to 0 uniformly as s(Q) approaches to 0 for all Q with cr(Q) = 1 , where s(Q) is defined to be the minimum of jb aj; jc bj; jd cj and ja d j . Similarly, a homeomorphism f of S1 is said to be symmetric on an open arc I on S1 if the previous convergence holds uniformly for all quadruple Q  I . Note that a symmetric homeomorphism f of S1 is quasisymmetric and that a local symmetric homeomorphism is a local quasisymmetric homeomorphism. Now we give the precise statements of the results proved in [21]. Theorem 3.16 ([21]). Let f be an orientation-preserving homeomorphism of S1 , p 2 S1 , and Ip an open arc on S1 containing p and symmetric with respect to p . If kf jIp kcr < 1 , then there exists an open hyperbolic half plane Up with Uxp \ S1 contained in Ip and symmetric with respect to p such that ln K(Φ(f )jUp )  C1 kf jIp kcr + C2 for two universal positive constants C1 and C2 , where K(Φ(f )jUp ) is the maximal dilatation of Φ(f ) on Up . Corollary 3.17 (Douady and Earle [7]). If f is a quasisymmetric homeomorphism of S1 , then Φ(f ) is quasiconformal. Theorem 3.18 ([21]). With the same notation introduced in the statement of Theorem 3.16, if f is symmetric on Ip , then Φ(f ) is asymptotically conformal on a neighborhood Vp of p in D , with Vxp \ S1  int(Ip ) . Corollary 3.19 (Earle, Markovic, and Sǎrić [11]). If f is a symmetric homeomorphism of S1 , then Φ(f ) is asymptotically conformal on D .

3.4. Extensions of diffeomorphisms In this subsection, we introduce a theorem, obtained in [25], on the differentiability of Φ(f ) on the boundary S1 if f is a diffeomorphism of S1 . Theorem 3.20 (differentiability at a boundary point, [25]). Let f be a homeomorphism of S1 and p 2 S1 . Assume that f is C 1 in a neighborhood of p on S1 and f 0 (p) ¤ 0 . If f is orientation-preserving, then the difference quotient Φ(f )(z) Φ(f )(p) z p @ converges to f 0 (p) as a point z of D approaches p , and furthermore @z Φ(f )(z) @ 0 and @¯z Φ(f )(z) converge to f (p) and 0 respectively as z approaches p .

7 Generalized conformal barycentric extensions

235

If f is orientation-reserving, then the difference quotient Φ(f )(z) Φ(f )(p) z¯ p¯ @ converges to f 0 (p) as a point z of D approaches p , and furthermore @z Φ(f )(z) @ 0 and @¯z Φ(f )(z) converge to 0 and f (p) respectively as z approaches p .

For each point z on the complex plane, let z  be the mirror image of z with respect to S1 ; that is, z  = 1/¯z . Then one can extend Φ(f ) to a map Φ(f ) b by letting Φ(f ) (z) = [Φ(f )(z  )] for each z defined on the Riemann sphere C x . We may call Φ(f ) the extended conformal barycentric outside the unit disk D extension of f . Then Theorem 3.20 implies the following theorem immediately.

1

1

1

Theorem 3.21 (local/global C 1 diffeomorphism, [25]). 1. Let f be a homeomorphism of S1 and p 2 S1 . If f is C 1 in a neighborhood of p on S1 and b. f 0 (p) ¤ 0 , then Φ(f ) is C 1 in a neighborhood of p on C

1

1

2. If f is a C 1 diffeomorphism of S1 , then Φ(f ) is a C 1 diffeomorphism b of C .

1

If f is only differentiable at a point p 2 S1 , then the differentiability of Φ(f ) at the point p may not hold, but the angular derivative of Φ(f ) (and hence of Φ(f ) ) at p exists in the following sense.

1

Theorem 3.22 (existence of angular derivative, [9]). Let f be an orientationpreserving homeomorphism of S1 . If f is differentiable at a point p 2 S1 and )(p) f 0 (p) ¤ 0 , then the difference quotient Φ(f )(z)z Φ(f converges to f 0 (p) as p @ a point z of D approaches p non-tangentially to S1 . Furthermore, @z Φ(f )(z) @ 0 and @¯z Φ(f )(z) converge to f (p) and 0 respectively as z approaches p nontangentially.

We finish this survey with the following remark. Remark 3.1. Earle proved in [8] that up to multiplication by a constant, there exists a unique operator to extend continuous vector fields V on S1 to continuous x in a conformally natural way. Interestingly, an operator vector fields L0 (V ) of D x studied in [35] is in fact a conformally natural extension extending V from S1 to D operator [15]. Furthermore, a quantitative relationship between the cross-ratio ¯ 0 (V ) is developed in [15] lately. distortion norm of V and the L1 norm of @L Acknowledgments. The author wishes to thank the editor for his invitation to write this survey, and his careful proof-reading. The author is partially supported by a CUNY fellowship leave grant in Spring 2018.

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Chapter 8

Higgs bundles and higher Teichmüller spaces Oscar García-Prada

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Higgs bundles . . . . . . . . . . . . . . . . . . . . . . . . . 3 A basic example: G = SL(2; R) . . . . . . . . . . . . . . . 4 The Hitchin map, split real forms and Hitchin components 5 Hermitian groups and maximal Toledo components . . . . . 6 SO(p; q) -Higgs bundles and higher Teichmüller spaces . . 7 Positivity and Cayley correspondence . . . . . . . . . . . . 8 Appendix: tables . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Given a compact surface S of genus g  2 , consider the moduli space of representations of the fundamental group of S in SL(2; R) . Attached to such a representation there is an integer invariant d , which by Milnor [57] satisfies the bound jd j  g 1 . In the maximal case d = g 1 (or d = 1 g ) the moduli space has 22g connected components, all of which are homeomorphic to R6g 6 and can be identified to the Teichmüller space of S (see [37]). All the representations in these components are Fuchsian, that is, discrete and faithful. When passing to the adjoint group PSL(2; R) all these components get identified. Higher Teichmüller spaces appear when one replaces SL(2; R) by certain real Lie groups of higher rank. These spaces have been studied from several points of view. In this paper we will focus on the role played by the theory of Higgs bundles. This powerful theory, introduced by Hitchin [46], is very useful in identifying these spaces and studying their topology, in particular counting the number of connected components. To do this, one chooses a complex structure on S , making it into a compact Riemann surface X . Given a semisimple real

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Lie group G , with a choice of a maximal compact subgroup H  G , one has a Cartan decomposition of the Lie algebra of G , given by g = h ˚ m , where h is the Lie algebra of H and m is the orthogonal complement of h with respect to the Killing form of g . A G -Higgs bundle over X is a pair (E; ') consisting of a holomorphic H C -bundle E over X and a holomorphic section ' of the bundle E(mC ) ˝ K , where E(mC ) is the vector bundle associated to E via the isotropy representation H C ! GL(mC ) , and K is the canonical line bundle of X —the holomorphic cotangent bundle of X . The non-abelian Hodge correspondence proved by Hitchin [46], Donaldson [24], Simpson [63] and [65], Corlette [21], and others (see [28]) establishes a homeomorphism between the moduli space M(G) of polystable G -Higgs bundles over X and the moduli space R(G) of reductive representations of the fundamental group of S in G . In [48] Hitchin constructed components of the moduli space of representations R(G) when G is the split real form of a complex simple Lie group G C which, like Teichmüller space, are homeomorphic to a real vector space, in this case of dimension dim G(2g 2) . Originally referred by him as generalized Teichmüller components, they are now called Hitchin components. Such components appear as sections of the Hitchin fibration for the moduli space of G C -Higgs bundles. This fibration is defined by the map hW M(G C ) ! B(G C ) sending a G C -Higgs bundle (E; ') to (p1 ('); : : : ; pr (')) , where p1 ; : : : ; pr is a basis L of the ring of G C -invariant homogeneous polynomials of gC . Hence B(G C ) = ri =1 H 0 (X; K di ) , where di is the degree of pi . These degrees are indepedent of the basis and depend only on the group. The Hitchin components can then be identified with the vector space B(G C ) which has half the dimension of M(G C ) . An alternative (equivalent) way of definining the Hitchin components is by directly considering the Hitchin map for the Lie group G . In fact this exists for any semisimple Lie group G . The Hitchin map in this case hW M(G) ! B(G) is obtained by applying a basis of the ring of H C -invariant polynomials of mC . One can construct in this general situation a section of h , but only when G is split does this define a component of M(G) —when G is split B(G) = B(G C ) (see [33]). The Hitchin components consist entirely of representations of 1 (S) in G that can be deformed to a representation that factors through a Fuchsian representation 1 (S ) ! SL(2; R) , where SL(2; R) ! G is the irreducible representation of SL(2; R) in G , which always exists when G is a split real form. One may need to consider PSL(2; R) if the group G is of adjoint type. In this case, like for PSL(2; R) , the Hitchin component is unique. It follows from the work of Labourie [55] and Fock and Goncharov [27] that, similarly to the Teichmüller component for PSL(2; R) , all the representations in a Hitchin component are discrete and faithful. In his paper [48] Hitchin already posed the question on the relation of his generalized Teichmüller components with geometric structures. A geometric interpretation of these components for G = SL(3; R) has been given by Choi and Goldman [17] and more recently for more general split groups by Labourie [55] Fock and Goncharov [27], and Guichard and Wienhard [40].

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The group SL(2; R) is of course a split real form of SL(2; C) but it is also a group of Hermitian type. For such a group G , the symmetric space G/H is Kähler. The symmetric space SL(2; R)/ SO(2) can be identified with the Poincaré upper-half plane. Another direction in which Teichmüller space can be generalized is by considering a higher rank non-compact simple Lie group G of Hermitian type with finite centre. The Lie algebra of a group of this type is one of the following: su(p; q) , so (2n) , sp(2n; R) , so(2; n) , e6 14 and e7 25 (we are using Helgason’s notation [44]). In this situation the centre z of h is isomorphic to R , and the adjoint action of a special element J 2 z defines an almost complex structure on m = To (G/H ) , where o 2 G/H corresponds to the coset H , making the symmetric space G/H into a Kähler manifold. The almost complex structure ad(J ) gives a decomposition mC = m+ + m in ˙i -eigenspaces, which is H C -invariant. An immediate consequence of this decomposition for a G -Higgs bundle (E; ') is that it gives a bundle decomposition E(mC ) = E(m+ ) ˚ E(m ) and hence the Higgs field decomposes as ' = (' + ; ' ) , where ' + 2 H 0 (X; E(m+ ) ˝ K) and ' 2 H 0 (X; E(m ) ˝ K) . Another important ingredient of G -Higgs bundles for this class of groups is the Toledo invariant. This is a topological invariant attached to a G -Higgs bundle (E; ') , in fact attached to E . It is defined by considering a special character T of hC called the Toledo character. If this lifts to a character ˜T of H C , we consider the associated line bundle E(˜T ) , and define the Toledo invariant  of (E; ') as  =  (E) := deg(E(˜T )):

Otherwise one can show that there is a rational number qT such qT T lifts to a character ˜T , and one can define  =  (E) := q1 deg(E(˜T )) . A crucial fact T is given by the Milnor–Wood inequality: the moduli space M(G) is empty unless j j  rk(G/H )(2g

2):

This was proved on a case by case basis for the classical groups (see [46], [38], [7], [8], [9], [11], and [29]) and in general in [5]. The Toledo invariant of (E; ') coincides with the classical Toledo invariant of a representation of the fundamental group in G , which is defined by integrating over X the pull-back of the Kähler form of the Bergman metric of G/H , via a  -equivariant map f W Xz ! G/H determined by  . The map f can be taken to be harmonic if the representation is reductive, and hence corresponding to a polystable G -Higgs bundle. In the context of representations the inequality j j  rk(G/H )(2g 2) , goes back to Milnor [57], who studies the case G = SL(2; R) , as mentioned above, and was proved in various cases in [70], [25], [22], and [18], and in general by Burger, Iozzi, and Wienhard [14]. The higher Teichmüller spaces appear when the Toledo invariant is maximal, i.e., when j j = rk(G/H )(2g 2) , and the symmetric space G/H is of tube type. A geometric characterization of the tube type condition is given in terms of the Shilov boundary of the Harish-Chandra bounded symmetric domain realization of G/H .

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We say that G/H is of tube type if the Shilov boundary of the corresponding bounded symmetric domain is a symmetric space H /H 0 (of compact type). If this is the case, this domain is biholomorphic to a ‘tube’ over the symmetric cone Ω := H  /H 0 —the non-compact symmetric space dual of the Shilov boundary. The simplest example of this is given by G = SL(2; R) Š SU(1; 1) , where the bounded symmetric domain is the Poincaré disc and the tube realization is the Poincaré upper half plane (these are of course related by the classical Cayley transform). Sometimes we will say that the group G is of tube type if the symmetric space G/H is of tube type. The groups of tube type have Lie algebra given by su(p; q) with p = q , so (2n) with n even, sp(2n; R) , so(2; n) and e7 25 . While those of non-tube type are su(p; q) with p ¤ q , so (2n) with n odd, and e6 14 . The subvariety Rmax (G)  R(G) of maximal representations (and corresponding subvariety Mmax (G)  M(G) ) in the tube case has special significance since, as proved in [15] and [13], it consists entirely of discrete and faithful representations, like the Hitchin components in the split case. As in this case, all the representations in Rmax (G) are Anosov in the sense of Labourie [55] and [13], and are related to geometric structures of various kinds [41]. From the Higgs bundle point of view the tube-type condition plays a fundamental role in constructing a bijective correspondence—called Cayley correspondence—between the moduli space of maximal Higgs bundles Mmax (G) and the moduli space of K 2 -twisted H  -Higgs bundles over X , where H   H C is the non-compact dual of H , described above. These are defined like Higgs bundles except that the twisting is by K 2 instead of K . An immediate consequence of this correspondence is the existence of new topological invariants for maximal Higgs bundles which are hidden a priori, and which play a crucial role in determining the number of connected components of Mmax (G) . Up to a finite cover, there is only one group which is both split and of Hermitian type, and in fact of tube type. This is the symplectic group Sp(2n; R) . The tube realization is the Siegel upper half space. The Hitchin representations are actually maximal but, except for the case n = 1 , there are maximal components that are not Hitchin components (see [38], [7], and [29]). Maximal Higgs bundles, and hence maximal representations, in the non-tube case present also very interesting phenomena. It turns out that in this case the dimension of the moduli space of maximal G -Higgs bundles is smaller than expected, since up to a compact factor every maximal representation reduces to a representation in a maximal subgroup of tube type. This rigidity phenomenon is very rare in the context of surface groups (see [36], [68], [45], [8], [14], [9], [11], and [5]), and it is conjectured to happen only for Hermitian groups of non-tube type [50]. For a while, split real groups and Hermitian groups were the only cases providing examples of higher Teichmüller spaces. However, gradually there was evidence that there may be other groups. Groups under scrutiny were the special orthogonal groups of signature (p; q) . Of course, for certain values of p and q the

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group is split or Hermitian, but for most values is neither. One such evidence came from the work of Guichard and Wienhard [42]. They defined a notion of positivity for certain Lie groups and for surface group representations into those groups. The notion of positivity for a representation, which refines Labourie’s Anosov condition, is open and they conjecture that is also closed (see [39] and [42]), hence detecting connected components of the moduli space of representations. They showed that apart from the split real forms and the real forms of Hermitian type, the only other non-exceptional groups which allow positive representations are the groups locally isomorphic to SO(p; q) for 1 < p < q . Indeed, Collier [19] has proved existence of special components containing positive representations for SO(n; n + 1) . Although this group is split, the Collier components are different from the Hitchin ones. More recently the existence of special components for SO(p; q) with 1 < p < q containing positive representations has been shown in [1] and [2]. After describing the basic ingredients in the theory of Higgs bundles and the three different situations where higher Teichmüller spaces emerge (split real forms, real forms of Hermitian type and SO(p; q) ), we finish the paper with some comments on the conjectural general picture. This seems to relate positive representations, and hence higher Teichmüller spaces, to components of the moduli space for which there exists a Cayley correspondence that generalizes the one for Hermitian groups of tube type and the one for the special components in the moduli space of SO(p; q) -Higgs bundles. Indeed there is also a Cayley correspondence for the Hitchin components. This seems to be the way in which positivity is manifesting from the point of view of Higgs bundle theory. A full study of this is carried out in [6], including the exceptional groups for which there is a notion of positivity. As shown in [42] these are real forms of F4 , E6 , E7 , and E8 whose restricted root system is of type F4 . As mentioned above, we have focused here on the point of view provided by the theory of Higgs bundles, and offering just a very partial picture. For other approaches, and additional references, one may see the very nice survey papers [16] and [69]. We have not touched at all, the case of non-compact surfaces for which there is also a very rich higher Teichmüller theory. This situation is studied for example in [27] and [15]. The Higgs bundle approach for the case of surfaces with punctures is very little developped. This involves the theory of parabolic Higgs bundles. In [4] a non-abelian Hodge correspondence is established between parabolic G -Higgs bundles and representations of the fundamental group of the punctured surface with fixed conjugacy classes around the puntures, where G is any real form of a complex reductive Lie group. This generalizes the work of Simpson [64] when G = GL(n; C) . Although the split and the Hermitian cases are briefly described in [4], much work has to be done along the lines of the compact case to study the higher Teichmüller spaces.

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2. Higgs bundles 2.1. Basic definitions Following Knapp (see [51], VII.2), we define a real reductive group is a 4 -tuple (G; H; ; B) where 1. G is a real Lie group with reductive Lie algebra g ; 2. H < G is a maximal compact subgroup; 3.  is a Lie algebra involution of g inducing an eigenspace decomposition g=h˚m

where h = Lie(H ) is the (+1) -eigenspace for the action of  , and m is the ( 1) -eigenspace; 4. B is a  - and Ad(G) -invariant non-degenerate bilinear form, with respect to which h ?B m and B is negative definite on h and positive definite on m; 5. the multiplication map H  exp(m) ! G is a diffeomorphism; 6. G acts by inner automorphisms on the complexification gC of its Lie algebra via the adjoint representation. In this section we fix a real reductive Lie group (G; H; ; B) . We have [m; m]  h , [m; h]  h . Complexifying the isotropy representation H ! Aut(m) , we obtain the representation AdW H C ! Aut(mC ) . When G is semisimple we take B to be the Killing form. In this case B and a choice of a maximal compact subgroup H determine a Cartan decomposition. Let X be a compact Riemann surface of genus g . A G -Higgs bundle on X consists of a holomorphic principal H C -bundle E and a holomorphic section ' 2 H 0 (X; E(mC ) ˝ K) , where E(mC ) is the associated vector bundle with fibre mC via the complexified isotropy representation, and K is the canonical line bundle of X . If G is compact, H = G and m = 0 . A G -Higgs bundle is hence simply a holomorphic principal G C -bundle. If G = H C , where H is a maximal compact subgroup of G , m = i h , and a G -Higgs bundle is a principal G -bundle together with a section ' 2 H 0 (X; E(g) ˝ K) , where E(g) is the adjoint bundle. This is the original definition for complex Lie groups given by Hitchin in [47]. Let G 0 be a reductive subgroup of G . A maximal compact subgroup of G 0 is given by H 0 = H \ G 0 and we can take a compatible Cartan decomposition, in the sense that h0  h and m0  m . Moreover, the isotropy representation of H 0 is the restriction of the isotropy representation Ad of H . We say that the structure group of a G -Higgs bundle (E; ') reduces to G 0 when there is a reduction of the structure group of the underlying H C -bundle to H 0 C , given by a subbundle E , and the Higgs field ' 2 H 0 (X; E(mC ) ˝ K) belongs to H 0 (X; E (m0 C ) ˝ K) .

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2.2. Stability of G -Higgs bundles There is a notion of stability for G -Higgs bundles (for details see [28]). To explain this we consider the parabolic subgroups of H C defined for s 2 i h as Ps = ¹g 2 H C W e t s ge

ts

is bounded as t ! 1º:

C

When H is connected every parabolic subgroup is conjugate to one of the form Ps for some s 2 i h , but this is not the case necessarily when H C is non-connected. A Levi subgroup of Ps is given by Ls = ¹g 2 H C W Ad(g)(s) = sº; and any Levi subgroup is given by pLs p 1 for p 2 Ps . Their Lie algebras are given by ps = ¹Y 2 hC W Ad(e t s )Y is bounded as t ! 1º; ls = ¹Y 2 hC W ad(Y )(s) = [Y; s] = 0º:

We consider the subspaces ms = ¹Y 2 mC W Ad(e t s )Y is bounded as t ! 1º; m0s = ¹Y 2 mC W Ad(e t s )Y = Y for every t º:

One has that ms is invariant under the action of Ps and m0s is invariant under the action of Ls . Remark 2.1. The subspace ms is the non-compact part of the parabolic subalgebra of gC defined by s 2 ih . Define p˜s = ¹Y 2 gC j Ad(e t s )Y is bounded as t ! 1º: We have that ps = p˜s \ hC and ms = p˜s \ mC . Analogously, define ˜ls = ¹Y 2 gC j Ad(e t s )Y = Y for every t º: Then ls = ˜ls \ hC and m0s = ˜ls \ mC . An element s 2 i h defines a character s of ps since hs; [ps ; ps ]i = 0 . Conversely, by the isomorphism (ps /[ps ; ps ]) Š zs , where zs is the centre of the Levi subalgebra ls , a character  of ps is given by an element in zs , which gives, via the invariant metric, an element s 2 zs  ih . When ps  ps , we say that  is an antidominant character of ps . When ps = ps we say that  is a strictly antidominant character. Note that for s 2 ih , s is a strictly antidominant character of ps . Let now (E; ') be a G -Higgs bundle over X , and let s 2 ih . Let Ps be defined as above. For  2 Γ(E(H C /Ps )) a reduction of the structure group of E from H C to Ps , we define the degree relative to  and s , or equivalently to  and s , as follows. When a real multiple s of the character exponentiates to a character ˜s of Ps , we compute the degree as 1 deg(E)(; s) = deg(E (˜s )):  This condition is not always satisfied, but one shows (see [28], §4.6) that the antidominant character can be expressed as a linear combination of characters of

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the centre and fundamental weights, and there exists an integer multiple m of the characters of the centre and the fundamental weights exponentiating to the group, so we can define the degree. An alternative definition of the degree can be given in terms of the curvature of connections using Chern–Weil theory. This definition is more natural when considering gauge-theoretic equations as we do below. For this, define Hs = H \ Ls and hs = h \ ls . Then Hs is a maximal compact subgroup of Ls , so the inclusions Hs  Ls is a homotopy equivalence. Since the inclusion Ls  Ps is also a homotopy equivalence, given a reduction  of the structure group of E to Ps one can further restrict the structure group of E to Hs in a unique way up to homotopy. Denote by E0 the resulting Hs principal bundle. Consider now a connection A on E0 and let FA 2 Ω2 (X; E0 (hs )) be its curvature. Then s (FA ) is a 2 -form on X with values in i R , and Z i deg(E)(; s) := s (FA ): 2 X

We define the subalgebra had as follows. Consider the decomposition h = z + [h; h] , where z is the centre of h , and the isotropy representation ad = adW h ! End(m) . Let z0 = ker(adjz ) and take z00 such that z = z0 + z00 . Define the subalgebra had := z00 + [h; h] . The subindex ad denotes that we have taken away the part of the centre z acting trivially via the isotropy representation ad. Remark 2.2. For groups of Hermitian type, z0 = 0 since an element both in z and ker(ad) belongs to the centre of g , which is zero, as g is semisimple. Hence had = h . With Ls ms and m0s defined as above. We have the following. Definition 2.1.

Let ˛ 2 iz  zC . We say that a G -Higgs bundle (E; ') is

• ˛ -semistable if for any s 2 ih and any holomorphic reduction  2 Γ(E(H C /Ps )) such that ' 2 H 0 (X; E (ms ) ˝ K) , we have that deg(E)(; s) s (˛)  0 ; • ˛ -stable if for any s 2 i had and any holomorphic reduction  2 Γ(E(H C /Ps )) such that ' 2 H 0 (X; E (ms ) ˝ K) , we have that deg(E)(; s) s (˛) > 0 ; • ˛ -polystable if it is ˛ -semistable and for any s 2 ihad and any holomorphic reduction  2 Γ(E(H C /Ps )) such that ' 2 H 0 (X; E (ms ) ˝ K) and deg(E)(; s) s (˛) = 0 , there is a holomorphic reduction of the structure group L 2 Γ(E (Ps /Ls )) to a Levi subgroup Ls such that ' 2 H 0 (X; EL (m0s ) ˝ K)  H 0 (X; E (ms ) ˝ K) .

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Remark 2.3. We may define a real group GLs = (Ls \ H ) exp(m0s \ m) with maximal compact subgroup the compact real form Ls \ H of the complex group Ls and m0s \ m as isotropy representation. Thus, an ˛ -polystable G -Higgs bundle reduces to an ˛ -polystable GLs -Higgs bundle since ' belongs H 0 (X; EL (m0s )˝K) . Remark 2.4. We can replace K in the definition of G -Higgs bundle by any holomorphic line bundle L on X . More precisely, an L -twisted G -Higgs bundle (E; ') consists of a principal H C -bundle E , and a holomorphic section ' 2 H 0 (X; E(mC ) ˝ L) . We reserve the name G -Higgs bundle for the K -twisted case. The stability criteria are as in Definition 3.5, replacing K by L . Remark 2.5. For G semisimple, the notion of ˛ -stability with ˛ ¤ 0 only makes sense for groups of Hermitian type, since ˛ belongs to the centre of h , which is not zero if and only if the centre of a maximal compact subgroup H is non-discrete, i.e., if G is of Hermitian type. For reductive groups which are not of Hermitian type, ˛ -stability makes sense, but there is only one value of ˛ for which the condition is not void. This value is fixed by the topology of the principal bundle (see [31] for details). Remark 2.6. When H C is connected, as mentioned above, every parabolic subgroup of H C is conjugate to one of the form Ps for s 2 i h . In this situation, we can formulate the stability conditions in Definition 2.1 in terms of any parabolic subgroup P  H C , replacing s by s , for any antidominant character  of the Lie algebra of P . Two G -Higgs bundles (E; ') and (E 0 ; ' 0 ) are isomorphic if there is an isomorphism f W E ! E 0 such that ' 0 = f  ' , where f  is the map E(mC ) ˝ K ! E 0 (mC ) ˝ K induced by f . The moduli space of ˛ -polystable G -Higgs bundles M˛ (G) is defined as the set of isomorphism classes of ˛ -polystable G -Higgs bundles on X . When ˛ = 0 we simply denote M(G) := M0 (G) . Remark 2.7. Similarly, we can define the moduli space of ˛ -polystable L -twisted ˛ G -Higgs bundles which will be denoted by ML (G) . These moduli spaces have the structure of a complex analytic variety, as one can see by the standard slice method, which gives local models via the Kuranishi map (see, e.g., [52]). When G is algebraic and under fairly general conditions, the moduli spaces M˛ (G) can be constructed by geometric invariant theory and hence are complex algebraic varieties. The work of Schmitt [62] deals with the construction of the moduli space of L -twisted G -Higgs bundles for G a real reductive Lie group. This construction generalizes the constructions of the moduli space of G -Higgs bundles done by Ramanathan [60] when G is compact ( G -Higgs bundles in this case are simply G C -principal bundles), and by Simpson (see [66] and [67]) when G is a complex reductive algebraic (see also [58] for G = GL(n; C) ).

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Remark 2.8. If G = GL(n; C) , we recover the original notions of Higgs bundle and stability introduced by Hitchin [46]. A G = GL(n; C) -Higgs bundle is given by a holomorphic vector bundle V := E(Cn ) —associated to a principal GL(n; C) -bundle E via the standard representation—and a homomorphism ΦW V

! V ˝ K:

The Higgs bundle (V; Φ) is stable if deg V 0 deg V < 0 rank V rank V for every proper subbundle V 0  V such that Φ(V 0 )  V 0 ˝ K . The Higgs bundle L (V; Φ) is polystable if (V; Φ) = i (Vi ; Φi ) where (Vi ; Φi ) is a stable Higgs bundle and deg Vi /rank Vi = deg V /rank V . If we consider G = SL(n; C) we have to add that det(V ) Š O (hence deg V = 0 ) and Tr(Φ) = 0 . The notion of stability emerges from the study of the Hitchin equations. The equivalence between the existence of solutions to these equations and the ˛ -polystability of Higgs bundles is known as the Hitchin–Kobayashi correspondence, which we state below. Theorem 2.1. Let (E; ') be a G -Higgs bundle over a Riemann surface X with volume form ! . Then (E; ') is ˛ -polystable if and only if there exists a reduction h of the structure group of E from HC to H , that is a smooth section of E(H C /H ) , such that Fh ['; h (')] = i˛! 1;0 C 0;1 where h W Ω (E(m )) ! Ω (E(mC )) is the combination of the anti-holomorphic involution in E(mC ) defined by the compact real form at each point determined by h and the conjugation of 1 -forms, and Fh is the curvature of the unique H -connection compatible with the holomorphic structure of E (the Chern connection). This theorem was proved by Hitchin in the case of SL(2; C) , by Simpson when G is complex, and in [12] and [28] for a general reductive real Lie group G Remark 2.9. There is a theorem similar to Theorem 2.1 for L -twisted G -Higgs bundles (see Remark 2.4 and [28]) for an arbitrary line bundle L . If (E; ') is such a pair, one fixes a Hermitian metric hL on L , and looks for a reduction of structure group h of E from H C to H satisfying Fh 0

C

['; h (')]! = 0

C

i˛!;

where now h W Ω (E(m ) ˝ L) ! Ω (E(m ) ˝ L) is the combination of the antiholomorphic involution in E(mC ) defined by the compact real form at each point determined by h and the metric hL .

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2.3. Higgs bundles and representations Fix a base point x 2 X . A representation of 1 (X; x) in G is a homomorphism 1 (X; x) ! G . After fixing a presentation of 1 (X; x) , the set of all such homomorphisms, Hom(1 (X; x); G) , can be identified with the subset of G 2g consisting of 2g -tuples (A1 ; B1 ; : : : ; Ag ; Bg ) satisfying the algebraic equation Qg [A ; B i i ] = 1 . This shows that Hom(1 (X; x); G) is an algebraic variety. i =1 The group G acts on Hom(1 (X; x); G) by conjugation: (g
)( ) = g( )g

1

;

where g 2 G ,  2 Hom(1 (X; x); G) and 2 1 (X; x) . If we restrict the action to the subspace Hom+ (1 (X; x); G) consisting of reductive representations, the orbit space is Hausdorff. We recall that a reductive representation is one whose composition with the adjoint representation in g decomposes as a direct sum of irreducible representations. This is equivalent to the condition that the Zariski closure of the image of 1 (X; x) in G is a reductive group. Define the moduli space of representations of 1 (X; x) in G to be the orbit space R(G) = Hom+ (1 (X; x); G)/G:

This is a real algebraic variety. For another point x 0 2 X , the fundamental groups 1 (X; x) and 1 (X; x 0 ) are identified by an isomorphism unique up to an inner automorphism. Consequently, R(G) is independent of the choice of the base point x . Given a representation W 1 (X; x) ! G , there is an associated flat principal G -bundle on X , defined as E = Xz  G; where Xz ! X is the universal cover and 1 (X; x) acts on G via  . This gives in fact an identification between the set of equivalence classes of representations Hom(1 (X ); G)/G and the set of equivalence classes of flat principal G -bundles, which in turn is parametrized by the (non-abelian) cohomology set H 1 (X; G) . We have the following. Theorem 2.2. Let G be a semisimple real Lie group. Then there is a homeomorphism R(G) Š M(G) . The proof of Theorem 2.2 is the combination of Theorem 2.1 and the following theorem of Corlette [21], also proved by Donaldson [24] when G = SL(2; C) . Theorem 2.3. Let  be a representation of 1 (X ) in G with corresponding flat G -bundle E . Let E (G/H ) be the associated G/H -bundle. Then the existence of a harmonic section of E (G/H ) is equivalent to the reductiveness of  . Remark 2.10. Theorem 2.2 can be extended to reductive groups replacing 1 (X ) by its universal central extension.

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We can assign a topological invariant to a representation  given by the characteristic class c() := c(E ) 2 1 (G) corresponding to E . To define this, z be the universal covering group of G . We have an exact sequence let G z !G !1 1 ! 1 (G) ! G which gives rise to the (pointed sets) cohomology sequence c z ! H 1 (X; G) ! H 1 (X; G) H 2 (X; 1 (G)):

(1)

Since 1 (G) is abelian, we have H 2 (X; 1 (G)) Š 1 (G);

and c(E ) is defined as the image of E under the last map in (1). Thus the z -bundle, and hence class c(E ) measures the obstruction to lifting E to a flat G z to lifting  to a representation of 1 (X ) in G . For a fixed c 2 1 (G) , we can consider the subvariety of the moduli space of representations Rc (G) defined by Rc (G) := ¹ 2 R(G)W c() = cº:

Similarly, we can assign a topological invariant to a G -Higgs bundle (E; ') . Assuming that G is connected (and hence H is connected), topologically, H C -bundles E on X are classified by a characteristic class c = c(E) 2 1 (H C ) = 1 (H ) = 1 (G) . Again, this comes from considering the exact sequence

e

1 ! 1 (H C ) ! H C

! HC ! 1

and the corresponding (pointed sets) cohomology sequence

e

c

H 1 (X; H C ) ! H 1 (X; H C ) ! H 2 (X; 1 (H C )):

For a fixed such class c , the subvariety Mc (G)  M(G) is defined as the set of isomorphism classes of polystable G -Higgs bundles (E; ') such that c(E) = c . If G is semisimple, of course, the homeomorphism in Theorem 2.2 restricts to give a homeomorphism Rc (G) Š Mc (G):

Proposition 2.4. Let G be a connected semisimple real Lie group. Then 1. there is a map 0 (R(G)) Š 0 (M(G)) ! 1 (G); 2. if G is compact or complex, the homomorphism in (1) is a bijection. The proof of (2) in Proposition 2.4 is due J. Li [56]. An alternative proof using Higgs bundles, and including the case of non-connected reductive Lie groups is given in [31]. The homomorphism in (1) of Proposition 2.4 for a real form of a complex semisimple Lie group may be neither surjective nor injective, as we will see in §3.

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3. A basic example: G = SL(2; R) We consider G = SL(2; R) . Let X be a compact Riemann surface of genus g  2 . Given a representation W 1 (X ) ! SL(2; R) , the invariant associated to  , defined in §2.3 is an integer d 2 1 (SL(2; R) Š Z . This is the Euler class d 2 Z of the corresponding flat SL(2; R) -bundle E . We can then define the subvarieties Rd := ¹ 2 R(SL(2; R))W with Euler class d º:

By Milnor [57], Rd is empty unless jd j  g

1:

A maximal compact subgroup of SL(2; R) can be identified with U(1) , and its complexification C can be embedded in SL(2; C) as the matrices of the form   a 0 ; 0 a 1 with a 2 C . On the other hand, in this case mC coincides with the matrices   0 b ; c 0 with b; c 2 C . An SL(2; R) -Higgs bundle is then simply a triple (L; ˇ; ) , where L is a holomorphic line bundle over X and ˇ 2 H 0 (X; L2 K) and 2 H 0 (X; L 2 K) . It can also be viewed as the SL(2; C) -Higgs bundle (V; Φ) obtained by extension of structure group given by V = L ˚ L 1 , and   0 ˇ Φ= :

0 The topological invariant attached to the Higgs bundle (L; ˇ; ) is then deg L , the degree of L (its first Chern class). As for representations, we can define the subvariety Md  M(SL(2; R)) as the SL(2; R) -Higgs bundles with fixed deg L = d . Here the natural gauge transformations are C -transformations, that is, those of L . Allowing SL(2; C) -transformation naturally identifies Md and M d inside M(SL(2; C)) . Now, the semistability condition gives a constraint on the possible degrees that L may have, namely, we must have jd j  g

1:

This can be seen very easily ([46]). Assume that d  0 (similar argument for d  0 ). Suppose that d > g 1 . Then = 0 and L is a Φ -invariant line subbundle of V . By the semistability of (V; Φ) we must have that d  0 which gives a contradiction. Considering the homeomorphism Rd Š Md , this gives a Higgs bundle proof of the Milnor inequality [57]. The moduli space of representations of 1 (X ) in SL(2; R) was studied by Goldman [37], who showed that for d satisfying d = g 1 (same for d = 1 g ) there are 22g isomorphic connected components that can be identified with Teichmüller

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space, and showed that for d such that jd j < g 1 there is only one connected component. This was also proved by Hitchin [46], who also gave a very explicit description of each component. Namely, for 0 < jd j < g 1 , Md is the total space of a holomorphic complex vector bundle of rank g + 2jd j 1 over a 22g -fold covering of the 2g 2 2jd j -symmetric power of X Sym2g

2 2jd j

X:

To see this, assume that d > 0 (the case d < 0 is similar), then the pair (L; ) defines an element in Sym2g 2 2d (X ) , given by the zeros of . But if L0 is a line bundle of degree 0 of order two, that is, L20 = O , then the element (LL0 ; ) defines also the same divisor in Sym2g 2 2d (X ) . Hence the set of pairs (L; ) gives a point in the 22g -fold covering of Sym2g 2 2d (X ) . The section ˇ now gives the fibre of the vector bundle. Notice Riemann–Roch implies that H 1 (X; L2 K) = 0 . The case of d = 0 is a bit more involved (see [46], [34], and [35]) From this we conclude the following. Proposition 3.1. 1. dim Md = 3g 3 . 2. Md is connected if jd j < g 1 . 3. Md has 22g connected components if jd j = g 1 , each isomorphic to 3g 3 C (the fibre of a rank 3g 3 vector bundle over a 22g -fold covering of a point!). Point (3) is clear since if deg L = g 1 , the line bundle L 2 ˝ K is of zero degree and hence has a section (unique up to multiplicaton by a scalar) if and only if L 2 ˝ K Š O , i.e. if L is a square root of K . For each of the 22g choices of square root L = K 1/2 , one has a connected component which is parametrized by ˇ 2 H 0 (X; K 2 ) . Each of these components is diffeomorphic to Teichmüller space (see [37]).

4. The Hitchin map, split real forms and Hitchin components For details on this section, see [33].

4.1. Maximal split subgroup and Chevalley map Let g be a reductive real Lie algebra with a Cartan involution  decomposing g as g = h ˚ m: Given a maximal subalgebra a  m it follows that it must be abelian, and one can easily prove that its elements are semisimple and diagonalizable over the real numbers (cf. [51], Chapter VI), note that Knapp proves it for semisimple Lie algebras, but for reductive Lie algebras it suffices to use invariance of the centre and the semisimple part of [g; g] ) under the Cartan involution. Any such

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subalgebra is called a maximal anisotropic Cartan subalgebra of g . By extension, its complexification aC is called a maximal anisotropic Cartan subalgebra of gC (with respect to g ). A maximal anisotropic Cartan subalgebra a can be completed to a  equivariant Cartan subalgebra of g , namely, a subalgebra whose complexification is a Cartan subalgebra of gC . Indeed, define d=t˚a

where t  ch (a) := ¹x 2 hW [x; a] = 0º is a maximal abelian subalgebra ([51], Proposition 6.47). Cartan subalgebras of this kind (and their complexifications) are called maximally split. The dimension of maximal anisotropic Cartan subalgebras of a real reductive Lie algebra g is called the the real (or split) rank of g . This number measures the degree of compactness of real forms: indeed, a real form is compact (that is, its adjoint group is compact) if and only if rkR (g) = 0 . On the other hand, a real form is defined to be split if rkR (g) = rkgC . Note that the split rank depends on the involution  associated with the real form, when g is not semisimple. In [54], Kostant and Rallis give a procedure to construct a  -invariant subalgebra ˆ  g such g ˆ  (ˆ g g)C is a split real form, whose Cartan subalgebra is a and such that z(ˆg) = z(g) \ m . Their construction relies on the following notion. A three dimensional subalgebra (TDS) sC  gC is the image of an injective morphism sl(2; C) ! gC . A TDS is called normal if dim sC \ hC = 1 and dim sC \ mC = 2 . It is called principal if it is generated by elements ¹e; f; xº , where e and f are nilpotent regular elements in mC , and x 2 hC is semisimple. A set of generators satisfying such relations is called a normal triple. A subalgebra gˆ  g generated by a and sC \ g , where sC is a principal normal TDS invariant by the involution defining g inside of gC is called a maximal split subalgebra. Remark 4.1. Let gC be a complex reductive Lie algebra, and let (gC )R be its underlying real reductive algebra. Then, the maximal split subalgebra of (gC )R is isomorphic to the split real form gsplit of gC . It is clearly split within its complexification and it is maximal within (gC )R with this property, which can be easily checked by identifying (gC )R Š gsplit ˚ i gsplit . We say that a real Lie group is split if its Lie algebra is split. Let (G; H; ; B) ˆ B) y H y ; ; y , where G y  G is the analytic be a real reductive Lie group. The tuple (G; ˆ H, subgroup correspoding to the maximal split subalgebra gˆ  g , Hy := exp(i h) ˆ y and  and B are obtained by restriction, is a reductive Lie group to which we refer as the maximal split subgroup of (G; H; ; B) . The restriction to a of the adjoint representation of g yields a decomposition of g into a -eigenspaces M g= g ; 2Λ(a)

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where Λ(a)  a is called the set of restricted roots of g with respect to a . The set Λ(a) forms a root system (see [51], Chapter II, §5), which may not be reduced (that is, there may be roots whose double is also a root). The name restricted roots is due to the following fact: extending restricted roots by C -linearity, we obtain Λ(aC )  (aC ) , also called restricted roots. We define the restricted Weyl group of gC associated to aC , W (aC ) , to be the group of automorphisms of aC generated by reflections on the hyperplanes defined by the restricted roots  2 Λ(aC ) . ˆ B) y H y ; ; y < Proposition 4.1. Let (G; H; ; B) be a real reductive Lie group and (G; (G; H; ; B) be its maximal split subgroup. Let a = dim aC . Then,

1. restriction induces an isomorphism C

C)

C[mC ]H Š C[aC ]W (a

yC

y C ]H ; Š C[m

C

2. C[aC ]W (a ) is generated by homogeneous polynomials of degrees d1 ; : : : ; da , canonically determined by (G;  ); 3. the degrees of the homogeneous polynomials for (G; H; ; B) and ˆ B) y H y ; ; y coincide (G; We thus have an algebraic morphism, called the Chevalley map W mC  mC //H C Š aC /W (aC )

(2)

where the double quotient sign // stands for the affine GIT quotient.

4.2. The Hitchin map Let (G; H; ; B) be a reductive real Lie group. Consider the Chevalley morphism (2): W mC ! aC /W (aC ):

This map is C -equivariant. In particular, it induces a morphism hW mC ˝ K ! aC ˝ K/W (aC ):

The map  is also H C -equivariant, thus defining a morphism hW M(G) ! B(G) := H 0 (X; aC ˝ K/W (aC )):

The map h is called the Hitchin map, and the space B(G) is called the Hitchin base. In more concrete terms Let p be an H C -invariant homogeneous polynomial of degree d on mC . Then, if (E; ') is a G -Higgs bundle, the evaluation of p on ' gives an element of H 0 (X; K d ) , so p induces a map pW M(G) ! H 0 (X; K d ):

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Let p1 ; : : : ; pr with r = rk(G) be a basis of the ring of H C -invariant homogeneous polynomials on mC . This basis defines a map M(G) !

r M

H 0 (X; K di )

i =1

where, for i = 1; 2; : : : ; r , di is the degree of pi . These degrees are independent of the bases and they only depend on G . This is the Hitchin map and hence the Hitchin base is r M B(G) = H 0 (X; K di ): i =1

If G is complex and E is a stable G -bundle, one has (see [47]) that dim B(G) = dim H 0 (X; E(g) ˝ K) and hence the dimension of B(G) coincides with the dimension of the moduli space of G -bundles, which is half the dimension of M(G) . In [47] Hitchin shows that M(G) is a completely algebraically integrable system and that, for a classical group, the generic fibre is either a Jacobian or a Prym variety of a curve covering X , which is called the spectral curve. The description of the fibres for a general complex reductive Lie group is given by Donagi and Gaitsgory [23] in terms of a cameral cover of the curve. The description of the fibration for general real forms is not yet fully understood (for some partial results see [59], [61], [49], and [32]).

4.3. Hitchin components for SL(n; R) -Higgs bundles We will illustrate first how Hitchin constructed in [48] special components of the moduli space of SL(n; R) -Higgs bundles as the image of certain sections of the Hitchin map. To explain this, recall that Cartan decomposition of sl(n; R) is given by sl(n; R) = so(n) + m; where m = ¹symmetric real matrices of trace 0º . A SL(n; R) -Higgs bundle is thus a pair (E; ') , where E is a principal holomorphic SO(n; C) -bundle over X and the Higgs field is a holomorphic section ' 2 H 0 (E(mC ) ˝ K):

Using the standard representations of SO(n; C) in Cn one can associate to E a holomorphic vector bundle V of rank n with det V = O together with a nondegenerate quadratic form Q 2 H 0 (S 2 V  ) . A SL(n; R) -Higgs bundle is then in correspondence with a triple (V; Q; ');

where the Higgs field is a symmetric and traceless endomorphism 'W V ! V ˝ K . The simplest case is to consider the complex Lie group SL(2; C) and its split real form SL(2; R) . The Lie algebra sl(2; C) has rank 1 and the algebra

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of invariant polynomials on it is generated by p2 of degree 2 obtained from the characteristic polynomial det(x1

A) = x 2 + p2 (A)

of a trace-free matrix. We are going to define a section of the Hitchin map pW M ! H 0 (K 2 ); (E; ') 7 ! p2 (');

where M denotes the moduli space of polystable SL(2; C) -Higgs bundles. This section gives an isomorphism between the vector space H 0 (K 2 ) and a connected component of the moduli space M(SL(2; R))  M of polystable SL(2; R) -Higgs bundles. To construct the section, we consider the elements        1 0 0 1 0 0 x= ;e= ; e˜ = Š sl(2; C) 0 1 0 0 1 0 that satisfy [x; e] = 2e , [x; e˜] =

2˜ e and [e; e˜] = x;

where x is an element of the Cartan subalgebra (a semisimple element) and e , e˜ are nilpotent. The pair    0 ˛ K 1/2 ˚ K 1/2 ; ' = e˜ ˛e = ; 1 0 where ˛ 2 H 0 (K 2 ) , is a SL(2; R) -Higgs bundle. In the vector bundle K 1/2 ˚K 1/2 we have the orthogonal structure Q = ( 01 10 ) and the Higgs field is symmetric with respect to this orthogonal form. The section is finally defined by    0 ˛ s(˛) = K 1/2 ˚ K 1/2 ; ' = : 1 0 That is, the pairs ² K 1/2 ˚ K

1/2

 ;' =

˛ 0

0 1

³ ˛ 2H 0 (K 2 )

form a connected component of M(SL(2; R)) of dimension 6g 6 , and there are 22g connected components isomorphic to this one—the number of possible choices of the square K 1/2 . These are precisely the components with maximal integer invariant given by the Milnor inequality, described in §3. Now consider the general case SL(n; R) which is the split real form of SL(n; C) . The Lie algebra sl(n; C) has rank n 1 and a basis for the invariant polynomials on sl(n; C) is provided by the coefficients of the characteristic polynomial of a trace-free matrix, det(x

A) = x n + p1 (A)x n

2

+


+ pn

1 (A);

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where deg(pi ) = i + 1 . Ehe Hitchin map hW M(SL(n; C)) !

n 1 M

H 0 (K i +1 )

i =1

is defined by h(E; ') = (p1 ('); : : : ; pn 1 (')); where M(SL(n; C)) is the moduli space of polystable SL(n; C) -Higgs bundles. We are going to defineL a section of this map that will give an isomorphism n 1 0 i +1 between the vector space ) and a connected component of the i =1 H (K moduli space M(SL(n; R))  M(SL(n; C)) of polystable SL(n; R) -Higgs bundles. A nilpotent element e 2 sl(n; C) is called regular if its centralizer is (n 1) -dimensional. In sl(n; C) a regular nilpotent element is conjugate to an element X e= X˛ ; ˛ 2Π

where Π = ¹˛i = ei ˛i , that is,

ei +1 ; 1  i  n

1º and X˛i = Ei;i +1 is a root vector for

1 0 1 0





0 B :: C B : 0 1 0


0 C B C B :: :: :: C :: B : : : : C C: e=B B : : C :: B :: :: : 0 C B C B : : C @ :: :: 1 A 0 0








0 Any nilpotent element can be embedded in a 3 -dimensional simple subalgebra hx; e; e˜i Š sl(2; C) , where x is semisimple, e and e˜ are nilpotent, and they satisfy 0

[x; e] = 2e;

[x; e˜] =

2˜ e;

[e; e˜] = x:

The adjoint action hx; e; e˜i Š sl(2; C) ! End(sl(n; C)) of this subalgebra breaks up the Lie algebra sl(n; C) as a direct sum of irreducible representations n 1 M sl(n; C) = Vi ; i =1

with dim(Vi ) = 2i + 1 . That is, each Vi is the irreducible representation S 2i C2 , where C2 is the standard representation of sl(2; C) , and the eigenvalues of ad x on Vi are 2i; 2i + 2; : : : ; 2i 2; 2i . The highest weight vector of Vi , defined as a vector ei 2 Vi that is an eigenvector for the action of x and is in the kernel of ad(e) , has eigenvalue 2i for ad x . We take V1 = hx; e; e˜i and e = e1 .

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Given (˛1 ; : : : ; ˛n Hitchin component by

1)

2

Ln

1 i =1

H 0 (K i +1 ) , we define the Higgs field in the

' = e˜1 + ˛1 e1 +


+ ˛n

1 en 1 ;

and the vector bundle is given by Sn

1

(K

1/2

˚ K 1/2 ) = K

(n 1)/2

˚K

(n 3)/2

˚


˚ K (n

3)/2

˚ K (n

1)/2

:

The field ' is given in the following order of the basis, K (n

1)/2

˚ K (n

3)/2

˚


˚ K

(n 3)/2

˚K

(n 1)/2

:

Hitchin [48] generalized this construction to the split real form of any complex semisimple Lie group, proving the following. Theorem 4.2. Let S be a compact oriented surface of genus g > 1 . Let G C be a complex semisimple Lie group and let G be the split real form of G C . Then the moduli space R(G) of representations of the fundamental group of S in G has a connected component homeomorphic to a Euclidean space of dimension dim G(2g 2) . Moreover if G C is of adjoint type this component is unique. When G = PSL(2; R) , this component can be identified with Teichmüller space, and this is why Hitchin calls these generalized Teichmüller components. They are now referred as Hitchin components.

4.4. The Hitchin–Kostant–Rallis section Hitchin’s construction of a section of the Hitchin map for the moduli space Higgs bundles for a complex group can be generalized to the moduli space of G -Higgs bundles for any real reductive Lie group [33]. This construction relies on the Kostant–Rallis construction of a maximal split subalgebra explained in §4.1. An element x 2 mC is said to be regular if dim cmC (x) = dim aC , where cmC (x) = ¹y 2 mC W [y; x] = 0º . Denote the subset of regular elements of mC by C mC reg : Regular elements are those whose H -orbits are maximal dimensional, so this notion generalises the classical notion of regularity of an element of a complex reductive Lie algebra. Note that the intersection mC \ gC reg is either empty or C C the whole of mreg . Here greg denotes the set of elements of gC with maximal dimensional G C -orbit. C A real form g  gC is called quasi-split if mC \ gC reg = mreg . These include split real forms, and the Lie algebras su(p; p) , su(p; p + 1) , so(p; p + 2) , and e6(2) . Quasi-split real forms admit several equivalent characterizations: g is quasi-split if and only if cg(a) is abelian—which holds if and only if gC contains a  -invariant Borel subalgebra—and if and only if mC \ gC reg = mreg . Consider the group Q , satisfying (Ad(G)C ) = Q Ad(H C ) . It is a finite group whose cardinality we denote by N . The following theorem is proved in [33].

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259

ˆ B) y H y ; ; y Theorem 4.3. Let (G; H; ; B) be a reductive real Lie group, and let (G; be its maximal split subgroup. Then, the choice of a square root of K determines N inequivalent sections of the map hW M(G) ! B(G):

Each such section sG satisfies the following properties: 1. if G is quasi-split, sG (B(G)) is contained in the stable locus of M(G) , and in the smooth locus if Z(G) = ZG (g); 2. if G is not quasi-split, the image of the section is contained in the strictly polystable locus; 3. for arbitrary groups, the Higgs field is everywhere regular; y 4. the section factors through M(G); C 5. if Gsplit < G is the split real form satisfying z(gC ) \ i u  m , sG is the factorization of the Hitchin section through M(Gsplit ) .

5. Hermitian groups and maximal Toledo components In this section we will assume that G is a connected, non-compact real simple Lie group of Hermitian type with finite centre. We fix a maximal compact subgroup H  G , with Cartan decomposition g = h + m . The centre Z(H ) of H is isomorphic to U(1) . For details on this section see [5].

5.1. Hermitian symmetric spaces and Cayley transform The homogenous space G/H is an irreducible Hermitian symmetric space of non-compact type. The same symmetric space G/H is given by all the finite coverings of the group Ad(G) = G/Z(G) , where Z(G) is the centre of G . The vector subspace m is isomorphic to the tangent space To G/H at the point o = eH . Let H C , hC , and mC be the complexifications of H , h , and m respectively. The almost complex structure J0 on m = To (G/H ) , where o 2 G/H corresponds to the coset H , is induced by the adjoint action of an element J 2 z(h) , so J0 = ad(J )jm . Since J02 = Id , we decompose mC into ˙i -eigenspaces for J0 : mC = m+ + m . Both m+ and m are abelian, [hC ; m˙ ]  m˙ , and there are Ad(H ) -equivariant isomorphisms m Š m˙ given by X 7! 12 (X  iJ0 X ) . Consider a maximal abelian subalgebra t of h . Its complexification tC gives a Cartan subalgebra of gC , for which the root system ∆ = ∆(gC ; tC ) P we consider C C C and the decomposition g = t + ˛ 2∆ g˛ . Since ad(tC ) preserves hC and mC , C C C gC or in mC . If gC (resp. gC ˛ must lie either in h ˛ h ˛  m ) we say that the root ˛ is compact (resp. non-compact). We choose an ordering of the roots in such a way that m+ (resp. m ) is spanned by the root vectors corresponding to the non-compact positive (resp. negative) roots.

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We denote by h
;
i the Killing form on gC . For each root ˛ 2 ∆ , let H˛ 2 i t be the dual of ˛ , i.e., for Y 2 i t:

˛(Y ) = hY; H˛ i

˛ Define, as usual, h˛ = hH2H 2 it , and e˛ 2 gC ˛ such that [e˛ ; e ˛ ] = h˛ . Two ˛ ;H˛ i roots ˛; ˇ 2 ∆ are said to be strongly orthogonal if neither ˛ + ˇ nor ˛ ˇ is a root (equivalently [g˛ ; g˙ˇ ] = ¹0º ). A system of strongly orthogonal roots is a maximal set of strongly orthogonal positive non-compact roots. It has a number of elements equal to the rank r = rk(G/H ) of the symmetric space G/H , i.e., the maximal dimension of a flat, a totally geodesic submanifold of G/H . Moreover, for two strongly orthogonal roots ¤ 0 we have

[e˙ ; e˙ 0 ] = 0;

[e˙ ; h 0 ] = 0:

For a strongly orthogonal system of roots Γ , consider X X X xΓ = x ; y Γ = y ; eΓ = e ;

2Γ

2Γ

c = exp

 4

2Γ



iyΓ 2 U  G C ;

where G C is the simply connected Lie group with Lie algebra gC and U is its compact real form (with Lie algebra h ˚ i m ). We define the Cayley transform as the action of the element c on the Lie algebra gC by Ad(c)W gC ! gC . The Cayley transform Ad(c) satisfies Ad(c 8 ) = Id , Ad(c) ı  =  ı Ad(c 1 ) for the Cartan involution  , and consequently Ad(c 4 ) preserves h and m , even though Ad(c) does not preserve g . It is well known that a Hermitian symmetric space of non-compact type G/H can be realized as a bounded symmetric domain. For the classical groups this is due to Cartan, while the general case is given by the Harish-Chandra embedding G/H ! m+ which defines a biholomorphism between G/H and the bounded symmetric domain D given by the image of G/H in the complex vector space m+ . Now, for any bounded domain D there is the Shilov boundary of D which is defined as the smallest closed subset S{ of the topological boundary @D for which x and holomorphic on D satisfies that every function f continuous on D jf (z)j  max jf (w)j w 2 S{

for every z 2 D:

The Shilov boundary S{ is the unique closed G -orbit in @D . The simplest situation to consider is that of the hyperbolic plane. The Poincaré disc is its realization as a bounded symmetric domain. However, we know that the hyperbolic plane can also be realized as the Poincaré upper-half plane. There are other Hermitian symmetric spaces that, like the hyperbolic plane, admit a realization similar to the upper-half plane. These are the tube type symmetric spaces.

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Let V be a real vector space and let Ω  V be an open cone in V . A tube over the cone Ω is a domain of the form TΩ = ¹u + iv 2 V C ; u 2 V; v 2 Ωº:

A domain D is said of tube type if it is biholomorphic to a tube TΩ . In the case of a symmetric domain the cone Ω is also symmetric. An important characterization of the tube type symmetric domains is given by the following. Proposition 5.1. Let D be a bounded symmetric domain corresponding to the Hermitian symmetric space G/H . The following are equivalent: i. D is of tube type; i i. dimR S{ = dimC D ; i i i. S{ is a symmetric space of compact type; i v. Ad(c 4 ) = Id . There is a generalization of the Cayley map that sends the unit disc biholomorphically to the upper-half plane. Let D be the bounded domain associated to a Hermitian symmetric space G/H . Acting by a particular element in G , known as the Cayley element, one obtains a map cW D ! m+

which is called the Cayley transform. A relevant fact for us is the following. Proposition 5.2. Let D be the symmetric domain corresponding to the Hermitian symmetric space G/H . Let D be of tube type. Then the image by the Cayley transform c(D) is biholomorphic to a tube domain TΩ where the symmetric cone Ω is the non-compact dual of the Shilov boundary of D . In fact the Shilov boundary is a symmetric space isomorphic to H /H 0 for a certain subgroup H 0  H , and Ω = H  /H 0 is its non-compact dual symmetric space. Proposition 5.3. (1) The symmetric spaces defined by Sp(2n; R) , SO0 (2; n) are of tube type. (2) The symmetric space defined by SU(p; q) is of tube type if and only if p = q. (3) The symmetric space defined by SO (2n) is of tube type if and only if n is even. For a tube type classical irreducible symmetric space G/H , Table 5 indicates the Shilov boundary S{ = H /H 0 , its non-compact dual Ω = H  /H 0 , the isotropy representation space m0 and its complexification m0 C , corresponding to the Cartan decomposition of the Lie algebra Lie(H  ) = h0 + m0 of H  . The vector space m0 has the structure of a Euclidean Jordan algebra, where the cone Ω is realized. The following lemma will be important for our Cayley correspondence.

262

Lemma 5.4.

Oscar García-Prada

There are Ad(H C ) -equivariant isomorphisms ad(eΓ )W mT ! m0 C

and

ad(eΓ )W m0 C ! m+ T:

The study of certain problems in non-tube type domains can be reduced to the tube type thanks to the following. Proposition 5.5. Let G/H be a Hermitian symmetric space of non-compact type. z  G such that G/ z H z  G/H is a maximal isometrically There exists a subgroup G z is a maximal compact embedded symmetric space of tube type, where Hz  G subgroup.

5.2. The Toledo character We introduce the Toledo character associated to a simple Lie algebra g of Hermitian type as the character on the Lie algebra hC given as follows. Definition 5.1. The Toledo character T W hC ! C is defined, for Y 2 hC in terms of the Killing form, by T (Y ) =

1 h iJ; Y i; N

where N is the dual Coxeter number. Since J is in the center, T vanishes on [hC ; hC ] , hence determines a character. We study now when the Toledo character lifts to a character of the group H C . Note that this depends on the choice of the pair (G; H ) defining the same symmetric space. Let Z0C denote the identity component of Z(H C ) . Proposition 5.6. Define oJ to be the order of e 2J and ` = jZ0C \ [H C ; H C ]j . For q 2 Q , the character qT lifts to H C if and only if q is an integral multiple of `N qT = : oJ dim m The value of qT in the standard examples is given in Table 3. Note in particular that qT = 12 for all classical groups except SO , for which qT = 1 . So for all classical groups the Toledo character lifts to H C . For the adjoint group, oJ = 1 `N so qT = dim . In the tube case N = dimr m so this gives qT = r` . The values of m qT in the non-tube case are given in Table 4. Finally, the following lemma will prove later that the Toledo invariants defined from the two points of view of Higgs bundles and representations coincide.

8 Higgs bundles and higher Teichmüller spaces

Lemma 5.7. by

263

The Toledo character T defines a symmetric Kähler form on G/H

!(Y; Z) = iT ([Y; Z]); for Y; Z 2 m; with minimal holomorphic sectional curvature 1 .

We define now for G of tube type a determinant polynomial, det , on m+ , whose degree equals the rank of the symmetric space. This determinant is a familiar object in Jordan algebra theory [26], but it can be introduced in an elementary way 0 + as follows (see [53], Lemma Lr 2.3): it is the unique H0 -invariant polynomial on m + which restricts on a = 1 Ce i to det

r X

i e i =

1

r Y

i :

1

Here H00 is the identity component of H 0 . The existence comes from the Chevalley theorem on invariant polynomials, since the Weyl group acts exactly by all permutations on the (e i ) (see again [53]). The main useful property for us is the following equivariance: Lemma 5.8.

Let G be of tube type. For h 2 H C and x 2 m+ we have det(Ad(h)x) = ˜T (h) det(x);

where ˜T is the lifting of T to H C . Note that we implicitly assumed here that the lifting ˜T exists, otherwise the same identity remains true after taking power qT . + We define a notion of rank for Lr an element in m for G+ of Hermitian type + (tube or non-tube). Choose a = 1 Ce i . Any element of m is conjugate under H C to an element of a+ . P Definition 5.2. Let x 2 m+ , and y = i e i 2 a+ be conjugate to x under H C . Then we say that x has rank r 0 if y has exactly r 0 non-zero coefficients. This is well defined because the Weyl group acts only by permutations on a+. Also, in the tube case, one can give a more intrinsic interpretation using the determinant: polarize the determinant to get an r -linear map C on m+ such that C (x; : : : ; x) = det(x) ; then the rank of x is the maximal integer r 0 such that the (r r 0 ) -form C (x; : : : ; x;
; : : : ;
) is not identically zero, which is clearly an invariant notion. Remark 5.1. In the case of SU(p; q) the rank on m+ specializes to the notion of rank for a rectangular matrix q  p . For Sp(2n; R) , the rank on m+ is the rank for an element of S 2 (Cn ) seen as an endomorphism. The next proposition plays an important role in what follows.

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Proposition 5.9. Let 1  r 0  r . The group H C acts transitively on the set of elements of rank r 0 in m+ . In particular, the set of regular (that is maximal rank) elements in m+ is H C /H 0 C .

5.3. Toledo invariant and Milnor–Wood inequality Let X be a compact Riemann surface and let (E; ') be a G -Higgs bundle over X . The decomposition mC = m+ + m gives a vector bundle decomposition E(mC ) = E(m+ ) ˚ E(m ) and hence the Higgs field has two components: ' = (' + ; ' ) 2 H 0 (X; E(m+ ) ˝ K) ˚ H 0 (X; E(m ) ˝ K) = H 0 (X; E(mC ) ˝ K):

When the group G is a classical group, or more generally when H is a classical group, it is useful to take the standard representation of H C to describe a G -Higgs bundle in terms of associated vector bundles. This is the approach taken in [8], [30], [9], [10], [11], and [29]. Let (E; ') be a G -Higgs bundle. Consider the Toledo character T . Up to an integer multiple, T lifts to a character ˜T of H C . Let E(˜T ) be the line bundle associated to E via the character ˜T . Definition 5.3.

We define the Toledo invariant  of (E; ') by  =  (E) := deg(E(˜T )):

If ˜T is not defined, but only ˜Tq , one must replace the definition by deg E(˜qT ): We denote by M˛ (G) the subspace of M˛ (G) corresponding to G -Higgs bundles whose Toledo invariant equals  . For ˛ = 0 we simplify our notation setting M (G) := M0 (G) The following proposition relates our Toledo invariant to the usual Toledo invariant of a representation, first defined in [68]. 1 q

Proposition 5.10. Let W 1 (X ) ! G be reductive and let (E; ') be the corresponding polystable G -Higgs bundle given by Theorem 2.2. Let f W Xz ! G/H be the corresponding harmonic metric. Then Z 1  (E) = f  !; 2 X

where ! is the Kähler form of the symmetric metric on G/H with minimal holomorphic sectional curvature 1 , computed in Lemma 5.7. In particular,  (E) is the Toledo invariant of  .

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The Toledo invariant is related to the topological class of the bundle E defined as an element of 1 (H ) . To explain this, assume that H C is connected. The topological classification of H C -bundles E on X is given by a characteristic class c(E) 2 1 (H C ) as follows. From the exact sequence

e

1 ! 1 (H C ) ! H C

! HC ! 1

we obtain a long exact sequence in cohomology and, in particular, the connection map c

H 1 (X; H C ) ! H 2 (X; 1 (H C ));

where H C is the sheaf of local holomorphic functions in X with values in H C . The cohomology set H 1 (X; H C ) (not necessarily a group since H C is in general not abelian) parametrizes isomorphism classes of principal H C -bundles over X . On the other hand, since dimR X = 2 , by the universal coefficient theorem and the fact that the fundamental group of a Lie group is abelian, H 2 (X; 1 (H C )) is isomorphic to 1 (H C ) . Moreover, 1 (H C ) Š 1 (H ) Š 1 (G) since H is a deformation retract for both H C and G . This map thus associates a topological invariant in 1 (H ) to any G -Higgs bundle on X . By the relation between the fundamental group and the centre of a Lie group, the topological class in 1 (H ) is of special interest when H has a non-discrete centre, i.e., when G is of Hermitian type. In this case, 1 (H ) is isomorphic to Z plus possibly a torsion group (among the classical groups, SO0 (2; n) is the only one with torsion). Very often (see for example [9]), the Toledo invariant of a G -Higgs bundle (E; ') is defined as the projection of c(E) defined by (5.3) on the torsion-free part, Z . The general relation is the following. Proposition 5.11. Let (E; ') be a G -Higgs bundle, and d 2 Z the projection on the torsion-free part of the class c(E) defined by (5.3). Then d is related to the Toledo invariant by d = : qT Definition 5.2 gives the ranks of ' + and ' at a point x 2 X . The space of elements of m˙ with rank at most  is an algebraic subvariety of m˙ , so the ranks of ' + and ' are the same at all points of X except a finite number of points where it it smaller. We therefore have a well defined notion of rank of ' + and ' : Definition 5.4. The generic value on X of the rank of ' + is called the rank of ' + and denoted rk ' + . Analogously we define the rank rk ' of ' . One has the following.

266

Oscar García-Prada

Theorem 5.12. Let (E; ' + ; ' ) be a semistable G -Higgs bundle. Then, the Toledo invariant of E satisfies rk(' + )(2g

2)    rk(' )(2g

2):

In particular, we obtain the familiar Milnor–Wood inequality j j  rk(G/H )(2g

and equality holds if and only if ' case  < 0 (resp.  > 0 ).

+

2);

(resp. ' ) is regular at each point in the

Theorem 5.12 was proved on a case by case basis for the classical groups [46], [38], [8], [9], [11], and [29]. In these references, the bound given is for the integer d 2 1 (H ) Š 1 (H C ) Š Z associated naturally to the H C -bundle E . This differs from the Toledo invariant by a rational multiple. From Table 3 and Proposition 5.11 combined with Theorem 5.12 we obtain the Milnor–Wood inequalities given in [9] for the classical Hermitian groups. Our intrinsic general approach covers of course the exceptional groups and quotients and covers of classical groups that have not been studied previously. A polystable G -Higgs bundle (E; ') is, by Theorem 2.2, in correspondence with a reductive representation W 1 (X ) ! G , and from Proposition 5.10 the Toledo invariant of (E; ') coincides with the Toledo invariant of a representation of the fundamental group in G . In the context of representations the inequality j j  rk(G/H )(2g 2) , goes back to Milnor [57], who studies the case G = PSL(2; R) , and was proved in various cases in [70], [25], [22], and [18], and in general in [14]. We should point out that the Higgs bundle approach gives the Milnor–Wood inequality for an arbitrary representation, as the other approaches do, since such a representation can always be deformed to a reductive one.

5.4. Hermitian groups of tube type and Cayley correspondence We define a polystable Higgs bundle (E; ') to be maximal if its Toledo invariant  attains one of the bounds of the inequality i.e.,  = ˙r(2g 2) , where r = rk(G/H ) . We denote max = rk(G/H )(2g 2) . Let H  be the non-compact dual of H as defined in Proposition 5.2. In this section we establish a bijective correspondence between maximal G -Higgs bundles over X and K 2 -twisted H  -Higgs bundles over X , as defined in Remark 2.4, where K 2 is the square of the canonical line bundle. Suppose that (E; ') is a polystable maximal G -Higgs bundle, and choose for example  = r(2g 2) . By Theorem 5.12, the field ' + has rank r at each point. Let Z0C ' C be the connected component of the identity of the center of H C . There is an exact sequence 1 ! Z0C ! H C ! H C /Z0C ! 1;

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267

so there is an action of Z0C -bundles on H C -bundles that we will denote by ˝ : in this way, if  is a line bundle over X , we can define E ˝  (here we are identifying the line bundle  with its corresponding C -bundle). If  is an oJ -root of K , where oJ is the order of e 2J , then ' + defines a reduction of the H C -bundle E ˝  to the group H C . Of course, such  exists only if oJ divides 2g 2 . We will now suppose that  exists and is fixed. Denote by E 0 the reduction of E ˝  to H C . As we have seen, ' + 2 H 0 (X; E 0 (m+ )) , and similarly ' 2 H 0 (X; E 0 (m ) ˝ K 2 ) . From Lemma 5.4, we have an isomorphism ad ' + W E 0 (m ) ! E 0 (m0 C );

(3)

so that we can define a Higgs field ' 0 = [' + ; ' ] 2 H 0 (X; E 0 (m0 C ) ˝ K 2 ):

The data (E 0 ; ' 0 ) is a K 2 -twisted H  -Higgs bundle. Conversely, from a K 2 -twisted H  -Higgs bundle (E 0 ; ' 0 ) we can reconstruct (E; ') in the following way. The bundle is E = E 0 ˝  1 . Observe that for the H C -bundle E 0 we have canonical section eΓ 2 H 0 (X; E 0 (m+ )) corresponding to the element eΓ 2 m+ fixed by H 0 , which becomes by Lemma 5.4 a section ' + 2 H 0 (X; E(m+ ) ˝ K) . Finally, ' is reconstructed from (3) as (ad ' + ) 1 (' 0 ) . Therefore,  being fixed, we obtain a complete correspondence between maximal G -Higgs bundles and K 2 -twisted H  -Higgs bundles. We refer to (E 0 ; ' 0 ) as the Cayley partner of (E; ') . One has the following. Theorem 5.13 (Cayley correspondence). Let G be a connected non-compact real simple Hermitian Lie group of tube type with finite centre. Let H be a maximal compact subgroup of G and H  be the non-compact dual of H in H C . Let J be the element in z (the centre of h ) defining the almost complex structure on m . If the order of e 2J 2 H C divides (2g 2) , then there is an isomorphism of complex algebraic varieties Mmax (G) Š MK 2 (H  )

given by (E; ') 7! (E 0 ; ' 0 ) as above. Remark 5.2. The condition oJ j(2g 2) is always satisfied for a group of adjoint type, since in this case oJ = 1 . Table 3 shows that the oJ divides (2g 2) for the classical and exceptional groups. This may not happen for coverings of these groups, where oJ may be bigger.

6. SO(p; q) -Higgs bundles and higher Teichmüller spaces Details on this section can be found in [1] and [2].

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Oscar García-Prada

6.1. SO(p; q) -Higgs bundles and topological invariants An SO(p; q) -Higgs bundle on X is equivalent to a triple (V; W; ) where V and W are respectively rank p and rank q vector bundles with orthogonal structures such that det(W ) ' det(V ) , and  is a holomorphic bundle map W W ! V ˝ K . The cases for p  2 are somewhat special. For p > 2; rank p orthogonal bundles on X are classified topologically by their first and second Stiefel– Whitney classes, sw1 2 H 1 (X; Z2 ) and sw2 2 H 2 (X; Z2 ) . These primary topological invariants are constant on connected components of the moduli space M(SO(p; q)) . Since det(W ) ' det(V ) , it follows that sw1 (V ) = sw1 (W ) . The components of the moduli space M(SO(p; q)) are thus partially labeled by triples (a; b; c) 2 Z2g 2  Z2  Z2 , where a = sw1 (V ) 2 H 1 (X; Z2 ); b = sw2 (V ) 2 H 2 (X; Z2 ); c = sw2 (W ) 2 H 2 (X; Z2 ):

Using the notation Ma;b;c (SO(p; q)) to denote the union of components labeled by (a; b; c) , we can thus write a M(SO(p; q)) = Ma;b;c (SO(p; q)): (4) 2g

(a;b;c)2 Z2  Z2  Z2

Stability for SO(p; q) -Higgs bundles implies that a Higgs bundle (V; W; ) with  = 0 is polystable if and only if V and W are both polystable orthogonal bundles. This leads to the immediate identification of one connected component in each space Ma;b;c (SO(p; q)) . We use the subscript “top” to designate these components, which contain SO(p; q) -Higgs bundles with vanishing Higgs field. Proposition 6.1. Assume that 2 < p  q . For every (a; b; c) 2 Z2g 2  Z2  Z2 a;b;c the space M (SO(p; q)) has a non-empty connected component, denoted by Ma;b;c ( SO (p; q)) , in which every point can be continuously deformed to the top isomorphism class of an SO(p; q) -Higgs bundle of the form (V; W;  = 0) where V and W are polystable orthogonal bundles. We define Mtop (SO(p; q)) =

a

Ma;b;c top (SO(p; q)):

a;b;c

Remark 6.1. When p = 2 it is no longer true that Ma;b;c top (SO(p; q)) is non-empty for all (a; b; c) . In particular, if a = 0; then V = L ˚ L 1 which (a) is polystable if deg L = 0 and (b) has sw2 (V ) = deg L mod 2 . Thus M0;b;c top SO(2; q)) is empty if b ¤ 0 .

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269

The main result in [1] and [2] is that the moduli space M(SO(p; q)) has additional ‘exotic’ components disjoint from the components of Mtop (SO(p; q)) . These exotic components are identified as products of moduli spaces of so-called L -twisted Higgs bundles, where in each factor L is a positive power of the canonical bundle K . Let L be a fixed holomorphic line bundle on X . An L -twisted SO(1; n) -Higgs bundle on X is equivalent to a triple (I; W0 ; ) , where W0 is a O(n; C) -bundle, I is the rank one orthogonal bundle det(W0 ) and W W0 ! I ˝ L is a holomorphic bundle map. We get a decomposition similar to (4), namely a a;c MK p (SO(1; n)) = MK p (SO(1; n)); 2g

(a;c)2 Z2  Z2 a;c where here MK p (SO(1; n)) denotes the component in which the SO(1; n) -Higgs bundles are of the form (I; W0 ; ) , with a = sw1 (W0 ) and c = sw2 (W0 ) .

6.2. The case 2 < p < q

1

We can now state the main result in [1] and [2]. Theorem 6.2. Fix integers (p; q) such that 2 < p < q 1 . For each choice of a 2 Z2g and c 2 Z2 , the moduli space M(SO(p; q)) has a connected component 2 disjoint from Mtop (SO(p; q)) . This component is isomorphic to a;c MK p (SO(1; q

p + 1))  MK 2 (SO0 (1; 1)) 


 MK 2p

2

(SO0 (1; 1));

(5)

and lies in the sector M˛;0;c (SO(p; q)) where ˛ = a if p is odd and ˛ = 0 if p is even. Moreover, M(SO(p; q)) has no other connected components. Remark 6.2. The group SO0 (1; 1) is the connected component of the identity in SO(1; 1) , and MK 2j (SO0 (1; 1)) can be identified with H 0 (K 2j ) . Thus, we can replace (5) with p M1 a;c MK p + 1))  H 0 (K 2j ): p (SO(1; q j =1

Remark 6.3. The existence of the exotic components described by (5) was proven for p = 2 in [9]. They are the exotic components with maximal Toledo invariant arising from Cayley correspondence (see §6.5). In particular, Theorem 6.2 can be viewed as a generalized Cayley correspondence. Contrary to the cases p > 2 , there are components of M(SO(2; q)) which are not in the family described by the theorem and also not in Mtop (SO(2; q)) . These are the components with non-maximal and non-zero Toledo invariant.

270

Oscar García-Prada

Corollary 6.3. For 2 < p < q 1 , the moduli space M(SO(p; q)) has 3  22g +1 connected components, 22g +1 of which are exotic components disjoint from Mtop (SO(p; q)) .

6.3. The case q = p + 1 a;c a;c If q = p + 1 , then MK p + 1)) = MK p (SO(1; q p (SO(1; 2)) , which is not always 0;c connected. Indeed, if a = 0 , then the Higgs bundles represented in MK p (SO(1; 2)) can be taken to be of the form (O; L ˚ L 1 ; ) , where L is a non-negative degree d line bundle. Stability considerations impose a bound on d so that a 0;c d MK MK (6) p (SO(1; 2)): p (SO(1; 2)) = 0d p(2g 2) d =c(mod 2)

Moreover, Collier [19] has shown that for each integer d 2 (0; 2g 2] , the moduli d space MK 1 over the p (SO(1; 2)) is diffeomorphic to a vector bundle of rank d + g (2g 2 d ) th -symmetric product Sym2g 2+d (X ) . In particular, the components d MK p (SO(1; 2)) are smooth and connected. The moduli spaces M(SO(p; p + 1)) have been analyzed by Collier [19]. It was shown there that the topological invariants for SO(p; p + 1) -Higgs bundles, i.e. the triples (a; b; c) , do not distinguish all connected components. Two families of exotic components were identified. The components in the first family are labeled by an integer, d , in the range 0  d  p(2g 2) , while those in the second family are labeled by a pair (a; c) 2 (Z2g ¹0º)  Z2 . Though not described in this way in [19], these families can be identified as follows: • in the the family labeled by d , each member is isomorphic to d MK p (SO(1; 2))  MK 2 (SO0 (1; 1)) 


 MK 2p

2

(SO0 (1; 1));

0;c d where, as in (6), MK p (SO(1;2)) is one of the components of MK p (SO(1;2));

• in the family labeled by (a; c) , each member is isomorphic to a;c MK p (SO(1; 2))  MK 2 (SO0 (1; 1)) 


 MK 2p

2

(SO0 (1; 1)):

The components are thus precisely those identified by Theorem 6.2 in the case q = p + 1 . The component count in this case is, however, different from the case q > p + 1. Corollary 6.4. For p > 2 , the moduli space M(SO(p; p + 1)) has 3  22g +1 + 2p(g 1) 1 connected components. Among those, there are 22g +1 + 2p(g 1) 1 ‘exotic’ components which are disjoint from Mtop (SO(p; p + 1)) .

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6.4. The case q = p In this case, MK p (SO(1; q p+1)) = MK p (SO(1; 1)) . A K p -twisted SO(1; 1) -Higgs bundle consists of a triple (I; I; ) where I is a square root of the trivial bundle O and  2 H 0 (K p ) . Such Higgs bundles are labeled by a single Stiefel–Whitney class, namely a = sw1 (I ) , so that a a MK p (SO(1; 1)) = MK p (SO(1; 1)): a 2H 1 (X;Z2 )

With q = p; Theorem 6.2 thus gives 22g exotic components of M(SO(p; p)) isomorphic to the moduli spaces a MK p (SO(1; 1))  MK 2 (SO0 (1; 1)) 


 MK 2p

2

(SO0 (1; 1)):

a 0 p For each a , we can identify MK p (SO(1; 1)) with H (K ) . Thus, each exotic L p 1 component is isomorphic to H 0 (K p ) ˚ j =1 (H 0 (K 2j ) . This recovers the Hitchin component in M(SO0 (p; p)) when a = 0 .

6.5. The case p = 2 < q An SO(2; q) -Higgs bundle is defined by a triple (V; W; ) in which V is an O(2; C) -bundle. If sw1 (V ) = 0 , i.e. if the structure group of V reduces to SO(2; C) , then V can be assumed to be a direct sum of line bundles of the form V = L ˚ L 1 , with orthogonal structure given qV = ( 01 10 ) in this splitting. Note that the second Stiefel–Whitney class of the orthogonal bundle L ˚ L 1 is given by sw2 = d (mod 2) where d = deg(L)  0 . For the groups SO(2; q) , the connected components of the identity are isometry groups of Hermitian symmetric spaces of non-compact type. As explined in §5, the Higgs bundles have an associated Toledo invariant which, up to a normalization constant, is integer-valued but subject to a Milnor–Wood bound. For an SO0 (2; q) -Higgs bundle (L ˚ L 1 ; W; ) , the Toledo invariant is basically the degree d of L and the Milnor–Wood bound is 0  d  2g 2 . We thus get a M0;b;c (SO(2; q)) = Md ;c (SO0 (2; q)); 0d 2g 2 d =b(mod 2)

where Md ;c (SO(2; q)) denotes the component in which deg(L) = d and sw2 (W ) = c . The components where d = 2g 2 specializes further because in these components: 1. L has to be isomorphic to KI where I 2 = O ; 2. W decomposes as W = I ˚ W0 where W0 is a rank q bundle with sw1 (W0 ) = I .

1 orthogonal

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As shown in [9] and [20], an SO(2; q) -Higgs bundle with L = KI , W = I ˚ W0 and  = [q2 ; ˇ]W I ˚ W0 ! KI is defined by a K 2 twisted SO(1; q 1) -Higgs bundle (I; W0 ; ˇ) together with a quadratic differential q2 . Denoting M2g 2;c (SO(2; q)) by Mcmax (SO(2; q)) it follows that a a;c Mcmax (SO(2; q)) = MK 2 (SO(1; q 1))  H 0 (K 2 ) a 2H 1 (X;Z2 )

=

a

a;c MK 2 (SO(1; q

1)  SO0 (1; 1))

a

where a = sw1 (I ) . We thus get M0;0;c (SO(2; q)) a a a;c = Md ;c (SO(2; q)) t MK 2 (SO(1; q 0d 0  U Θ . The existence of the semigroup U>0 gives a well defined notion of positively oriented triples of pairwise transverse points in G/P Θ : This notion allows one to define a positive Anosov representation. If the pair (G; P Θ ) admits a positive structure, then a P Θ -Anosov representation W 1 (S) ! G is called positive if the Anosov boundary curve W @1 1 (S) ! G/P Θ sends positively ordered triples in @1 1 (S) to positive triples in G/P Θ : Guichard, Labourie, and Wienhard (see [42] and [39]) conjecture that if (G; P Θ ) admits a notion of positivity, then the set P Θ -positive Anosov representations is an open and closed subset of R(G) . In particular, the aim of this conjecture is to characterize the connected components of R(G) which are not labeled by primary topological invariants as being connected components of positive Anosov representations. This is what defines the higher Teichmüller components.

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When G is a split real form and Θ = ∆ , the corresponding parabolic is a Borel subgroup of G . In this case, the connected component of the identity of the Levi + rk(G ) factor is L∆ and each simple root space uˇi is one dimensional. 0 Š (R ) ∆ The L0 -invariant acute convex cone in each simple root space uˇi is isomorphic to R+ : The set of P ∆ -positive Anosov representations into a split group is exactly that consisting of Hitchin representations. When G is a Hermitian Lie group of tube type and P is the maximal parabolic associated to the Shilov boundary of the Riemannian symmetric space of G , the pair (G; P ) also admits a notion of positivity, see [14] and [15]. In this case, the space of maximal representations into G are exactly the P -positive Anosov representations. In particular, the above conjecture holds in these two cases. In general, the group SO(p; q) is not a split group and not a group of Hermitian type. Nevertheless, if p ¤ q , then SO(p; q) has a parabolic subgroup P Θ which admits a positive structure. Here P Θ is the stabilizer of the partial flag V1  V2 


 Vp 1 ; where Vj  Rp +q is a j -plane which is isotropic with respect to a signature (p; q) inner product with p < q: Here the Θ Θ subgroup LΘ pos  L  SO(p; q) which preserves the cones cˇj is isomorphic + l to LΘ p + 1) with l = p 1 , We refer the reader to [42] pos Š (R )  SO(1; q and [19] for more details. Using work by Collier [19], the following is shown in [2].

Proposition 7.1. Let P Θ  SO(p; q) be the stabilizer of the partial flag V1  V2 


 Vp 1 ; where Vj  Rp +q is a j -plane which is isotropic with respect to a signature (p; q) inner product with p < q . If q > p + 1 , then each connected component of Rex (SO(p; q)) from (10) contains P Θ -positive Anosov representations. When q = p + 1 , this was shown in [19] for the Collier components mentioned in §6.3. Proposition 7.1 gives further evidence for Guichard, Labourie, and Wienhard conjecture, and it is thus natural to expect that all representations in the connected components from Theorem 6.5 are positive Anosov representations. Indeed, this would follow from the conjecture and Proposition 7.1. Moreover, if the conjecture is true, then the connected components of Theorem 6.5 correspond exactly to those connected components of R(SO(p; q)) which contain positive Anosov representations.

7.2. General Cayley correspondence As we have seen in §5.4, if G is a Hermitian group of tube type, subject to a certain topological constraint (always satisfied if G is of adjoint type, for example), there is a bijective correspondence between maximal G -Higgs bundles and K 2 -twisted H  -Higgs bundles, where H  is a non-compact group determined by the Shilov

8 Higgs bundles and higher Teichmüller spaces

277

boundary. In fact H  , is the Levi factor of the parabolic subgroup P which defines positivity and determines the Shilov boundary as G/P . In particular, Theorem 5.13 states that the moduli space of maximal G -Higgs bundles is isomorphic to the moduli space of K 2 -twisted H  -Higgs bundles. We have also seen in Theorem 6.5 that for the components of the moduli space of SO(p; q) -Higgs bundles containing positive representations, for the positive structure defined above, there is also a generalized Cayley correspondence. Although this seems a bit more involved that the Hermitian case, as explained in [2], the structure of the Cayley partner is entirely determined by the parabolic subgroup of SO(p; q) defining positivity. When G is a split real form the Hitchin components of M(G) admit a similar interpretation. The positivity condition in this case is defined by a minimal parabolic subgroup P . The Levi factor is L = (R )rk(G) and the identity component is L0 = (R+ )rk(G) . Recall from §4.3 that the Hitchin base is given L B(G) = ri =1 H 0 (X; K di ) , where r = rk(G) . Now, the summand H 0 (X; K di ) can be interpreted as the moduli space of K di -twisted R+ -Higgs bundles and the Hitchin components are given by MK d1 (R+ ) 


 MK dr (R+ ):

From these three situations it seems natural to conjecture that, in general, higher Teichmüller components, that is, those consisting of positive representations for a certain pair (G; P ) , are in correspondence with what one can call Cayley components, i.e., components of M(G) for which there is a Cayley correspondence. Moreover, the structure of the Cayley partner is closely related to the parabolic subgroup defining positivity. Work in this direction is being pursued in [6], building upon results in [3]. This includes the exceptional groups for which there is a notion of positivity. As shown in [42] these are real forms of F4 , E6 , E7 and E8 whose restricted root system is of type F4 .

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Oscar García-Prada

8. Appendix: tables We use the following notation for Table 2: • ∆˙ 10 are the half-spinor representations of the group Spin(10; C) . They are 16 -dimensional. • M and M  are the irreducible 27 -dimensional representations of E6 , which are dual to each other. • r is the representation r W C ! C given by z 7! z r .

Table 1. Maximal split subalgebras Type AI AII AIII BI CI CII BDI DI DII

g

ˆ g

sl(n; R)

sl(n; R)

su (2n)

sl(n; R)

su(p; q); p < q su(p; p)

so(p; p + 1) sp(2p; R)

so(2p; 2q + 1); p  q

so(2p; 2p + 1)

sp(2n; R)

sp(2n; R)

sp(p; q) p < q sp(p; p)

so(p; p + 1) sp(2p; R)

so(p; q) p + q = 2n; p < q

so(p; p + 1)

so(p; p)

so(p; p)

so (4p + 2) p < q so (4p)

so(p; p + 1) sp(2p; R)

Type EI EII EIII EIV EV EVI EVII EVIII EIX FI FII G

g

ˆ g

e6(6)

e6(6)

e6(2)

f4(4)

e6(

14)

so(3; 2)

e6(

26)

sl(3; R)

e7(7)

e7(7)

e7(

5)

f4(4)

e7(

25)

sp(6; R)

e8(8)

e8(8)

e8(

f4(4)

24)

f4(4)

f4(4)

f4(

sl(2; R)

20)

g2(2)

g2(2)

Table 2. Irreducible Hermitian symmetric spaces G/H H

HC

mC = m+ + m

SU(p; q)

S(U(p)  U(q))

S(GL(p; C)  GL(q; C))

Hom(Cq ; Cp ) + Hom(Cp ; Cq )

Sp(2n; R)

U(n)

GL(n; C)

S 2 (Cn ) + S 2 (Cn  )

SO (2n)

U(n)

GL(n; C)

Λ2 (Cn ) + Λ2 (Cn  )

SO0 (2; n)

SO(2)  SO(n)

SO(2; C)  SO(n; C)

Hom(Cn ; C) + Hom(C; Cn )

E6 14

Spin(10) Z4 U(1)

Spin(10; C) Z4 C

3 ∆+ 10 ˝  + ∆10 ˝ 

E7 25

E6 78 Z3 U(1)

E6 Z3 C

M ˝ 2 + M  ˝ 

3 2

Table 3. Toledo character data for the classical and exceptional groups H

N

dim m

`

o(e 2J )

SU(p; q)

S(U(p)  U(q))

p+q

2pq

lcm(p; q)

gcd(p;q)

Sp(2n; R)

U(n)

n+1

n(n + 1)

n

2

1/2

SO (2n)

U(n)

n(n

n

2

1

SO0 (2; n)

SO(2)  SO(n)

n

2n

1

1

1/2

E6 14

Spin(10) Z U(1)

12

32

4

3

1/2

E7 25

E6 78 Z3 U(1)

18

54

3

2

1/2

2(n

4

1)

1)

p +q

qT 1/2

279

G

8 Higgs bundles and higher Teichmüller spaces

G

280

Table 4. Toledo character data for adjoint groups of non-tube type G

H

N

dim m

`

o(e 2J )

qT

PSU(p; q)

P S(U(p)  U(q))

p+q

2pq

gcd(p; q)

1

p +q 2 lcm(p;q)

P SO (2n = 4m + 2)

U(n)

n

1

2

E6 14 /Z3

Spin(10) Z4 U(1)

4

1

3/2

2(n

1)

n(n

12

1)

32

G

H

H

H0

S{ = H /H 0

m0

mC

SU(n; n)

S(U(n)  U(n))

¹A 2 GL(n; C) j det(A)2 2 R+ º

¹A 2 U(n) j det(A)2 = 1º

U(n)

Herm(n; C)

Mat(n; C)

Sp(2n; R)

U(n)

GL(n; R)

O(n)

U(n)/ O(n)

Sym(n; R)

Sym(n; C)

SO (2n) n = 2m

U(n)

U (n)

Sp(n)

U(n)/ Sp(n)

Herm(m; H)

Skew(n; C)

SO0 (2; n)

SO(2)  SO(n)

E7 25

E6 78 Z3 U(1)

SO0 (1; 1)  SO(1; n

E6 26 ËR

1)

O(n

1)

F4 Z2

U(1)  S n Z2

1

E6 78
U(1) F4

R  Rn

1

Herm(3; O)

C  Cn

1

Herm(3; O) ˝ C

Oscar García-Prada

Table 5. Irreducible Hermitian symmetric spaces G/H of tube type

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281

Acknowledgments. The author would like to thank Athanase Papadopoulos for his kind invitation to contribute to this volume with this paper, and his patience in the delivery. He also wishes to thank his collaborators in the works on which this paper is based, and Nigel Hitchin for inspiration, suggestions and many discussions over the years. The author was partially supported by the Spanish MINECO under the ICMAT Severo Ochoa grant № SEV-2015-0554, and under grant № MTM2016-81048-P.

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Part B Essays on the early works on quasiconformal mappings

Chapter 9

A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings Athanase Papadopoulos

Contents 1 Introduction . . . . . . . . . . . . . . . . . . 2 Biographical note on Tissot . . . . . . . . . 3 On the work of Tissot on geographical maps References . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Darboux, starts his 1908 ICM talk whose title is Les origines, les méthodes et les problèmes de la géométrie infinitésimale (The origins, methods and problems of infinitesimal geometry) with the words: “Like many other branches of human knowledge, infinitesimal geometry was born in the study of practical problems,” and he goes on explaining how problems that arise in the drawing of geographical maps, that is, the representation of regions of the surface of the Earth on a Euclidean piece of paper, led to the most important developments in geometry made by Lagrange, Euler, Gauss and others. The theory of quasiconformal mappings has its origin in problems related to the drawing of geographical maps. Teichmüller, in the last part of his paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings and quadratic differentials), published in 1939 [34], which is the main paper in which he develops the theory that became known as Teichmüller theory, makes some comments on this origin, mentioning the work of the French mathematician and geographer Nicolas-Auguste Tissot (1824–1897). Grötzsch, in his paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions) [18], published in 1930 and whose translation is contained in the present volume, mentions several times the name Tissot, referring to the Tissot indicatrix which he represents in the figures contained

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in his article. The directions of the major and minor axes of these ellipses constitute are important element in some of his results. A geographical map is the image of a mapping—henceforth called a projection—from the surface of the Earth, considered as a sphere or spheroid, onto the Euclidean plane. The Tissot indicatrix is a device introduced by Tissot, who called it the indicating ellipse (“ellipse indicatrice”), which was used by geographers until the middle of the twentieth century. It is a field of ellipses drawn on the geographical map, each ellipse representing the image by the projection—assumed to be differentiable—of an infinitesimal circle1 at the corresponding point on the sphere (or spheroid) representing the surface of the Earth. Examples of Tissot indicatrices are given in figure 1. In figure 2, we have reproduced drawings from a paper of Grötzsch in which are represented the Tissot indicatrix of the maps he uses. Although the work of Tissot is closely related to the theory of quasiconformal mappings, his name is never mentioned in the historical surveys of this subject, and the references by Grötzsch and by Teichmüller to his work remained unnoticed. In this note, I will give a few indications on this work. I will survey the work of Tissot in §3. Before that, I will start, in §2, with a short biographical note on him.

2. Biographical note on Tissot Nicolas-Auguste Tissot was born in 1824, in Nancy, which was to become, 26 years later, the birthplace of Henri Poincaré (whom we shall mention soon). Tissot entered the École Polytechnique in 1841. He started by occupying a career in the Army2 and defended a doctoral thesis on November 17, 1851; cf. [35]. On the cover page of his thesis, he is described as “Ex-Capitaine du Génie.” Tissot became later a professor at the famous Lycée Saint-Louis in Paris, and at the same time examiner at the École Polytechnique, in particular for the entrance exam. He eventually became an assistant professor (répétiteur) in geodesy at the École Polytechnique.

1 The expression “infinitesimal circle” means here, as is usual in the theory of quasiconformal mappings, a circle on the tangent space at a point. In practice, it is (as in the representation of the Tissot indicatrix) seen as a circle drawn on the surface which has a “tiny radius.” In the art of geographical map drawing, these circles, on the domain surfaces, are all supposed to have the same small size, so that the collection of relative sizes of the image ellipses becomes also a meaningful quantity. 2 The reader should note that the École Polytechnique was, and is still, a military school.

9 A note on Nicolas-Auguste Tissot

Figure 1. Four geographical maps on which the field of ellipses (Tissot indicatrix) are drawn. The maps are extracted from the book Album of map projections [32]. These are called, from left to right, top to bottom, the stereographic [32], p. 180, Lagrange [32], p. 180, central cylindrical [32], p. 30, and equidistant conical projections [32], p. 92. The first two projections are conformal and not area-preserving. The last two are neither conformal nor area-preserving.

Figure 2. Two figures from Grötzsch’s paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche [18]. Grötzsch drew the Tissot indicatrices of his quasiconformal mappings. (In each drawing, the major and minor axes of the ellipses are shown.)

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After having published, in the period 1856–1858, several papers and Comptes Rendus notes on cartography, in which he analyzed the distortion of some known geographical maps (see [37], [38], and [39]), Tissot started developing his own theory, on which he published three notes, in the years 1859–1860, see [40], [41], [42], and then a series of others in the years 1865–1880 [44], [45], [46], [47], [48], [49], and [50]. He then collected his results in the memoir [51], published in 1881, in which he gives detailed proofs. In a note on p. 2 of this memoir, Tissot declares that after he published his first Comptes Rendus notes on the subject, the statements that he gave there without proof were reproduced by A. Germain in his Traité des projections des cartes géographiques [17] and by U. Dini in his memoir Sopra alcuni punti della teoria delle superfici [11]. He notes that Germain and Dini gave their own proofs of these statements, which are nevertheless more complicated than those he had in mind and which he gives in the memoir [51]. He also writes that Dini showed that the whole theory of curvature of surfaces may be deduced from the general theory that he had developed himself. In fact, Dini applied this theory to the representation of a surface on a sphere, using Gauss’s methods. Tissot also says that his ideas were used in astronomy, by Hervé Faye, in his Cours d’astronomie de l’École Polytechnique [15]. The texts of the two Comptes rendus notes [42] and [39] of Tissot are reproduced in Germain’s treatise [17]. Besides his work on geographical maps, Tissot wrote several papers on elementary geometry. We mention incidentally that several preeminent French mathematicians of the nineteenth and the beginning of the twentieth centuries published papers on this topics. We mention Serret, Catalan, Laguerre, Darboux, Hadamard and Lebesgue; see e.g. [43], [36], and [52]. On the title page of Tissot’s memoir [51] (1881), the expression Examinateur à l’École Polytechnique follows his name, as he was in charge of the entrance examination. In his Éloge historique de Henri Poincaré [10], Darboux relates the following episode about Tissot, examining Poincaré:3 Before asking his questions to Poincaré, Mr. Tissot suspended the exam during fourty-five minutes: we thought it was the time he needed to prepare a sophisticated question. Mr. Tissot came back with a question from the Second Book of Geometry. Poincaré drew a formless circle, marked the lines and the points indicated by the examiner, and then, after wandering some time in front of the blackboard, with his eyes fixed on the ground, concluded loudly: “It all comes down to proving the equality AB = CD . This is a consequence of the theory of reciprocal polars, applied to the two lines.” 3 Poincaré entered the École Polytechnique in 1873. In the French system of oral examinations, which is still in use, a student is given a question or a set of questions which he is asked to prepare while another student (who had already been given some time to prepare his questions) is explaining his solutions at the blackboard, in the same room. Thus, it is not unusual that at such an examination, some students listen to the examinations of others.

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Mr. Tissot interrupted him: “Very good, Sir, but I would like a more elementary solution.” Poincaré started wandering again, this time not in front of the blackboard, but in front of the examiner’s table, facing him, almost unconscious of his acts; then suddenly he developed a trigonometric solution. “I would like you to stay in Elementary Geometry,” objected Mr. Tissot, and almost immediately after that, the examiner of Elementary Geometry was given satisfaction and warmly congratulated the examinee, announcing to him that he deserved the maximal grade.4

Poincaré kept a positive momory of Tissot’s examinations. He expresses this in a letter to his mother sent on May 6, 1874, opposing them to the 10-minute examinations (known as “colles”) that he had to take regularly at the École Polytechnique and which he considered as pitiful. He writes:5 “When I think about the exams of Tissot and others, I can not help but take pity of these 10 minutes little colles where one puts in danger his future with an expression which is more or less exact or a sentence which is more or less well crafted, and where a person is judged upon infinitesimal differences.”6

3. On the work of Tissot on geographical maps Tissot studied at the École Polytechnique, an engineering school where the students had a high level of mathematical training and at a period where the applications of the techniques of differential geometry to all the domains of science were an integral part of the curriculum. His work is part of a well-established tradition where mathematical tools are applied to the craft of map drawing. This tradition passes through the works of preeminent mathematicians such as Ptolemy [30] 4 Avant d’interroger Poincaré, M. Tissot suspendit l’examen pendant trois quarts d’heure : le temps de préparer une question raffinée, pensions-nous. M. Tissot revint avec une question du deuxième Livre de Géométrie. Poincaré dessina un cercle informe, il marqua les lignes et les points indiqués par l’examinateur; puis, après s’être promené devant le tableau les yeux fixés à terre pendant assez longtemps, conclut à haute voix: Tout revient à démontrer l’égalité AB = CD . Elle est la conséquence de la théorie des polaires réciproques, appliquée aux deux droites. “Fort bien, Monsieur, interrompit M. Tissot; mais je voudrais une solution plus élémentaire.” Poincaré se mit à repasser, non plus devant le tableau, mais devant la table de l’examinateur, face à lui, presque inconscient de ses actes, puis tout à coup développa une solution trigonométrique. “Je désire que vous ne sortiez pas de la Géométrie élémentaire,” objecta M. Tissot, et presque aussitôt satisfaction fut donnée à l’examinateur d’élémentaires, qui félicita chaleureusement l’examiné et lui annonça qu’il avait mérité la note maxima. 5 [29], letter №. 62. 6 Quand je pense aux exams de Tissot et autres, […] je ne puis m’empêcher de prendre en pitié ces petites colles de 10 minutes où on joue son avenir dans une expression plus ou moins exacte ou sur une phrase plus ou moins bien tournée et où on juge un individu sur des différences infinitésimales.

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and [2], Lambert [21] and [22], Euler [12], [13], and [14], Lagrange [19] and [20], Gauss [16], Chebyshev [4] and [5], Beltrami [1], Liouville (see the appendices to [25]), Bonnet [3], Darboux [7], [8], and [9], and there are others. It was known since antiquity that there exist conformal (that is, angle-preserving) projections from the sphere to the Euclidean plane.7 But it was noticed that these projections distort other quantities (length, area, etc.), and the question was to find projections that realize a compromise between these various distortions. For instance, one question was to find the closest-to-conformal projection among the maps that are area-preserving. Hence, the idea of “closest-to-conformal” projection came naturally. Among the mathematicians who worked on such problems, Tissot came closest to the notion of quasiconformality. Let us summarize a few of his results on this subject. An important observation made by Tissot right at the beginning of his memoir [51] (p. 1) is that finding the most appropriate mode of projection depends on the shape of the region—and not only its size, that is, on the properties of its boundary. Finding maps of small “distortion” (where, as we mentioned, this word has several possible meanings) was the aim of theoretical cartography. Tissot discovered that in order for the map to minimize an appropriately defined distortion, a certain function  , defined by setting d 2 = (1 + )2 ds 2 ;

must be minimized in some appropriate sense, where ds and d are the line elements at the source and the target surfaces respectively. The minimality of  may mean, for example, that the value of the gradient of its square must be the smallest possible. In fact, Tissot studied mappings between surfaces that are more general than those between subsets of the sphere and of the Euclidean plane. He started by noting that for a given mapping between two surfaces, there is, at each point of the domain, a pair of orthogonal directions that are sent to a pair of orthogonal directions on the image surface. Unless the mapping is angle-preserving at the given point, these pairs of orthogonal directions are unique. The orthogonal directions at the various points on the two surfaces define a pair of orthogonal foliations preserved by the mapping. Tissot calls the tangents to these foliations principal tangents at the given point. They correspond to the directions where the ratio of lengths of the corresponding infinitesimal line elements attains its greatest and smallest values. Using the foliations defined by the principal tangents, Tissot gave a method for finding the image of an infinitely small figure drawn in the tangent plane of the first surface. In particular, for a differentiable mapping, the images of infinitesimal circles are ellipses. In this case, he gave a practical way of finding the major and 7 Ptolemy, in his Geography, works with the strereographic projection, see [2]. See also Ptolemy’s work on the Planisphere [31].

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minor axes of these ellipses, and he provided formulae for them. This is the basis of the theory of the Tissot indicatrix. From the differential geometric point of view, the Tissot indicatrix gives information on the metric tensor obtained by pushing forward the metric of the sphere (or the spheroid) by the projection mapping. We recall that in modern quasiconformal theory, an important parameter of a map is the quasiconformal dilatation at a point, defined as the ratio of the major axis to the minor axis of the infinitesimal ellipse which is the image of an infinitesimal circle by the map (assumed to be differentiable at the give point, so that its derivative sends circles centered at the origin in the tangent plane to ellipses). The Tissot indicatrix gives much more information than this quasiconformal dilatation, since it keeps track of (1) the direction of the great and small axes of the infinitesimal ellipse, and (2) the size of this ellipse, compared to that of the infinitesimal circle of which it is the image. Darboux got interested in the work of Tissot on geography, and in particular, in a projection described in Chapter 2 of his memoir [51]. He wrote a paper on Tissot’s work [9] explaining more carefully some of his results. He writes: “[Tissot’s] exposition appeared to me a little bit confused, and it seems to me that while we can stay in the same vein, we can follow the following method […].”8 In his work, besides the theoretical results, Tissot showed how to construct mappings that have minimal distortion. This work was considered as very important by cartographers. The American cartographer J. P. Snyder, in his book Flattening the earth: two thousand years of map projections [33], published in 1997 and which is a reference in the subject, after presenting the existing books on cartography, writes: “Almost all of the detailed treatises presented one or two new projections, they basically discussed those existing previously, albeit with very thorough analysis. One scholar, however, proposed an analysis of distortion that has had a major impact on the work of many twentieth-century writers on map projections. This was Tissot […].” Modern cartographers are still interested in the theoretical work of Tissot, see [23]. We mentioned several preeminent mathematicians who worked on the theory of geographical maps before Tissot. From the more recent era, let me mention Milnor’s paper titled A problem in cartography [24], published in 1969. The reader interested in the theory of geographical maps developed by mathematicians is referred to the papers [28], [26], and [27], which also contain more on the work of Tissot.

8 Son exposition m’a paru quelque peu confuse, et il m’a semblé qu’en restant dans le même ordre d’idées on pourrait suivre avec avantage la méthode suivante.

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[17] A. Adolphe. Germain, Traité des projections des cartes géographiques. Représentation plane de la sphère et du sphéroııde. Librairie de la Société de géographie. Paris, Arthus Betrand, 1866. R 292 [18] H. Grötzsch, Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche. Leipz. Ber. 82 (1930), 69–80. English translation by M. Karbe, On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 16, 375–385. JFM 56.0298.02 R 289, 291 [19] J.-L. de Lagrange, Sur la construction des cartes géographiques. Premier mémoire. Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, 1779. Reprint in Œuvres complètes. Tome 4, 637–664. https://gallica.bnf.fr/ark:/12148/bpt6k229223s/f639 R 294 [20] J.-L. de Lagrange, Sur la construction des cartes géographiques. Second mémoire. Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, 1779, Reprint in Œuvres complètes. Tome 4, 664–692. https://gallica.bnf.fr/ark:/12148/bpt6k229223s/f666 R 294 [21] J. H. Lambert, Beiträge zum Gebrauche der Mathematik und deren Anwendung. Four volumes. Im Verlage des Buchladens der Realschule, Berlin, 1765–1772. R 294, 297 [22] J. H. Lambert, Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten. Hrsg. von A. Wangerin, 1772. Ed. Engelmann, Leipzig, 1894. (This is part of [21].) JFM 25.1304.02 R 294 [23] P. L. Laskowski, The traditional and modern look at Tissot’s indicatrix. The American Cartographer 16 (1989), no. 2, 123–133. R 295 [24] J. Milnor, A problem in cartography. Amer. Math. Monthly 76 (1969), 1101–1112. MR 0254791 Zbl 0184.49602 R 295 [25] G. Monge, Applications de l’analyse à la géométrie. Cinquième édition, revue, corrigée et annotée par M. J. Liouville. Bachelier, Paris, 1850. R 294 [26] A. Papadopoulos, Nicolas Auguste Tissot: A link between cartography and quasiconformal theory. Arch. Hist. Exact Sci. 71 (2017), no. 4, 319–336. MR 3665710 Zbl 1381.01007 R 295 [27] A. Papadopoulos, Quasiconformal mappings, from Ptolemy’s geography to the work of Teichmüller. In L. Ji and S.-T. Yau (eds.), Uniformization, Riemann–Hilbert correspondence, Calabi–Yau manifolds & Picard–Fuchs equations. Papers based on the conference held at the Institute Mittag-Leffler, July 13–18, 2015. International Press, Somerville, MA, and Higher Education Press, Beijing, 2018, 237–315. MR 3822909 Zbl 1411.30001 R 295 [28] A. Papadopoulos, Euler and Chebyshev: from the sphere to the plane and backwards. In Proceedings of Cybernetics. A volume dedicate to the Jubilee of Academician Vladimir Betelin. Polythechnical Insitute. Surgut State University, Surgut, 2016, 55–69. http://jc.surgu.ru/attachments/article/206/7_ath.pdf R 295 [29] H. Poincaré, La correspondance de jeunesse d’Henri Poincaré. Les années de formation. De l’École Polytechnique à l’École des Mines (1873–1878). Edited by L. Rollet. Publications des Archives Henri-Poincaré. Birkhäuser/Springer, Cham, 2017. MR 3837567 Zbl 1369.01003 R 293 [30] C. Ptolémée (Ptolemy), Traité de géographie. Hranslated from the Greek into French by l’Abbé de Halma. Eberhart, Paris, 1828. https://gallica.bnf.fr/ark:/12148/bpt6k3973m.texteImage R 293

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[31] N. Sidoli and J. L. Berggren, The arabic version of Ptolemy’s Planisphere or lattening the surface of the sphere. Text, translation, commentary. Dual English–Arabic text. Translated and with a commentary by the authors. SCIAMVS 8 (2007), 37–139. MR 2376620 R 294 [32] J. P. Snyder and P. M. Voxland, An album of map projections. United States Government Printing Office, 1989. R 291 [33] J. P. Snyder, Flattening the Earth. Two thousand pears of map projections. University of Chicago Press, Chicago, 1997. R 295 [34] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 22 (1939), 1–197. Reprint in Gesammelte Abhandlungen. L. V. Ahlfors and F. W. Gehring, eds. Springer-Verlag, Berlin etc., 1982, 335–531. English translation by G. Théret, Extremal quasiconformal mappings and quadratic differentials. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. V. IRMA Lectures in Mathematics and Theoretical Physics, 26. European Mathematical Society (EMS), Zürich, 2016, 321–483. MR 0003242 JFM 66.1252.01 Zbl 0024.33304 Zbl 1344.30044 (translation) R 289 [35] N.-A. Tissot, Mouvement d’un point matériel pesant sur une sphère. Suivi de Sur la détermination des orbites des planètes et des comètes. Thèses présentées á la faculté des sciences de Paris. Bachelier, Imprimeur-libraire de l’École Polytechnique et du bureau des longitudes, Paris, 1852. R 290 [36] N.-A. Tissot, Sur les hélices. Nouv. Ann. 11 (1852), 454–457. R 292 [37] N.-A. Tissot, Sur la détermination des latitudes au moyen de la méthode de M. Babinet. C. R. Acad. Sci. Paris 42 (1856), 287–288. R 292 [38] N.-A. Tissot, Sur les altérations d’angles et de distances dans le développement modifié de Flamsteed. J. Ec. polytech. 21 (1858), 217–225. R 292 [39] N.-A. Tissot, Sur le développement modifié de Flamsteed. C. R. Acad. Sci. Paris 46 (1858), 646–648. R 292 [40] N.-A. Tissot, Sur les cartes géographiques. C. R. Acad. Sci. Paris 49 (1859), 673–676. R 292 [41] N.-A. Tissot, Sur les cartes géographiques. C. R. Acad. Sci. Paris 50 (1860), 474–476. R 292 [42] N.-A. Tissot, Sur les cartes géographiques. C. R. Acad. Sci. Paris 50 (1860), 964–968. R 292 [43] N.-A. Tissot, Démonstration nouvelle du théorème de Legendre sur les triangles sphériques dont les côtés sont très-petits relativement au rayon de la sphère. Nouv. Ann. (2) 1 (1862), 5–11. R 292 [44] N.-A. Tissot, Sur la construction des cartes géographiques. C. R. Acad. Sci. Paris 60 (1865), 933–934. R 292 [45] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2) 17 (1878), 49–55. R 292 [46] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2) 17 (1878), 145–163. R 292 [47] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2) 17 (1878), 351–366. R 292 [48] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2) 18 (1879), 337–356. R 292 [49] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2) 18, 385–397. R 292

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[50] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Nouv. Ann. (2), Supplément au tome 19 (1880), S3–S40. R 292 [51] N.-A. Tissot, Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Gauthier-Villars, Paris, 1881. R 292, 294, 295 [52] N.-A. Tissot, Formules relatives aux foyers des coniques. Nouv. Ann. (2) 13 (1894), 97–98. R 292

Chapter 10

Memories of Herbert Grötzsch Reiner Kühnau (Translated from the German by Annette A’Campo-Neuen)

Contents 1 Biographical notes . . . 2 Notes on the work . . . 3 Grötzsch and Koebe . . 4 Spatial almost conformal References . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (extremal quasiconformal) mappings . . . . . . . . . . . . . . . . . . . .

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301 305 309 310 311

1. Biographical notes Camillo Herbert Grötzsch was born on May 21, 1902 in Döbeln (Saxony) as the son of Emil Camillo Grötzsch (1874–1974) and Ludwiga Julie Grötzsch, née Hünich. Until 1945, his father was headmaster (Oberstudiendirektor) of the high school (Realgymnasium) at Crimmitschau (Saxony). From 1894 until 1899, he studied mathematics in Leipzig and then obtained a PhD in 1898 under the supervision of Sophus Lie. Herbert Grötzsch first went to primary school (Bürgerschule) and then to high school (Realgymnasium) in Döbeln and finally in Crimmitschau, where he graduated at Easter 1922. From 1922 until 1926 he studied mathematics and physics in Jena. He was greatly already influenced by Koebe. When Koebe moved to Leipzig in 1926, he followed him and continued his studies until 1927/28 in Leipzig. In 1927–1929, he held a position of a “substitute of an assistant” (Verwaltung einer Assistentenstelle) at the mathematical institute of Leipzig, then until 1930, he held half of an assistant position that he shared with Hubert Cremer. In 1929, he obtained his PhD with the Dissertation Über konforme Abbildung unendlich-vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten (On conformal mapping of infinitely-multiply connected schlicht domains with finitely many accumulation boundary components) [10]. Koebe wanted him then to write a habilitation in Tübingen where K. Knopp was

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professor. But after having given a talk there, he returned furiously to Leipzig because of Knopp’s condescending behaviour which in turn annoyed Koebe. He had a great aversion for big-noise persons. And he probably, at an early stage, was very much annoyed from the “referees” of his papers. This was the reason why he published his papers in “obscure journals” (Ahlfors). He also suggested to me to publish my papers in “our” journal, as in [36], [37], and [38]. That I did not follow his recommendation was probably a grief for him. Grötzsch then completed his habilitation 1931 in Giessen with his work Über die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche (On the displacement of a schlicht mapping of schlicht domains) [17]. The habilitation lecture entitled Der Gruppenbegriff und einige seiner Anwendungen in den mathematischen Wissenschaften (The notion of group and some of its applications in the mathematical sciences) was characterised as being “of great clarity and instructive and fascinating form.”1 On June 20, 1935, the Reich governor (Reichsstatthalter) in Hessen wrote to the rectorate: “I entrust you to immediately, at the latest on June 30, 1935, dismiss Privatdozent Dr. H. Grötzsch from his position as a scientific assistant at the mathematical seminar of the university of Giessen by October 1.” The cause of these events was the following. Grötzsch was member of the “Jungstahlhelm”2 (which at first still existed though in tension with the NSDAP).3 As such he was “transferred” to the SA,4 where however he refused to serve and resigned without having been put under oath. (He said to the captain (Sturmführer): “I cannot cooperate with people preaching violence,” he refused to sing the Horst-Wessel-Lied,5 etc.) He did not go to the NS-training camps for academic instructors and other than that, he may have provoked annoyance several times by his statements.6

1 The (early) talks of Grötzsch were extremely lively and convincing (by the way, in contrast to talks of H. Grunsky, who even had to repeat his habilitation talk, since he did not succeed in convincing E. Schmidt in the discussion). I learnt this from Willi Rinow in 1968 in Greifswald. Grötzsch was really able to be very eloquent. 2 [Editor’s note] The youth organisation of the Stahlhelm, a paramilitary organization that arose after the German defeat of World War I. 3 [Editor’s note] The Nazi Party: Nationalsozialistische Deutsche Arbeiterpartei, that is, National Socialist German Workers’ Party. 4 Sturmabteilung, that is, the Storm Detachment, which was the paramilitary wing of the Nazi Party. 5 [Editor’s note] A song used as the anthem of the Nazi Party between 1930 and 1945. It was written by Horst Wessel. 6 Prof. Hans Schubert (Halle), with whom Grötzsch had a very close relationship, told me about 40 years ago that the last inducement for the dismissal was an episode in a pub in Giessen: While Grötzsch, one evening, was sitting in the pub having a beer, the rector came around with a collecting box (for political purposes, which was very common at the time), and Grötzsch made a derogative comment.

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After that, and until 1939, Grötzsch was a member of the staff of the J. C. Poggendorffs Handwörterbuch in Leipzig. Because of these new living and working conditions, as far as I know, he never met O. Teichmüller in person. From August 27, 1939 and until his dismissal on May 12, 1944, Grötzsch was member of the armed forces (cannoneer, eventually lance corporal—Obergefreiter—on the Western and Eastern fronts). During a stay at a military hospital, he turned to graph theory since at that time this could still be done (almost) without references. Grötzsch was liberated from active military service in 1944 because of a decree arranging to use scientists in military research. Until the end of the war, he worked in Göttingen at the KW institute for flow research of Prof. Prandtl (calculations of jet engines). I have no information about results from this activity. Anyway, the time he spent in this activity was too short. Until about 60 years ago, he used to talk sometimes about a project of setting up a catalogue of conformal mappings. But he probably soon gave up. Grötzsch once told me that despite the very difficult travel conditions, he visited Paul Koebe who was in a hospital in Leipzig suffering from stomach cancer, shortly before his death on August 6, 1945. After the war, starting from November 16, 1945, Grötzsch got a position as a scientific tutor (“Wissenschaftliche Hilfskraft”) at the mathematical seminar of the University of Marburg. From July 1946 until the spring of 1948, he was lecturer (“Dozent,” and from July 1, 1947, “Diätendozent,” i.e. with a proper salary), and by the end of 1947 he was appointed as an “außerplanmäßiger Professor” (associate professor). In the articles [57], [58], and [59], Horst Tietz (1921–2012) who returned from the KZ7 in Buchenwald, reports on this time in Marburg. Grötzsch kept a close friendship with him all his life. From 1948 until 1965, Grötzsch was “Professor mit vollem Lehrauftrag” (associate professor with full teaching assignment) in Halle. It was only from 1965 until his retirement in 1967 that he was full professor (persönliches Ordinariat).8 I presume that this promotion would have been possible much earlier. But most likely he preferred his independence to a higher salary so that he contented himself with the role of a “Beta” in the mathematical institute. On June 8, 1959, he was elected as a member of Leopoldina, the German Academy of Scientists. In 1967, he also received the National Prize of Class II (after having refused this prize of Class III in 1961). In October 1951, he married Annemarie Jung (1921–2014). She was the daughter of the mathematician H. W. E. Jung (1876–1953) from Halle and a descendant of Caspar Cruciger who was a fellow campaigner of Martin Luther in Leipzig. Helmut Hasse, who obtained his first professorship 1925 in Halle, carried her on his arms as a child. Grötzsch always had a good relationship with Hasse. 7 [Editor’s note] Concentration camp. 8 In the reference [3], the matter of these positions is not clearly formulated; this was kindly observed by Dr. Günther Schiemann (Bitterfeld).

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On June 17, 1953, Grötzsch got in big trouble when he was arrested at a demonstration in Halle against the regime of the GDR9 [50]. The rector of the university was able to save him—I learned this from Grötzsch’s wife. But since this affair, he was extremely cautious with political statements. Probably this was also due to the cruel fate of Fritz Noether (the brother of Emmy Noether) and of Helmut Sonnenschein (a friend from his time in Leipzig, also a student of Koebe): Both were shot by the NKWD10 as alleged spies. Just once when a young “comrade” recommended him to make a request for traveling to the “West” (in fact, to Oberwolfach, from where he received an invitation), he burst out: “I certainly won’t beg for permission to travel!” He declined an acknowledgement as victim of fascism (“they aren’t any better than the Nazis” according to Mrs. Grötzsch, maybe also under the influence of the so-called June 17 affair, that is, the uprise in the GDR starting with the worker’s manifestations fought down by the GDR regime with the help of the Soviet military.) He also didn’t want a special salary (called “Einzelvertrag” at the time), and that was certainly rare; compare [40]. He was not one of the brave “afterwards-antifascists.”11 But after the reunification of Germany, this made it more difficult to calculate his pension and the widow pension of his wife respectively. These difficulties essentially could only be overcome by the great personal commitment of Director Ephemeridum of the Leopoldina. One can see that Grötzsch was not an apolitical person (though he appeared to be so at first to some people) from the observation that in the library among the newly arrived journals he first looked at the “Physical Letters” (“Physikalische Blätter,” which contained a lot of contemporary history at the time) and the International Mathematical Newsletter, and just afterwards at the Mathematical Reviews, etc. Grötzsch decided to decline the title “Dr. habil.” which was also used in the GDR, probably because this title was commonly used in the Nazi period. I was once present when our secretary (who did not know about this attitude of Grötzsch) had to rewrite a complete letter with such a title for the sender. He was of massive, baroque stature. And it was impressive to see him stamping quickly to his lecture in a white gown with a big box of chalk. He was known for his great commitment to the problems of the students. During the difficult period after the war, he even, anonymously, supported some students financially. Because of the distribution of office space at the university of Halle, from 1960 until his retirement in 1967, I had to share the office with him in the “Melanchthonianum.” Through the resulting close contact, I obtained the role of an “Eckermann”12 of Grötzsch. He once told me that after the war when there 9 [Editor’s note] The German Democratic Republic. 10 [Editor’s note] The state organism in the USSR in charge of fighting crime and maintaining public order. 11 [Editor’s note] People who just pretended afterwards to have resisted the Nazi regime to draw advantages from it. 12 [Editor’s note] Goethe’s secretary.

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was a shortage of heating material, his mother had burnt almost all his reprints (“a big pile,” and he made a corresponding gesture). So I am very proud that one day, from his small remains he brought me a complete collection of the works [6] to [35] provided with small (not very important) handwritten supplements. On each reprint, he had written a number with pencil, whose order corresponds precisely to the order in the bibliography below. By the way, according to what he said, he had written all his articles himself with a typing maching. He looked forward to his retirement for a long time announcing then to start working properly. However, after 1962 he did not succeed in publishing any further article, and I do not understand the reasons. The opening up of his assets (almost exclusively on graph theory) in the university and state library of lower Saxony in Göttingen (“Niedersächsische Universitäts- und Landesbibliothek in Göttingen”) is going to be extremely difficult. On the occasion of his 75th birthday in 1977, Lipman Bers gave a lecture. Unfortunately, Grötzsch then already had great difficulty leaving his appartment (because of problems with his knee), so that he could not come to the talk himself. Herbert Grötzsch died on May 15, 1993 in his appartment in Halle, Mozartstrasse 22. A detailed obituary is given in [48].

2. Notes on the work Now we restrict ourselves to the results leading to the Grötzsch–Teichmüller theory in geometric complex analysis (without aiming for completeness). For results on quasiconformal mappings, this has already been done in [47], [48], and [49].13 On Grötzsch’s results on graph theory, Horst Sachs has reported in [55]. If today one looks at how painfully Grötzsch arrives to his result in his first article [6] (Ahlfors: “Confusingly stated”), then it is hard to believe that this is indeed the starting point of his theory. He later notes in [9] that the Cauchy–Schwarz inequality is the basic principle. (Later, the arising fundamental inequality was often called Rengel’s inequality by mistake.) Anyhow, one has to admit that in his original proof in [6] the “inner reasons” become clearer. As Grötzsch himself once mentioned, he considered as his main achievement that with his method, for the first time, he could also solve extremal problems for multiply connected domains in a geometric complex analytical way. Up to now it has been less noted that he also solved extremal problems for conformal mappings of non-overlapping systems of domains in the same way. That’s in the nature of his surface strip method. 13 [Editor’s note] English translations of several papers of Grötzsch on conformal geometry and quasiconformal mappings are contained in the present volume, together with commentaries.

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Figure 1. Herbert Grötzsch (year unknown)

Figure 2. P. M. Tamrazov (Kiev), V. Ya. Gutlyanskii (Donetsk), H. Grötzsch, P. P. Belinskii (Novosibirsk). Photo taken in Grötzsch’s apartment in 1980.

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He considered his articles on what we call today “quasi-conformal mappings” to be part of differential geometry. Apart from Koebe and may be Bieberbach, the first persons who comprehensively recognised the importance of Grötzsch’s work were Ahlfors and Teichmüller. Later, Jenkins contributed crucially to the popularization of his work with his report [39]. For Grötzsch, this was the cause of delight and satisfaction; cf. a little remark added by Grötzsch in [41] at the end (concerning [39]): “Ist umfassender neuester Bericht über die Theorie der schlichten Funktionen (Zusatz von H. Grötzsch.)” (This is a comprehensive newest report about the theory of schlicht functions. Added by H. Grötzsch.) Comparing the articles of Grötzsch and Teichmüller, one can perceive different ways of mathematical thinking (or maybe better: different ways of feeling), both having their eligibility and value. There is the elementary intuition on the one hand (as in the early papers by Grötzsch) and the abstraction on the other hand (as in papers by Teichmüller who learned in Göttingen as a student also the abstract directions), leading to more clarity but being rarely at the origin. Grötzsch was more and more interested in “Ways of mathematical thinking.”14 However, that did not lead to very systematic presentations in his later lectures. Grötzsch’s way of thinking is also expressed in the following citations that I picked up at various occasions or that he noted on slips of paper. • “New notions should not be introduced unless there is a need for them (after Koebe).” • “The fact that a mathematical investigation is formally logically correct does not mean automatically that it is a valuable contribution to our mathematical knowledge.” • “We must get away from formalism in mathematics and get back to the contents and the concrete realization of mathematical notions by individual representatives.” • “Mathematics should get easier” (after Koebe). • “If you don’t feel it, you won’t catch it” (after Goethe). • In a spontaneous conversation with a small group: “Yeah, van der Waerden was no algebraist but a mathematician,” so that Grötzsch himself was startled by his own comment and some of the people present looked grimly. • “Unfortunately the time of beautiful accessible problems is over.” • “Probably the big problems will be solved but the seemingly small ones are the challenge.” 14 In his estate, there is a handwritten 6 -pages manuscript for a talk entitled “Experience and Thinking in Mathematics” (“Erfahrung und Denken in der Mathematik”) whose date is unclear (maybe from the time right after the war).

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By the way, though in mathematics he had clearly a very broad interest, he did not have any close relationship to physics. It is strange that today many people associate the name “Grötzsch” only to the ridiculous “Grötzsch-ring” (or the name “Koebe” to the ridiculous Koebe-function). In [25], Grötzsch started thinking about formulating ordering principles for his articles and extremal problems. He considers the number of normalizations. This corresponds to the possibility of introducing an order by counting the number of poles and zeros of a quadratic differential. This was done later, e.g., in [45] for the simplest cases of the disc and the projective plane as “support.”15 Grötzsch’s results were always purely “theoretical” (“geometric complexanalytic”) in the sense that in general they do not provide the extremal functions explicitly by analytical formulae nor the precise bounds for the considered functionals. This problem was stated in [27] as cura posterior (though Grötzsch was not particularly interested in it) and then later dealt with in [44]. Here, quadratic differentials for the description of extremal functions turn out to be useful. They give rise to a presentation of the extremal functions by a kind of Schwarz–Christoffel formula. However, the occurring parameter in general complicates the calculations. (Therefore, hitherto, one could not prove the Bieberbach conjecture with this method or the variational method.) The formulae that arise also fit into the general topic of “Identities for the mapping functions,” which was already studied comprehensively in [5]. By the way, Grötzsch may have mentioned such an identity for the first time in [20]: Simple relations between parallel slit mappings associated to different directions. Grötzsch was first surprised by my habilitation [44]. He himself had only made isolated attempts to find formulae for the final result of these distortion theorems, compare, e.g., [46]. Also in [27] the phenomenon of “bifurcation” (“Verzweigungserscheinung”) occurs. Here in certain cases (but not always) infinitely many (even an entire continuum) solutions of the corresponding extremal problem can occur; in [44] one can find a more detailed discussion of the simplest non-trivial case, compare also [47]. This bifurcation complicates the existence proof with the continuity method substantially. Let us remark that Grötzsch with respect to the problem of quadratic differentials often pointed out that even Koebe (without using the term “quadratic differential”) implicitly mentioned this analytic description of extremal slit domains, namely in [13]. The article [2] contains some remarks on the relationships between Ahlfors, Teichmüller, Strebel and others with Grötzsch, and biographical notes. Further remarks on Grötzsch and his work can be found among others in [1], [4], and [51]. Grötzsch had kept contact with the Russian literature at an early stage. Stimulated by his father, he had learnt some Russian even before the war. For example, he 15 “Träger,” this expression was introduced by Grötzsch for this purpose.

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exchanged reprints with M. A. Lawrentjew. A. W. Bizadse told me in April 1968 in Novosibirsk that Lawrentjew discussed the articles of Grötzsch in his seminar. Moreover, Lawrentjew, in his first article on quasi-conformal mappings, adopted Grötzsch’s notation AQ from [8].16 Grötzsch initiated the German translation of the Russian book by Golusin as well as the book by Lawrentjew and Shabat.

3. Grötzsch and Koebe Grötzsch took pride in being a student of Paul Koebe (1882–1945).17 Among others, the following important aspects are due to Koebe’s influence. • He inspired Grötzsch to also consider “non-conformal” mappings. • From him, Grötzsch learnt the method to construct his isothermal family of curves (i.e. the trajectories of quadratic differentials) with Riemannian manifolds, that were always considered here as given explicitly as triangulated manifold. • In his existence proofs for extremal functions, Grötzsch used Koebe’s continuity method competently. However, to deal with bifurcation as in [27] (already announced in [26]) in order to apply the continuity method, one needs to make difficult modifications since the needed mappings are not always unique under the normalizations (as, for example in the case of the parallel slit theorem). Grötzsch has discussed that very briefly only, leaving the details to the reader, therefore his ideas are difficult to follow. Jenkins told me once that he had problems with that. Those problems were the reason for which Udo Pirl asked a PhD student to work out the details (though for a different extremal problem) (see [53]). One must admit that it is easier to use the variational method here, even though again the unity question remains open. Grötzsch did not want to write an obituary for Koebe since though he would have been a natural candidate, he was not considered as Koebe’s successor in Leipzig. But he once said excitedly that if he would have to write an obituary for Koebe then it would be: “Obituary for Paul Koebe—he does not need it!!!” In 1961 or 1962, one day Grötzsch came into our office in the institute seemingly rather upset and put a piece of paper on my desk. He told me that the day before in the journal reading room of the university he had seen the article by Edgar Reich [54] referred to on the piece of paper: the author had found 16 As Lawrentjew told me 1968 in Novosibirsk, he could speak German very well since until 1914 he had lived in Göttingen with his father who was a physicist. At his return to Kazan in 1914, he was even more fluent in German than in Russian. [Editor’s note] A biography of Lavrentieff may be found in the present article, including information on his two-year stay in Europe (Göttingen and Paris) with his parents, see [52]. 17 By the way, also L. Bieberbach considered himself a student of Koebe.

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a mistake made by Koebe. Of course the piece of paper on my desk meant that he wanted me to investigate the matter and I did that immediately. This led to the article [43]. After a long and tedious discussion of the manuscript, I was relieved to find out the following: in the construction of a “minimal slit domain” Koebe had indeed overlooked that its boundary is not compact, but when adding the accumulation points the problem was solved directly. Grötzsch explained that this strange misunderstanding of Koebe might be due to the psychological situation at the end of the first world war. Through this “rescue” of Koebe, Grötzsch was very happy. Koebe privately subscribed to all mathematical journals of importance at the time, so that he may have had one of the largest private mathematical libraries. Grötzsch often complained melancholically: “Where may Koebe’s library be now?”18 Here is a last remark on Koebe. In his long articles he had a rather colloquial style which seems nowadays very unusual. We have the big Desideratum: How can we open up Koebe’s rich work? For example Grötzsch around 1960 drew my attention to an article of Koebe from 1936 where he already considers “circle packings” that have been rediscovered much later.

4. Spatial almost conformal (extremal quasiconformal) mappings Of course, Grötzsch also considered at an early stage the question of how in three-dimensional space the problem of extremal quasiconformal mappings could be dealt with. In Halle, after the war, Dr. Eduard Zimmermann,19 a refugee from Reichenberg, Sudetenland, had found a position at the University of Halle. Grötzsch wanted to help him obtaining a habilitation (which did not go through) and advised him to study this problem of spatial mappings leading to the articles [60] and [61]. Then I was able to show [42], that for spatial mappings, in general, the extremal quasiconformal mappings are not uniquely determined, in contrast to the planar case. I gave Grötzsch a first version of [42] in February 1961. But only in February 1962 we discussed about it and then the article [42] appeared in a much shortened version. He probably was very disappointed about the result. Certainly, this is also 18 Around 1997, Prof. Herbert Beckert (Leipzig) told me that during the war Koebe’s library for safety reasons was kept in a factory (HASAG) in Leipzig for bazookas. Therefore after the war it was transported to the Soviet union (together with German “specialists”). In 2002, I learnt from Prof. Thomas Riedrich (Dresden) that parts of Koebe’s library reappeared in Dresden. 19 By the way, he knew Karl Löwner from his time as a student in Prague. However, he dismissed him with the brief comment “O, this Löwner was very special.” But Grötzsch had greatest respect for the method of Löwner’s differential equation which for him was a “book with seven seals.”

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related to the fact that in the meantime he was thinking about very different topics (graph theory). Probably here one has to resign and follow the salomonic and nebulous remark of Hilbert (made in a different context) on the strange special role of dimension 2 : “Time is one-dimensional, space is three-dimensional, number, i.e. the perfect complex number, is two-dimensional” (compare [56], §19). Notes on the bibliography that follows Leipz. Ber. = Berichte über die Verhandlungen der Sächsischen Akad. d. Wiss. zu Leipzig, math.-phys. Klasse. SB preuss. Akad. = Sitzungsberichte d. preuss. Akad. d. Wiss., phys.-math. Klasse. WZ Halle = Wiss. Zeitschr. d. Martin-Luther-Univ. Halle-Wittenberg, Math.-Nat. Reihe. The papers of H. Grötzsch are listed in the order chosen by himself (most probably corresponding to the order of preparation). Further sources • Personalakte Dr. Herbert Grötzsch im Archiv d. Univ. Giessen (Sign.: PrA Phil № 10). • Marburg: Staatsarchiv Marburg, Best. 307d, Acc. 1966/10, № 115. • Military Government of Germany, Fragebogen MG/PS/G/9a (Rev: 15 May 45). • Personalakte Grötzsch im Archiv d. Univ. Halle (PA 24733). Acknowledgments. The author would like to thank Annette A’Campo (Basel) for the careful translation of this article into English. The German original can be requested from the author.

References [1] L. V. Ahlfors, Classical analysis: Present and future. In A. Baernstein II, D. Drasin, P. Duren, and A. Marden (eds.), The Bieberbach conjecture. Proceedings of the symposium on the occasion of the proof of the Bieberbach conjecture held at Purdue University, West Lafayette, IN, March 11–14, 1985. Mathematical Surveys and Monographs, 21. American Mathematical Society, Providence, RI, 1986, 1–6. MR 0875227 R 308 [2] N. van Arkel, Book review, Nieuw Archief voor Wiskunde IV (1986), Ser. 4, 51–58. R 308 [3] Complex Beauties 2017 (April for H. Grötzsch), a Calendar by E. Wegert, G. Semmler (TU Bergakademie Freiberg). R 303

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[10] H. Grötzsch, Über konforme Abbildung unendlich-vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten. Leipz. Ber. 81 (1929), Dissertation, 51–86. JFM 55.0792.03 R 301, 305 [11] H. Grötzsch, Über die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender schlichter Bereiche. II. Leipz. Ber. 81 (1929), 217–221. JFM 55.0794.01 R 305 [12] H. Grötzsch, Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche. Leipz. Ber. 82 (1930), 69–80. English translation by M. Karbe, On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 16, 375–385. JFM 56.0298.02 R 305 [13] H. Grötzsch, Über ein Variationsproblem der konformen Abbildung. Leipz. Ber. 82 (1930), 251–263. JFM 56.0298.03 R 305, 308 [14] H. Grötzsch, Zur konformen Abbildung mehrfach zusammenhängender schlichter Bereiche. (Iterationsverfahren). Leipz. Ber. 83 (1931), 67–76. JFM 57.0400.05 R 305 [15] H. Grötzsch, Zum Parallelschlitztheorem der konformen Abbildung schlichter unendlichvielfach zusammenhängender Bereiche. Leipz. Ber. 83 (1931), 185–200. JFM 57.0401.01 Zbl 0003.01401 R 305 [16] H. Grötzsch, Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche. Leipz. Ber. 83 (1931), 238–253. JFM 57.0401.02 Zbl 0003.26101 R 305

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[17] H. Grötzsch, Über die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche. Leipz. Ber. 83 (1931), Habilitationsschrift, 254–279. JFM 57.0402.01 Zbl 0003.26102 R 302, 305 [18] H. Grötzsch, Über die Verzerrung bei schlichter konformen Abbildung mehrfach zusammenhängender schlichter Bereiche. III. Leipz. Ber. 83 (1931), 283–297. JFM 57.0402.02 Zbl 0004.29903 R 305 [19] H. Grötzsch, Über Extremalprobleme bei schlichter konformer Abbildung schlichter Bereiche. Leipz. Ber. 84 (1932), 3–14. JFM 58.0364.01 Zbl 0005.06801 R 305 [20] H. Grötzsch, Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche. Leipz. Ber. 84 (1932), 15–36. JFM 58.0364.02 Zbl 0005.06802 R 305, 308 [21] H. Grötzsch, Über möglichst konforme Abbildungen von schlichten Bereichen. Leipz. Ber. 84 (1932), 114–120. English translation by M. Brakalova-Trevithick, On closest-to-conformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 17, 387–392. JFM 58.1093.02 Zbl 0005.06901 R 305 [22] H. Grötzsch, Über die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche. II. Leipz. Ber. 84 (1932), 269–278. JFM 58.0365.01 Zbl 0006.17103 R 305 [23] H. Grötzsch, Über zwei Verschiebungsprobleme der konformen Abbildung. SB preuss. Akad. 1933, 87–100. JFM 59.0349.01 Zbl 0006.17104 R 305 [24] H. Grötzsch, Die Werte des Doppelverhältnisses bei schlichter konformer Abbildung. SB preuss. Akad. 1933, 501–515. JFM 59.0350.01 Zbl 0007.21404 R 305 [25] H. Grötzsch, Über die Geometrie der schlichten konformen Abbildung. SB preuss. Akad. 1933, 654–671. JFM 59.0350.02 Zbl 0007.31204 R 305, 308 [26] H. Grötzsch, Verallgemeinerung eines Bieberbachschen Satzes. Jber. d. Deutsch. Math.Verein. 43 (1934), 143–145. JFM 59.0351.01 R 305, 309 [27] H. Grötzsch, Über die Geometrie der schlichten konformen Abbildung. Zweite Mitteilung. SB preuss. Akad. 1933, 893–908. JFM 59.1045.04 Zbl 0008.31902 R 305, 308, 309 [28] H. Grötzsch, Über die Geometrie der schlichten konformen Abbildung. Dritte Mitteilung. SB preuss. Akad. 1934, 434–444. JFM 60.0288.01 Zbl 0011.31301 R 305 [29] H. Grötzsch, Über Flächensätze der konformen Abbildung. Jber. d. Deutsch. Math.-Verein. 44 (1934), 266–269. JFM 60.0288.02 Zbl 0011.12202 R 305 [30] H. Grötzsch, Einige Bemerkungen zur schlichten konformen Abbildung. Jber. d. Deutsch. Math.-Verein. 44 (1934), 270–275. JFM 60.0289.01 Zbl 0010.30802 R 305 [31] H. Grötzsch, Zur Theorie der schlichten konformen Abbildung schlichter Bereiche. Leipz. Ber. 87 (1935), 145–158. JFM 61.0363.03 Zbl 0013.02802 R 305 [32] H. Grötzsch, Zur Theorie der schlichten konformen Abbildung schlichter Bereiche. (2. Mitteilung). Leipz. Ber. 87 (1935), 159–167. JFM 61.0363.03 Zbl 0013.02803 R 305 [33] H. Grötzsch, Eine Bemerkung zum Koebeschen Kreisnormierungsprinzip. Leipz. Ber. 87 (1935), 319–324. JFM 61.1156.01 Zbl 0014.16602 R 305 [34] H. Grötzsch, Zur Theorie der Verschiebung bei schlichter konformer Abbildung. Comment. Math. Helv. 8 (1935), no. 1, 382–390. MR 1509535 JFM 62.0385.02 Zbl 0014.26705 R 305 [35] H. Grötzsch, Zur Geometrie der konformen Abbildung. Hallische Monographien, 16. Max Niemeyer Verlag, Halle (Saale), 1950, 5–11. MR 0037911 Zbl 0038.23403 R 305 [36] H. Grötzsch, Konvergenz und Randkonvergenz bei Iterationsverfahren der konformen Abbildung. WZ Halle 5 (1956), 575–582. MR 0090644 Zbl 0071.29301 R 302

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[37] H. Grötzsch, Zum Häufungsprinzip der analytischen Funktionen. WZ Halle 5 (1956), 1095–1100. MR 0083559 Zbl 0071.07201 R 302 [38] H. Grötzsch, 16 “Mitteilungen” unter dem gemeinsamen Obertitel “Zur Theorie der diskreten Gebilde.” WZ Halle from 5 (1956) to 11 (1962). R 302 [39] J. A. Jenkins, Univalent functions and conformal mapping. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, 18. Moderne Funktionentheorie. Springer-Verlag, Berlin etc., 1958. MR 0096806 Zbl 0083.29606 R 307 [40] R. Jessen, Akademische Elite und kommunistische Diktatur. Die ostdeutsche Hochschullehrerschaft in der Ulbricht-Ära. Vandenhoeck & Ruprecht, Göttingen, 1999. R 304 [41] R. Kühnau, Berechnung einer Extremalfunktion der konformen Abbildung. WZ Halle 9 (1960), 285–287. MR 0151589 Zbl 0089.28901 R 307 [42] R. Kühnau, Elementare Beispiele von möglichst konformen Abbildungen im dreidimensionalen Raum. WZ Halle 11 (1962), 729–732. MR 0142.33301 R 310 [43] R. Kühnau, Über ein Koebesches Beispiel zur Theorie der minimalen Schlitzbereiche. WZ Halle 14 (1965), 319–321. MR 0199361 Zbl 0156.30102 R 310 [44] R. Kühnau, Über die analytische Darstellung von Abbildungsfunktionen, insbesondere von Extremalfunktionen der Theorie der konformen Abbildung. J. Reine Angew. Math. 228 (1967), 93–132. MR 0224795 Zbl 0176.37701 R 308 [45] R. Kühnau, Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene. Mathematische Forschungsberichte. 28. VEB Deutscher Verlag der Wissenschaften. Berlin, 1974. Zbl 0278.30022 R 308 [46] R. Kühnau, Zur ebenen Potentialströmung um einen porösen Kreiszylinder. Z. Angew. Math. Phys. 40 (1989), no. 3, 395–409. MR 1002317 Zbl 0688.76069 R 308 [47] R. Kühnau, Einige neuere Entwicklungen bei quasikonformen Abbildungen. Jahresber. Deutsch. Math.-Verein. 94 (1992), no. 4, 141–169. MR 1190209 Zbl 0758.30015 R 305, 308 [48] R. Kühnau, Herbert Grötzsch zum Gedächtnis. Jahresber. Deutsch. Math.-Verein. 99 (1997), no. 3, 122–145. MR 1466369 Zbl 0871.01014 R 305 [49] R. Kühnau, Some historical commentaries on Teichmüllers paper “Extremale quasikonforme Abbildungen und quadratische Differentiale.” In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VI. IRMA Lectures in Mathematics and Theoretical Physics, 27. European Mathematical Society (EMS), Zürich, 2016, 537–546. MR 3618200 Zbl 1350.01014 R 305 [50] H.-P. Löhn, Der 17. Juni 1953. In H.-J. Rupieper (ed.), Beiträge zur Geschichte der Martin-Luther-Univ. Halle-Wittenberg 1502–2002. Mitteldeutscher Verlag, Halle (Saale), 2002, 559–595. R 304 [51] M. Pinl, Kollegen in einer dunklen Zeit. Jber. Deutsch. Math.-Verein. 71 (1969), no. 1, 167–228. MR 0476356 Zbl 0182.00102 R 308 [52] A. Papadopoulos, A note about Mikhaïl Lavrentieff and his world of analysis in the Soviet Union. With an appendix by G. Sinkevich. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 11, 317–347. R 309 [53] W. Pohl, Aus Extremalproblemen über dem Wertebereich f 0 (P ) folgende Normalformen dreifach zusammenhängender Gebiete bezüglich einer Klasse schlichter Funktionen, die P und zwei Randkomponenten festhalten. Dissertation, Math.-Nat. Fakultät, HumboldtUniversität zu Berlin, 1974. R 309

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Chapter 11

A note about Mikhaïl Lavrentieff and his world of analysis in the Soviet Union Athanase Papadopoulos (with an appendix by Galina Sinkevich)

Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Family . . . . . . . . . . . . . . . . . . . . . . . . . . . Luzin . . . . . . . . . . . . . . . . . . . . . . . . . . . Back to Lavrentieff: his mathematics . . . . . . . . . . Siberia . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanics and engineering . . . . . . . . . . . . . . . . Appendix. Some clarifications on the history of Russian of the 20 th century (G. Sinkevich) . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mathematics . . . . . . . . . . . . . .

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317 318 320 330 335 336

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1. Introduction This is a one-sided report on the life and work of Mikhaïl Alekseïevich Lavrentieff 1 (1900–1980), one of the main foundeHe held many honorific positions, and some of them were with much responsibilityrs of the theory of quasiconformal mappings. Besides a few notes on Lavrentieff’s work on the theory of functions of a complex variable, the report includes information on on some aspects of his life and some of the persons who influenced him. Lavrentieff had a densely packed academic life, often with heavy administrative duties, and he interacted with a diverse collection of people in the Soviet Union. He held many honorific positions, and some of them were with much responsibility: he was Vice-President of the Ukrainian Academy of 1 Like all the Russian similar names, Lavrentieff may also be transliterated with a terminal v instead of ff. We are following Lavrentieff’s own transliteration of his name in his papers written in French, see [38], [39], [41], [43], [44], and [45] (there are several others). Lavrentieff was fluent in French.

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Sciences (1945–1948), Vice-President of the USSR Academy of Sciences, President of the Siberian Division of that Academy (1957–1975), and Vice-President of the International Mathematical Union (1966–1970). He played a preeminent role in the creation of Akademgorodok, the well-known science town in Siberia, which was founded in 1957. In more political settings, he was an elected deputy of the Supreme Soviet of the Ukraine SSR (1947–1951) and of the Supreme Soviet of the USSR (1958–1980). The present report will not do justice to Lavrentieff from the purely scientific point of view because his work encompasses much more than the topics we review here. In fact, Lavrentieff was one of the most preeminent 20 th century scientists of the Soviet Union. He had a brilliant career, as a mathematician, as a physicist, and as a leader in science organisation. In mathematics, he worked on functions of one and several complex variables, on conformal and quasiconformal geometries, on partial differential equations, on nonlinear problems, and on the calculus of variations, but perhaps his most important achievement was the broad use he made of mathematics in the other sciences. With his profound intuition, Lavrentieff applied the mathematical tools he developed (in particular the theory of quasiconformal mappings) to solve difficult problems in physics and engineering. The topics included flows around air wings,2 shock waves, the motion of a plate under the surface of a liquid (in particular the motions of diving planes of submarines) and the resistance of buildings to the flow of ground water. He was the first to understand that matter (metal) behaves like an incompressible fluid under the effect of an explosion, an idea which led to several technological applications of explosives, like the construction of dams by blasting. He was responsible for the construction of several networks of soil projects which permitted the regulation of lakes and rivers, preventing natural catastrophes in several regions of the USSR. We shall report briefly on some of these achievements. Beyond the mathematics and physics of Mikhaïl Lavrentieff, we have tried to give an idea of his creative style, of the incredible amount of energy that emanated from him, and on his ability to find connections between abstract mathematical theories and concrete physical situations. The personal life of Lavrentieff and his mathematical training are tied to an important chapter of the history of mathematics in the Soviet Union: the Luzin school and the “Luzin affair,” and I have included in this report a section on that subject.

2. Family The information we have on the history of the Lavrentieff family begins with the birth of Mikhaïl Lavrentieff’s father. Our description of this event and of the few 2 These were times where aircraft design was in a period of rapid development in the Soviet Union and elsewhere.

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years that followed it is based on the collection of papers The age of Lavrentieff [19] which appeared on the occasion of Lavrentieff’s hundredth anniversary.3 Mikhaïl’s father, Alexey Lavrentievich Lavrentieff, was born in 1875, as an illegitimate child, in Paris, where his mother, who was Russian, was sent to give birth. His names were given to him in the Russian orthodox church of Paris by the priest who baptized him, whose own name was Lavrentii (Lawrence). Deriabina [18] tells us that Alexey Lavrentievich never knew his mother and was brought up in a foster family. His childhood was so difficult that he never wanted to talk about it. He married, at the age of 20, a woman who was 12 years older than he was, and their son Mikhaïl Lavrentieff was born at the turn of the twentieth century. Alexey Lavrentievich studied mathematics and became a mathematics teacher at the technical school of Kazan, the capital of Tatarstan. In 1910, he passed the Master’s exam in mechanics at Kazan University. This is a provincial university, but it is also the university where Lobachevsky studied and spent his whole career. After obtaining his diploma, Alexey Lavrentieff was sent for two years of study in Göttingen and Paris, the two cities which were the world mathematical centers. In those times, it was common that a talented Russian young scientist go to France or to Germany, for a few years of training. Lavrentieff ’s family accompanied him. Mikhaïl Lavrentieff, who was ten years old in Göttingen, recalls in his memoirs [19] that at school, he was frequently harassed by his fellow pupils who used to point their fingers at him, shouting: “Russian, Russian.” Even the teachers were harsh; they hit him because of his lack of knowledge in German. His parents eventually stopped sending him to school; instead, each night his father read to him, in German, the tales of the Brothers Grimm. The Lavrentieffs socialized with the Russians living in Göttingen; accompanied by his wife, Nikolai Luzin was there, studying with Edmund Landau. The young Mikhaïl was fascinated by the conversations he heard at home concerning mathematical problems and new theories. On his return from Paris, Alexey Lavrentieff was appointed to a position on the Faculty of Mathematics and Physics of Kazan University. He remained in contact with Luzin, who, after his stay in Göttingen, spent a few months in Paris and then returned to Moscow. In 1921, Luzin suggested that the Lavrentieff family move to Moscow. In 1922, after obtaining his undergraduate diploma, Mikhaïl became a graduate student at the Institute for Mechanics and Mathematics of Moscow State University and was supervised by Luzin. Egorov was teaching there and in 1923, he became director of the Institute; he noticed Lavrentieff’s talent during an examination session and later assisted him and helped prepare him for his career as 3 The collective book The age of Lavrentieff contains valuable information about Mikhaïl Lavrentieff’s life and epoch. It is written by his former students and colleagues, and it also contains Lavrentieff’s own memories. Let me also mention the paper “Lavrentieff” [18] by Angelina Deriabina. I am grateful to Valerii A. Galkin and Mikhaïl Lavrentiev Jr. for providing these two references.

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a researcher. During his studies, Lavrentieff attended courses on function theory by Luzin, and seminars on topology by P. S. Alexandrov, who was Luzin’s student. The relation between Luzin and the Lavrentieff family continued. Mikhaïl Lavrentieff remained close to Luzin whose life was marred with tragic events; we shall outline some of them in the next section.

3. Luzin Nikolai Nikolaïevich Luzin (1883–1950) together with his mentor, Dmitri Fiodorovich Egorov (1869–1931),4 are the founders of the Moscow school of function theory. Luzin was one of the architects of “descriptive set theory,” a field closely related to topology whose goal was to set up firm foundations for the topological and set-theoretical methods of the theory of functions of a real variable. This field emerged as one of the most active research topics in the first quarter of the twentieth century. The leaders in the field included Cantor (who, by the way, was Russian-born), Borel, Baire, Hadamard, Lebesgue, Denjoy, Painlevé, Hausdorff and Hilbert. It is worth emphasizing here that some of the greatest geometers were reluctant to search for functions with very unusual behavior that this modern theory of functions led to. We can quote here Poincaré who wrote in his book Science et méthode [74] (§V), talking about this field: Logic sometimes generates monsters. The last half-century saw the emergence of a horde of bizarre functions whose endeavor seems to be that of having the least possible resemblance with the honest functions that are useful. No more continuity, or continuity but no derivative, etc. […] In the old days, when a new function was invented, it had some practical aim; today, we invent them intentionally in order to destroy our father’s reasonings, and this is all we can get from them.5

We also recall Hermite’s words in a well-known letter to Stieltjes, dated May 20, 1893 [4], p. 318: “Analysis takes away with one hand what was given with the other. I move away with horror from that dreadful plague of continuous functions that have no derivative.” 6 4 Demidov, in his paper On an early history of the Moscow school of theory of functions [13], considers that the Moscow school of theory of functions was born with Egorov’s Comptes Rendus note Sur les suites des fonctions mesurables [20]. 5 La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir une foule de fonctions bizarres qui semblent s’efforcer de ressembler aussi peu que possible aux honnètes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivée, etc. […] Autrefois, quand on inventait une fonction nouvelle, c’était en vue de quelque but pratique ; aujourd’hui on les invente tout exprès pour mettre en défaut les raisonnements de nos pères et on n’en tirera jamais que cela. 6 L’Analyse retire d’une main ce quelle donne de l’autre. Je me retourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n’ont pas de dérivées.

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Borel, Baire and other French mathematicians among those we mentioned spent some effort putting some order in the multitude of new functions that were discovered and whose properties were shaking the classical theory of functions. They introduced the language of set theory in analysis and they tried a classification of functions in terms of set-theoretic properties and limit operations. Set theory was an emerging field which, with its pseudo-logical apparatus, gives the impression of a firm ground. One may recall, as an example of a new object introduced, the notion of “Baire function,” obtained from continuous functions by a transfinite repetition of the operation of taking simple limits. A foundational crisis had arisen in set theory, and more generally, in mathematics; topology and the theory of functions were affected by it. In particular, Luzin was concerned with the difficulties that arise from the use of transfinite constructions and the continuum hypothesis, from the discovery of hierarchies between infinities, from the formulation of the axiom of choice,7 etc. The era was also marked by conflicting priority claims and other disputes, especially in France; for example, Lebesgue and Borel had noteworthy differences regarding the theories of measure and integration. There were other controversies concerning intuitionism, constructive vs. non-constructive mathematics, etc. The problem of the non-contradiction of the foundations of mathematics was seriously raised. In fact, in these years, mathematics was closely related with philosophy. This was not new; the relation between the two fields dates back to Greek Antiquity, but its appearance at the forefront of research was episodic. Constructions that use infinite processes were carefully revised, from the philosophical as well as from the mathematical viewpoint. It was during that period that Russell and Whitehead showed how to write an incoherent book on the foundations; see the comments by Grattan-Guinness in [26]. Religion was also present in the philosophico-mathematical debate, in France8 and at a much higher level of acuteness in Russia, where a group of mathematicians who were profoundly religious raised questions such as to what extent infinity—which, for them, is an attribute of God—can be used in human contexts, like the sciences. Set theory combined with the theory of functions were concerned with the general classification of sets and their behavior under various operations and mappings. Open and closed sets were the basic sets of topology, but these sets are not stable under simple operations. Gı - and Fı -sets were introduced as countable unions (respectively intersections) of open (respectively closed) sets. Other new 7 I wrote “formulation” instead of “discovery” because before Zemelo formulated this principle as an axiom, it was used without any special notice in mathematics. Baire, Borel, Hadamard and Lebesgue are among the mathematicians who refused the use of the axiom of choice, considering it as counter-intuitive. According to Lebesgue, in order for a mathematical object to exist, one has to able to explicitly “name” a property that defines it in a unique way. See the interesting historical account made by Bourbaki on this matter in [7], p. 53ff. 8 Hermite and Appell are representative of the religious stream in France.

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classes of sets arose: A-sets (A stands for “analytic”),9 B-sets (B stands for Borel),10 Baire first category sets and second category sets,11 Luzin sets,12 etc. Lavrentieff showed the topological invariance of Suslin sets and Borel sets. Lebesgue did not exclude the existence of sets that are neither finite nor infinite. Luzin introduced the notion of projective set, that is, a set obtained from a B-set in n -dimensional Euclidean space by operations of projection and taking complement. Many other classes of sets were given names. Some of these sets played an important role in the formulation of the bases of measure and integration theory. Notions of named sets (introduced by Lebesgue) and effective sets (sets whose definition does not use the axiom of choice) were also introduced. At the time Luzin was studying mathematics, he traversed a spiritual crisis that shaked what he called his “materialistic worldview,” an expression which, at that epoch, meant atheism.”13 The crisis lasted three years and, presumably, was eventually resolved with help of Pavel Florensky (1882–1937), an extremely talented fellow student in mathematics at the University of Moscow who had strong philosophical and mystic inclinations. The relation between the two men was decisive, and it led to Luzin’s recovery of a peace of mind and his return to mathematics. A few words about Florensky are in order. Florensky entered the mathematics division of Moscow University in 1900, one year after Luzin—this was also the year Lavrentieff was born. Among Florensky’s teachers were Nikolai Bugaev, an outstanding mathematician who studied under Weierstrass and Kummer in Berlin, and Liouville in Paris. Bugaev was also Egorov’s mentor. He belonged to a group of mathematicians with strong religious beliefs, those who later were accused of being “reactionaries” and “enemies of the people.” Bugaev was also the father of Boris Nikolaevich Bugaev, the writer and poet who published under the pseudonym Andrei Bely and who is sometimes considered as 9 Analytic sets were later called Suslin sets. Suslin was a student of Luzin who died at the age of 25, in 1919, from typhus, an epidemic which followed the Russian Civil War. The existence of these sets started with a remark he made, saying that a continuous image of a Borel set is not necessarily a Borel set, correcting a mistake made by Lebesgue. He was led therefore to define a class of sets which became these “analytic sets” (see the 4-page note [80] which is the only work he published during his lifetime). Luzin, who gave Suslin the motivation to introduce these sets, continued the latter’s work on these sets after his death. 10 Borel sets are subsets of an n -dimensional Euclidean space that are obtained from segments by repeated application of the operations of taking countable union and countable intersection. 11 A Baire first category set is a subset of a Baire topological space contained in a countable union of closed subsets which all have empty interior. A Baire second category set is a subset of a Baire topological space which is not a Baire first category set. 12 A Luzin set is a subset of the real numbers which is uncountable but whose intersection with every set of the first Baire category is countable. 13 In a letter sent to P. A. Florensky sent on May 1st , 1906, Luzin writes: “The worldviews that I have known up to now (material worldviews) absolutely do not satisfy me. I may be wrong, but I believe there is some kind of vicious circle in all of them, some fatal reluctance to accept the contingency of matter, some reluctance, which I find absolutely incomprehensible” ([23], p. 335, translated from [14]).

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the founder the Russian symbolist movement. From the religious/intellectual point of view, one of the themes that was promoted by these mathematicians was that religion is not subject to mathematical logic. They developed instead a line of thought called “paraconsistent logic,” where apparent inconsistencies and contradictions are allowed, and where the “principle of explosion” (ex contradictione quodlibet), saying that anything follows from a contradiction does not hold. Mathematically, this principle is the “reasoning by contradiction” which is at the basis of set theory, which, as we recalled, was a rapidly growing field. It is not easy for a mathematician to accept that the logical system that he is used to is not the unique possible. The position these mathematicians held is in fact close to Pascal’s point of view, that something else than “reason” is used when one talks about God (“La raison n’y peut rien déterminer […] il faut renoncer à la raison”, Pascal, Pensées). Leonid Sabaneeff,14 in an article on Florensky, writes that the latter “often spoke of the ‘many facets’ of any true thought and of the compatibility of contradictions on the deepest level. He even asserted that every perceived law ‘generates’ its own negation. […] He obviously regarded antinomy as the basic law of the universe, encompassing all others” [77], p. 314. In 1903, Florensky wrote a paper titled On symbols of the infinite, which is an exposition, intended for a general reader, of the basic notions of set theory newly developed by Cantor. At the same time, the paper contains philosophical and theological thoughts. Through his friendship with Andrei Bely, Florensky had entered the Russian symbolist poets movement, and instead of submitting his paper to a mathematical journal, he published it, in 1904, in the newly founded periodical of Symbolic poetry, Novyj Put’ (the New way). The same year, he graduated from the mathematics division of the University of Moscow with the highest grade, after submitting a thesis titled Singularities of algebraic curves. A concise overview of this thesis, with several interesting comments, is contained Demidov’s article [13]. Florensky turned down a graduate fellowship from Moscow University and entered the Moscow Theological Academy, from where he graduated in 1908 and became a priest. At the same time, he became a teacher at the theological seminary. In 1914 he defended a doctoral thesis, with a dissertation titled The pillar and foundation of truth [21], which was an expanded version of a pamphlet he wrote in 1908, the year he graduated from the mathematics and physics department, and which he had called On spiritual truth. The dissertation was published as a book which became a foundational text for Russian orthodox twentieth-century theology. In some sense, this new approach to theology was a return to the ancient so-called “apophatic (or negative) theology” which finds its roots in the third-century monks of the desert of Egypt. It asserts that mystical experience does not follow the laws of logical reasoning, and that paradoxical situations, like the truth of an assertion 14 Leonid Sabaneeff (1881–1968) studied music under Rimsky-Korsakov and Taneyev, and graduated in mathematics from the State University of Moscow. He became a composer and musicologist. He taught at the State University of Moscow and he was close to Luzin and Florensky, until the year 1926 where he left Russia.

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and its negation, are acceptable in theology. It was important at that time that a specialist in the mathematics of Cantor and Peano declares that these logical theories are irrelevant to Christian theology. Florensky also wrote a treatise on religious painting [22], and his writings confirmed his belonging to the Russian symbolist artistic movement. During the Soviet period, his books were translated into French and published in Paris, and they were used there in the teaching of the Russian theological seminary which replaced the theological schools that were closed in the Soviet Union. Egorov and Luzin corresponded with Florensky and read his books. Sabaneeff writes about Florensky: “It was clear to me that mathematics was his guide even in the area of mystic speculations: it helped him not only through the elementary language of numbers (as in the case of many earlier mystics) but by means of the whole panoply of the modern mathematical apparatus: analysis, the theory of sets, and all the latest theories on the boundary between physics and mathematics.” (See [77], p. 321.) Florensky was arrested briefly in 1928, and again in 1933, where, this time, he was sentenced to ten years of corrective labor and deported. In 1934, he was sent to the Soloveckij concentratsion camp which was opened in 1923 in the Solovetsky Islands, in the White sea of Northern Russia. Historically, these islands were the location of an ancient Orthodox monastery complex and they were transformed into a Gulag, where half-a-million people perished. Solzhenitsyn described the Solovski camp as the “mother of the Gulag.” Florensky is sometimes referred to as the Gulag’s Pascal and the Russian Leonardo. He was executed on December 8, 1937. For the relation between Luzin and Florensky, we refer the reader to the paper [23] by Charles Ford, which also contains excerpts of the correspondence between the two men, translated from the article [14]. This correspondence explains in part Luzin’s religious crisis which started when he experienced the bloody events of the 1905 political troubles in Russia which were part of a failed revolution that was later referred to by Lenin as “the great rehearsal” for the 1917 October Revolution. We now return to Luzin. In a tribute to him, published in 1974 [53], Lavrentieff writes that as a young boy, Luzin was found to be poor in mathematics, and a student tutor was brought to the house to help him. The tutor discovered that it was difficult for the young Nikolai to understand established theories, but that he was able to quickly solve difficult problems demanding originality, and often with unusual methods. The University of Moscow was temporarily closed after the 1905 events, and Luzin went to Paris, having obtained, with the support of Egorov, a scholarship to study there. This allowed him to follow courses by Borel, Poincaré, Hadamard, and other preeminent French mathematicians. The period was for Luzin one of profound crisis. In a letter he sent to Florensky from Paris, dated May 1st , 1906, he writes ([23], p. 335, translated from [14]): “You found me a mere child at the university, knowing nothing. I don’t know what happened, but I cannot be satisfied anymore

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with analytic functions and Taylor series…. To be precise, it happened about a year ago […] To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting, to wend my way through the Alexandrovsky Garden [on the side of the Kremlin nearest to the university], where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror—this is an unbearable sight. It is unbearable, having seen this, to study calmly (in fact, to enjoy) science…. I barely remember what happened to me. I could not work in science, and I seem to have begun to lose my mind as a result of the impossibility of living quietly and understanding where, where the truth is. Dmitri Fedorovich [Egorov], seeing me in such a state, sent me here to Paris. That is how I got here…. I have been here about 5 months, but have only recently begun to study. I lacked the self-awareness…. I now understand that “science,” in essence, is metaphysical and based on nothing…. At the moment my scholarly interests are in principles, symbolic logic, and set theory. But I cannot live by science alone.” Egorov arranged Luzin’s second stay abroad, in 1910–1914, in Göttingen and Paris. It is during that stay in Göttingen that he first met the Lavrentieff family. In Paris, Luzin attended the courses of Picard on the theory of functions and of Bôcher on second order differential equations. He also met Borel, Lebesgue and Denjoy, and he participated actively in Hadamard’s seminar; see [73]. In 1911 and 1912, Egorov and Luzin published a series of papers in the French Comptes Rendus which sealed the support of the French school of analysis to the Russian one, see [20], [58], and [59]. In particular, Egorov’s result in [20] on sequences of measurable functions was a substantial improvement of previous works by Borel and Lebesgue. It says that an arbitrary sequence of measurable functions can be made uniformly convergent up to a set of arbitrarily small measure. Luzin’s result in [59] is a response to a question raised by Baire. On his return to Moscow, Luzin joined the faculty of Mathematics of the University of Moscow. He was appointed professor there in 1917, just before the revolution. Lavrentieff’s memoirs published in [19] contain several pages on Luzin.15 We read there that Luzin’s Master’s dissertation, Integral and trigonometric series, which he had prepared in Paris and which he defended in 1915, on his return to Moscow, differed substantially from regular dissertations. Along with concrete results in each section, it contained formulations of new problems and statements that were unproved, with only sketches of evidence. Furthermore, the dissertation contained sentences such as: “It seems to me,” “I am sure that,” etc. This style did not fit into the classical tradition of mathematical writing, and the mathematicians at Saint Petersburg—the capital—who were in charge of examining the manuscript, were not convinced by its value. Lavrentieff reports that the academician V. A. Steklov made several ironic comments on the document, like: “It seems to him, but it does not seem to me,” “the author is a Göttingen chatter,” etc. 15 A translation of this part of Lavrentieff’s memoirs is also contained in Lavrentieff’s paper on Luzin [53].

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Egorov appreciated Luzin’s monograph and understood that it was a significant contribution to science. He made the rare recommendation to the Academic Council of Moscow University that the text be accepted as a doctoral (instead of Master’s) dissertation. The dissertation was accepted as such, the thesis defense took place soon after, and Luzin became a doctor of science. Lavrentieff wrote in his memoirs: “Today, we can see the great importance of the innovative style of the book, and it was especially valuable for the young mathematical generation. The book played a huge role in the formation of the Luzin school. The problems addressed in it by Luzin, and his hypothetical formulations, found a solution in the subsequent works of Luzin himself, as well as in the works of his students. The part ‘it seems’ and ‘I am sure that’ was not justified immediately, but it was justified 15 to 30 years later.” Luzin’s memoir contained several substantial results, one of them saying that any measurable function which takes finite values up to a set of measure zero can be represented by a trigonometric series which converges in the sense of Poisson and Riemann to the given function. In some sense, the memoir is a sequel to Riemann’s Habilitation dissertation on trigonometric series Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representability of a function by a trigonometric series) [76]. The period 1917–1921, which saw the October revolution and the civil war was a period of economic devastation in Russia. Luzin, with his students Menchov, Suslin and Khinchin, left Moscow and worked as a teacher at the newly-founded Polytechnic Institute of Ivanovo-Voznesensk, about 250 km North-East of Moscow. He came back to Moscow in 1922. Lavrentieff writes in [53] that about the years 1922–1926, Luzin introduced to the mathematical division of Moscow university a new style of unprepared lectures, far from the standard “good lecture,” and that the result was a brilliant, deep and fascinating way of teaching, encouraging the development of originality and independent thinking. He also reports that the essential characteristic of the Luzin school (which became known under the name Luzitania) was “the fostering of independent thought, the capacity to crack problems, to find new methods and to pose new problems.” Lavrentieff reports that Luzin prepared his lectures only in outline and that he was often late to his class. He adds that “nearly all his students would arrive on time and discuss problems while waiting in the corridor.” Luzin believed that having a strict schedule was not compatible with doing mathematics. In 1927, Luzin was elected corresponding member of the USSR Academy of Sciences. The following year, he was elected full member, but, surprisingly, in the Department of Philosophy. In 1928, he was Vice-President of the ICM (Bologna). He presented there a paper titled Sur les voies de la théorie des ensembles (On the paths of set theory) in which he discussed the problems that appeared in the theory of functions as a consequence of the questions that were raised in set theory regarding infinity and the definition of a real number.

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In 1930, Luzin published an important monograph on set theory, Leçons sur les ensembles analytiques et leurs applications [60], with an appendix by Sierpiński and a preface by Lebesgue. The book was published in Paris, in the series Monographies sur la théorie des fonctions founded by Emile Borel. In a review of this book in l’Enseignement mathématique, Adolphe Bühl writes: “It seems that this wonderful volume proves that set theory has undergone tremendous progress which ties it permanently with all mathematical disciplines. The philosophical aspect which emerges from the pages written by Mr. Luzin is of the same nature as the one which emerges from group theory or from some extremely general exposition of geometry.” The so-called “Luzin affair” erupted in July 1936. It constitutes one of darkest episodes of the history of Soviet mathematics, if not the darkest one. It is a mixture of a generation dispute and personal interests with a conflict on political, national, religious and ideological beliefs and it ended with a disaster. From the political point of view, the moment looked relatively calm; this was the period which preceded Stalin’s Great Terror or Purges, which started in May–July 1937 and ended in September–November 1938. These purges rippled almost all opposition to Stalin’s power. In particular, the scientific community and other independent thinkers were liquidated. About one million people were either executed or sent to concentration camps.16 In the summer of 1936, a violent political campaign started against Luzin, whose purpose was to expel him from the Academy of Sciences. The campaign was initiated with two articles in Pravda, Answer to Academician N. Luzin (July 2, 1936) and On enemies is Soviet masks (July 3, 1936). These articles were anonymous.17 In 1936, a Commission of the Presidium of the Academy of Sciences of the USSR concerning N. Luzin was created. It was headed by the Vice-president of the Academy, G. Krzhizhanovsky, and its members were the Academicians A. Fersman, S. Bernshtein, I. Vinogradov, O. Schmidt, A. Bakh, and the Corresponding members

16 About the relatively calm period that preceded the Great Terror, we mention a passage that André Weil wrote in his Commentaries to Volume 1 of his Collected Works [83], p. 535. Weil recalls an international conference on topology that was organized in Moscow, in September 1935, by P. S. Alexandrov and to which he participated. He says that the fact that the conference took place, with a selection of mathematicians from all over the world, seems to him retrospectively as a miracle, and that many mathematicians, including him, thought at that time that this was a sign of the beginning of a liberalization of the Soviet regime. He adds that the big trials that took place just after showed that this was only an illusion. (I am grateful to Yuri Neretin for this reference.) 17 According to Yuri Neretin (correspondence with the author), the first article, titled “Answer to Academician N. Luzin,” was signed by Shulyapin, a school director. According to Alexei Sossinsky (correspondence with the author), experts agree that the author of all these articles was Ernest Kol’man, a Marxist philosopher and party bureaucrat who was well known for his activities in the Soviet science community as chief ideological watchdog. Kol’man had a long history (starting in the 1920s) of attacks against Egorov, Luzin and other scientists who did not share his views on science organization. At the time of the Luzin affair, he was the head of the Science Section of the Moscow Committee of the Communist party.

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L. Shnirelman, S. Sobolev, P. Alexandrov, and A. Khinchin. Additionally, the Academician Nikolay Gorbunov was a member of the Commission, as a political representative in the Academia. Kolmogorov, and Lyusternik, were not members of the Commission, but were invited to attend the meetings. Pontryagin, visited some meetings, but his name is absent from the stenograph. He pronounced a heavy speech in Moscow University against both Luzin and Alexandrov. Most of the members of the Commission, including the invited members, were Luzin’s former students or student’s students. Gelfond (a student of Hinchin and Stepanov, who were members of the Luzitania) participated to several political attacks that took place at the meetings of the Commission. These meetings were also attended by Kol’man. The academic accusation claimed that Luzin, by the end of the 1920s, showed a lack of interest in other mathematical fields than the one he was working in, and in the mathematical schools that were emerging in the Soviet Union. It is probably true that Luzin considered problems that are unrelated to function theory and topology as secondary, although his ideas had an undeniable influence on all the schools that were appearing, some of them founded by his students, or his student’s students: Kolmogorov and Khinchin in probability, Pontryagin and Alexandrov in modern topology (to which Kolmogorov contributed as well), Gelfond and Shnirelman in number theory, Lyusternik and Shnirelman in functional analysis, etc. Luzin’s accusation also included claims made by some of his students saying that he was putting pressure on (other) students to include his name in their publications, or to make them acknowledge his help in their work. He was also blamed for the fact that he published his papers in French rather than Russian journals, pretending that this was a sort of disloyalty to the Soviet power. Shnirelman, Sobolev, Alexandrov, and Khinchin, Kolmogorov, Pontryagin and Lyusternik were actively against Luzin, whereas Bernshtein defended him, and Krzhizhanovsky was at first undecided. According to the Stenograph, Vinogradov was completely silent during the meeetings. Shnirelman, two times and without success, proposed to submit the case to NKVD, the “People’s Commissariat for Internal Affairs.”18 A number of scientists outside the Commission defended Luzin, in particular the applied mathematician A. Krylov, the mineralogist V. Vernadsky and the physicist and future Nobel Prize winner P. Kapitsa (the latter wrote a strongly worded letter attacking the Pravda article). Lavrentieff, P. Novikov and L. Keldysh refused to be members of the Commission and to come to the meetings, which apparently required a great deal of courage. Eventually, and surprisingly, Luzin was not condemned as being an “enemy of the people,” as was planned by his detractors. But as a consequence of this affair, Luzin was rejected by an important part of the mathematical establishment and the

18 This is the organisation which soon became responsible for mass extrajudicial executions of a large number of citizens of the USSR and which was the official administrator of the Gulag system of forced labour camps.

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case left a deep wound in the Russian mathematical community.19 Luzin continued working in isolation.20 Discussion of the Luzin case remained a taboo in the Soviet Union, until its re-examination several decades later. In 2012, the Russian Academy of Sciences reversed the decision that was taken against Luzin in 1936. The reconstruction of the minutes of the various trials that took place is reported on in the the book The case of Academician Nikolai Nikolaevich Luzin [17] by Demidov and Lëvshin. The reader may refer to the additional information contained in the appendix to the present article, written by Galina Sinkevich. For more details on these events, including an account of the dramatic fates of Egorov, Luzin, Florensky and others, the reader may consult Demidov’s article The Moscow school of the theory of functions in the 1930s [15] which constitutes an excellent concise report (in English) about the school founded by Luzin. The Lavrentieff memorial book [19] also contains a chapter dedicated to the Luzin affair. The book Naming infinity: A true story of religious mysticism and mathematical creativity by Graham and Kantor [25], despite many inaccuracies, contributed in the rising of interest among European readers in the Soviet history of mathematics. Egorov, who had not been politically active, fell into disfavor with the Soviet authorities six years before Luzin. One of the reasons is that he had defended the Church against Marxist attacks. As early as 1903, he had protested a pogrom against Jews. In 1929, after having been President of the Moscow Mathematical Society for seven years, he was dismissed from his position at the University. The next year he was jailed for being a “religious sectarian.” Little is known about this sad story and the information about the end of his life contains contradictions. Most probably, Egorov died after a hunger strike. According to Demidov, he died in the Hospital of the Institute for continuous education of doctors (a branch of the Kazan University), [16].

19 Let me quote here a passage from an email I received on April 29, 2019, from Alexei Sossinsky: “When the Commission concluded its work, it was expected that it would declare that Luzin is ‘an enemy of the people,’ which meant that he would be sent to the camps or even be given the death penalty. But it didn’t! Until a few years ago, we did not know why this happened. The publication of certain documents now gives us the correct story, which involves Kapitsa’s letter, the falling of Kol’man into disfavor, Krzhizhanovsky, Molotov, and Stalin, the final decision apparently being taken in a tête-a-tête between Krzhizhanovsky and Stalin.” 20 I learned from Neretin that in 1937–39 Luzin made an ingenious work on some types of bendings of two-dimensional surfaces solving a problem which had been discussed during 50 years, see [61]. Unfortunately for him, the solution was negative and his work closed the topic and had no direct continuation. We refer the reader to the interesting review of this problem by Sabitov in [79]. The author writes there: “a truly dramatic end to this direction of geometry came with Luzin’s paper [61].” In 1946, Luzin published a 900 pages textbook on calculus for technological universities. In 1953 (the year Stalin died), 300 000 copies of the book were already sold.

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4. Back to Lavrentieff: his mathematics After he studied at the Physics and Mathematics Faculty of Kazan State University, Lavrentieff taught mathematics at the Moscow School of Engineering, from 1921 to 1929. During the years 1923–1926, he was affiliated to the Institute of Mathematics and Mechanics of Moscow State University as a postgraduate student, working on topology and the theory of functions, with Luzin as an advisor. In the period 1924–1925, he wrote four papers on the theory of functions (see [33], [34], [35], and [36]) and a paper on differential equations [37]. The papers were published in French, Polish and German journals, and Lavrentieff became known internationally. The paper [37] contains a celebrated example of a first-order differential equation of the form y 0 = f (x; y) , with f continuous, such that the uniqueness result on integral curves is violated everywhere: through every point in the domain of definition, at least two integral curves pass. In 1927, Lavrentieff was sent to Paris for six months as a science researcher.21 He heard lectures on function theory by Goursat, Montel, Borel, Lebesgue and Julia, and he attended Hadamard’s seminar. He was interested in conformal geometry. One should also note that Luzin was also interested in this field; together with Privalov, he proved in 1924 that if a disc is sent conformally onto a domain with rectifiable boundary, then the map induced on the boundary of the domains sends a set of measure zero to a set of measure zero. In 1925 Luzin and Privalov obtained several results on boundary values of conformal mappings known as the Luzin–Privalov theorems [62]. The results were later generalized by Lavrentieff. During his stay in Paris, Lavrentieff wrote two Comptes Rendus notes on the boundary correspondence induced by a conformal mapping, see [38] and [39]. The first one concerns the boundary correspondence of a conformal mapping between two simply connected regions, and more precisely the behavior under such a map of ratios the lengths of subarcs of the boundaries of the two regions. Among the results he obtained, we note the following two: 1.— Let D and D 0 be two domains in the plane, bounded by two simple closed curves Γ and Γ0 with bounded curvature. Consider a conformal correspondence between D and D 0 . Then the ratio of lengths of corresponding arcs of Γ and Γ0 is bounded. 2.— Let D and D 0 be two domains in the plane, bounded by two simple closed curves Γ and Γ0 which have continuously varying tangents. 21 D. E. Menchov, in an interview with A. P. Youchkevich whose French translation is published in 1985 [65], gives some details on the visits to Paris made at that time by Luzin, Lavrentieff and himself. He reports that the three of them used to stay in a small hotel, Parisiana, situated 4 rue Tournefort, near the Panthéon. (The hotel does not exist anymore.) He recalls that Luzin and Denjoy were close friends, but that at that time Denjoy was not very active in research. He gives several details about Hadamard’s seminar at the Collège de France. He also says that Montel and Lavrentieff were close to each other and had several common interests, and that Borel and Lebesgue were no more involved in research.

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Consider a conformal correspondence between D and D 0 . If ı and ı 0 denote the lengths of corresponding arcs on Γ and Γ0 , then we have K1 ı 1



> ı 0 > K2 ı 1+

where K1 and K2 are constants that depend only on  .

The second paper, [39], has a sequel carrying the same name, and which Lavrentieff published two years later [40]. It concerns sequences of analytic functions. The basic question addressed there is the following: for a pointwise convergent sequence of analytic functions fn defined on an open subset Ω of the complex plane, find an open dense subset Ω0 of Ω on which fn converges uniformly on compact subsets to an analytic function. The problem started with a question of Montel who showed that the limit f of fn on Ω may be continuous and not analytic [68], and later proved that fn converges uniformly on compact sets to f on an open dense subset Ω0 of Ω . Montel was interested in the structure of the complement of Ω0 , and considered the question of whether it is possible to have f analytic on Ω0 , see [69]. Lavrentieff solved completely this problem. He later treated this question in a comprehensive manner in his monograph Sur les fonctions d’une variable complexe représentables par des séries de polynômes published in 1936 [44], after he gave a necessary and sufficient condition for a set to be the set of nonuniform convergence points of a sequence of polynomials converging everywhere in a given domain. This book is in fact concerned with the extension to the complex plane of the Baire–Weierstrass theory of approximation of functions of a real variable by series of polynomials. Lavrentieff gave a complete characterization of closed connected sets on which arbitrary continuous functions can be approximated by polynomials. The monograph also contains extensions of the same theory to harmonic sequences, and results on the boundary correspondence of conformal mappings. In 1928, Lavrentieff gave an address, as an invited speaker, at the ICM (analysis section), which was held in Bologna [41]. The talk he gave there is reviewed in another chapter in the present volume [2]. He writes in his memoirs that during that conference, he met Leonida Tonelli and that the two men realized they had close interests. In the paper [6], written in 1930 in collaboration with P. Bessonoff, Lavrentieff studied again boundary mappings of conformal representations. This subject occupied him during several years. In a series of papers written in collaboration with his student Mstislav Vsevolodovich Keldysh,22 Lavrentieff studied problems related to domains on which continuous functions may be approximated by entire functions or by polynomials, including the case of dimension three; cf. [27], [28], and [29]. 22 Mstislav Vsevolodovich Keldysh was the brother of Ludmila Keldysh, who was member of the Luzitania and who became the wife of Piotr Sergueïevitch Novikov. Sergei Novikov, the Fields medalist, is their son.

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At the end of 1927, Lavrentieff was appointed assistant professor at Moscow University and he began his teaching with a course on the theory of conformal mappings. In 1929, he received the title of professor at the Moscow Institute of Chemical Engineering and he was elected chairman of the department of mathematics there. In 1931, he was appointed professor at Moscow University. During the decade that followed his return to Moscow, Lavrentieff published a series of papers on the theory of functions of a complex variable, including results on conformal mappings onto canonical domains and variational problems leading to extremal domains. His papers during that period contained also results on boundary correspondence of conformal representations, Bloch’s constant, approximation of complex-valued functions by polynomials, distance geometry and measure theory, and many other topics. In his paper Sur une classe de représentations continues [45] published in 1935, he developed the theory of “almost analytic” functions, a class of functions very close to that of quasiconformal mappings. (Later on, Lavrentieff called his functions “quasiconformal.”) Herbert Grötzsch introduced his own version of quasiconformal mappings in 1928. Whereas the latter’s methods were purely geometric, Lavrentieff had a more analytic approach. In his 1946 paper [49], a quasiconformal mapping is defined as a mapping which satisfies a partial differential system of equations which is more general than the Cauchy–Riemann system. According to Migirenko, Lavrentieff began investigating “almost analytic” functions in 1927 [66], p. 2. We refer the reader to Lavrentieff’s 1935 paper [45], translated in this volume, which concerns exclusively almost analytic functions, and to the commentary [2]. In the 1930s, Lavrentieff became gradually the leader of the Soviet school of theory of functions. At the same time, he was working in physics, and in particular on incompressible fluid dynamics and on shock waves. In some sense, he was revisiting the interplay that was so productive in the development of both mathematics and physics. It is good to recall that Riemann himself, the main founder of the theory of conformal mappings, was more involved in physics than in mathematics. His study of electric and magnetic fields was his motivation for his development of the theory of abelian integrals. His use of the Dirichlet Principle is just the statement that a harmonic function minimizes the energy in an electrical field. The electrical potential or voltage in a source-free domain satisfies Laplace’s equation as does the basic statement of incompressible fluid flow—both are formulations of conservation laws. Shocks in fluid flows can arise in different ways (one may think of the sonic boom created by a plane crossing Mach 1). The relevant equation changes from elliptic to hyperbolic (wave) and the transition creates a shock. In the period 1929–1935, Lavrentieff worked as a senior engineer at the Theoretical Department of the Russian Institute of aerodynamics. His work on conformal and quasiconformal mappings was of great help in that domain. In his memoirs [19], he writes that in 1929, he was given the task of determining the velocity field of the fluids in a problem related to thin wings, and that he wanted

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in some way to “justify the mathematics.” In a lapse of time of six months, he managed, on the basis of variational principles using conformal mappings, to find a number of estimates for the desired solution. These estimates allowed him to identify a class of functions, among which the solution had to be found. It turned out that the theory of conformal mappings, which was already used in aerodynamics, could not fully meet the needs of this field, because flight speeds have increased, with the possibility that they might exceed the speed of sound. It was also necessary to take into account the compressibility of air. Thus, the classical Cauchy–Riemann equations satisfied by conformal mappings were not sufficient any more, and the theory of conformal mappings needed to be extended to a wider class of functions, satisfying a certain nonlinear system of partial differential equations. This is how his theory of quasi-conformal mappings was born. It is worth noting here that Lipman Bers’s work on partial differential equations was also motivated by fluid dynamics. He introduced the notion of pseudo-analytic function by studying elliptic systems that arise in the study of subsonic flows; see the historical details in the articles by Abikoff and Sibner [1] and Nirenberg [70]. In 1934, Lavrentieff was awarded a doctorate in engineering by Moscow State University and in the following year he obtained his doctorate in mathematical sciences, without having to defend a thesis. Mathematics and physics were for him one and the same field of research. In 1934, Lavrentieff published a paper on two extremal problems [42], one of them mathematical and the other one on fluid mechanics. The first problem concerns the conformal representation of the annulus 1 < jzj < r in the complex plane onto a doubly connected region which is the complement of the union of the closed disc jwj  1 and a simple path contained in the complement of that disc and converging to infinity. The problem asks for the maximization or the minimization of a certain quantity involving the derivative of the conformal representation of the annulus. An extremal image domain (that is, a domain realizing the maximum or the minimum of the quantity involved) is found in both cases to be the complement of the union of the closed disc jwj  1 and a straight line going to infinity. The second problem concerns a simple arc of class C 1 under the action of the flow of an incompressible fluid, and the extremal problem asks for the maximization of the sum of the pressure forces exerted on the arc. Although the two problems have different characters, they are treated in the same paper because they use the same preliminary lemmas. It is also possible that Lavrentieff, in publishing the two results in the same paper, wanted to pont out a parallel between a purely geometric problem and a problem in fluid dynamics. The article [30] (1937), written in collaboration with Keldysh, is again concerned with the boundary correspondence induced by the Riemann mapping. The main result gives a negative response to the following natural question, which bears

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the name Smirnov problem:23 suppose that w = f (z) realizes the conformal representation of a simply connected domain D bounded by a rectifiable Jordan curve, and let z = (w) be the inverse function. Is it possible to represent the harmonic function log j 0 (w)j defined on the unit disc jwj < 1 by means of the Poisson integral of its limit values on the circle jwj = 1 ? In the paper [46], written in 1938, Lavrentieff uses quasiconformal mappings to describe the stationary flow of a gas. The paper [47], written the same year, is the first paper where higher-dimensional quasiconformal mappings are studied. The definition given by Lavrentieff involves the ratio of the greatest axis to the smallest axis of ellipsoids that are images of infinitesimal spheres. Lavrentieff studied several properties of these mappings. In particular, he obtained a result saying that a locally injective quasiconformal mapping of three-dimensional space is injective. A proof in all dimensions was given 30 years later by V. A. Zorich [84]. Higher-dimensional quasiconformal mappings are further studied by Lavrentieff in the papers [52] and [51], and others. In his paper [5] written with Belinskij, Results of Martio–Rickman–Väisälä and Gehring are used to prove normality of families of Q -quasiconformal local homeomorphisms in dimension  3 . For a report on the current state of the study of higher dimensional quasiconformal mappings, the reader can refer to the article by Gaven Martin in Volume IV of the present Handbook [64]. One may also note here that Grötzsch had also ideas on quasiconformal mappings in dimension three (see §4 of the biographical article by Kühnau in the present volume [32]). In the paper [31] (1939), also written with Keldysh, Lavrentieff gives a characterization of Carleman sets, that is, closed subsets E of the complex plane such that for any continuous function f which is analytic in the interior of E and for any positive continuous function (r) , r  0 , there exists an entire function F (z) satisfying jf (z) F (z)j  (jzj) . Such sets are used extensively in the theory of analytic approximation. In 1947, Lavrentieff began to develop a new theory of non-linear quasiconformal mappings which he called “strongly elliptic,” to be used in hydrodynamics; see the exposition in [66], p. 20, where the author writes that this turned out to be the widest generalization of the Cauchy–Riemann system of equations that preserves the basic geometric properties of conformal functions. He extended to this general class several properties of conformal mappings. The paper [54] (1963) contains an exposition of a variational principle that Lavrentieff developed and which he called Lindelöf’s principle (the principle is an extension of a principle due to Lindelöf), for the solution of boundary value 23 The name refers to Vladimir Ivanovich Smirnov (1887–1974), another preeminent Russian mathematician of the same generation as Luzin who experienced in his student years the same period of political unrest. Smirnov was born in Saint Petersburg, and he founded there the department of Theory of Functions of a Complex Variable. He spent his career at the university of his hometown whose name changed to Leningrad. For a concise biography of Smirnov, we refer the reader to the article by Apushkinskaya and Nazarov [3].

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problems for the Laplace equation and for more general elliptic equations. The approach is again based on the use of conformal and quasiconformal mappings and their geometric properties. The paper contains applications to hydrodynamics, e.g. problems concerning free stream lines and gravity waves (like waves at the air-sea interface, where gravity force tries to restore equilibrium), and to subsonic compressible gas flows. Lavrentieff’s mathematical ideas continued to have relevance also in the West (see Courant [10] (1950) which contains a section on Lavrentieff’s methods). M. V. Keldysh, with whom Lavrentieff wrote several papers, became President of the Academy of Sciences of the USSR in the period 1961–1975, a period during which Lavrentieff was Vice-President of that academy. Bill Abikoff informed the author of these lines that he learned from Jewish mathematicians that at the time where they were experiencing discrimination, Laventieff, with his position of Vice-President, protected them heroically.

5. Siberia Lavrentieff contributed in a substantial way to the transformation of the science landscape in the USSR when, together with the mathematician Sobolev and the physicist Christianovich, he convinced Khrushchev to approve the creation of the science city of Akademgorodok, a few kilometers from Novosibirsk. The place became the host of several research centers devoted to various branches of science, including mathematics, physics, chemistry, geology, economics, history and others. Novosibirsk State University and the headquarters of the Siberian Division of the Russian Academy were naturally hosted in Akademgorodok. For many years, Lavrentieff was the director of the Institute of Hydrodynamics there. He convinced many mathematicians and other scientists to move there from the European part of Russia. It is possible that among the scholars that settled in Novosibirsk, many found in Siberia some kind of intellectual freedom that was inexistent in other parts of the big cities of the Soviet Union where they used to work. In a few years, and due in large part to the immense energy of Lavrentieff, his courage and his talent as a science organiser, Siberia was transformed from a scientifically underdeveloped region into a new world center of science. Lavrentieff also implemented there a new way of selecting students and of training them, organizing olympiads in mathematics and physics and creating specialized boarding schools for talented students. As a matter of fact, Lavrentieff created the first physico-mathematical boarding school. (A year later, Kolmogorov created a second one.) One of Lavrentieff’s favorite sentences was: “There can be no scientists without science students” [66], p. 7. De Gaulle made a spectacular state visit to the Soviet Union, from June 20 to July 1st , 1966, which was part of his plan to enhance the ties between France and the Soviet Union, and to make his country less dependent on the United States.

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Akademgorodok was among the places he visited. He gave a famous speech there in which he expressed his admiration for this science city, which at that time was comprised of 20 research institutes, and for the people who were working there, including 6000 researchers. He communicated his hope that French mathematicians, physicists, chemists, biologists, philosophers, anthropologists and historians establish close relations and collaborations with their Soviet colleagues. He declared that he was touched by the cordial welcome he received “from those who live in this city, this strange and courageous land of Siberia,” quoting Teilhard de Chardin: “a land where people give their life to know, rather than to obtain.”24 Jean Leray, in his memorial article on Lavrentieff, [55] recounts that de Gaulle, who was very tall, during his visit to Akademgorodok, found himself surrounded by a group of men as tall as him; one of them was Lavrentieff.

6. Mechanics and engineering In the period 1929–1969, Lavrentieff wrote several papers in which he applied mathematical methods, including the theory of conformal and quasiconformal mappings in dimension  2 , to fluid mechanics and aerodynamics. The fields of application includes aircraft, hydrofoil and underwater wing theories, the study of vibrating plates of infinite extent, of turbine propellers, of the impact of a body on water and of jet flows of an ideal liquid. At the same time, he developed theories as varied as those of long waves, of dynamical stability of buckling elastic systems and of the movement of grass snakes and fish. He built mathematical models for discontinuous flow patterns and for flows by stream of finite breadth around obstacles. He studied problems concerning directional explosions that were used for dam constructions and he obtained results on the welding of metals by explosions utilized in the design of canals and funnels. A spectacular result of Lavrentieff’s ideas was the construction of the gigantic dam that protects Almaty, the largest city of Kazakhstan,25 at the center of Eurasia. The dam was constructed by a series of 5 directed blasts which involved the displacement of a total of 5 million cubic meters of rock. The last explosion used 3600 tons of ammonium nitrate based explosive. Before the dam was constructed, the city was periodically devastated by gigantic mudflows from the Meteu Valley. 24 De Gaulle cites the following excerpt from Teilhard de Chardin: “Une Terre où les télescopes géants et les broyeurs d’atomes absorberont plus d’or et susciteront plus d’admiration spontanée que toutes les bombes et tous les canons. Un Terre où, non seulement pour l’armée groupée et subventionnée des chercheurs, mais pour l’homme de la rue, le problème du jour sera la conquête d’un secret et d’un pouvoir de plus arrachés aux corpuscules, aux astres ou à la matiére organisée. Une Terre où, comme il arrive déjà, c’est pour savoir et être plutôt que pour avoir, qu’on donnera sa vie” [81], p. 281f. De Gaulle’s discourse is contained in [11]. 25 During the Soviet period, the city was called Alma-Ata and was the Capital of Kazakhstan.

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Figure 1. Lavrentieff welcoming de Gaulle at Novosibirsk airport, July 1, 1966

The most severe one in the twentieth century occurred in 1921, it killed 500 people and destroyed a large part of the city. Lavrentieff also worked on artillery and on the improvement of gunpowder, upgrading the production of bullets using ideas from fluid dynamics. He helped in the conception of the famous multiple weapon launcher called the Katyusha that was used by the Soviet army during World War II. It is conceivable that the applications of his ideas and discoveries in the military field played a non-negligible role in the Soviet Union’s victory on Nazi Germany. His work on explosive material was extended after World War II. Migirenko writes that in the years 1951 to 1953, “[Lavrentieff] frequently became engaged in experiments which he literally conducted with his own hands, and he youthfully expressed delight with his success and disappointment when his expectations were not confirmed. With the help of young scientists he set charges under water and equipped fire hoses with grains of TNT or so-called unconditioned powder, in which he was extremely interested at the time. He could have simply burned powder according to the existing canons, but Mihail Alexeevič could not accept the possibility that ‘property was being needlessly wasted’.” (See [66], p. 5.)

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One may safely assert that Lavrentieff’s involvement in the military technology was crucial in the decision of the Soviet heads of the state to increase their support of science, creating new mathematical centers and hiring a large number of scientists. Lavrentieff also studied metal welding by explosions, and he developed hydraulic jet pulse techniques with applications to the breaking down of hard rocks. He conceived a device which became the prototype of modern hydraulic jet guns, used for pulsed high-energy forming of metals and other works. He also studied the phenomena of high-velocity collision with various kinds of obstacles (metallic, porous, etc.), together with the question of detonation of high-speed explosives. He found new ways for using the underground hot water, with practical methods for dealing with tsunami waves and insuring the security of coal mines. One may note here that Grötzsch, the other discoverer of quasiconformal mappings worked also, by the end of World War II, worked on jet engines at a institute for flow reearch. (see the biographical article by Kühnau in the present volume [32]). Starting in the 1940s, Lavrentieff worked on solitary waves. These are waves whose envelope has a unique peak. They decay away from this peak and their shape and amplitude stay constant in time. They arise as ocean water waves and in optical fibers where intensity increases from dark to light. Such waves have been used in the modelisation of tsunamis. Solitons, which are waves of big amplitude and small spatial extension and which can travel for long distances without deformation, are special cases of solitary waves. From the mathematical point of view, solitary waves are solutions of certain non-linear partial differential equations (Korteweg-de Vries and nonlinear Schrödinger equations). Seen the range of interests of Lavrentieff, it is not surprising that he was implicated in this research. The theory of solitary waves is also a classical subject. Their existence under special assumptions was proved by Boussinesq (1871) [8], Lord Rayleigh (1876) [75], Korteweg and de Vries (1895) and Levi-Civita (1911) [56]. Lavrentieff was the first to give a mathematical proof of the existence of general solitary waves. Using approximation techniques by Stokes waves, he established their existence as limits of periodic waves with indefinitely increasing wave length; cf. [48] and [50]. Starting from 1956, Lavrentieff worked on the question of spiraling detonation of gases and on problems related to the formation of turbulent vortex rings in a viscous fluid, including the transfer of impurities under such phenomena. His results led to a complete re-organization of large earth-works in the Soviet Union. In 1946, while he was Vice-President of the Ukrainian Academy of Science and Director of the Mathematics Institute of Kiev, Lavrentieff wrote to Stalin, urging him to accelerate the development of computers, emphasizing their utility for military purposes. Stalin responded positively and authorized him to create a new laboratory dedicated to computer science. In 1948, Lavrentieff was selected by the government to develop modern computing technology, and in 1951 he was appointed director of the Institute of Precision Mechanics and Computer Technology in Moscow. He remained director of this institute until 1953; see [66], [24], and [63].

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Lavrentieff writes in his memoirs [19]: “I learned for myself personally, first, the experience of applying pure mathematics to important engineering problems, and secondly, a clear understanding that in solving such problems new ideas and approaches are born in mathematical theories themselves.” He was responsible for projects using geothermal energy, in particular in the Eastern part of Russia bordering the volcanoes of Kamchatka. He was particularly concerned by the effect of extreme cold on constructions in Siberia, by the pollution of rivers and lakes (in particular, that of the Lake Baïkal, located in Southern Siberia and which is the largest freshwater reserve by volume in the world). Lavrentieff received the most prestigious prizes awarded in the Soviet Union, among them the Lenin Prize (1962),26 the Hammer and Sickle gold star (1967), which is the highest honor in Soviet Union for nonmilitary achievement, and the Lomonosov gold medal (1977). He was also a member of several foreign academies of sciences In his article on Lavrentieff [55], Leray described half a dozen works of the latter as Herculean works. In the same article, Leray compared him to Peter the Great, for the breadth of his achievements and his vision on the present and the future of Russian science. The main street in Akademgorodok is naturally called Lavrentieff Prospekt. Acknowledgments. Bill Abikoff read carefully a first version of this article and sent me a number of suggestions that led to a thorough revision. Yuri Neretin and Alexei Sossinsky read a second version and made several suggestions and corrections. Galina Sinkevich sent me very useful suggestions that amend some of my statements concerning the Moscow School of analysis in which Lavrentieff was trained and shed a different light on some facts that I relate. I asked her to write an appendix explaining her ideas. The appendix follows this article in the present volume. I would like to thank Abikoff, Neretin, Sossinsky and Sinkevich for their care and invaluable help.

26 Some of the other recipients of the Lenin prize are the mathematicians Bogolyubov (1958), Novikov (1967), Pogorelov (1962), Arnold (1965), Kolmogorov (1965), Keldysh (1957) and Manin (1967). The prize was also awarded to the physicist Andrei Sakharov (1956), to the composers Prokofiev (1957), Shostakovich, (1959) and Khachaturian (1959) and to the panist Sviatoslav Richter (1961). Other recipients of the Lomonosov gold medal include Vinogradov (1970), Denjoy (1970), Sobolev (1988, posthumously), Leray (1988), Hirzebruch (1996), Carleson (2002), Faddeev (2013) and Lax (2013).

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7. Appendix. Some clarifications on the history of Russian mathematics of the 20 th century Galina Sinkevich In this note, I would like to bring a few additions that clarify some points in A. Papadopoulos’s article related to the biographies of P. Florensky and N. Luzin. My additions may shed a different light on this complicated period of the history of mathematics in the Soviet Union. According to B. Mlodzeevsky, it was characteristic for Moscow mathematicians of the turn of the 19 th and 20 th centuries, “to explain mathematics through the worldview, and worldview through the mathematics” [67], p. 185. The leading role in this orientation of the Moscow school belonged to N. V. Bugaev, whose theory of discontinuous functions was considered in a philosophical context and had a wide resonance among Moscow mathematicians [12], although his colleagues did not appreciate his mathematical results and lectures [82], pp. 165–175. In 1897, at the First Congress of Mathematicians in Zurich, Bugaev made a report On the influence of mathematics on the worldview [9]. According to him, world laws are explained in the language of the theory of continuous functions. This gave rise to determinism, while in the natural and human sciences many phenomena cannot be subordinate to the laws of mathematical analysis. Using continuity, only part of the world events can be explained. At the same time, a new science emerges from number theory, a theory of discontinuous functions, called arhythmology. In the realm of philosophy proper, arhythmology is refracted into monadology. The arhythmological point of view complements the analytical worldview. Combining both worldview approaches, analytic (using mathematical analysis) and arhythmological (using the theory of discontinuous functions), and adding probability theory in necessary cases, Bugaev obtains a scientific world view that allows us to explain mechanically world phenomena. Finally, whenever phenomena do not obey the correct laws, probability theory applies. From the cumulative application of all these departments of mathematics, a true scientific and philosophical outlook was formed. N. Luzin and P. Florensky entered Moscow University with a difference of one year (1901 and 1900 respectively). They soon became friends. First a mathematics student and then a priest, Florensky was a very ambiguous figure. No doubt he was very talented. In his student years, he was fascinated by Bugaev’s ideas. He then became interested in Cantor’s set theory. Florensky was the first to publish a competent statement of Cantor’s ideas in Russian (On the symbols of Infinity, 1904). But then his interests shifted towards theology. After graduating from university he entered as a student the Moscow Theological Academy. He needed mathematics only as a basis for building the philosophical foundation of the universe. His philosophical and religious works, The pillar and foundation of truth (1914), Reverse perspective (1919), Imaginaries (imaginary numbers) in geometry (1922)

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are widely known. In particular, in his last work, Florensky interprets the complex plane as a two-sided object for the rehabilitation of the Ptolemaic model of the Universe. The statement made by Athanase Papadopoulos in his article that Florensky was extremely talented in mathematics is debatable. Luzin and his teacher D. Egorov estimated the mathematical success of the student Florensky low. In 1908, Luzin wrote to his wife about Florensky: “As soon as he showed his work in mathematics, again the old opinion began to stir in me: all his works have no value in the field of mathematics. Hints, beautiful comparisons are something that revels and promises, is tantalizing, alluring and ineffectual. And in the end I stopped understanding what Florensky is. Or is it a precursor of the new, the petrel, or a capable person with subconscious hellish self-love, who, because of the desire to be the best of all, has retired here?” [14], p. 150. In 1904–1907, Luzin experienced a spiritual crisis which was so severe that he wanted to stop practicing mathematics, and even wanted to take his own life. One reason for the crisis was the dramatic atmosphere that preceded the first revolution. His teacher Egorov hastily sent him from to France to prepare him for a professorship. At this time, Luzin was under the strong influence of Florensky. On the ambiguity of Florensky’s personality and his negative charisma, L. Sabaneev wrote in his memoirs (1915): “Very black and very thin, for some reason he always looked down and slightly sideways, he did not like to show his eyes. He never smiled. It was a strange thing—he had many students, apparently, he taught them not only classical theological subjects common to the spiritual sciences, but also gave them esoteric knowledge and habits. Three of his students committed suicide—powerful vibes emanated from him, and I myself felt it, felt that not all fluids were good, there were very demonic among them. I don’t remember exactly who, but speaking of him, one of the Russian ‘neo-Christian’ group called him ‘a clever and cruel Lavra priest.’ In any case, he was an absolutely extraordinary person, and I am very grateful to the fate that brought him together with me, although not for long” [78]. The relations between Luzin and Florensky were uneven and the influence of Florensky on the mathematical research of Luzin is problematic. In 1915, Luzin presented his master’s thesis Integral and trigonometric series, with so strong results that it was qualified as a doctor’s thesis. Starting in 1917, Luzin began to teach at Moscow University; a group of talented students gathered around him. This circle was named Luzitania. It was a happy period of the relationship between a teacher and equally talented students. Years passed, the students themselves became leaders of scientific schools, and in the meantime, the Stalinist terror gained momentum in the Soviet Union. Trials began, the search for “enemies of the people” was going on. Under the blow was the intelligentsia, the old professorship. Much has been written about this in Russian literature. The Soviet government sought to subjugate the Academy of Sciences, to “domesticate” the old scientists, orienting them to the tasks of communist construction, re-educate

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Soviet scientists in the spirit of communist ideology. For this, any contradiction among scientists was used. The conflict between Luzin and his former disciples in 1936 served as a convenient pretext. About this tragic story, see above all [17]. Despite this fact, it is impossible to perceive the persecution of Luzin as a direct manifestation of Stalin’s terror. It was part of a more general process. Stalin did not set himself the goal of punishing Luzin and did not even know about the beginning of the persecution. The reason was different: the happy period of Lusitania ended, the former disciples became independent scientific leaders, and rivalry began. A tangle of very complex contradictions both in the mathematical community and in society as a whole, multiplied by the difficult character of Luzin, who in his own way understood his role in the students’ successes; denunciations written on Luzin by the pro-Communists E. Kolman and V. Molodshij; the clash of traditional ethics of old scientists with the new ethics of the younger scientific generation; the atmosphere of fear cultivated among the intelligentsia by the Stalinist search for “enemies of the people”: all this inspired this persecution. The case was accompanied by violent attacks on Luzin in the newspapers. Nevertheless, it did not go beyond the academic environment, and a month later it was completed by the Decree of the Presidium of the Academy of Sciences of 08/05/1936 on Academician Luzin with the following wording: “Given the importance of N. N. Luzin as a major mathematician, and weighing the full force of social impact, what had revealed in such a broad, unanimous and fair condemnation of the behavior of N. N. Luzin, and based on the desire to give Luzin the opportunity to restructure his whole behavior and work, we consider it possible to limit the warning to N. N. Luzin that in the absence of a decisive change in his future behavior, the Presidium will have to urgently raise the question of expelling N. N. Luzin from the academic ranks” [85]. Luzin escaped the terrible machine of Stalin’s repression, which many wellknown scientists got into, including Florensky who was shot in 1937. See also (in Russian) S. Novikov (Jr.) [72], a very deep study of Yu. Neretin [71], and (in English) A. E. Levin [57]. Luzin recovered with difficulty after the persecution. He was ill for a long time, then continued his scientific work, but he no longer made major discoveries.

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[21] P. A. Florensky, The pillar and foundation of truth, Put, Moscow, 1914 In Russian. French translation by C. Andronikov, La colonne et le fondement de la vérité. L’Âge d’homme, Lausanne, 1975. R 323 [22] P. A. Florensky, La perspective inversée, suivi de L’Iconostase et autres écrits sur l’art, transl. Françoise Lhœst, L’Âge d’homme, Lausanne, 1992. R 324 [23] C. Ford, The influence of P. A. Florensky on N. N. Luzin. Historia Math. 25 (1998), no. 3, 332–339. Reprined in Y. Sinai (ed.) Russian mathematicians in the 20 th century. World Scientific, River Edge, N.J., 2003. MR 1649961 Zbl 0932.01029 R 322, 324 [24] L. Graham, Lonely ideas: Can Russia compete? MIT Press, Cambridge, MA, 2013. R 338 [25] L. Graham and J.-M. Kantor, Naming infinity. A true story of religious mysticism and mathematical creativity. The Belknap Press of Harvard University Press, Cambridge, MA, 2009. MR 2526973 Zbl 1206.01036 R 329 [26] I. Grattan-Guinness, Algebras, projective geometry, mathematical logic, and constructing the world: intersections in the philosophy of mathematics of A. N. Whitehead. Historia Math. 29 (2002), no. 4, 427–462. MR 1936800 Zbl 1046.00003 R 321 [27] M. V. Keldysh and M. A. Lavrentieff, Sur les suites des polynômes harmoniques. Comptes Rendus Acad. Sci. Paris 202 (1936), 1145–1151. R 331 [28] M. V. Keldysh and M. A. Lavrentieff, Sur les suites convergentes de polynômes harmoniques. Trav. Inst. Math. Tbilissi 1 (1937), 165–184. Zbl 0017.20701 R 331 [29] M. V. Keldysh and M. A. Lavrentieff, Sur le problème de Dirichlet. Comptes Rendus Acad. Sci. Paris 204 (1937), 1788–1790. Zbl 1788-1790 R 331 [30] M. V. Keldysh and M. A. Lavrentieff, Sur la représentation conforme des domaines limites par des courbes rectifiables. Ann. Sci. École Norm. Sup. (3) 54 (1937), 1–38. MR 1509363 JFM 63.0299.04 Zbl 0017.21702 R 333 [31] M. V. Keldysh and M. A. Lavrentieff, Sur un problème de M. Carleman. C. R. (Dokl.) Acad. Sci. URSS (2) 23 (1939), 746–748. JFM 65.1227.04 Zbl 0021.33502 R 334 [32] R. Kühnau, Memories of Herbert Grötzsch. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 10, 301–315. R 334, 338 [33] M. A. Lavrentieff, Sur la recherche des ensembles homéomorphes. Comptes Rendus Acad. Sci. Paris 178 (1924), 187–190. JFM 50.0134.02 R 330 [34] M. A. Lavrentieff, Sur la représentation des fonctions mesurables B par les séries transfinies de polynômes. Fund. Math. 5 (1924), 123–129. JFM 50.0182.02 R 330 [35] M. A. Lavrentieff, Contribution à la théorie des ensembles homéomorphes. Fund. Math. 6 (1924), 149–160. JFM 50.0143.04 R 330 [36] M. A. Lavrentieff, Sur les sous-classes de la classification de M. Baire. Comptes Rendus Acad. Sci. Paris 180 (1925), 111–114. JFM 51.0166.04 R 330 [37] M. A. Lavrentieff, Sur une équation différentielle du premier ordre. Math. Z. 23 (1925), no. 1, 197–209. MR 1544737 JFM 51.0332.04 R 330 [38] M. A. Lavrentieff, Sur la représentation conforme. Comptes Rendus Acad. Sci. Paris 184 (1927), 1407–1409. JFM 53.0324.01 R 317, 330 [39] M. A. Lavrentieff, Sur un problème de M. P. Montel. Comptes Rendus Acad. Sc. Paris 184 (1927), 1634–1635. JFM 53.0282.02 R 317, 330, 331 [40] M. A. Lavrentieff, Sur un problème de M. P. Montel. Comptes Rendus Acad. Sci. Paris 188 (1929), 689–691. JFM 55.0181.05 R 331

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[41] M. A. Lavrentieff, Sur une méthode geométrique dans la représentation conforme. In Atti del Congresso Internazionale dei Matematici (Bologna 3–10 settembre 1928). Tomo III. Bologna 1930, 241–242. JFM 56.0295.03 R 317, 331 [42] M. A. Lavrentieff, Sur deux questions extrémales. Rec. Math. Moscou 41 (1934), 157–165. JFM 60.1030.03 Zbl 0009.36103 R 333 [43] M. A. Lavrentieff, Sur une classe de représentations continues. Comptes Rendus Acad. Sci. Paris. 200 (1935), 1010–1013. JFM 61.0300.02 Zbl 0012.21501 R 317 [44] M. A. Lavrentieff, Sur les fonctions d’une variable complexe représentables par des séries de polynômes. Actualités Scientifiques et Industrielles, 441. Hermann, Paris, 1936. JFM 62.1205.01 Zbl 0017.20602 R 317, 331 [45] M. A. Lavrentieff, Sur une classe de représentations continues. Mat. Sbornik 42 (1935), 407–424. English translation by V. Alberge and A. Papadopoulos, On a class of continuous representations. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 19, 417–439. JFM 61.1131.04 Zbl 0014.31905 R 317, 332 [46] M. A. Lavrentieff, Sur une classe de transformations quasi-conformes et sur les sillages gazeux. C. R. (Dokl.) Acad. Sci. URSS (2) 20 (1938), 343–345. JFM 64.1075.06 Zbl 0019.40305 R 334 [47] M. A. Lavrentieff, Sur un critére différentiel des transformations homéomorphes des domaines à trois dimensions. C. R. (Dokl.) Acad. Sci. URSS 20 (1938), 241–242. JFM 64.0708.01 Zbl 0019.40306 R 334 [48] M. A. Lavrentieff, On the theory of long waves. C. R. (Dokl.) Acad. Sci. URSS 41 (1943), 275–277. In Russian. English translation, Am. Math. Soc. Transl. 102 (1954) 51–53. Zbl 0061.46105 R 338 [49] M. A. Lavrentieff, The general problem of quasi-conformal mappings of plane regions. Rep. Acad. Sci. Ukr. SSR 3-4 (1946), p. 3–10. R 332 [50] M. A. Lavrentieff, A contribution to the theory of long waves. Dokl. Akad. Nauk Ukrainskoi RSR 8 (1947), 13–69. English translation, Am. Math. Soc. Transl. 102 (1954) p. 3–50. In Ukranian. MR 0048985 R 338 [51] M. A. Lavrentieff, Sur la théorie des représentations quasi-conformes. Ann. Acad. Sci. Fenn. Ser. A. I 250/18 (1958) 8 pp. MR 0095247 Zbl 0080.28902 R 334 [52] M. A. Lavrentieff, On the theory of quasi-conformal mappings of three-dimensional domains. J. Analyse Math. 19 (1967), 217–225. MR 0222290 R 334 [53] M. A. Lavrentieff, Nikolai Nikolaevich Luzin (on the 90 th Anniversary of His Birth). Usp. Mat. Nauk 29 (1974) 5(179), 177–182. In Russian. English transtalion, Russ. Math. Surv. 29 (1974), no. 5, 173–178. The content is part of Lavrentieff’s memoirs published in [19]. Zbl 0299.01017 Zbl 0315.01020 (translation) R 324, 325, 326 [54] M. A. Lavrentieff, Variational methods for boundary value problems for systems of elliptic equations. Translated from the Russian by J. R. M. Radok. Second edition. Dover Books on Advanced Mathematics. Dover Publications, New York, 1989. MR 1032518 Zbl 0743.35019 R 334 [55] J. Leray, Sur Mikhaïl Alexievich Lavrentiev. Comptes rendus Acad. Sci. Paris, Vie Académique, 296 (suppl. 16) (1983), 59–62. R 336, 339 [56] T. Levi-Civita, Sulle onde progressive di tipo permanente. Rend. Acad. Lincei (5) 16 (1907) 777–799. JFM 38.0745.02 R 338 [57] A. E. Levin, Anatomy of a public campaign “Academician Luzin’s case” in Soviet political history. Slavic Review 49 (1990), no. 1, 90–108. R 342

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[58] N. N. Luzin, Sur les propriétés des fonctions mesurables. Comptes Rendus Acad. Sci. Paris 154 (1912), 1688–1690. JFM 43.0484.04 R 325 [59] N. N. Luzin, Sur un problème de M. Baire. Comptes Rendus Acad. Sci. Paris. 158 (1914), 1258–1261. JFM 45.0632.02 R 325 [60] N. N. Luzin, Leçons sur les ensembles analytiques et leurs applications. Collection de Monographies sur la théorie des fonctions. Gauthier-Villars, Paris, 1930. JFM 56.0085.01 R 327 [61] N. N. Luzin, Proof of one theorem in the theory of bendings. Izv. Akad. Nauk SSSR, Div. Techn. Sci. 1939, no. 2, 81–106; no. 7, 115–132; no. 10, 65–84. In Russian. Zbl 0023.26704 R 329 [62] N. N. Luzin and I. I. Privalov, Sur l’unicité et la multiplicité des fonctions analytiques. Ann. Sci. École Norm. Sup. (3) 42 (1925), 143–191. MR 1509265 JFM 51.0245.01 R 330 [63] B. N. Malinovsky, Pioneers of Soviet computing. A. Fitzpatrick (ed.), E. Aronie (transl.), and K. Maldonado (ed. consultant), First ed., Kiev, 1999. 2 nd ed. 2010. Electronic book. https://www.sigcis.org/malinovsky_pioneers R 338 [64] G. J. Martin, The theory of quasiconformal mappings in higher dimensions, I. In A. A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. IV. IRMA Lectures in Mathematics and Theoretical Physics, 19. European Mathematical Society (EMS), Zürich, 2014, 619–677. MR 3289712 Zbl 1319.30014 R 334 [65] D. E. Menchov, Impressions sur mon voyage à Paris en 1927. In R. Taton and P. Dugac (eds.), Cahiers du séminaire d’histoire des mathématiques. 6. Institut Henri Poincaré, Paris, 1985, 55–59. MR 0771485 Zbl 0571.01024 R 330 [66] G. S. Migirenko, Mihail Alekseevič Lavrentiev. Amer. Math. Soc. Transl. (2) 104 (1976) 1–39. Zbl 0329.01014 R 332, 334, 335, 337, 338 [67] E. Mioduszewski, Mathematicians and philosophers. Transl. from Polish and comments by G. Sinkevich. Al’manah “Russkij mir.” Prostranstvo i vremya russkoj kul’tury. Russkaya kultura 7 (2012), 179–199. In Russian. R 340 [68] P. Montel, Sur les séries de fonctions analytiques. Bull. Soc. Math. France (2) 30 (1906), 189–192. JFM 37.0412.03 R 331 [69] P. Montel, Leçons sur les séries de polynômes à une variable complexe. Gauthier-Villars, Paris, 1910. JFM 41.0277.01 R 331 [70] L. Nirenberg, Lipman Bers and partial differential equations. In J. Dodziuk and L. Keen (eds.), Lipa’s legacy. Proceedings of the 1st Bers Colloquium held at the City University of New York, New York, October 19–20, 1995, 455–461. Contemporary Mathematics, 211. American Mathematical Society, Providence, R.I., 1997. Reprint in L. Keen , I. Kra and R. Rodriguez, Lipman Bers, a life in Mathematics Amercian Mathematical Society, Providence, R.I., 2015, 47–54. MR 1477002 Zbl 1477002 Zbl 1361.35006 (reprint) R 333 [71] Yu. A. Neretin, Nikolay Luzin, his students, adversaries, and defenders. Notes on the history of Moscow mathematics, 1914–1936. Preprint, 2017. arXiv:1710.10688 [math.HO] R 342 [72] S. Novikov, Jr., My stories. In Russian. http://www.mi.ras.ru/~snovikov/Mem.pdf R 342 [73] E. R. Phillips, Nikolai Nicolaevich Luzin and the Moscow school of the theory of functions. Historia Math. 5 (1978), 275–305. R 325 [74] H. Poincaré, Science et méthode. Flammarion, Paris, 1908. R 320 [75] J. W. S. Rayleigh, On waves. Phil. Mag. 1 (1876), 257–279. JFM 08.0613.03 R 338

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[76] B. Riemann, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1867. JFM 01.0131.03 R 326 [77] L. L. Sabaneeff, Pavel Florensky—Priest, scientist and mystic. The Russian Review 20 (1961), no. 4, 312–325. R 323, 324 [78] L. L. Sabaneeff, Memories of Russia Classica, Moscow, 2005. http://www.belousenko.com/books/memoirs/sabaneev_vosp_o_rossii.htm R 341 [79] I. Kh. Sabitov, The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research. Tr. Mosk. Mat. O.-va 77 (2016), no. 2, 184–218. In Russian. English translation, Trans. Moscow Math. Soc. 2016, 149–175. MR 3643969 Zbl 1360.01027 R 329 [80] M. Ya. Suslin, Sur une définition des ensembles mesurables B sans nombres transfinis. Comptes Rendus Acad. Sci Paris 164 (1917), 88–91. JFM 46.0296.01 R 322 [81] P. Teilhard de Chardin, Le phénoméne humain. Seuil, Paris, 1970. R 336 [82] M. Ya. Vygodskij, Mathematics and its leaders at Moscow University in the second half of the 19 th century. Istoriko-matematicheskie issledovaniya 1 (1948), 141–181. R 340 [83] A. Weil, Œuvres Scientifiques. Collected Papers. Vol. I (1926–1951). Springer Verlag, Berlin etc., 1979. MR 0537937 Zbl 0424.01027 R 327 [84] V. A. Zorich, The theorem of M. A. Lavrenteff on quasiconformal mappings in space. Mat. Sb. 74 (1967), 417–433. In Russian. Zbl 0181.08701 R 334 [85] The Decree of the Presidium of the Academy of Sciences of 08/05/1936 about Academician N. N. Lusin. http://www.math.nsc.ru/LBRT/g2/english/ssk/ARTICLES/2012-1-11.pdf R 342

Part C Sources

Chapter 12

A letter Oswald Teichmüller (translated from the German by Annette A’Campo-Neuen)

In Norway, May 15, 1940 Dear Professor, I hope that in the meantime you have received a reprint of my article “Extremal quasiconformal mappings and quadratic differentials.” During my furlough in the fall, I have written down the application of quadratic differentials to extremal problems of conformal geometry that follow at the end of that article. It shall appear in “Deutsche Mathematik.” In both articles the proofs are missing, and there are many details still to be worked out (e.g. the problem of coefficients). Here I would like to inform you briefly into which direction I imagine a continuation of these investigations since I do not know whether I will still have the occasion to carry them out. Apart from the “ordinary” principal domains that are considered in the Academy article, there are “higher” principal domains arising from the ordinary ones by moving distinguished points together or from the interior to the boundary. For those the metric of the space R of classes of conformally equivalent topologically determined principal domains has no meaning because I cannot find extremal quasiconformal mappings for higher principal domains. But for ordinary principal domains, in R we have the (unsymmetric) notion of orthogonality of two (affine) d vectors a , b at a point P : a ? b means that d˛ k b + ˛ a k˛ =0 = 0 . The double dashes denote the convex Finsler metric for infinitely small scale. I would suppose that at least this relation a ? b also for higher principal domains has some meaning. Example: A sphere with 4 distinguished points that all coincide. Here one can think of passing to the limit from ordinary principal domains, however I am not yet able to do that. But maybe one can proceed more directly. For in the case of ordinary principal domains, at the end of this holiday paper, I could endow the relation a ? b in certain cases with a complex-analytic meaning, and it must be possible to generalize that to higher principal domains. Namely, if for an extremal problem as in the papers I referred to, the class represented by P of conformally equivalent, topologically determined principal domains is extremal,

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then the proof of this extremal property is obtained by integration along the lines d  2 > 0 with the metric jd  2 j , where d  2 ¤ 0 is a regular quadratic differential of the principal domain that is determined up to a constant positive factor. To this d  2 corresponds in R a vector b at the point P which is only determined up to a positive factor. The a with a ? b form a hyperplane, i.e. a tangent direction at P . And this is the tangent direction of the subset of R , defined by the extremal problem of conformal geometry, having P as a boundary point (we will ignore vertices and edges). This connection should be developed further, in particular with respect to higher principal domains. This leads to another further reaching problem. Let a (firstly ordinary) principal domain be given and a regular quadratic differential d  2 or a point P of R and at P a direction b as above. I imagine that to these two one can associate a somehow regularly bounded subset M (P; b ) of R . So far I cannot define this subset, and I just want to note some properties. M (P; b ) is closed and has P as boundary point and is unbounded. M (P; b ) and M (P; b ) have no common point except P . If  = 2 and  = 1 , (i.e., a ring surface without distinguished points or a sphere with 4 distinguished points), then most likely M (P; b ) is the interior of the limit circle on whose boundary P lies and such that b points into it orthogonally. For c > 0 , we have M (P; b ) = M (P; c b ) . The common tangent direction of M (P; b ) and M (P; b ) at the point P is given by the a with a ? b . Let Q(P; b ; s) be the geodesic through P in direction b , with arc length s starting from P as parameter. Then Q(P; b ; s) , for s  0 lies in M (P; b ) and for s  0 lies in M (P; b ) . In my last article, I considered the set of all points S where for some s  0 (and hence also for all greater s ) we have [S; Q(P; b ; s)]  s . This will be a subset of M (P; b ) , maybe it even equals M (P; b ) . The open kernel of M (P; b ) is the set of all S in R having the property that the fact that the topologically determined principal domains belonging to P and S cannot be mapped conformally onto each other may be proven by the method of Grötzsch and Ahlfors using the quadratic differential d  2 . This is a terribly inexact definition that needs to be replaced by an analytic inequality. If one here allows equality then we obtain M (P; b ) . Hence we ask for an investigation of how far the method of Grötzsch and Ahlfors (within finiteness) reaches. It will be possible to describe M (P; b ) by several definitions whose equivalence is the scientific benefit that I hope for. The statement above of whether Q(P; b ; s) belongs to M (P; b ) , subordinates to our “definition” as you can see from the explicit form of the geodesic in my Academy article. Maybe one can always map conformally the principal domain H belonging to P to the one belonging to S 2 M (P; b ) such that H is sliced along the lines d  2 < 0 and then put together differently or something similar, and it should be possible to define M (P; b ) in this way. That would be the converse of the fact that in Grötzsch arbitrarily many slits do not make any difference. But the number of necessary slits is most probably bounded.

12 A letter

353

I suppose that the Finsler metric in general does not belong to a quadratic form. Otherwise daring limit processes to the unit disc with only distinguished boundary points and beyond would lead to contradictions. Heil Hitler! Yours, Oswald Teichmüller

Editor’s note. A scan of the letter (originally hand-written in Sütterlin handwriting) was provided to us by Reinhard Kühnau. The letter was given to Kühnau by Kurt Strebel about 30 years ago. The name of the recipient is unknown.

Chapter 13

On some extremal problems of the conformal mapping Herbert Grötzsch (Leipzig) presented to the academy by Mr. Koebe (translated from the German by Annette A’Campo-Neuen) Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse, Bd. 80 (1928), 367–376.

Contents I II III IV

Two lemmas . . . . . . . . . . . . . . . . Application of the two lemmas . . . . . . The problem of the closest boundary point The problem of the closest boundary point

. . . . for for

. . . . . . . . . . domains domains

. . . . Kr K0

. . . . . . . . . . ır and K ı0 and K

. . . .

355 358 359 360

The following investigation contains some new theorems and some extensions of known results on extremal bijective conformal mappings. The proofs are all based on the same elementary principle which is expressed in lemmas 1 and 2.

I. Two lemmas Lemma 1. In the concentric circular annulus 0 < r  jzj  1 , let Sk (for k = 1; 2; 3; : : : ) be finitely or infinitely many non-overlapping planar strips meeting at most along boundary curves. Suppose that the Sk are placed in the annulus as bijective conformal images of rectangles Rk such that the vertices of Rk correspond to points Pk and Pk0 on jzj = r , and to points Pk00 and Pk000 on jzj = 1 , and such that everywhere else the mappings of the boundaries onto each other are regular. Let ak denote the length of the side of Rk corresponding to the arc Pk Pk0 , and let bk denote P1 theaklength of a side from the other pair of sides of Rk . Then for the modulus k =1 bk the following holds Original title: Über einige Extremalprobleme der konformen Abbildung.

356

Herbert Grötzsch 1 X ak 2  ; bk log 1r k =1

and equality occurs if and only if all the Sk arise via radial slicing of the annulus, and moreover the total area of all Sk equals (1 r 2 ) . (Such a decomposition into strips Sk shall be called for short a maximal subdivision of the annulus.) Proof. By choosing a suitable decomposition of the Sk into finitely or infinitely many new substrips, one can achieve that each of the new numbers abkk is of the P1 ak form n1k (for nk = 1; 2; 3; : : : ) without changing the sum k =1 bk . We now assume that the decomposition has been chosen in this way. In a  -plane, we place similar images of the Rk —for simplicity also denoted by Rk —next to each other such that all the Rk are contained in a rectangle R with vertices 0 , 2 i , 2 i M , M (for some real M > 0 ) and all the sides of length ak corresponding to the arcs Pk00 Pk000 lie on the segment from 0 to 2 i , and such that the sum over the area of all Rk equals 2M . Then we have 1 X

ak = 2;

1 1 X X ak 1 2 = = : bk nk M

bk = M;

k =1

k =1

k =1

The annulus, cut up along one of the boundary curves of S1 say, to obtain a simply connected domain, can be mapped via  = log z . The rectangle Rk is decomposed () () into n0k congruent rectangles Rk (  = 1; 2; : : : ; n0k ) with sides of length ak = nak0 k

()

()

and bk = M . Each Rk is divided into nk
n0k congruent squares. The number n0k is chosen big enough so that the image of every square of Rk under k = fk () () differs arbitrarily little from the shape of a square. Let Lk be the length of the () () image of a side of Rk lying in the interior of R , let Jk be the area of this image, j∆jk = n Mn0 the length of the side of a square of the square decomposition k k

of Rk , and let

(;) k

be the vertices of the square (for  = 1; 2; : : : ; nk n0k ). Then: nk n0k

() Jk

=

() c˜k




nk n0k (;) 2 j fk0 (k )j j∆j2k

X

=

() c˜˜k




() (Lk )2

=1

X

(;) 2

(qk

) ;

=1

where

nk n0k (;) j fk0 (k )j


j∆jk =

() Lk




(;) qk

and

X

(;)

qk

= 1:

=1

The minimum of (;)

Pnk n0k =1

(;) 2

(qk

) is then

1 nk n0k

() (Lk )2 nk n0k

, and it is only attained for equal

() () () . Hence Jk  c˜˜k
, where c˜˜k after sufficiently fine subdivision of Rk differs arbitrarily little from 1 for all  , if n0k is sufficiently big. If the mapping of Rk via k = fk () is not a similarity mapping, then for at () most one  , exceptionally Lk = log 1r . By possibly further subdividing all Rk ,

qk

=

1 nk n0k

13 On some extremal problems of the conformal mapping

357

this case can be excluded. Thus, unless Rk is mapped by a similarity mapping, we can set 1 () Lk > ck
log for some ck > 1; for all : r Hence we obtain 1 1   X 1 1 2 X 1 1 2 2 2 log  Jk  log
= log
; r r nk r M k =1

k =1

and thus 2 2  : 1 M log r

Equality holds if and only if all Rk are mapped congruently.

□

If instead of Rk one considers smaller rectangles contained in Rk and with some common boundary with Rk , one can see that the assumption of regularity of the mapping at the boundary may also be omitted.1 If the annulus is cut up along arbitrarily finitely many concentric arcs with 0 as center, and if the individual Sk are put together from finitely many pieces such that possibly two consecutive pieces of Sk are connected by identifying boundaries 2 along the same cut, then the argument above leads to the same maximal value log 1 r for the modulus and the same maximal decomposition where the assignment of boundaries is the identity (here we ignore the exchange of parts of two different strips). Lemma 2. Now instead of the strips Sk that cross the annulus, we consider strips enclosing the inner circle jzj = r . If we imagine these strips being cut up such that after cutting they can be mapped onto a rectangle Rk being faithful on the vertices with abkk as small as possible, then we obtain analogously that the P1 ak log 1r maximal value of the modulus k =1 bk is 2 , and the maximal decomposition is given by a decomposition into circles with 0 as center.2 The same is true for the circular annulus 0 < r < jzj < 1 that is cut up along finitely many radial slits where the assignment of boundaries is the identity as above. 1 The principle of my proof was unfolded to me by Mr. G. Faber’s method of treating the assignment of boundaries under a conformal mapping, see Münch. Ber. 1922, pp. 96–99. Compare also a similar argument in P. Koebe, Math. Ann. 69 (1910), pp. 20–21. 2 This proposition is related to the following result: “If within the circular annulus r  jzj  1 (the annulus Kr ) there are finitely many non-overlapping, doubly connected planar strips that enclose the circle jzj = r , then the product of the moduli of these strips is  r ( = modulus of Kr ), and equality holds if and only if Kr is decomposed into strips by concentric circles.” This theorem was mentioned to me by Mr. P. Koebe as a result known to him but not yet published by him that could easily be proven with Dirichlet’s minimal property of potential functions.

358

Herbert Grötzsch

For non-analytically bounded domains, the maximal value of the modulus is given as the limit of the respective moduli of exhausting analytically bounded approximating domains. It is generally an invariant with respect to conformal mappings.

II. Application of the two lemmas Let us denote the circular annulus 0 < r < jzj < 1 for short by Kr , and the surface ır . Kr , slit along finitely many arcs with 0 as center by K ır is mapped bijectively conformally such that jzj = r Theorem. If the domain K is mapped entirely to jwj = r and jzj = 1 is mapped entirely to jwj = R > r , then3 R  1 , and R = 1 only holds if the mapping is a rotation.

Proof. Suppose that R < 1 . Then from spreading out a maximal cross-stripdecomposition of the annulus r < jwj < R in the image and assigning boundaries in radial direction, we would obtain a decomposition of the annulus Kr whose 2 modulus would be greater than the maximal modulus log 1 . If R = 1 , then every r radial slit would be mapped to a radial slit and the concentric circles as orthogonal trajectories would be mapped to concentric circles. That means that the mapping would be a rotation.4 □ Applying lemma 1 or 2 yields the following theorem: Let 1 ; : : : ; n be the slits ır , and suppose that K ır is mapped by wk = fk (z) bijectively along arcs of a domain K conformally such that jzj = r corresponds to jwj = r , the first k slits 1 ; : : : ; k are mapped to radial slits, the remaining k +1 ; : : : ; n are mapped to arcs with center 0 and jzj = 1 is mapped to jwj = Rk > r , then 1 < R1 < R2 <


< Rn . ır where jzj = r is sent to jwj = r , For every bijective conformal mapping of K and jzj = 1 is sent to jwj = R > r , and the k are mapped to arbitrary boundary curves, we have R  Rn , and R = Rn only holds if all the k are mapped to radial slits.

3 Translator’s note: Correction of typing error in the German original. 4 Concerning conformal mappings onto slit domains, see P. Koebe, Abhandlungen zur Theorie der konformen Abbildung IV, Acta Math. 41, pp. 305–344.

13 On some extremal problems of the conformal mapping

359

III. The problem of the closest boundary point ˆr for domains Kr and K ır to Theorem. Consider a bijective conformal mapping of a domain Kr or K ˆ a domain Br or Br respectively, not containing 1 in its interior, such that jzj = r is sent entirely to jwj = r and no point jwj < r occurs in the image. Let ˆr . This has distance d  dr jzj = 1 correspond to the outer boundary of Br or B to w = 0 , where dr is the corresponding distance for one of those mappings A of Kr sending jzj = r to jwj = r and jzj = 1 to a ray to 1 on whose extension lies the point w = 0 .

Proof. For a mapping A there is a maximal subdivision where for every Sk there is an associated strip Sk 0 which is symmetric with respect to the bounding ray. The strips Sk and Sk 0 contribute the same amount n1k = n1 0 to the maximal modulus k 2 . Together they form a strip Sk + Sk 0 which is the bijective conformal image log 1 r

of a rectangle with side ratio 2n1k , where every maximal subdivision line of Sk together with its mirror image in Sk 0 forms a maximal subdivision line of Sk + Sk 0 . ˆr Now suppose d < dr . Then on the outer boundary of the domain Br or B respectively, there would be a point at distance dr to w = 0 . We place a maximal strip subdivision of a mapping A which is symmetric with respect to the boundary slit leading to 1 , into the w -plane in such a way that the closest boundary point of the mapping A coincides with the observed point. If the strip Sk and its mirror ˆr Sk 0 are chosen sufficiently narrow then a piece of the outer boundary of Br or B respectively, will cross the strip Sk + Sk 0 either completely within Sk or completely within Sk 0 , or it will decompose it again into Sk and Sk 0 . In the first case, the 1 newly occurring strips Szk and Szk 0 contribute the amount nk +x + nk 1x 0 > n2k to k k the modulus, since then xk ¤ 0 and xk 0 gets arbitrarily close to xk . In the last case, Szk and Szk 0 contribute the amount n2k . Thus we would obtain a decomposition E ˆr respectively, with greater modulus than the maximal modulus 21 , in Br or B log

r

unless all strips Sk + Sk 0 are decomposed by the outer boundary precisely into Sk and Sk 0 . Hence this must be the case here. The decomposition E is maximal, the boundaries of all possible strips Szk and Szk 0 are images of radial slits. Every curve which is an image of a radial slit under A , is also the image of a radial slit via the given mapping. Therefore being an orthogonal trajectory, every curve occurring as the image of a circle jzj =  under A , is also the image of a circle jzj = 0 under the given mapping (and even  = 0 ). The outer boundary of Br ˆr is thus reduced to the slit provided by the mapping A , and this contradicts or B the assumption that d < dr .

360

Herbert Grötzsch

In the case d = dr it follows that the map considered is one of the mappings A . ır one needs to use a uniqueness For a domain Kr this is immediate and for a domain K theorem on slit mappings.5 □ The same method of proof yields the following theorem. Suppose that a domain ır is mapped, bijectively and conformally by ∗ w = f (z) , to a domain not Kr or K containing 1 in its interior, in such a way that jzj = r is mapped entirely to jwj = r , that no point jwj < r occurs as image point and that jzj = 1 corresponds to the outer boundary of the image domain. Suppose that there are n points on the outer boundary that are closest to the point w = 0 , and that they form the vertices of a regular n -gon with w = 0 as center, and that moreover j f (z)j  M > 1 . Then the minimal distance in question, of the outer boundary of the image domain to w = 0 , is attained by mapping Kr to a domain bounded by jzj = r , jzj = M and with n radial slits emanating from the points ˛
r;M;n
e

2 i k n

(r < r;M;n < M; j˛j = 1; k = 0; 1; : : : ; n

1)

to jwj = M and containing 0 in their prolongations. Here M = 1 is also possible. ır .) The minimum is only taken (This mapping is also the extremal mapping for K by the described extremal functions.6 †

IV. The problem of the closest boundary point ˆ0 for domains K0 and K ı0 the unit disc slit along finitely Let K0 denote the unit disc in the z -plane, and K many arcs with center 0 . ı0 be mapped bijectively and conformally by w = f (z) to Theorem. Let K0 or K ˆ0 not containing 1 in its interior such that f (0) = 0 and a domain B0 or B 0 ˆ0 corresponds to jzj = 1 j f (0)j = 1 . Suppose that the outer boundary of B0 or B and that there are n boundary points at shortest distance d to the point w = 0 forming the vertices of a regular n -gon with center 0 . Then r n 1 d : 4 5 See P. Koebe, Abh. zur Theorie der Konf. Abb. IV, Acta Math. 41, pp. 305–344. The uniqueness theorem also follows directly from lemmas 1 and 2. ∗ [Translator’s note] Added for clarity. 6 Mr. G. Szegö has posed this problem in Berichte der Deutschen Math. Ver. 31, 1922, p. 42, Exercise 2. † [Editor’s note] The reference should rather be G. Szegö, Über eine Extremalaufgabe aus der Theorie der schlichten Abbildungen. Sitzungsberichte der Berliner Mathematischen Gesellschaft 22 (1923), pp. 38–47.

13 On some extremal problems of the conformal mapping

The value

q

1 4

p z (1+z n )2

. This function q 2 i k maps B0 to the w -plane slit along the n rays emanating from the points n 14
e n ( k = 0; 1; 2; : : : ; n 1 ) to 1 . 7 q q Proof. Suppose that d < n 14 . We consider the function w(z) z = d˛
n 14
f (z) , n

is attained by the extremal function w =

361

n

e

ˆ z 0 or B where ˛ with j˛j = 1 is determined such that the image B q 0 of K0 2 i k ı0 respectively under w(z) or K z has the n closest boundary points n 14
e n . A sufficiently small circle jzj = r is sent by w(z) z to an image curve lying outside z of the circle jwj = Q
r , for fixed Q > 1 . The function np(1+z sends this circle n )2 jzj = r to a curve in the interior of jwj = Q
r . If as above one now considers maximal strips Sk and Sk 0 of a maximal decomposition that are symmetric with z respect to the radial slits as given by the function np(1+z , then one finds that in n )2

e

ˆ0 by adding the image of jzj = r under z 0 or B the new domain obtained from B w(z) z as new boundary, there would be a decomposition into strips whose modulus q

2 n 1 would be greater than the greatest possible modulus log . 1 . Therefore d  4 r Now we q need a special consideration in order to prove that all extremal functions

with d =

n

1 4

, are of the form

p ˛z (1+ˇ z n )2

n

(for some j˛j = jˇj = 1 ).

q 2 i k First we treat the mappings of K0 . We assume that the points n 14
e n z are the n closest points on the outer boundary of B0 . The function np(1+z , n )2 for small r , maps the circle jzj = r into a curve which is almost a circle. The maximal decomposition of the image of r  jzj < 1 is symmetric with respect 2 i k to every straight line through 0 and e n . We again consider any two maximal strips Sk and Sk 0 , that are symmetric with respect to such a straight line and that are joining to a unique strip which emanates from the image of jzj = r and returns back to it. The boundary part of B0 corresponding to the circle jzj = 1 via w = f (z) again either decomposes all Sk + Sk 0 into Sk and Sk 0 , or after sufficiently fine subdivision, there are certain strips Sk + Sk 0 that are decomposed into two new strips Szk and Szk 0 in such a way that the decomposing curve is either contained completely in Sk or in Sk 0 . In the first case, the outer boundary of B0 consists either only of the n slits extending to 1 —then we have identified f (z) as being of the form given above—or in addition to these slits there is at least one more slit occurring as the z image under np(1+z of a radial slit ending in jzj = 1 . The unit disc with this n )2 slit can be mapped by an elementary function to the unit disc such that 0 goes to 0 and the dilatation factor at 0 is greater than 1 . Hence we would have mapped the 7 Mr. G. Szegö has posed this problem in Berichte der Deutschen Math. Ver. 31, 1922, p. 42, Exercise 2.

362

Herbert Grötzsch

unit disc in the z -plane bijectively conformally to a proper or improper subdomain of another unit disc such that 0 is sent to 0 and the dilatation factor there is > 1 . This leads to a contradiction concerning the maximal modulus as above (without using Schwarz’s lemma). In the second case, by the following argument we find that the maximal modulus of the image of r  jzj < 1 for sufficiently small r , would be greater than log2 : (1/r) For sufficiently small jzj = r we have j log j f (z)jj = j log jz(1 + ˛2 z +


)jj < log

1 +c
r r

for fixed positive real c . For the image of r  jzj < 1 we obtain a strip decomposition with modulus >

2 2a a + 1 1 log r + cr log r + cr

b

+

log

1 r

a ; + cr + b

for fixed positive real a and b and possibly a new c . We arrive at a contradiction if for sufficiently small r we find
2a log 1r + cr 2 2a 2 + > :
2 1 1 2 log r + cr log 1r log + cr b r

We consider: 2

2a



2a



2a +

1+

2a 1

(log

b2 1 2 r +cr)

1 1

 cr  ? 2 1 + ; log 1r



?

2cr ; log 1r



>

2ab 2 2cr ? ; (log 1r + cr)2 log 1r

b2 2 (log 1 r +cr)

and 1+

1 1

(log

b2 1 2 r +cr)

for very small r . Now since limr !0 r
log 1r = 0 , indeed the following holds if r is sufficiently small:   1 cr  2ab 2 > 2cr log + cr 1 + : r log 1r Finally, one can see in a way closely related to the previous considerations that ı0 the functions of the form np ˛z also for a domain K with j˛j = jˇj = 1 are (1+ˇ z n )2 the only extremal functions. □

13 On some extremal problems of the conformal mapping

363

If in the extremal problem just considered one adds the condition j f (z)j < M , then one obtains as the only extremal functions those easily determined elementary functions that send jzj < 1 according to the conditions to a domain bounded by 2 i k jwj = M and by n radial slits emanating from M;n
˛
e n to jwj = M ( j˛j = 1 , k = 0; 1; : : : ; n 1 , M;n < M ).

Chapter 14

On some extremal problems of the conformal mapping II Herbert Grötzsch (Leipzig) presented to the academy by Mr. Koebe (translated from the German by Melkana Brakalova-Trevithick) Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse, Bd. 80 (1928), 497–502

Contents V The exact bounds for jf (z)j in the case of normalized schlicht mappings from Rr and R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 VI The exact bounds for jf 0 (z)j for normalized schlicht mapping of Rr and R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

The following investigations are an extension of my communication “Über einige Extremalprobleme der konformen Abbildung” (in short cited as Extremalprobleme I.) in the Berichte, vol. 80, 1928, pp. 367–376. We will prove again here the known distortion theorems concerning jf (z)j and jf 0 (z)j for normalized schlicht conformal mappings w = f (z) of the unit disk in the z -plane, denoted in short by R0 ; and extend them to normalized schlicht conformal mappings of the concentric circular ring r  jzj < 1; denoted by Rr : A schlicht conformal mapping w = f (z) , which maps R0 ; will be considered normalized if f (0) = 0; jf 0 (0)j = 1 , and if 1 is not an image point; in the case when w = f (z) maps Rr ; we call that mapping normalized if the circle jzj = r is mapped onto the circle jwj = r; jwj = jf (z)j  r and if 1 does not belong to the interior of the image. The function-geometric way of treating extremal problems of the conformal mapping which takes place here is based, as in Extremalprobleme I., on the solution of an Original title: Über einige Extremalprobleme der konformen Abbildung. II.

366

Herbert Grötzsch

elementary variational problem of the conformal mapping with the help of Lemma 1 from Extremalprobleme I., p. 367. If one assumes jf (z)j < M; one obtains also bounds for jf (z)j and for the minimum of jf 0 (z)j for the mappings from Rr and R0 ; for R0 these are the results of Mr. G. Pick found in his work Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet (Wiener Berichte vol. 126, 1917, as long as they are formulted without using the circle-geometric measure condition.1)

V. 2 The exact bounds for jf (z)j in the case of normalized schlicht mappings from Rr and R0 Let Er (z) be the function which maps Rr schlicht and conformally onto a domain bounded by jwj = r and an infinite slit along the positive real axis such that the point z = +r is contained in the backwards prolongation of the slit. Let Mr () be the maximum of jEr (z)j; and mr () be the minimum of jEr (z)j; for jzj = : Then the following distortion formula holds on jzj =  : mr ()  jf (z)j  Mr ();

where an equality sign holds only for the extremal functions ˛
Er (ˇz); with j˛j = jˇj = 1; in such a way that each occurs at a unique point on jzj = : The proofs of the inequalities mr ()  jf (z)j and jf (z)j  Mr () given below can be used to verify directly that the absolute value of Er (z) attains its maximum and minimum on jzj =  exactly at z =  and z = ; respectively, so that one does not necessarily derive these facts from the analytic expression of Er (z):

a) Proof of the inequality mr ()  jf (z)j: The domain Rr will be slit along the segment from z0 to jzz00 j ( r < jz0 j < 1 ) to obtain a domain R0r : For the non-overlapping strips in the interior of R0r that connect jzj = r to the outer boundary of R0r there exists, in the sense of Lemma 1 from Extremalproble I, pp. 367–369, a symmetric with respect to the line through 0 and z0 strip subdivision with (finite) maximum module M 0 : For a domain R00r obtained from Rr ; in a similar manner, by a slit starting at a point 1 With regard to the literature on the distortion problem, reference is usually made to the encyclopedia article II C 4 by Mr. L. Bieberbach “Neuere Untersuchungen über Funktionen von komplexen Variablen,” in particular number 64 and 65, pp. 510–513; see also the citation on conformal mapping in the encyclopedia article II C 3 by Mr. L. Lichtenstein: “Neuere Entwickelung der Potentialtheorie. Konforme Abbildung,” in particular p. 311, footnote 462, and p. 321, footnote 494. 2 This numbering follows the one in Extremalprobleme I.

14 On some extremal problems of the conformal mapping II

367

˛
z0 ; (r < j˛z0 j < 1; j˛j < 1); the corresponding maximum module M 00 is greater than M 0 . If f (z) were different from any of the functions ˛
Er (ˇz); with j˛j = jˇj = 1; and such that jf (z0 )j  mr () for jz0 j = ; then one would conclude, in analogy to the closest boundary point problem from Extremalproble I, pp. 371f, which uses the strip subdivision in the image domain of R0r under the extremal function
f (z0 ) E jzz00jz , that in the image domain the maximum value of the module would jf (z0 )j r be > M 0 ; though M 0 is the maximum.

b) Proof of the inequality jf (z)j  Mr (): The domain Rr will be slit along the segment s 0 from jzz00
r to jzz00j to obtain j a new domain R0r . We consider not self-intersecting and mutually non-overlapping strips lying inside R0r , such that they originate at the segment s 0 and the circle jzj = 1 and end at both sides of the segment from zjz00
rj to z0 : One such maximal strip subdivision with a maximum finite module M 0 is symmetric with respect to the line through 0 and z0 in a way that after reflection along this line, each strip is mapped onto a corresponding strip. Then one maps the domain R0r , slit along the straight line from z0 to jzz00 j , onto a rectangle so that the twice counted points rz0 and jzz00 j are mapped onto the vertices of the rectangle, and then one applies jz0 j the method of Lemma 1 in Extremalprobleme I. If one considers, instead of z0 ; a point a
z0 with jaj > 1; then the maximum module M 00 of the domain R00r ; obtained in a similar manner, is greater than M 0 : Now, if for z0 = ; jf (z0 )j > Mr () , one would consider once more, in analogy to what was done earlier, the conformal image in the w -plane of a maximal subdivision of a suitable domain R00r ; through the extremal function
f (z0 )
Er jzz00 jz ; one would conclude that the maximum module of the constructed jf (z0 )j image of R0r under f (z) would be  M 00 ; therefore > M 0 ; though it is = M 0 : If jf (z0 )j = Mr (); then the resulting strip subdivision of the image domain
of R0r by the extremal function jff(z(z00)j)
Er jzz00 jz is a maximal subdivision. From here it follows that the segment connecting z0 to jzz00 j corresponds to the radial ray connecting f (z0 ) to 1 and not containing 0; and from here, that the image domain is symmetric with respect to the line through 0 and f (z0 ); and that the rf (z0 ) segment from z0 to r jzz00 j is transformed into the segment from f (z0 ) to jf : (z0 )j z0 r z0 Therefore the image of the segment from jz0 j to jz0 j lies on the radial ray arg w = arg jff(z(z00)j) ; and the same is true for the entire boundary in the image domain corresponding to jzj = 1 since the resulting strip devision is a maximal one. Thus f (z) proves to be an extremal function.

368

Herbert Grötzsch

c) If R0 is mapped, then z   ; m0 () = ; M0 () = ; (1 + z)2 (1 + )2 (1 )2 and the known distortion formula holds:    jf (z)j  for jzj = : 2 (1 + ) (1 )2 E0 (z) =

  The impossibility of the inequalities jf (z0 )j < (1+) 2 and jf (z0 )j > (1 )2 for jz0 j =  follows from the observation that for sufficiently small r a very small circle jzj = r is transformed into a curve which deviates less and less from the circle jwj = r; however instead of this curve in the image domain one uses the circle jwj = r  inscribed in it to show that jf (z0 )j < m0 () is not possible, and then also uses the smallest circle jwj = r  circumscribed around the curve to show that the inequality jf (z0 )j > M () is not possible. If f (z) is different from any of the extremal functions ˛
E0 (ˇz) (j˛j = jˇj = 1); the impossibility of the equality jf (z0 )j = m0 () for jz0 j =  is a result of a straightforward transfer of the conclusions applied in Extremalprobleme I., pp. 374–376. Furthermore, a consideration similar to the process used on planar strips, as in §Vb, finally shows that if f (z) is not an extremal function then the equality jf (z0 )j = M0 () (jz0 j = ) cannot hold.

d) If one adds the condition jf (z)j < M (M > 1); then one can apply the given approach one more time in order to conclude that both extremal bounds for jf (z)j on jzj =  can be obtained only from those extremal functions which map Rr onto a schlicht domain which is bounded by jwj = r; jwj = M and by a straight-line slit running from a suitable point w0 w0 (r < jw0 j < M ) to jw
M: Under the condition jf (z)j < M the same applies 0j for appropriate schlicht mappings on3 R0 .

VI. The exact bounds for jf 0 (z)j for normalized schlicht mapping of Rr and R0 Let m0r () be the minimum of jEr0 ()j on jzj =  and let Mr0 () be the maximum. One has m0r () = jEr0 (+)j and Mr0 () = jEr0 ( )j . These equations need not 3 Mr. G. Pick is the first to derive the stated exact bounds for jf (z )j for such mappings from R0 : See G. Pick: “Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet,” (Wiener Berichte Bd. 126, 1917, pp. 247–263); with regard to numerical calculation, reference is made to this work.

14 On some extremal problems of the conformal mapping II

369

necessarily be taken from the analytic expression for Er0 (z); as in §V above, they are given by the following proof, when applied to the extremal mapping itself. The distortion formula m0r ()  jf 0 (z)j  Mr0 ()

holds on jzj = ; where equality holds only for the functions ˛
Er (ˇz) (j˛j = jˇj = 1); at a suitable point on jzj = :

a) Proof of the inequality m0r ()  jf 0 (z0 )j: We construct a small circle jz z0 j = r  ; with center z0 ; which lies entirely within Rr ; as well as two straight-line slits s1 , connecting jzrz0 j0 to jzz00j ; and s2 ; connecting jzz00 j (jz0 j + r  ) to jzz00 j : We consider non–intersecting planar strips, lying inside Rr , which do not intersect the circle jz z0 j = r  ; starting at the slits s1 and s2 ; as well as jzj = 1; and going to both sides of the segment from rz0 to jzz00 j (jz0 j r  ): Through an appropriate auxiliary mapping onto a rectangle, jz0 j similar to §Vb, one recognizes that the maximal such strip subdivision with (finite) maximum module M is symmetric with respect to the line through 0 and z0 and so that for each such strip corresponds a symmetric one with respect to this line. Let f (z) be neither one of the functions ˛
Er (ˇz) (j˛j = jˇj = 1) . Then, according to §Va, jf (z0 )j > mr () for jz0 j = : If now jf 0 (z0 )j  m0r (); after considering a sufficiently small circle jz z0 j = r  whose image is almost circular, since jf (z0 )j > mr (); one would obtain a strip subdivision in the image domain which has module > M , while yet M is the maximum module.

b) Proof of the inequality jf 0 (z)j  Mr0 (): We construct again a sufficiently small circle jz z0 j = r  and consider a symmetric, with respect to the line through 0 and z0 ; strip subdivision of non-intersecting strips with maximum module M , connecting jz z0 j = r  to jzj = r both sides of the segment from jzrz0 j0 to jzz00j ; and to the boundary jzj = 1: Now, if f (z) is neither of the extremal functions ˛
Er (ˇz) (j˛j = jˇj = 1); by §Vb jf (z0 )j < Mr () (jz0 j = ); if it were jf 0 (z0 )j  Mr0 (); when one uses a sufficiently small circle jz z0 j = r  whose image is close to a circle, there would have been a strip subdivision of the image domain with module > M though jf (z0 )j < Mr (); but this is impossible.

c) If R0 is mapped, then the extremal function is E0 (z) =

z ; (1+z)2

1  1+ 0 thus m00 () = (1+) 3 and M0 () = (1 )3 : For a function which differs from each of the functions ˛
E0 (ˇz) (j˛j = jˇj = 1) the following known distortion formula holds for jzj   : 1  1+ < jf 0 (z)j < : 3 (1 + ) (1 )3

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Herbert Grötzsch

This can be proved, in analogy to sections §VIa and §VIb, when one observes that the inequality m0 () < jf (z0 )j < M0 () is already derived.

d) If one adds the condition jf (z)j < M; one can show, using one of the methods employed in the proofs here, that only the extremal functions considered in §Vd for jf 0 (z)j on jzj =  provide the exact lower bounds. For mappings from R0 ; this result is derived by Mr. G. Pick in the work cited in footnote in §Vd.

Chapter 15

On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem Herbert Grötzsch (Leipzig) presented to the academy by Mr. Koebe (translated from the German by Melkana Brakalova-Trevithick) Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse, Bd. 80 (1928), 503–507

Contents I A notion of planar AQ mapping . . . II Properties of the schlicht AQ . . . . . III The big Picard theorem for mappings distortion AQ . . . . . . . . . . . . . .

. . . . of . .

. . . . . . . . . . bounded . . . . .

. . . . . . . . . . . . . . . . . . . . infinitesimal . . . . . . . . . .

371 372 374

The following study follows my two communications: “Über einige Extremalprobleme der konformen Abbildung” (these Berichte, Bd. 80, 1928, pp. 367–376)—cited as Extremalprobleme I—and “Über einige Extremalprobleme der konformen Abbildung” (these Berichte, Bd. 80, pp. 497–502)—cited as Extremalprobleme II.

I. A notion of planar AQ mapping A single valued, invertible mapping of a planar domain G in the z -plane onto a surface, stretched as a Riemann surface over the w -plane, is called an AQ mapping of bounded infinitesimal distortion Q; if the following conditions hold: Original title: Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes.

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Herbert Grötzsch

1. the mapping is continuous everywhere; 2. in general, i.e. apart from at most infinite countably many points which can only accumulate at the boundary of G , the mapping is schlicht in a neighborhood of each point; 3. at the regular points, where the mapping is also schlicht, in the sense of a differential-geometric approach, an infinitesimal circle is transformed onto an infinitesimal ellipse, whose major axes quotient ab satisfies the inequality 1  ab  Q; with the same Q at all regular points z ; Q 4. at the exceptional points mentioned in 2., the mapping is continuous, and a schlicht neighborhood of such a point z is mapped onto an n -sheeted ( n -finite) branched surface, so that z correspond to the branch point; 5. if one sets z = x + iy; w = f (z) = u + iv; in a neighborhood of a regular point z , the partial derivatives @u ; @u ; @v ; @v are continuously differentiable. @x @y @x @y For Q = 1 the mapping is conformal.

II. Properties of the schlicht AQ With regard to the distortion of schlicht mappings, the more general mappings AQ of bounded infinitesimal distortion have to a certain extent properties analogous to the schlicht conformal mappings; the boundedness of the infinitesimal distortion is connected to the boundedness of the distortion at large. We consider schlicht AQ mappings of a schlicht domain G . Based on Mr. G. Faber’s studies, going back1 to Extremalprobleme I, pp. 367–369, if one chooses an arbitrary small "; the image of an arbitrary, yet sufficiently narrow rectangle in an interior subdomain of G , satisfies the inequality: J >

1

"2 L2
; Q n

where J is the area of the image, L is the length of the image of the long rectangle side, and n1 denotes the ratio ( < 1 ) of the sides of the rectangle. We only need to observe that an infinitesimal square is mapped to an infinitesimal parallelogram, and that the area of the latter, with lengths of one of its sides s , is between the 2 bounds (1 + 2 )Q
s 2 and 1 Q
s 2 ; where  can be chosen arbitrarily small. Using this inequality, we can derive several elementary facts about the mapping. a) If a rectangle with sides of lengths a and b is mapped by an AQ mapping onto a rectangle with sides of lengths a and kb , so that vertices are mapped onto 1 See the footnote on p. 369 in Extremalprobleme I.

XV. On the distortion of schlicht non-conformal mappings

373

corresponding vertices, and sides of length a correspond to sides of length a; it follows: 1  k  Q: Q Equality sign holds only for certain easily specified extremal affine mappings. b) After using the function log z; where a composition of a, AQ mapping with a conformal mapping results in an AQ mapping, the following theorem holds: If two concentric circular rings r < jzj < 1 and r  < jzj < 1 are mapped bijectively and single-valued onto each other by an AQ mapping, then it holds: 1

rQ  r  r Q ;

and equality sign hold only for certain extremal mappings, each one of them arising from rotation and reflection. The theory of boundary correspondence for schlicht AQ mappings (of prescribed infinitesimal distortion Q ) from the domain jzj < 1 onto a simply connected domain bounded by a Jordan curve holds exactly as for conformal mappings, based on a proof by Mr. G. Faber for conformal mappings, given in Münch. Berichte 1922, in particular, pp. 96–99, which can be generalized immediately to the one considered.2 The methods, used in the proofs in Extremalprobleme II to study distortion properties of schlicht conformal mappings, can be carried over, as far as possible, to orientation-preserving schlicht AQ mappings of bounded infinitesimal distortion, when one observes that after applying such a mapping to a planar region with 4 marked boundary points, the image has module, which is at most Q times and at 1 least Q times the module of the original planar region. Skipping the details for more precise bounds, one obtains the following theorem: The concentric circular ring r  jzj < 1 will be mapped by a continuous mapping w = u + iv = f (x + iy); with continuous partial derivatives @u ; @u ; @v ; @v ; @x @y @x @y so that jzj = r is carried to jwj = r; j f (z)j  r holds and 1 is not an image point. If Q is the maximum infinitesimal distortion ratio (= ratio of the major axes  1 of the ellipse which is the image of an infinitesimal circle) at all points z which satisfy the condition r  jzj   + 1 2  < 1; then there exist two functions s(Q; r; ) and S(Q; r; ); depending on Q; r; ; such that: s(Q; r; ) < j f (z)j

r < S (Q; r; );

for jzj = :

(1)

Moreover, for fixed Q; r; lim S (Q; r; ) = 0: There exist also two functions  !r

s ? (Q; r; ; j∆zj) and S ? (Q; r; ; j∆zj); such that for jzj =  and r < jz + ∆zj <  + 1 2  it holds: ˇ ˇ ˇ f (z + ∆z) f (z) ˇ ? ˇ ˇ < S ? (Q; r; ; j∆zj): s (Q; r; ; j∆zj) < ˇ (2) ˇ ∆z 2 Cf. for clarification of this question a simple figure in “Vorlesungen über ausgewählte Gegenstände der Geometrie”, issue 2, p. 45, fig. 7, by Messrs. E. Study and W. Blaschke (Teubner, Leipzig 1913).

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Herbert Grötzsch

In addition, j f (z + ∆z) f (z)j < "; as long as j∆zj < than a value ı(Q; r; ); depending on Q; r; and : If these are schlicht mappings of the punctured disk 0 < jzj < 1; then we consider mappings that fix two points, perhaps 0 and +r (r < 1); and do not take on the value 1: Then, the corresponding distortion statements, analogous to statements (1) and (2), hold.

III. The big Picard theorem for mappings of bounded infinitesimal distortion AQ That both the small and the big Picard theorems hold for mappings AQ of bounded infinitesimal distortion is obtained in the following way: If an AQ mapping w = f (z) has an essential singularity at a point, which we assume to be z = 0; then it has the property that for some sequence z ; lim z = 0; lim f (z ) = ˛ and for a second sequence z0 , lim z0 = 0;  !1

 !1

lim f (z0 ) = ˇ; where ˛ ¤ ˇ:

 !1

 !1

Suppose that w = f (z) does not assume three values in 0 < jzj  r: By a fundamental theorem for the conformal mapping, the resulting planar image of 0 < jzj  r; under the AQ mapping, can be mapped, conformally and schlicht, onto the circular ring r  < jzj < r: By the big Picard theorem for conformal mappings, r  > 0: Consequently, 0 < jzj < r would be mapped to r  < jzj < r single-valued and bijectively by an AQ mapping except for infinite countably many points, accumulating only at 0 . However, this is impossible since one cannot fit in r  < jzj < r a sequence of doubly connected non-overlapping strips with an arbitrary large sum of modules R (the module in this case = log ; where R is the ratio of the radii > 1 of the 2 conformally equivalent to the strips circular rings), each of which wraps around the circle jzj = r  : This follows immediately when one applies the inequality 2 J > 1 Q"
L2
n1 to sufficiently narrow concentric circular rings, instead of to sufficiently narrow rectangles. Comment on I and II. The same direction, as the one communicated in this study, despite all the differences in the execution, is followed in the work of Mr. K. Szilárd: “Untersuchungen über die Grundlagen der Funktionentheorie” (Math. Zeitschr. vol. 26, 1927, pp. 653–671). Compare especially Theorem 10 on p. 670. Comment on III. Mr. S. Bernstein has proved closely related theorem to the one proved here in “Sur une généralisation des théorèmes de Liouville et de M. Picard” (Comptes rendus vol. 151, 1910, p. 636).

Chapter 16

On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions Herbert Grötzsch (Leipzig) presented to the academy by Mr. Koebe (translated from the German by Manfred Karbe) Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse, Bd. 82 (1930), 69–80.

Contents I II III IV

Concept of a mapping AQ . . . . . . . . . . . . . . Normalized mappings AQ . . . . . . . . . . . . . . . Theorems about normalized mappings AQ . . . . . . Theorems about mappings AQ normalized differently

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

375 376 377 382

I. Concept of a mapping AQ Let B be a schlicht, finite region of a z -plane.∗ We consider bijective continuous mappings of B onto a schlicht image region Bx of a w -plane that satisfy the following conditions: 1. Up to a finite number of interior points (exceptional points) of B , in the neighbourhood of any ordinary interior point of B the mapping shall be approximated uniformly continuous by an affine mapping, which implies certain differentiability characteristics of the pair of functions giving rise to the mapping, as one is accustomed to assume in differential-geometric investigations. Original title: Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche. ∗ [Translator’s note] There is a typo in the German original.

376

Herbert Grötzsch

2. Each ratio a/b of the principal axes of an infinitesimal ellipse which is the image of an infinitesimal circle shall satisfy the condition 1/Q  a/b  Q , where Q  1 is independent of the particular choice of the ordinary point of B just considered; that is, the distortion of the shape of Tissot’s indicatrix (distortion ellipse) of the mapping is generally kept within fixed bounds. A schlicht mapping of B satisfying these two conditions is briefly called a schlicht mapping AQ of B . We put z = x + iy and w = u + iv , and denote by w = f (z) the mapping function,† where w is generally not a complex-analytic function of z . Only in the case of Q = 1 , the mapping is necessarily directly or indirectly conformal, thus w is an analytic function of z or z¯ ( z¯ = x iy ). In general, a mapping AQ is either a direct or an indirect one, depending on whether or not the positive orientation of an infinitesimal circle, say, is preserved in the image. These mappings AQ belong to the differential-geometric mappings, in contrast to the more general bijective and continuous topological mappings. The fact that the mapping AQ is still continuous in one of the finitely many exceptional points in which the mapping does not show a differential-geometric behavior, that is, that a point must necessarily correspond to the exceptional point, is shown in Leipz. Ber. 80 (1928), p. 507, where (pp. 503–507) I have already investigated certain mappings AQ . More generally, under a schlicht mapping AQ , image boundary lines remain lines, as is the case with boundary points. Note that the inverse mapping of a mapping AQ is again a mapping AQ . Also observe that the composition of a schlicht mapping AQ and a schlicht conformal mapping, or vice versa, again yields a schlicht mapping AQ .

II. Normalized mappings AQ We consider a schlicht, finitely-connected, namely (n + 1) -fold connected ( n = 1; 2; 3; : : : ), region Bn+1 of the z -plane, which hereinafter is supposed to be a fixed normalized region, i.e., fully contained in the unit circle jzj = 1 , and the latter is the exact outer boundary of Bn+1 . Let R1 , R2 , …, Rn be the remaining n boundary lines or boundary points of Bn+1 . We consider normalized mappings AQ of Bn+1 , i.e., bijective mappings AQ to a likewise normalized schlicht region Bzn+1 of the w -plane, where, in addition, the outer boundary jwj = 1 of Bzn+1 as a whole is the image of the outer boundary jzj = 1 of Bn+1 as a whole. Here Q  1 is assumed to be fixed for the normalized mappings AQ under consideration. Denote the mapping function by w = f (z) ; let Rz1 , Rz2 , …, Rzn be the images of R1 , R2 , …, Rn . † [Translator’s note] Here and in the following, Grötzsch uses the German expression “Abbildungsfunktion” ( = “mapping function”).

XVI. On the distortion of non-conformal schlicht mappings

377

In the following, we shall ask for those normalized maps AQ of a given normalized region Bn+1 that provide the exact extreme value for certain simple geometric quantities to be considered here, whereby all possible normalized mappings AQ are permitted to compete, or occasionally only those which satisfy a meaningful boundary condition. The theorems (Theorems 1, 2, 3) in this respect are, in part, the generalization of some theorems on normalized conformal mappings, which I have stated in Leip. Ber. 81 (1929), pp. 220 and 221, and, in part, the generalization of some theorems (Theorems 5, 7, 8), which I have laid down in Leipz. Ber. 80 (1928), pp. 371–72 (§III) and pp. 498–500 (§V a and §V b). The proofs given here are all based on the method employed in the papers cited, which is to consider the moduli of more suitable surface strips, perhaps made connected only by means of boundary assignments. Compared with conformal mappings the following is essentially new: If M is the conformal modulus of a simply-connected schlicht surface strip F , punctured in four boundary points (corner points) (modulus in this case depending on the puncturing = one of the two side ratios of the uniquely determined rectangle to which the surface strip is conformally mapped, with corner points mapping faithfully), and M 0 is the corresponding conformal modulus of the image surface strip, punctured in the four image boundary points, resulting from F by a mapping AQ , the following 1 relation holds: Q
M  M 0  Q
M . Furthermore, if V is the conformal modulus of a schlicht doubly-connected surface strip F , bounded by two lines (conformal modulus in this case = radius ratio > 1 of the uniquely determined, schlicht concentric circular ring to which F is bijectively and conformally mapped), and V 0 is the modulus of the image surface strip resulting from F by a mapping AQ , 1 we have: V Q  V 0  V Q . For the proof of these facts, I refer to my explanations in Leipz. Ber. 80 (1928), pp. 504–05 (§II). Now the method, used for conformal mappings, to consider the moduli of suitably progressing surface strips can be transferred in a meaningful manner. The possible presence of a finite number of exceptional points of the mapping AQ makes no problem since, because of the continuity of the mapping AQ in the exceptional points, these latter as well as their images are not an obstacle to the construction of surface strips, as they are required in extremal proofs.

III. Theorems about normalized mappings AQ Theorem 1. For a normalized mapping AQ of Bn+1 , the area of the domain ‡ (more precisely, the outer surface area consisting of the points of Rzk and the points enclosed by Rzk ) enclosed by Rzk (which is assumed to be non-point-like (“nicht punktförmig”) attains its maximum value if and only if ‡ [Translator’s note] A region (“Bereich” in German) is a non-empty open subset of C ; a domain (“Gebiet” in German) is a connected region.

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Herbert Grötzsch

1. 2. 3. 4.

zk is a circle with center w = 0 , R the remaining Rzp (if present) are circular-arc slits with center w = 0 , the minor axes of all distortion ellipses are directed radially towards w = 0 , the ratio of principal axes of each distortion ellipse is constant = Q (see figure 1).

All extremal mappings arise from one of them by rotation around w = 0 and reflection in a straight line through w = 0 (uniqueness theorem).

Figure 1. The position of the principal axes of a distortion ellipse is shown schematically

Proof. We first deal with the case of the conformal mapping Q = 1 . From the theory of the conformal mapping, the existence of one of the asserted mappings Ùn+1 is known. Let B Ùn+1 be situated in a z -plane with onto a circular-arc region B inner circle radius r . We now assume that, for a certain normalized conformal Ùn+1 onto a normalized region Bzn+1 , the area enclosed mapping w = f (z) of B by Rzk , which corresponds to jzj = r , is   r 2 , and w = f (z) is different from Ùn+1 resulting from a rotation around the zero point and reflection any mapping of B in a straight line through the zero point. For the sake of simplification, it is assumed Ùn+1 to Bzn+1 is still regular and schlicht, at first that the conformal mapping of B even including the boundary of Rk and a piece beyond it. Then the length of Rzk is  2 r because of the isoperimetric property of the circle. A sufficiently narrow circular ring r  jzj  r +  will surely be mapped to a surface strip whose area is > (r + )2  r 2 . This can be seen by using the inequality J >

1

"2 L2
Q n

of the work already cited and published in Leipz. Ber. 80 (1928), p. 504. Here, if jf 0 (z)j is not constant on jzj = r , this inequality must be replaced by L2 (1 + p); n where p is a fixed number > 0 . Because of the isoperimetric property of the circle, the image line of jzj = r +  therefore has length > 2(r + ) . By continuing this J >1


XVI. On the distortion of non-conformal schlicht mappings

379

procedure one recognizes that the area of Bzn+1 would be >   r 2 , while it is in fact    r 2 . It is easy to see how one can remove the assumption on the mapping on jzj = r made above, and how the uniqueness proof for the case Q = 1 can be obtained from what has already been said earlier. From the extremal mapping of Bn+1 onto Bzn+1 for the case Q = 1 we can easily derive a mapping with all the properties expressed in Theorem 1 for1 Q > 1 . 1 The radius of the inner image circle is then r Q . Now if some normalized map AQ of Bn+1 onto Bzn+1 is given, then by Theorem 1 for Q = 1 , the area bounded by Rzk is increased by a conformal extremal mapping of Bzn+1 unless Bzn+1 is not bounded by circles and circular arcs in the manner indicated. Let Bzn+1 be bounded in such a way and let r¯ be 1 the radius of the inner bounding circle. If r¯ > r Q , then it would be impossible to place inside Bzn+1 doubly-connected, schlicht surface strips which do not overlap each other and entangle the circle of radius r¯ , and whose product of conformal
1 Ùn+1 by a mapping AQ , in which moduli is  1r Q . Bzn+1 is formed from B correspondingly running surface strips with module product 1r can be specified. By means of AQ , however, the image strips of these surface strips have module
1
1 product  1r Q , contradicting that such a product must be < 1r Q . 1 In the case of r¯ = r Q , if we consider divisions of strips with maximal module Ùn+1 , and hence that of Bn+1 , onto an extremal region has product, the mapping of B the properties expressed in Theorem 1. The uniqueness theorem also follows. □ The proof method given here must be regarded as typical for the proofs of the following theorems. Theorem 2. For a normalized mapping AQ of Bn+1 , the diameter of Rzk (which is assumed to be non-point-like) attains its maximum value if and only if the concentric circular ring area formed by the two circles of the image region of an extremal map of Bn+1 according to Theorem 1 is conformally mapped in such a normalized way that the inner circle goes to a rectilinear slit going through and bisected by the zero point (see figure 2). Proof. If, for a normalized mapping of Bn+1 given by w = f (z) , the diameter  D of the boundary Rk of the image region Bn+1 corresponding to Rk is greater than the corresponding diameter D for one of the maps in Theorem 2, then we employ a normalized conformal mapping to map the region bounded by Rk and  jwj = 1 onto a region Bn+1 bounded by jwj = r  and jwj = 1 . Now r  would 1 be strictly larger than the number r Q in the proof of Theorem 1, which was shown impossible. 1 By the formula:  = 
j j( Q 1

1)

.

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Herbert Grötzsch

Figure 2. The position of the principal axes of Tissot’s indicatrix is shown schematically

If D  = D , then w = f (z) is one of the extremal mappings mentioned in Theorem 2. □ Let a schlicht doubly-connected region B of the z -plane, bounded by two linear (“linienhaft”) boundaries, be bijectively and conformally mapped onto a schlicht concentric circular ring r < jwj < 1 . The lines of B to which a circle jwj =  ( r    1 ) corresponds may briefly be called module-lines of B . Let z1 and z2 be two arbitrary but fixed interior points of B . Theorem 3. For a normalized mapping AQ of Bn+1 given by w = f (z) , the maximum of the quantity jf (z2 ) f (z1 )j is attained if and only if 1. f (z1 ) and f (z2 ) are the endpoints of a rectilinear path  bisected by w = 0, 2. all boundaries Rzk are slits on the module-lines of the doubly-connected zn+1 defined by jwj = 1 and  (where some or all of the R zk region Bx may also lie on  ), 3. the directions of the semi-major axes of the distortion ellipses all fall into zn+1 , the corresponding module-line of Bx 4. the ratio of semi-major to semi-minor axis of each distortion ellipse is constant = Q (see figure 3). All extremal mappings arise from one of them by rotations around w = 0 and reflection in the line through f (z1 ) and f (z2 ) (uniqueness theorem). At the points z1 and z2 the differential-geometric character of the extremal map is necessarily lost for Q > 1 . Proof. The proof of existence for mappings with the above-mentioned properties is easily obtained if the existence of such a map is established in the case Q = 1 of the conformal map. This special case is dealt with in Leipz. Ber. 81 (1929), p. 220 and p. 221 (extremal mapping 20 ). The proof of the extremal property and of the uniqueness theorem for mappings AQ ( Q > 1 ) is analogous to the previous considerations by analogously using surface strips running as in the case

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381

Figure 3. The principal axes of a distortion ellipse are shown schematically

of a conformal mapping. It is also easy to see that for Q > 1 it is not correct that an infinitesimal circle with center z1 or z2 goes to an infinitesimal ellipse with center f (z1 ) or f (z2 ) . □ Now we suppose that Bn+1 has z = 0 as interior point. We consider only 0 those normalized maps AQ of Bn+1  Bn+1 , briefly called normalized mappings 0 AQ , that transfer z = 0 into w = 0 . Again z1 is an arbitrary but fixed interior 0 point of Bn+1 which is different from z = 0 . As a counterpart to Theorem 3, we have 0 Theorem 4. For a normalized mapping A0Q of Bn+1 given by w = f (z) , the minimum of the quantity jf (z1 )j is attained if and only if z0 1. all Rz are orthogonal trajectories of the module-lines of the region Bx k

n+1

bounded by jwj = 1 and the line segment  from 0 to f (z1 ) (some or all of the Rzk can cross or impinge on the line segment  ), 2. the direction of the semi-minor axis of each distortion ellipse belonging to 0 zn+1 a point w falls into the module-line of Bx passing through that point. 3. the maximum of the two ratios of principal axes of each distortion ellipse is constant = Q for all points (see figure 4).

Figure 4. The principal axes of a distortion ellipse are shown schematically

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Herbert Grötzsch

All extremal mappings arise from one of them by rotations around w = 0 and reflection in the straight line through 0 and f (z1 ) . For Q > 1 , the extremal mapping A0Q loses its differential-geometric character at the points z = 0 and z = z1 . Proof. The proof of the extremal property and of the uniqueness theorem is to be conducted in analogy with the earlier considerations. The existence proof of the asserted mapping follows easily for Q > 1 from the case Q = 1 . This latter proof of existence can be given on the basis of the mapping provided by Mr. P. Koebe in Leipz. Ber. 81 (1929), footnote on p. 47, of an area consisting of two copies of 0 Bn+1 stitched together along  . Instead of the points 0 and 1 in the footnote cited, two concentric circles appear, and radial slits instead of the concentric circular arc slits. □

IV. Theorems about mappings AQ normalized differently In this section we consider schlicht, finitely connected regions, namely (n + 1) -fold connected regions Bn+1 , briefly again called normalized regions Bn+1 , which are bounded inside by a circle jzj = r , which lie entirely outside jzj = r , and which are bounded outside by a linear boundary L , separating Bn+1 from 1 or heading towards 1 , as well as by n 1 further boundaries Rk lying between jzj = r and L . We apply to such a region Bn+1 of the z -plane, which is fixed as is the number r , schlicht mappings A0Q normalized by w = f (z) . A mapping AQ of Bn+1 is now called normalized if it is bijectively and schlicht and the image region Bzn+1 of the w -plane is also normalized, jzj = r as a whole goes to jwj = r , and z limiting Bzn+1 to 1 . Thus 1 is not an image L goes to the linear boundary L z point. Let Rk be the boundary of Bn+1 corresponding to Rk . Theorem 5. For a normalized mapping AQ of Bn+1 , the distance of the boundary z from the point w = 0 attains its minimum value if and only if Bn+1 is mapped L z to a normalized region Bzn+1 which is bounded by jwj = r , by a half-ray L corresponding to L , running to 1 and containing jwj = 0 on its extension, and z , and by (n 1) slits on module-lines of the region bounded by jwj = r and L if at the same time the directions of the major axes of all distortion ellipses fall into these module-lines, and the larger of the two ratios of principal axes of each distortion ellipse is constant = Q for all points (see figure 5). All extremal mappings arise from one of them by directly or indirectly conformal mapping into itself of the doubly-connected region bounded by jwj = r and the z , passing the circle jwj = r into itself. half-ray L

XVI. On the distortion of non-conformal schlicht mappings

383

Figure 5. The principal axes of a distortion ellipse are shown schematically

Proof. By means of the conformal transformations w1 = wr and w2 = w11 , the z of an extremal region Bzn+1 of Theorem 5 is transformed into a rectilinear half-ray L slit s which has w2 = 0 as one end point. If the surface of the w2 -unit circle is conformally mapped into itself such that s goes to a rectilinear slit bisected by w2 = 0 , then Bzn+1 has changed into a region of exactly the same structure as in Theorem 2. Thus one can see how Theorem 5 is also proved by Theorem 2. □ For conformal mappings I have dealt with this problem of the next boundary point in Leipz. Ber. 80 (1928), pp. 371–72. The counterpart to Theorem 5 is Theorem 6. For a normalized schlicht mapping AQ of Bn+1 , the distance of z to w = 0 attains its maximum value if and only if Bn+1 is the outer boundary L mapped, in a normalized way, onto a region Bzn+1 which is bounded by jwj = r , jwj = R (this corresponds to L ) and (n 1) radial slits between these two circles, where the major axes of the distortion ellipses are all directed radially towards w = 0 and the larger principal axis ratio has constant value Q everywhere (see figure 6).

Figure 6. The principal axes of a distortion ellipse are shown schematically

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All extremal mappings arise from one of them by rotation around w = 0 and reflection in a straight line through w = 0 . Proof. The correctness of the theorem for case Q = 1 follows from the considerations in Leipz. Ber. 80 (1928), p. 370 and p. 371 (§II). The existence of such a conformal mapping is proved in the work of Mr. P. Koebe in Acta Math. 41 (1918), pp. 305–344. The proof of Theorem 5 for Q > 1 makes no difficulty according to what has been said earlier. □ Now let z1 be an arbitrary but fixed interior point of Bn+1 . Theorem 7. For a normalized mapping AQ of Bn+1 given by w = f (z) , the minimum of the quantity jf (z1 )j is attained if and only if Bn+1 is mapped z onto a region Bzn+1 bounded inside by jwj = r , outside by a radial half-ray L corresponding to L , first containing f (z1 ) , then 0 on its backward extension and running to 1 , and, moreover, by (n 1) slits on module-lines of the doublyz extended backwards up connected region bounded by jwj = r and the half-ray L to f (z1 ) (some or all of the Rzk may also lie on the straight line through 0 and z ), where, in addition, the direction of the f (z1 ) , namely between f (z1 ) and L major axis of each distortion ellipse falls into the corresponding module-line, and the larger principal axis ratio has constant value Q (see figure 7).

Figure 7. The principal axes of Tissot’s indicatrix are shown schematically

All extremal mappings arise from one of them by rotation around w = 0 and reflection in the straight line through w = 0 and w = f (z1 ) . For Q > 1 , the extremal mapping loses its differential-geometric character at the point z = z1 . Proof. It will suffice to prove the existence of the asserted mappings for Q = 1 . We had reduced the existence proof in Theorem 5 to that of Theorem 2 through certain linear transformations. By analogous linear transformations of a slit region Bzn+1 , we obtain the reduction to the extremal map 2 (where one of the boundaries zk , R zl is a point) in Leipz. Ber. 81 (1929), p. 220. R □

XVI. On the distortion of non-conformal schlicht mappings

385

Theorem 8. For a normalized schlicht mapping AQ of Bn+1 given by w = f (z) , the maximum of the quantity jf (z1 )j is attained if and only if Bn+1 is mapped z onto a region Bzn+1 bounded inside by jwj = r , outside by a radial half-ray L corresponding to L , running to 1 and first containing 0 , then f (z1 ) on its backward extension, and, moreover, by (n 1) slits on orthogonal trajectories of the module-lines of the doubly-connected region that is bounded by jwj = r and the radial half-ray h˜ running from f (z1 ) to 1 and not containing 0 (some or all of these slits may possibly traverse h or hit it), where, in addition, at each point the direction of the minor axis of the distortion ellipse falls into the corresponding module-line, and the larger ratio of the two principal axes has constant value Q at each point (see figure 8).

Figure 8. The principal axes of a distortion ellipse are shown schematically

All extremal mappings arise from one of them by rotation around w = 0 and reflection in a straight line through w = 0 . At the point z = z1 the extremal mapping loses its differential-geometric character in the general case Q > 1 . Proof. It is only necessary to give the proof of existence for conformal maps A1 . If we apply linear transformations analogous to the proof of Theorem 7, we obtain a related mapping problem, which can be treated, for example, according to the model of the proof of Theorem 4. □ Remark. For theorems 7 and 8, I have dealt with the case of the conformal mapping of a normalized region B2 in Leipz. Ber. 80 (1928), pp. 498–500 (§V a and §V b).—Theorems 7 and 8 also hold in the case where L degenerates to the point 1 , which is fixed under the mappings.

Chapter 17

On closest-to-conformal mappings Herbert Grötzsch (Leipzig) presented to the academy by Mr. Koebe (translated from the German by Melkana Brakalova-Trevithick) Berichte über die Verhandlungen der sächsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse, Bd. 84 (1932), 114–120

In the present note, which follows closely two other earlier communications,1 we treat some elementary differential-geometric mapping problems, solutions of which are implicitly contained in these communications. A schlicht, simply connected region B1 of the z -plane, bounded by a closed, maybe analytic curve, can be mapped conformally, one-to-one, orientation preserving or reversing, to any similarly formed region Be1 of the w -plane, and this mapping is uniquely determined when three different boundary points of B1 are assigned to three arbitrarily specified boundary points of Be1 . If it is required that a one-to-one differential-geometric mapping2 of B1 onto Be1 assigns to four distinct, cyclically ordered boundary points of B1 four arbitrary, distinct, cyclically ordered boundary points of Be1 , such a mapping may not be, in general, made orientation preserving or reversing conformal mapping. From the point of view of differential geometry and proof theory therefore arises the question: which differential-geometric mapping Original title: Über möglichst konforme Abbildungen von schlichten Bereichen. 1 “Uber die Verzerrung bei schlichten nichtkonformen Abbildungen and über eine damit

zusammenhängende Erweiterung des Picardschen Satzes” in Leipz. Ber. 80, 503–507, 1928 and “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zussammenhängender Schlichter Bereiche” in Leipz. Ber. 82, 69–80, 1930. 2 Differential-geometric here is a mapping, for which in a neighborhood of each point, apart from possibly finitely many interior points, or countably infinitely many interior points that accumulate at the boundary, the pair of real functions that constitute the mapping satisfy the common requirements for continuity and differentiability; allowing such exceptional points is convenient, in a sense that at most at these points the determinant can become zero; one defines analogously the meaning of the concept of differential-geometric behavior of a mapping at the boundary of the domain.

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that maps one-to-one the interior of B1 onto the interior of Be1 ; realizes the correspondence between quadruples of boundary points and is either differentiable or just continuous on the boundary, is closest-to-conformal (in case such closest-toconformal mapping exists at all), that is for which such mapping is the deviation from conformality the smallest, where an appropriate measure for such deviation from conformality must be introduced beforehand. In addition to the question just stated, which is treated in Example 1, a few additional elementary examples of mappings of simply and doubly connected domains will be treated, based on similar considerations. In a neighborhood of a point z where the mapping is differential-geometric, an infinitesimal circle with center z is transformed onto an infinitesimal ellipse with the image of z as center, for which we let ab  1 denote the ratio of the larger over the smaller axis; let Q be the upper bound of all such ratios of axes ab at varying inner points z , in whose neighborhoods the mapping is differential-geometric. The number Q 1 is the measure of deviation from conformality for the mapping denoted3 by w = w(z) . The inverse mapping z = z(w) has the same deviation from conformality, Q 1; as w = w(z) . If one applies conformal mappings  = (z) and  = (w); then the new differential-geometric mapping  = () has also the same deviation Q 1 from conformality as w = w(z): In a neighborhood of a point z , where the mapping is differential-geometric, an infinitesimal square q has as an image an infinitesimal parallelogram q˜ . Let ˜s be the length of any one of the sides of q˜ , I˜ be the area of q˜: For any sufficiently small choice of q , which can lie anywhere in the neighborhood of z considered, due to the assumptions of continuity and differentiability, one can easily see that the following relationship holds 1 2 (1 ")

˜s < I˜ < (1 + ")
Q
˜s 2 ; (1) Q where " > 0 can be arbitrarily small.

Example 1 The mapping problem which we consider here goes beyond schlicht conformal mappings and is about mapping rectangles onto each other so that vertices are mapped onto vertices. Let the vertices of the rectangle in the z -plane, R , be 0; p; p + 2 i; 2 i ( p > 0 is real), the corresponding vertices of the other rectangle z in the w -plane be 0; p; ˜ p˜ + 2 i; 2 i ( p˜ > 0 is real), and, in addition, let us R assume that p˜  p: Let Q be the upper bound of the ratio of the axes ab  1 3 The notation w = w (z ) , z = z (w ) , etc., at z = x + iy , w = u + iv with real x; y; u; v etc., will be used as abbreviated notation for the pairs of real functions u = u(x; y ) , v = v (x; y ) and x = x (u; v )y = y (u; v ) etc. of the given mapping. ( x , y , etc. are the rectangular Cartesian parallel coordinates.)

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of a differential-geometric mapping of R onto Rz that maps the vertices onto the vertices. We consider the images of very narrow rectangular strips  in R; created through lines parallel to the real axis. The sum of their modules (the value of the module of a single rectangular strip =ratio of the length of the shorter to the length of the longer side, where the latter has length p ) is 2 . The images of  p ˜ and, after the usual consideration applied to (1), it follows cover the area 2 p; 1 that 2 p˜ > (1 ")
Q
p˜2
2 , where " > 0 can be chosen arbitrarily small, as p long as  is sufficiently narrow. Therefore Q > (1

")


p˜ p

i.e. Q 

p˜ : p

The equality sign can therefore occur only when, for all points z in R; the value of ab is exactly equal to Q = pp˜ ; which is when the mapping of R onto Rz is an affine mapping, which takes vertices onto vertices. The smallest deviation from conformality for such mapping is Q 1 .

Example 2 Two arbitrary, schlicht, doubly connected domains, B2 in the z -plane and Be2 in the w -plane, each bounded by two curves, are to be mapped one onto another, one-to-one and closest-to-conformally. If not by a conformal mapping, the problem is reduced to the case of mapping onto each other and closest-to-conformally, two non-degenerate annuli of the z -, resp. w -plane, centered at z = 0 , resp. w = 0 . Through the transformations  = log z and  = log w; one can reduce the question to closest-to-conformal mapping of two rectangles, each made doubly connected through assignment of the boundaries, to which the study from Example 1 can be immediately transferred, and the result yields that the relation between log z and log w , in analogy to Example 1, is closest-to-conformal, if the mapping is the considered affine one. If one of the two doubly connected regions, resp. annuli, has one boundary consisting of exactly one point, while the other doubly connected region, resp. annulus, is not degenerate then our problem cannot be solved since under any considered differential-geometric map, the upper boundary of ab is infinite, as is the case, for example, for all geographic maps for which the North pole, resp. the South pole, corresponds to a line.4 4 This is also proved using the strip method, check Leipz. Ber. 80, 1928, p. 507.

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Example 3 Two schlicht, simply connected domains, each bounded by one curve, are to be mapped onto each other, one-to-one, closest-to-conformally, and so that two distinct prescribed interior points of one domain correspond to two arbitrarily prescribed interior points of the other domain. If not by a conformal mapping, one can assume that the domain jzj < 1 is mapped one-to-one and onto the domain jwj < 1; so that z = +r and z = r correspond to w = +˜r and w = r˜; resp. ( r; r˜ are real and > 0 ). One cuts jzj < 1 along the interval s; connecting the point z = r to z = r; and, by analogy, one cuts jwj < 1 along the interval ˜s connecting the points w = r˜ and w = +˜r . The two doubly-connected regions so obtained will be mapped, following Example 2, one-to-one and closest-to-conformally, so that, in addition, z = r and z = r correspond to w = r˜ and w = +˜ r; which is possible. This mapping also immediately provides a mapping of jzj < 1 onto jwj < 1; which is differential-geometric at every point s between r and +r; it is differential geometric at z = r and z = +r only when the closest-to-conformal mapping is conformal; because if the latter is not the case, the magnification ratios of the line elements, at both z = r and z = r; resp., in outgoing real directions are different, so that an infinitesimal circle with r; resp. +r; as a midpoint, can not be mapped onto an infinitesimal ellipse with r˜; resp., +˜r as a midpoint. If Q is the constant value of ab for the closest-to-conformal mapping, it follows easily from the strip method that the upper bound ab must be bigger than Q for every other differential-geometric mapping. Therefore, when Q > 1; the closest-to-conformal mapping has two exceptional points z = r and z = r; it is not difficult to realize that it can be interpreted as a limit of suitable everywhere differential-geometric mappings, without exceptional points, whose corresponding Q  to the upper bound ab converge in a decreasing fashion toward Q .

Example 4 Two simply connected domains are to be mapped closest-to-conformally, one onto the other, so that two arbitrarily chosen distinct boundary points and one arbitrarily chosen interior point of one correspond to two arbitrarily chosen distinct boundary points and one arbitrarily chosen interior point of the second domain. By means of a conformal mapping, one can assume that the domain jzj < 1 is mapped onto the domain jwj < 1 one-to-one and closest-to-conformally, so that z = 0 is mapped onto w = 0 and e i ˛ and e i ˛ are mapped onto e iˇ and e iˇ ; respectively, ( ˛; ˇ are real, 0 < ˛ < ; 0 < ˇ <  ). If the mapping has to be orientation preserving, which might be assumed, then the arc of jzj = 1 determined by e i˛ and e i ˛ that contains the point 1 must correspond to the arc of jwj = 1 determined by e iˇ and e iˇ and containing the point 1:

17 On closest-to-conformal mappings

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To obtain the closest-to-conformal mapping, one cuts jzj < 1 along a straight line from 0 to 1 , and does the same for jwj < 1 . This gives rise to two simply connected domains with four boundary points, namely e i ˛ ; e i ˛ and the doubly counted boundary point 1; on the one hand, and correspondingly e iˇ ; e iˇ and the doubly counted boundary point 1; on the other hand. Using Example 1, these two domains can be mapped closest-to-conformally, by preserving the correspondence of the vertices, so that e i ˛ goes into e iˇ ; e i ˛ goes into e iˇ ; and the two boundary points 1 go to the two boundary points 1: This mapping yields immediately a mapping from jzj < 1 onto jwj < 1; which is differential-geometric for every interior point on the line connecting 0 to 1; with the exception of the point z = 0; unless the closest-to-conformal mapping is again, after all, conformal. At every interior point the number ab is identically equal to the upper bound Q: The fact that the mapping so obtained is, indeed, the closest-to-conformal, and is the only orientation preserving closest-to-conformal mapping, can be easily seen from the strip method. One constructs the closest-to-conformal orientation reversing mapping from jzj < 1 onto jwj < 1; for which 0; e i ˛ ; e i ˛ are mapped onto 0; e iˇ ; e iˇ , by analogy. This mapping has a constant value ab which is not, in general, equal to the previous Q .

An additional remark The following remark is about a differential-geometric transfer of the Schwarz lemma. We consider one-to-one differential-geometric AQ mappings of jzj < 1 , i.e., mappings, for which the differential-geometric behavior of the mapping at every point, the number ab , does not exceed a fixed value Q  1 . The mapping does not need to be necessarily schlict, but rather to take values on a Riemann surface, with several sheets. To an exceptional point of the mapping corresponds then necessarily only one point on the Riemann surface, which can be a finite order branch point. One such mapping, which we denote by w = w(z); satisfies only the assumptions that w(0) = 0 and jw(z)j < 1 for jzj < 1: Let z1 ¤ 0 be an arbitrarily chosen but then fixed point in jzj < 1: For Q = 1 , the Schwarz lemma states that jw(z1 )j  jz1 j for each conformal mapping of jzj < 1 satisfying the conditions. If Q > 1 then the following, yet to be shown, differential-geometric transfer states that jw(z1 )j  f (jz1 j) < 1 holds, where the bound f (jz1 j) is valid for all permissible AQ mapping, for fixed Q and z1 , and is achieved only by certain schlicht extremal mappings. The Riemann surface onto which jzj < 1 is mapped by the AQ mapping, will be mapped conformally, one-to-one to the full schlicht interior of jj = 1 by  = (w); so that (0) = 0: By Schwarz’s lemma, if we denote by w1 = w(z1 ); we have jw1 j = jw(z1 )j  j(w1 )j: The relationship so established between z and  is

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a one-to-one schlicht A?Q mapping. In this case, from earlier,5 j(z1 )j  f (jz1 j) < , where f (jz1 j) with fixed Q and fixed z1 does not depend any more on the schlicht A?Q mapping, and the equality sign holds only for certain AQ mappings, for which a is equal to the constant Q . Thus jw(z1 )j  f (jz1 j) < 1; which is what had to b be shown. By the way, one can derive this differential geometric transfer of the Schwarz lemma, without using the latest, through a suitable application of the strip method.

Addition Here I would like to draw attention to a printing error in my announcement “Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche” (Leipz. Ber. 84, pp. 15–36). On p. 31 there, line 3 from below, instead of: [Eigeonschaft ( ? )] one should read [Eigenschaft ( ? )2 ) ]

5 This follows from Leipz. Ber. 82, 1930, p. 75, Theorem 3, when there one sets z2  0 , and for the considered there schlicht AQ mapping, the point z2  0 is a fixed point, and Rk are non existent or, what amounts to the same, reduce to a single point. It should be pointed out that generally the indicated properties of the extremal maps in the theorems from Leipz. Ber. 82, 1930, pp. 69–80 are characteristic for these more recent ones, which follows from the proof methods used there. One can compare to the differential-geometric extension, resp. perception of Picard’s theorem in Leipz. Ber. 80, pp. 506–507, 1928.

Chapter 18

On five papers by Herbert Grötzsch Vincent Alberge and Athanase Papadopoulos

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conformal representations . . . . . . . . . . . . . . . . . . . . . . . . 3 Two lemmas from the paper “Über einige Extremalprobleme der konformen Abbildung” . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Four theorems from the paper “Über einige Extremalprobleme der konformen Abbildung” . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Two theorems from the paper “Über einige Extremalprobleme der konformen Abbildung. II.” . . . . . . . . . . . . . . . . . . . . . . . . 6 On the content of Grötzsch’s paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterungdes Picardschen Satzes” . . . . . . . . . . . . . . . . . . . 7 Some comments on Grötzsch’s paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterung des Picardschen Satzes” . . . . . . . . . . . . . . . . . . 8 The paper “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche” . . . . 9 The paper “Über möglichst konforme Abbildungenvon schlichten Bereichen” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Examples of closest-to-conformal mappings (extremal quasiconformal mappings) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393 395 398 399 402

403

405 406 409 410 412

1. Introduction Herbert Grötzsch is the main founder of the theory of quasiconformal mappings. In a series of papers written between 1928 and 1932, he introduced these mappings as a natural generalization of conformal mappings and he developed their main properties. He saw that several problems in conformal geometry naturally lead to problems on quasiconformal mappings. In this chapter, we review five of his papers

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that show the progress of his work from conformal to quasiconformal geometry. The five papers are the following: 1. “Über einige Extremalprobleme der konformen Abbildung” (On some extremal problems of the conformal mapping), published in 1928 [12]; 2. “Über einige Extremalprobleme der konformen Abbildung. II.” (On some extremal problems of the conformal mapping II), published in 1928 [13]; 3. “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes” (On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem) [14]; published in 1928. 4. “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche” (On the distortion of nonconformal schlicht mappings of multiply-connected schlicht regions), published in 1930 [15]; 5. “Über möglichst konforme Abbildungen von schlichten Bereichen” (On closest-to-conformal mappings of schlicht domains), published in 1932 [16]. Translations of these five papers into English are included in the present volume. We shall present the main results of these papers, providing introductory remarks, establishing the connections between them, and explaining the background. We shall also indicate some relations with works of Lavrentieff and Teichmüller. The first two papers are concerned with conformal representations of multiplyconnected domains. Grötzsch uses there, for the first time, the so-called length-area method, a powerful method which is a direct application of the Schwarz inequality. In the third paper, he introduces the notion of non-conformal (“nichtkonform”) mapping, which he also calls mapping of bounded infinitesimal distortion (“Abbildung von beschränkter infinitesimales Verzerrung”). This notion is very close to that of quasiconformal mapping, as it is understood today. The results he proves are all valid for quasiconformal mappings (with the same proofs) and for this reason we shall refer to Grötzsch’s non-conformal mappings as quasiconformal mappings. His aim in this paper is to show that several properties known for conformal mappings remain true in this more general setting. He proves a distortion theorem for the new class of mappings—an analogue of a distortion theorem of Koebe—and a generalization of the big Picard theorem, one of the very classical results on meromorphic functions. The fourth paper is concerned with quasiconformal representations of multiply-connected domains. The results obtained in this paper are generalizations of results known for conformal representations. The fifth paper is concerned with maps that Grötzsch calls closest-to-conformal (“möglichst konforme”). These are the mappings that we call today “extremal quasiconformal mappings.”

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2. Conformal representations The problem of mapping conformally domains of the complex plane C onto “canonical” domains was inaugurated by Riemann in his doctoral dissertation [24] (1851), in which he proved that any simply-connected open subset of the plane which is not the whole plane can be conformally mapped in a one-to-one manner onto the unit disc. He also proved that such a mapping is unique up to post-composition by a conformal automorphism of the disc (a Möbius transformation). This is the wellknown Riemann Mapping Theorem. Riemann proved it using the so-called Dirichlet principle, which he formulated and used extensively in his works on Riemann surfaces. Generalizing the Riemann Mapping Theorem to non-simply-connected planar domains is a broad subject that occupied generations of mathematicians after Riemann. We recall the following two well-known facts. 1. Two connected open subsets of the complex plane that have the same connectivity are not necessarily conformally equivalent. For instance, two circular annuli (that is, annuli in the complex plane bounded by two concentric circles) are conformally equivalent if and only if they have the same module, that is, if and only if the ratio of the two radii is the same for both annuli. 2. There is no natural “canonical” class of domains for multiply-connected domains. Circular annuli may be considered as some kind of “standard” domains for doubly-connected regions, but there are other possibilities. We now discuss some of these standard multiply-connected domains. The unit disc slit along an interval of the form [0; r] ( r < 1 ) is a useful object in conformal geometry, and it was employed by Grötzsch in his work on conformal and quasiconformal geometry. It is known in the classical literature on quasiconformal mappings under the name Grötzsch domain. Such a domain appears in particular as a solution of an extremal quasiconformal problem described in Theorem 9 below, which is Theorem 3 of Grötzsch’s paper [15]. The same extremality property of this domain is proved in the book by Lehto and Virtanen, Quasiconformal mappings in the plane [23], p. 52. In the same book, the Grötzch domain is shown to be the solution of the following extremal problem for conformal mappings: if (r) denotes the module of this domain, then, the module of any doubly-connected domain separating the points 0 and r from the circle jzj = 1 is at most equal to (r) ; see [23], p. 54. In Ahlfors’ book on conformal invariants, the expression Grötzsch annulus is used for the complement, in the complex plane, of the closed unit disc and a segment of the form [R; 1] for some positive R [3], p. 72. It is possible to write explicitly a conformal homeomorphism between the last two domains. Teichmüller, in his paper Untersuchungen über konforme und quasikonforme Abbildung (Investigations on conformal and quasiconformal mappings), translated in the present volume, calls a Grötzsch extremal region the complement of the

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unit disc in the Riemann sphere C [ ¹1º cut along a segment of the real axis joining a point P > 0 to the point 1 , see §2.1 of [26]. He gives estimates for this domain and he uses it in his investigations. Another “standard model” for doubly-connected domains is the Riemann sphere C [ ¹1º slit along two intervals of the form [ r1 ; 0] and [r2 ; 1) where r1 and r2 are positive numbers. This domain is known under the name Teichmüller extremal domain [23], p. 52. Teichmüller uses this domain in his study of extremal properties of topological annuli, in §2 of his paper [26]; see in particular §2.6. This domain satisfies the following extremal property: its module is an upper bound for the module of any doubly-connected domain that separates the pair 0 and r1 < 0 from the pair r2 > 0 and 1 . Teichmüller proved this characterization in §2.4 of [26], using Koebe’s one-quarter theorem and Koebe’s distortion theorem. His proof is reproduced in Ahlfors’ book [3], p. 72; see also [23], p.q 55. The module of
1 the Teichmüller extremal domain is equal to the quantity 2 r1r+r (where  2 denotes as above the module of the Grötzsch extremal domain). Teichmüller, in his paper [26], also works with other “standard” domains, e.g. the circular annulus 1 < jzj < P2 cut along the segment joining z = P1 to z = P2 , where P1 is a point on the real axis satisfying 1 < P1 < P2 , see §2.4 of [26]. He proves several extremal properties of these domains. He uses them in a geometric proof which simplifies and generalizes a distortion result that Ahlfors obtained in his thesis [1]; see also the discussion in Ahlfors’ book [3], p. 76. Among the other “standard” doubly-connected domains, we mention the Mori domain, called so in the monograph [23]. Before Grötzsch, Koebe studied extensively conformal mappings of finitelyconnected domains of the plane onto circle domains, that is, multiply-connected domains in the plane whose boundary components are all circles (which may be reduced to points), cf. [18] and [19]. He proved that every finitely-connected domain in the plane is conformally equivalent to a circle domain. This generalizes the Riemann Mapping Theorem, which says that a simply-connected domain which is not the entire plane is conformally equivalent to the unit disc. We mention by the way that the question of whether there is a result analogous to Koebe’s theorem which is valid for an open subset of the plane having an infinite number of boundary components is open, and is known as the Kreisnormierungsproblem, or the Koebe uniformization conjecture. Koebe formulated this conjecture in his 1908 paper [17], p. 358. We refer the reader to the recent paper [8] by Bowers in which the author surveys many developments of this conjecture, including the works on circle packings done by Koebe, Thurston and others. Grötzsch, who was a student of Koebe, was naturally led to study problems connected with conformal representations. In the paper [12], he works with two classes of domains which were already considered by Koebe: 1. Annuli with circular slits. These are circular annuli from which a certain number (  0 ) of circular arcs (which may be reduced to points) centered at the center of the annulus, have been removed, see figure 2 (left).

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2. Annuli with radial slits. These are circular annuli from which a certain number (  0 ) of radial arcs (which may be reduced to points) have been removed. A radial arc is a segment in the annulus which, when extended, passes through the center of the annulus, see figure 2 (right). Koebe, in the paper [12], showed that any finitely-connected domain of the complex plane which is not the whole plane admits a conformal representation (which is essentially unique) onto a domain of one of the two kinds above.

0

1

1

0

2

∞

Figure 1. A Grötzsch extremal domain (left) and a Teichmüller extremal domain (right), called so in the book of Lehto and Virtanen [23]. In the figure to the right, the doubly-connected domain is the complement in the Riemann sphere of the union of the segment and the ray indicated.

Figure 2. An annulus with circular slits (left) and an annulus with radial slits (right)

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3. Two lemmas from the paper “Über einige Extremalprobleme der konformen Abbildung” Grötzsch’s paper Über einige Extremalprobleme der konformen Abbildung [12] (the first paperin the above list) starts with two lemmas that give a subadditivity result for the modules of (finitely or infinitely many) disjoint subdomains of a circular annulus in the complex plane. In the first lemma, the subdomains are disjoint topological quadrilaterals, each having two opposite sides on the boundary circles of the circular annulus. In the second lemma, the subdomains are disjoint topological annuli that are homotopy-equivalent to the circular annulus. In each case, the lemma says that the sum of the modules of the subdomains is bounded above by the module of the ambient circular annulus. In particular, if we divide the interior of a circular annulus into two annuli by a simple closed curve homotopic to the boundary components, then the sum of the modules of the two resulting annuli is not greater than the modules of the ambient annulus. Grötzsch also obtains a characterization of the equality case: equality occurs, in the first case, if and only if the rectangles are obtained by radial slicing of the circular domain, and in the second case, if and only if the decomposition into subdomains is obtained by a slicing of the circular domain by circles concentric to the boundary. Furthermore, in the two equality cases, the subdomains must fill out the ambient circular annulus. The proofs of these module inequalities are based on the so-called length-area method. Regarding this method, Ahlfors writes, in his collected papers edition [5], commenting on his first two published papers on the asymptotic values of entire functions of finite order, which in fact are those of his doctoral dissertation: “[…] The early history of this method is obscure, but I knew it from and was inspired by its application in the well-known textbook of Courant and Hurwitz1 to the boundary correspondence in conformal mapping.” Talking about Nevanlinna and himself, he adds: “None of us was aware that only months earlier H. Grötzsch had published two important papers on extremal problems in conformal mapping in which the same method is used in a more sophisticated manner […] The method that Grötzsch and I used is a precursor of the method of extremal length.” In fact, Ahlfors applied this method in his thesis [1] in order to prove what became later known as the Ahlfors distortion theorem. Teichmüller, in his papers [26], [27], and [28], uses extensively the length-area method, and he calls it the Grötzsch–Ahlfors method. The module inequalities proved in [12] are considered today as classical results. They were also obtained (with different methods) and widely generalized by Teichmüller in his paper Untersuchungen über konforme und quasikonforme Abbildung (Investigations on conformal and quasiconformal mappings) (1938), translated in the present volume (§2.4 and §2.5 of [26]). 1 Courant and Hurwitz, in their book Funktionentheorie (Springer, Berlin, 1922), used the length-area method if their proofs of results of Carathéodory on the boundary values of the Riemann mapping, which the latter has published in then three papers [9], [10], and [11].

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4. Four theorems from the paper “Über einige Extremalprobleme der konformen Abbildung” In [12], Grötzsch uses the two lemmas of the previous section in his proof of four theorems which we now state. Theorems 3 and 4 answer questions posed by S. Szegö, see [25]. Following Grötzsch, for 0 < r < 1 , we denote by Kr the annulus r < jzj < 1 ˆr an annulus with circular slits obtained from Kr by removing a finite and by K number of circular arcs centered at the origin (figure 2, left). ˆr Theorem 1 ([12], §2). Let R be a real number satisfying R  r . If a domain K is mapped conformally and bijectively onto a domain having jzj = r and jzj = R as boundary components with the circle jzj = r sent onto the circle jwj = r and the circle jzj = 1 sent onto the circle jwj = R , then R  1 . Furthermore, the case R = 1 holds if and only if the conformal mapping is a rotation.

Theorem 1 implies in particular that if a circular annulus with circular slits and with boundary circles jzj = r and jzj = 1 is mapped conformally and bijectively onto a domain in the complex plane that has the same circles jzj = r and jzj = 1 as boundary components, then the second domain is also an annulus with circular slits and the conformal map is a rotation. ˆr is mapped conformally and In the next two theorems, a domain Kr or K ır ) bijectively by a mapping w = f (z) onto a domain called Br (respectively B of the extended complex plane C [ ¹1º not containing 1 in its interior, such that the circle jzj = r is sent to the circle jwj = r and such that the image of Kr ˆr ) is contained in the subset jwj  r of C [ ¹1º . (respectively K The image by w of the circle jzj = 1 is called the outer boundary of Kr ˆr ). (respectively K Let d be the shortest distance from a point on the outer boundary of Br ır ) to the point w = 0 . (respectively B

Theorem 2 ([12], §3). Consider a conformal mapping of Kr sending jzj = r to jwj = r and jzj = 1 to a ray whose extension contains the point w = 0 and joining a point q to 1 . Then, d  jqj . In Theorem 2, the domain represented in figure 3 appears as an extremal domain. This theorem is also known under the name Grötzsch module theorem, and it is usually stated under a form which uses the modulus of an annulus instead of the distance.

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Figure 3. An annulus with circular boundary with the other boundary being an infinite ray. This extremal domain appears in the second theorem.

ır . In the next theorem, we use the above notation Br and B

Theorem 3 ([12], §3). Assume there exist n points on the outer boundary of the ır ) realizing the distance of this outer boundary image domain Br (respectively B to the point w = 0 , assume that these points are the vertices of a regular n -gon having w = 0 as center and that there exists a positive constant M ( possibly equal to +1 ) such that jw(z)j  M for all z . Then a map that realizes the shortest distance of the outer boundary of the image domain to the point w = 0 sends Kr ˆr respectively to a domain bounded by jzj = r , jzj = M and which has n or K radial slits whose extensions contain 0 and which join the points ˛
r;M;n
e

2 i k n

(r < r;M;n < M; j˛j = 1; k = 0; 1; : : : ; n

1)

to points on the circle jwj = M . Furthermore, the mappings satisfying the stated properties are the unique ones that realize the required minimum property. The extremal region that appears in Theorem 3 is represented in figure 4.

Figure 4. An extremal domain for Theorem 3, with k = 4 . The inner slits admit a rotational symmetry.

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ˆ0 denote respectively the unit disc in C and In the next theorem, K0 and K the unit disc slit along finitely many circular arcs centered at the origin. When we ˆ0 by a conformal map f , we shall say, in analogy with map the domain K0 or K the previous cases, that the image by w of the circle jzj = 1 is the outer boundary ˆ0 ) . of the image f (K0 ) or f (K ˆ0 ) is mapped conformally Theorem 4 ([12], §4). Suppose that K0 (respectively K ı0 ) not containing and bijectively by w = f (z) onto a domain B0 (respectively B 0 1 in its interior satisfying f (0) = 0 and jf (0)j = 1 . Suppose furthermore that ı0 ) that are the there exist n points on the outer boundary of B0 (respectively B vertices of a regular n -gon centered at 0 and whose distance to the point w = 0 is the shortest distance from a point on the outer boundary to that point. Then r n 1 d ; 4 z and this extremal value is attained by the mapping f (z) = np(1+z . This n )2 mapping sends B0 to the w –plane slit along the n rays emanating from the points q n

1 4


e

2 i k n

( k = 0; 1; 2; : : : ; n

1 ) to 1 (figure 5).

Figure 5. An extremal domain for Theorem 4. The inner slits admit a rotational symmetry

Remarks 4.1. 1.— Theorem 4 is a generalization of Keobe’s quarter theorem (for n = 1 ). 2.— An extremal result similar to the one that appears in Theorem 2 was obtained by Lavrentieff in his paper [21] (1934). Lavrentieff attributes the result to Grötzsch, and he uses completely different methods (analysis instead of geometry).

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3.— Extremal problems similar to those studied by Grötzsch that we mentioned in this section are analyzed by Teichmüller in his paper [26] (1938), in particular §2.2 and §2.3. Teichmüller attributes the results he obtains to Grötzsch.

5. Two theorems from the paper “Über einige Extremalprobleme der konformen Abbildung. II.” Über einige Extremalprobleme der konformen Abbildung. II. [13] by Grötzsch is a sequel to the paper [12] and is concerned with the same subject. The proofs are also based on the lemmas proved in the first paper. The notation is as follows (we use Grötzsch’s notation so that the reader can easily compare with the paper). • The unit disc in the complex plane is denoted by R0 . • For r > 0 , Rr is the subset of the complex plane defined by r  jzj < 1 . • A conformal mapping w = f (z) defined on R0 is said to be normalized if it satisfies f (0) = 0 and jf 0 (0)j = 1 , and if 1 is not in the image. • A conformal mapping w = f (z) defined on Rr is said to be normalized if the circle jzj = r is sent to itself and if 1 is not in the image. Grötzsch establishes distortion theorems for the modules jf (z)j and jf 0 (z)j of normalized injective conformal mappings defined on Rr ( r  0 ). The formulation of the result uses a conformal mapping Er (z) defined as follows. For r > 0 , Er (z) is the conformal map that maps Rr onto a domain of the w -plane bounded by the circle jwj = r and a suitable slit contained in the positive real axis going to infinity and whose extension in the finite direction contains the point +r on the real line. We set mr () = min jEr (z)j jzj=

and Mr () = sup jEr (z)j: jzj=

Theorem 5. We have, for all jzj =  : mr ()  jf (z)j  Mr ()

with the two inequalities being equalities holding if and only if f (z) = ˛Er (ˇz) with j˛j = jˇj = 1 .

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For R0 , a similar statement hods, and in this case an explicit formula for the mapping E0 and the extremal values m0 () and M0 () are given. In fact, the resulting map is the so-called Koebe map, and the values of m0 () and M0 () are those that are given by Koebe’s distorsion map. In the next theorem, Grötzsch gives estimates on the derivative jf 0 (z)j . Here, the notation is as follows: m0r () = min jEr0 (z)j jzj=

and Mr0 () = sup jEr0 (z)j: jzj=

The result is then the following: Theorem 6. We have m0r ()  jf 0 (z)j  Mr0 ()

for jzj =  , with equality holding only for the functions ˛Er (ˇz) as above. In the case of R0 , explicit formulae for m0r () and Mr0 () are given. These results generalize Koebe’s distortion theorems.

6. On the content of Grötzsch’s paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterungdes Picardschen Satzes” The paper [14] is the first which contains Grötzsch’s notion of quasiconformal mappings. This is a mapping with bounded infinitesimal distortion Q . He introduces such a class of mappings in the first section of the paper and he denotes them by AQ . As a matter of fact, the term nichtkonformen (which we translate by non-conformal) is only used in the title. It was Ahlfors, in [2] who used for the first time the term quasikonform (“quasiconoformal”); cf. Ahlfors’ comments at the beginning of Volume 1 of his Collected papers [5]. For Q  1 , Grötzsch defines in [14] a mapping in the class AQ to be a bijective mapping of a domain G of the z -plane onto a surface which covers the Riemann sphere or a domain in it (Grötzsch uses the term “spread as a Riemann surface over a w -plane,” which is close to the terminology used by Riemann) which satisfies the following three properties:

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1. f is continuous on G ; 2. up to a set with at most countably many interior points (called exceptional points) of G , the mapping f is a local diffeomorphism which sends an infinitesimal circle to an infinitesimal ellipse whose ratio of minor to major 1 axis is between Q and Q ; 3. in the neighborhood of each exceptional point, the map f is a finite branched covering. Such a mapping from a domain of the Riemann sphere onto another domain of the Riemann sphere is not necessarily bijective, but it can be lifted to a bijective map from a domain of the Riemann sphere onto a Riemann surface which is a ramified covering of the sphere. Furthermore, the mapping is not assumed to be sense-preserving. Lehto and Virtanen call these “nicht-conformal maps” of [14] “regular quasiconformal” (see [23], p. 17). In the second section of the paper, Grötzsch establishes the property known today as the geometric definition of quasiconformality, saying that the image of a quadrilateral of module M by a bijective mapping in AQ is a quadrilateral whose module M 0 satisfies the inequality 1 M  M 0  QM: Q

(1)

The proof that Grötzsch gives of this property is based on the length-area method, which he already used in the papers [12] and [13]. After setting the inequality (1), Grötzsch writes that in the case of rectangles, equality in (1) holds “only for certain immediately determinable extremal affine mappings.” This result is now called the solution of the Grötzsch Problem; cf. Ahlfors’ book [4], p. 8. The solution was given by Grötzsch in the article [16] which we review below. Right after the relation (1), Grötzsch deduces the corresponding inequalities for annuli. More precisely, using the logarithm function and then the exponential function, he obtains the fact that if there exists a bijective mapping in the class AQ from the annulus with inner radius r and outer radius 1 onto the annulus with inner radius r˜ and outer radius 1 , then 1

r Q  r˜  r Q :

(2)

As in the case of rectangles, Grötzsch says that equalities hold only for certain simple extremal maps which consist of rotations and reflections. By using logarithms, we can deduce from the double inequality (2) the geometrical definition of quasiconformal mappings for doubly-connected domains. In the same section of the paper, Grötzsch notes that a result of Carathéodory is still valid for bijective AQ mappings, namely, any bijective AQ mapping from a simply-connected domain bounded by a Jordan curve onto a domain of the plane can be extended continuously to a homeomorphism.

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To conclude this section, Grötzsch states two distortion inequalities that hold for an arbitrary mapping f in the class AQ defined on the annulus 0  r < jzj < 1 , sending the circle of radius r onto the circle of radius r and such that for any r  jzj < 1 , we have jf (z)j  r . The first inequality concerns the behaviour of the distance between the circle of radius r and a point in the image by f of any circle of radius  with r <  < 1 . The second inequality concerns the behaviour of the rate of change of f . In the last section, Grötzsch gives an extension of the (big) Picard theorem for mappings in the class AQ . More precisely, he proves that for a given r , any AQ mapping defined on the punctured disc 0 < jzj < r for which 0 is an essential singularity and which omits at most two points in the plane (or three points in the Riemann sphere) is constant. (An essential singularity is a point at which the function does not admit any finite or infinite limit.) The idea of Grötzsch’s proof is quite elementary. Indeed, he takes an AQ mapping f defined on the punctured disc 0 < jzj < r , extending continuously to the set jzj = r with an essential singularity at 0 and which omits three values in the Riemann sphere, that we can assume to be 0 , 1 and 1 . By definition of an AQ map, we can lift f to a bijective map f˜ from 0 < jzj < r onto a Riemann surface Sf equipped with a holomorphic map onto a subset of the Riemann sphere such that f =  ı f˜ . Since Sf is doubly-connected, it is biholomorphic by a mapping ' to an annulus a < jzj < b . Since 0 is an essential singularity, we have a > 0 , because otherwise the map  ı ' 1 would be a non-constant holomorphic map from 0 < jzj < b to C n ¹0; 1º with an essential singularity at 0 , which is impossible by the big Picard theorem (for meromorphic functions). Thus, we have an AQ mapping ' ı f˜ from 0 < jzj < r onto a < jzj < b , which is impossible because of relation (2) and the supperadditivity for the module of an annulus. In conclusion, Grötzsch in this short paper gives a certain number of results which show that conformal and quasiconformal mappings share several properties.

7. Some comments on Grötzsch’s paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterung des Picardschen Satzes” 1.— Picard’s big theorem says that if a holomorphic function has an essential singularity, then in any punctured neighborhood of this essential singularity, the function, takes infinitely often all possible values in the Riemann sphere, with at most two exceptions. Even though the statement that refers to Picard in Grötzsch’s paper is not an exact analogue of that theorem, what is proved gives easily such a result. Indeed, if the quasiconformal mapping omits at most two points in the plane,

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we can lift it to the associated Riemann surface which is necessarily a punctured disc (since otherwise we would have a quasiconformal mapping onto an annulus), and thus, we get a holomorphic function with one essential singularity. Therefore (by Picard’s theorem) the quasiconformal mapping takes all values infinitely often with the exception of at most two. 2.— The idea of a class of functions which generalize conformal mappings is also the subject of Lavrentieff’s paper [22] in which this author gives another proof of the big Picard theorem for such maps. 3.— The use of supperadditivity in §2 of Grötzsch’s paper, along with the double inequality (2), was used by Teichmüller in [26] in order to prove that quasiconformal mappings preserve the type of a simply connected Riemann surfaces. 4.— Grötzsch continued to investigate similarities between conformal and quasiconformal mappings in his paper [15] which we review next, by solving geometrical extremal problems for quasiconformal mappings, after he studied such problems in the setting of conformal mappings in the papers [12] and [13].

8. The paper “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche” Grötzsch’s paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche [15] published in 1930 is a sequel to the two papers [12] and [13]. Here, instead of working with conformal representations, Grötzsch considers representations of multi-connected domains by quasiconformal mappings, which he calls “nichtkonformen.” The terminology is close to te one he uses in his paper [14]. Here, Grötzsch means by this word a mapping defined on a region B of the z -plane which can be approximated uniformly, in the neighborhood of each point except for finitely many points in B —called exceptional points—by an affine mapping, and such that in the neighborhood of each non-exceptional point the ratio a/b of the great axis to the small axis of the infinitesimal ellipse which is the image of an infinitesimal circle is uniformly bounded by two constants: 1  a/b  Q Q where Q is independent of the choice of the non-exceptional point. As we did for the mappings introduced by Grötzsch in his paper [14], we shall use here the terminology AQ mapping for such a mapping. Grötzsch’s results in this paper are based again on the length-area method, which he used extensively in his previous papers in the setting of conformal

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mappings, together with the two double inequalities (1) and (2) which he proved in his paper [14]. The method, as in the case of conformal mappings, consists in taking appropriate sequences of surface strips. Grötzsch’s goal is to determine, under some normalization conditions, the mappings in the class AQ that realize extremal values to certain distortion quantities. He obtains six theorems which we state below. Before stating them, we introduce some notation (which is Grötzsch’s notation). Bn+1 is an (n + 1) -connected open region of the complex plane. It is said to be normalized if the circle jzj = 1 is one of its boundary components and if it is contained in the interior of that circle. All the regions Bn+1 that we consider are normalized. We say that jzj = 1 is the outer boundary of Bn+1 , and we let R1 ; : : : ; Rn be its remaining boundary components (they may be reduced to points). Grötzsch gives the solutions of eight extremal theorems. The question, in each case, is to find a mapping in the class AQ of normalized mappings for which some geometrically defined quantity is extremal. For each of these problems, any AQ mapping which is a solution turns out to have constant dilatation Q . In the first 4 theorems, a normalized mapping AQ of Bn+1 is a bijective mapping from Bn+1 onto a normalized region Bzn+1 in the complex plane sending the outer boundary jzj = 1 of Bn+1 to the outer boundary jwj = 1 of Bzn+1 . We let Rz1 ; : : : ; Rzn be the images of the boundary components R1 ; : : : ; Rn . In the first theorem, the quantity that is maximized is the area of the domain enclosed by Rzk . Theorem 7 (Theorem 1 of [15]). In the set AQ of normalized mappings of Bn+1 , the area of the domain enclosed by Rzk attains its maximum if and only if 1. Rzk is a circle centered at w = 0 ; 2. the other Rzp (if they exist) are circular arc slits centered at w = 0 ; 3. at each point of the image, the minor axis of all the distortion ellipse (which Grötzsch calls the Tissot indicatrix) is directed radially toward the origin w = 0 ; 4. the distortion is everywhere constant and equal to Q . Furthermore, any two such extremal mappings differ from each other by at most a rotation around w = 0 and a reflection in a straight line through w = 0 . The proof of this theorem, like the one of the theorems that follow, is first given in the case of conformal mappings ( Q = 1 ), and then for AQ mappings, using the two distortion results for quadrilaterals and annuli that we recalled. The remaining seven theorems are of the same sort. Instead of restating them, we indicate, for each one, the quantities that are minimized. In each case, Grötzsch gives a description of the extremal image domain.

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Theorem 8 (Theorem 2 of [15]). In the set AQ of normalized mappings of Bn+1 , to find the mappings for which the maximum value of the diameter of Rzk is attained. In this case, the extremal image domain has a rectilinear slit passing through the origin w = 0 and bisected by that point. Theorem 9 (Theorem 3 of [15]). In the set AQ of normalized mappings w = f (z) of Bn+1 and for a given pair (z1 ; z2 ) in Bn+1 , to find the mappings for which the maximum of the quantity jf (z1 ) f (z2 )j is attained. In this case, in the extremal image domain, the images f (z1 ) and f (z2 ) are the endpoints of a rectilinear segment bisected by the origin w = 0 . Theorem 10 (Theorem 4 of [15]). Assume that Bn+1 has z = 0 as interior point. In the set AQ of normalized mappings w = f (z) of Bn+1 that satisfy f (0) = 0 , and for a given z1 in Bn+1 , to find the mappings for which the maximum of the quantity jf (z1 )j is attained. In the remaining four theorems, a different normalization of the mappings in AQ is used. Here, an (n + 1) -connected region Bn+1 is said to be normalized if it is bounded (“from inside”) by a circle jzj = r and lies entirely outside jzj = r , and which is bounded (“from outside”) by a linear boundary L , either separating Bn+1 from 1 (which means the curve is closed) or heading towards to 1 , and by n 1 other boundaries Rk lying between jzj = r and L . Here, a mapping AQ of Bn+1 is called normalized if the image region Bzn+1 of the w -plane is also normalized, if the circle jzj = r is sent to jwj = r , and if z joining Bzn+1 to 1 . (In particular, 1 is not L is sent to the linear boundary L an image point). Let Rzk be the boundary of Bn+1 corresponding to Rk . The next three theorems are stated in the form of problems of which Grötzsch gives the solution. Theorem 11 (Theorem 5 of [15]). Among the normalized mappings of Bn+1 , to z from the point find those AQ mappings for which the distance of the boundary L w = 0 attains its minimum value. Theorem 12 (Theorem 6 of [15]). Among the normalized mappings of Bn+1 , to z from the point find those AQ mappings for which the distance of the boundary L w = 0 attains its maximum value. Theorem 13 (Theorem 7 of [15]). Among the normalized mappings w = f (z) of Bn+1 , and for a given point z1 in Bn+1 , to find those AQ mappings for which the minimum of the quantity jf (z1 )j is attained. Theorem 14 (Theorem 8 of [15]). Among the normalized mappings w = f (z) of Bn+1 , and for a given point z1 in Bn+1 , to find those AQ mappings for which the maximum of the quantity jf (z1 )j is attained.

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9. The paper “Über möglichst konforme Abbildungen von schlichten Bereichen” The paper Über möglichst konforme Abbildungen von schlichten Bereichen [16] by Grötzsch was published in 1932 and is concerned with maps he calls möglichst konforme (closest-to-conformal). These are maps that we call today “extremal quasiconformal mappings.” Grötzsch starts by recalling that (by the Riemann Mapping Theorem) one can map conformally and in a one-to-one way an arbitrary simply-connected open subset B1 of the plane which is not the whole plane onto another such subset Bz1 , and that this map is completely determined if one asks that the image of three arbitrarily chosen distinct points on the boundary of B1 are sent to three arbitrarily chosen distinct points on the boundary of Bz1 . He then remarks that if one takes, on the boundary of each domain, four cyclically ordered points instead of three, then, generally speaking, one cannot find a conformal mapping between the two domains tat send distinguished points to distinguished points. In this case, one looks for mappings whose deviation from conformality is the smallest. The question is then to introduce an appropriate measure for this deviation from conformality and to investigate the existence of such closest-to-conformal mappings. The question is also relevant for domains of the plane which are not simplyconnected. In the case of multiply-connected domains, the same question appears naturally, with or without distinguished points on the boundary. In the rest of the paper, Grötzsch gives several examples of closest-to-conformal mappings between domains of the plane, some of them simply-connected and others multiply-connected. Right at the beginning of the investigation, Grötzsch says that one has to allow the maps to have singularities at isolated points in the interior, possibly countably many and converging to points on the boundary. At the non-singular points (which he calls “differential-geometric” and which we call “regular”), the maps are assumed to be differentiable, and the defect in conformality is measured by the ratios of the great axis to the small axis of infinitesimal ellipses which are images of infinitesimal circles, as in the previous papers, [14] and [15]. Likewise, the quantity Q  1 is defined to be the upper bound of such ratios, taken over all the regular points. The measure of deviation of the mapping is taken to be the quantity Q 1 . Grötzsch notes that this quantity is invariant by (pre- and post-) composition of the mapping with conformal mappings. He then remarks that in a neighborhood of a regular point, infinitesimal squares are sent to infinitesimal parallelograms, and he gives a result on the distortion of sufficiently small parallelograms in the form of a double inequality bounding the quotient of the area over the square of the length of one of the sides in terms of the upper bound Q of the mapping’s dilatation. More precisely, the result says the following (we use Grötsch’s notation; formula (1) in his paper):

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Proposition. For every  > 0 and at every regular point, there exists a neighborhood of this point such that the image of an infinitesimal square is an infinitesimal parallelogram satisfying the following: if ˜s is the length of any one of its sides, and I˜ its area, then 1 2 (1 ")

˜s < I˜ < (1 + ")
Q
˜s 2 ; (3) Q where " > 0 can be arbitrarily small. After this proposition, Grötzsch gives examples of closest-to-conformal mappings. He calls the problem of finding such mappings “the mapping problem,” a term that suggests the Riemann Mapping Theorem.

10. Examples of closest-to-conformal mappings (extremal quasiconformal mappings) Example 1: rectangles The mapping problem is considered between two Euclidean rectangles in the plane, where the distinguished points are the vertices. The rectangles may be assumed to have the same height, and their lengths are denoted by p and p˜ respectively, with p  p˜ . Grötzsch proves that we always have the inequality Q  pp˜ , with equality holding if and only if the map is affine, in which case the value of Q is equal, at every point, to the ratio of the two axes of the image infinitesimal ellipse. In his proof, Grötzsch uses the strip method, a method that is a form of the length-area method. It is also based on inequalities that relate lengths of curves in a certain family and the area of the domain that the family occupies. For this method, he refers to his paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche [15] (Grötzsch calls it so in Example 3, where he uses it again).

Example 2: doubly connected domains Grötzsch remarks that any annulus can be mapped conformally onto a Euclidean annulus bounded by two circles. By using logarithms, the problem of finding closest-to-conformal mappings between Euclidean annuli is reduced to that of finding closest-to-conformal mappings between Euclidean rectangles, and hence to Example 1. The unique closest-to-conformal mapping is therefore, in this case too, the affine map. Grötzsch notes that in the case where the boundary component of one annulus (and not the other one) is reduced to a point, then we cannot find a closest-to-conformal mapping between the two (the quantity denoted by Q , that is, the upper bound of the major to the minor axes of infinitesimal ellipses is always infinite).

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Example 3: simply-connected domains with two distinguished interior points Grötzsch considers the problem of finding closest-to-conformal mappings between two simply-connected domains with two distinguished points in the interior. Using the Riemann Mapping Theorem, one may assume that each domain is the open unit disc in the complex plane and that in each such disc, the distinguished points lie on the real axis and are symmetric with respect to the origin. By cutting each of the two discs along the segment joining the two distinguished points, we obtain annuli, and the mapping problem is reduced to that of Example 2. Grötzsch notes that the closest-to-conformal mapping obtained is C 1 at the two distinguished points only in the case where it is conformal.

Example 4: simply-connected domains with two distinguished points on the boundary and one distinguished point in the interior Grötzsch considers the problem of finding closest-to-conformal mappings between two simply-connected domains having two distinguished points on the boundary and one distinguished point in the interior. Like in the Example 3, by composing with a conformal mapping, each of the two domains can be assumed to be the unit disc in C , with the interior distinguished point being the center of the disc and the distinguished points on the boundary being the points e ˙i ˛ and e ˙iˇ in the first and the second disc respectively, where ˛ and ˇ are real numbers satisfying 0 < ˛ <  and 0 < ˇ <  . We may also assume that the mapping sought for (supposing it is orientation-preserving) sends the arc of circle whose extremities are the two points e ˙i ˛ and containing the point 1 to the arc of circle containing the two points e ˙iˇ and containing the point 1 . One then cuts the two domains along the segment joining 0 to 1 , obtaining in each case a simply-connected domain with four distinguished points on the boundary. We are then reduced to finding the closest-to-conformal map between two simply-connected domains with four boundary points on the boundary, and we proceed as in Example 1. The closest-to-conformal map obtained between the two initial discs is C 1 except at the origin, and it is C 1 there only if it is conformal. Remark. A few years after Grötzsch’s papers appeared, Teichmüller considered the existence and uniqueness problems of closest-to-conformal mappings mappings (which were already called extremal quasiconformal mappings) for arbitrary surfaces: compact or not, orientable or not, with an arbitrary number of distinguished interior or boundary points. See the paper [27] Extremale quasikonforme Abbildungen und quadratische Differentiale (1939) and the commentary in [7]. In 1944, he published a paper, Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem for quasiconformal mapping) [29], in which he considers a new type of existence problem for extremal mappings, in which each point on the boundary

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is a distinguished point. In other words, he studies the question finding and describing the extremal quasiconformal mapping from the unit disc with the origin as a distinguished point to the unit disc with some distinguished point in the interior which is the identity on the boundary (that is, each point on the unit circle is also considered as a distinguished point). We refer the reader to the paper [6] where the developments of this problem are also discussed. In the last section of the paper [16], called Additional remark, Grötzsch establishes an extension of Schwarz’s lemma to the class of differentiable quasiconformal mappings, that is, to the mappings that belong to the class denoted by AQ in the paper [14] discussed above. The extension is valid for branched coverings of the unit disc. The proof of this extension involves Riemann surfaces which cover the sphere and it uses the classical Schwarz lemma for conformal mappings. This result is in the trend of previous results of Grötzsch whose aim is the extension to the class of quasiconformal mappings of results that are known to hold for conformal mappings; see e.g. the extension of Picard’s theorem in the paper [14] we reviewed above. In a biographical note on Grötzsch published in the present volume [20], Reiner Kühnau writes: “It is strange that today many people associate the name Grötzsch only to the ridiculous ‘Grötzsch-ring’.” We hope that after publishing here the English translations of these five papers of Grötzsch, his work on the theory of quasiconformal mappings will be better known.

References [1] L. V. Ahlfors, Üntersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen. Acta Soc. Sci. Fenn. (2), A, 1 9 (1930), 1–40. Reprint in Collected papers. Vol. 1. 1929–1955. Edited with the assistance of R. M. Shortt. Contemporary Mathematicians. Birkhäuser, Boston, MA, 1982, 18–55. JFM 56.0984.02 R 396, 398 [2] L. V. Ahlfors, Zur Theorie der Überlagerungsflächen. Acta Math. 65 (1935), no. 1, 157–194. Reprint in Collected papers. Vol. 1. 1929–1955. Edited with the assistance of R. M. Shortt. Contemporary Mathematicians. Birkhäuser, Boston, MA, 1982, 214–251. MR 1555403 JFM 61.0365.03 Zbl 0012.17204 R 403 [3] L. V. Ahlfors, Conformal invariants: Topics in geometric function theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York etc., 1973. MR 0357743 Zbl 0272.30012 R 395, 396 [4] L. V. Ahlfors, Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura, and J. H. Hubbard. University Lecture Series, 38. American Mathematical Society, Providence, R.I., 2006. MR 2241787 Zbl 1103.30001 R 404 [5] L. V. Ahlfors, Collected papers. Vol. 1. 1929–1955; Vol. 2. 1954–1979. Edited with the assistance of R. M. Shortt. Contemporary Mathematicians. Birkhäuser, Boston, MA, 1982. MR 0688648 (Vol. 1) MR 0688649 (Vol. 2) Zbl 0497.01025 R 398, 403

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[6] V. Alberge, A commentary on Teichmüller’s paper “Ein Verschiebungssatz der quasikonformen Abbildung.” In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VI. IRMA Lectures in Mathematics and Theoretical Physics, 27. European Mathematical Society (EMS), Zürich, 2016, 613–629. MR 3560242 Zbl 1345.30020 R 412 [7] V. Alberge, A. Papadopoulos, and W. Su, A commentary on Teichmüller’s paper “Extremale quasikonforme Abbildungen und quadratische Differentiale.” In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. V. IRMA Lectures in Mathematics and Theoretical Physics, 26. European Mathematical Society (EMS), Zürich, 485–531. R 411 [8] P. L. Bowers, Combinatorics encoding geometry: The legacy of Bill Thurston in the story of one theorem. In V. Alberge, K. Ohshika, and A. Papadopoulos (eds.), In the tradition of Thurston Springer Verlag, Berlin etc. in press. R 396 [9] C. Carathéodory, Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten. Math. Ann. 72 (1912), no. 1, 107–144. MR 1511688 JFM 43.0524.01 R 398 [10] C. Carathéodory, Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis. Math. Ann. 73 (1913), no. 2, 305–320. MR 1511735 R 398 [11] C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Math. Ann. 73 (1913), no. 3, 323–370. MR 1511737 JFM 44.0757.02 R 398 [12] H. Grötzsch, Über einige Extremalprobleme der konformen Abbildung. Leipz. Ber. 80 (1928), 367–376. English translation by A. A’Campo-Neuen, On some extremal problems of the conformal mapping. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 13, 355–363. JFM 54.0378.01 R 394, 396, 397, 398, 399, 400, 401, 402, 404, 406 [13] H. Grötzsch, Über einige Extremalprobleme der konformen Abbildung. II. Leipz. Ber. 80 (1928), 497–502. English translation by M. Brakalova-Trevithick, On some extremal problems of the conformal mapping II. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 14, 365–370. JFM 54.0378.01 R 394, 402, 404, 406 [14] H. Grötzsch, Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterung des Picardschen Satzes. Leipz. Ber. 80 (1928), 503–507. English translation by M. Brakalova-Trevithick, On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 15, 371–374. JFM 54.0378.02 R 394, 403, 404, 406, 407, 409, 412 [15] H. Grötzsch, Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche. Leipz. Ber. 82 (1930), 69–80. English translation by M. Karbe, On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 16, 375–385. JFM 56.0298.02 R 394, 395, 406, 407, 408, 409, 410 [16] H. Grötzsch, Über möglichst konforme Abbildungen von schlichten Bereichen. Leipz. Ber. 84 (1932), 114–120. English translation by M. Brakalova-Trevithick, On closest-to-conformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA

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[17] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung). Gött. Nachr 1908, 337–358. JFM 40.0467.01 JFM 39.0489.02 R 396 [18] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung. IV. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche. Acta Math. 41 (1918), 305–344. JFM 46.0545.02 R 396 [19] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung. V. Abbildungen mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche. Math. Z. 2 (1918), 198–236. JFM 46.0546.01 R 396 [20] R. Kühnau, Memories of Herbert Grötzsch. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 10, 301–315. R 412 [21] M. A. Lavrentieff, Sur deux questions extrémales. Rec. Math. Moscou 41 (1934), 157–165. JFM 60.1030.03 Zbl 0009.36103 R 401 [22] M. A. Lavrentieff, Sur une classe de représentations continues. Mat. Sbornik 42 (1935), 407–424. English translation by V. Alberge and A. Papadopoulos, On a class of continuous representations. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 19, 417–439. JFM 61.1131.04 Zbl 0014.31905 R 406 [23] O. Lehto and L. Virtanen, Quasiconformal mappings in the plane. Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, 126. Springer-Verlag, Berlin etc., 1973. Original German edition, Quasikonforme Abbildungen. Springer-Verlag, Berlin etc., 1965. MR 0344463 MR 0188434 (original) Zbl 0267.30016 Zbl 0138.30301 (original) R 395, 396, 397, 404 [24] B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Groesse. Göttingen, 1851. In Gesammelte mathematische Werke (Collected mathematical works, scientific posthumous works and supplements). According to the edition by H. Weber and R. Dedekind, newly edited by R. Narasimhan, Springer-Verlag, Berlin, 1990, 3–48. R 395 [25] G. Szegö, Über eine Extremalaufgabe aus der Theorie der schichten Abbildungen. Sitzungsberichte der BMG 22 (1923), 38–47; corrigendum, ibid. 23 (1924), 64. R 399 [26] O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung. Deutsche Math. 3 (1938), 621–678. Reprint in Gesammelte Abhandlungen (L. V. Ahlfors and F. W. Gehring, eds.), Springer-Verlag, Berlin etce, 1982, 205–262. English translation by M. Brakalova-Trevithick and M. Weiss, Investigations on conformal and quasiconformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 22, 463–531. JFM 64.0313.06 Zbl 0020.23801 R 396, 398, 402, 406 [27] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 22 (1939), 1–197. Reprint in Gesammelte Abhandlungen. L. V. Ahlfors and F. W. Gehring, eds. Springer-Verlag, Berlin etc., 1982, 335–531. English translation by G. Théret, Extremal quasiconformal mappings and quadratic differentials. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. V. IRMA Lectures in Mathematics and Theoretical Physics, 26. European Mathematical Society (EMS), Zürich, 2016, 321–483. MR 0003242 JFM 66.1252.01 Zbl 0024.33304 Zbl 1344.30044 (translation) R 398, 411

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[28] O. Teichmüller, Über Extremalprobleme der konformen Geometrie. Deutsche Math. 6 (1941), 50–77. Reprint in Gesammelte Abhandlungen (L. V. Ahlfors and F. W. Gehring, eds.), Springer-Verlag, Berlin etc., 1982, 554–582. English translation by M. Karbe, On extremal problems of conformal geometry. In A. Papadopoulos (ed.), Handbook of Teichmüller Theory. Vol. VI. IRMA Lectures in Mathematics and Theoretical Physics, 27. European Mathematical Society (EMS), Zürich, 2016, 567–594. MR 0005764 MR MR3560242 (translation) JFM 67.0289.01 Zbl 0025.32301 Zbl 1345.30066 (translation) R 398 [29] O. Teichmüller, Ein Verschiebungssatz der quasikonformen Abbildung. Deutsche Math. 7 (1944), 336–343. English translation by M. Karbe, A displacement theorem for quasiconformal mapping. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VI. IRMA Lectures in Mathematics and Theoretical Physics, 27. European Mathematical Society (EMS), Zürich, 2016, 605–612. MR 0018761 Zbl 0060.23401 R 411

Chapter 19

On a class of continuous representations Mikhaïl Lavrentieff (translated from the French by Vincent Alberge and Athanase Papadopoulos)

Contents 1 Preliminary propositions 2 Existence theorems . . . 3 Analytical applications . 4 Geometrical applications Note . . . . . . . . . . . . .

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419 427 430 434 439

In this article,1 I propose to prove by the methods of function theory a proposition on the existence of the conformal representation of the two-dimensional Riemannian space onto a domain of the Euclidean plane e : given the quadratic differential form ds 2 = E dx 2 + 2F dx dy + G dy 2 ;

EG

F 2 > 0;

where the functions E , F , and G are continuous, there exists a conformal representation of the corresponding Riemannian space to a domain2 of the plane e . By applying the linear transformation and Theorem 2, it is easy to find an “almost” conformal representation from the Riemannian space onto a domain of e . To verify the passage to the limit, I study a class of continuous representations. Definition. We shall say that a function w = f (z) of the complex variable z is almost analytic in a domain D if this function enjoys the following properties:

Original title: S̈ur une classe de représentations continues. Mat. Sb. 42 (1935), 407–424. 1 The statements of the results below are published in the “Comptes rendus,” t. 200, p. 1010,

1935. 2 In the work of L. Lichtenstein, we find the additional condition: E , F , G have to satisfy the Hölder condition.

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1. The function f (z) is defined, uniform and continuous in the domain D . 2. With the exception of a countable set of points z , the function w = f (z) realises a homeomorphic correspondence between the sufficiently small neighborhoods of the points z0 and w = f (z0 ) ; if in the same neighborhood of z0 , z describes a circle in the positive sense, then the point w = f (z) describes a simple closed curve in the positive sense. 3. There exist two real-valued functions p(z)  1 and (z) of the complex variable z such that: a. With the exception of a set E consisting of a finite number of analytic arcs, p(z) is continuous, and (z) is continuous for p(z) ¤ 1 . b. p(z) is uniformly continuous in each domain ∆ that does not contain points of E and whose frontier is an analytic simple curve; under the same conditions on ∆ , if ∆ and the frontier of ∆ do not contain points p(z) = 1 , (z) is uniformly continuous in ∆ . c. Let z0 be an arbitrary point that does not belong to E . Let us construct the ellipse E : z0 is the center of E , the angle between the major axis of E and the real axis is equal to (z0 ) ; a and b being the axes of E , we have 1  ab = p(z0 ) . This being set, we have: ˇ ˇ ˇ f (z1 ) f (z0 ) ˇ ˇ=1 lim ˇˇ a !0 f (z2 ) f (z0 ) ˇ where z1 and z2 are the points of E for which the expression jf (z) f (z0 )j attains respectively its maximum and its minimum. The functions p(z) and  (z) are called characteristic functions of the almost analytic3 function f (z) . The problem of the conformal representation of a Riemannian space onto a domain in the Euclidean plane is equivalent to the problem of constructing a univalent almost analytic function having given characteristic functions p and  . We consider this question in §2 and §4. §1 contains some preliminary propositions. Lemmata 1, 2, and Theorem 1 give some properties of a particular class of univalent almost analytic functions. Lemmata 2 and 3 are not new. The theorem gives a solution of a problem on the conformal representation of the complementary domains; such a proposition is essential in the proof of the fundamental theorem, and the same proposition can be applied to the uniformization problem. 3 Mr. Stoïloff noted (Comptes rendus, t. 200, p. 1519) that we can give a less restrictive definition, and by applying some results from Mr. Stoïloff and Mr. Menchoff, prove that we obtain the same class of functions. If we consider the class of almost analytic functions having the characteristic p (z ) bounded on their set, we obtain a class of functions analogous to a class of functions studied by Mr. Grötzsch (“Ber. der Sächs. Acad. Wiss.,” 80, p. 503, 1928). Some special classes of almost analytic functions were studied by myself (“Atti del Congresso Int. Mat.,” 1928, t. III, p. 241; “Rec. math.,” t. 38, 1931).

19 On a class of continuous representations

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§3 contains some applications of the fundamental theorem to the study of arbitrary almost analytic functions.

1. Preliminary propositions Let us consider a particular class of almost analytic functions. We shall say that a function w = f (z) = u + iv which is almost analytic on a domain D is of class @v @v C 1 in this domain if its partial derivatives @u , @u , @x , @y are equally continuous @x @y in each domain where the characteristic function p(z) is equally continuous and if the functional determinant ˇ @u @u ˇ ˇ ˇ ˇ ˇ ˇ @x @y ˇ ˇ ˇ ˇ @v @v ˇ ˇ ˇ ˇ ˇ @x @y is positive. Lemma 1. Let w = f (z) , f (0) = 0 , be an almost analytic function of class C 1 for 0 < jzj < 1 . Assume that w = f (z) realises a homeomorphic representation from the disc jzj  1 onto the disc jwj  1 . Let us denote by q(r) the maximum of p(z) for jzj = r < 1 . We have:  jf (z)j < q R : 1 dr 2 jzj=1 rq(r) Indeed, let Sr be the image of the annulus  < jzj < r and let r be the image of the circle jzj = r . Let us denote by (r) the area of Sr and by L(r) the length of r . Let us set jf 0 (z)j = lim sup h!0

f (z + h) h

f (z)

:

We have ˇ @u ˇ ˇ ˇ @x ˇ ˇ @v ˇ ˇ @x

@u ˇˇ ˇ @y ˇ ˇ = 1 jf 0 (z)j2 : @v ˇˇ p(z) ˇ @y

Therefore, Zr (r) =

Z2 dr



0

1 jf 0 (z)j2 d'  p(z)

Zr 

rdr q(r)

Z2 0

f 0 (re i' )d':

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On the other hand, Z2 L(r) 

jf 0 (re i' )jrd':

0

Applying the Schwarz inequality, we obtain ² Z2 ³2 Z1 Z1 2 rdr 1 1 L (r) 0 i'  >  (1)  jf (re )jd' > dr: q(r) 2 2 rq(r) 



0 i'

But for any ' , L(r) > 2jf (e )j ,  < r ; therefore Z1 2 dr i' 2  > jf (e )j  rq(r) 

which proves the proposition. Lemma 2. 4 Assume that the function w = f (z) , f (0) = 0 , satisfies the condition of Lemma 1 and that q(r) < q0 = const . Then r q0 1 jf (z2 ) f (z1 )j < 5 p 2 j log jz2 z1 jj where z1 and z2 are two points of the disc jzj < 1 . Let us extend the homeomorphic correspondance, w = f (z) , to the discs jzj  2 , jwj  2 . The function w = f (z) is defined for jzj  1 , then for jzj > 1 we set5 1 f (z) =
: f z1¯ Let us introduce two complex variables z and ! : z1 2z z=2 ; 2 z1 z (1) w1 2! w=2 ; w1 = f (z1 ); 2 w1 ! the relations (1) give conformal representations of the discs jzj < 2 , jwj < 2 onto the discs jzj < 1 and j!j < 1 respectively. To the points z = z1 and w = w1 correspond the points z = 0 and ! = 0 .

4 Lemma 2 is contained in a theorem of Mr. Grötzsch (loc. cit.); the lemmata analogous to lemmata 2 and 3 are used in my notes quoted. ¯ the complex conjugate of the number a . 5 We denote by a

19 On a class of continuous representations

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Let ! = F (z) be the function defined by the relations (1) and the relation w = f (z) . The function F (z) is almost analytic for jzj < 1 , F (0) = 0 , and it realises a homeomorphic representation of the disc jzj  1 onto the disc j!j  1 . The characteristic function p(z) for F does not exceed q0 , therefore, by Lemma 1, we have: r  q0 1 jF (z)j < q R = : (2) p 1 2 dr j log jzjj 2 jzj=1

q0 r

Let us denote by z2 the point corresponding to the point z2 and by !2 the point corresponding to w2 = f (z2 ) . We have: jz2 j  M jz2

z1 j;

j!2 j  mjz2

z1 j;

ˇ dz ˇ ˇ for where M and m are respectively the maximum and the minimum of ˇ dz 2 6 1 jzj  1 , M = 3 < 1 , m = 25 > 5 . Applying (2), we obtain: r r  q0 1 q0 1 jf (z2 ) f (z1 )j < < 5 : p p m 2 j log M jz2 z1 jj 2 j log jz2 z1 jj

Lemma 3. Let w = f (z) , f (0) = 0 be an almost analytic function of z , jzj < 1 , which satisfies the conditions of Lemma 1 and such that: 1. f (1) = 1 , 2. 1  p(z)  1 +  ,  > 0 , where p(z) is the first characteristic function of f . Then, jf (z)

zj < ()

where () depends only on  , lim !0 () = 0 . Let C be an arbitrary analytic simple closed arc situated in the disc jzj < 1 . Let us compute the integral Z f (z)dz: C

For that purpose, let us make a coverage by squares of the domain bounded by the curve C , let Ck , k = 1; 2; : : : ; n be squares of this coverage. Assuming that the Ck are infinitely small and neglecting the infinitesimals, we have: Z XZ f (z)dz = f (z)dz: C

k C k

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Let zk = xk + iyk be the center of the square Ck . Then, by neglecting infinitesimals of highest orders, we have, for any z contained in Ck , f (z) = f (zk ) + k (x cos

y sin ) + ik (x sin + y cos )

+ k ¹˛1 x + ˇ1 y + i (˛2 x + ˇ2 y)º

where k = jf0 (zk )j = lim suph!0 j˛1 j < ;

f (zk +h) f (zk ) h

j˛2 j < ;

and , ˛ , ˇ are constants,

jˇ1 j < ;

jˇ2 j < :

The integral of the first three terms of f is zero, therefore ˇZ ˇ ˇ ˇ ˇ f (z)dz ˇ < 4k ı 2 ˇ ˇ Ck

where ı is the length of the side of Ck . On the other hand, X X 2k ı 2  (1 + ); ı 2 < : k

Applying Schwarz’s inequality, we obtain ˇZ ˇ XˇZ ˇ X ˇ ˇ ˇ ˇ p ˇ f (z)dz ˇ  ˇ f (z)dz ˇ < 4 k ı 2  4 1 + : ˇ ˇ ˇ ˇ k

C

Ck

From Lemma 2, the family of functions f (z) , for 0 <  < 1 , is a family of equally continuous functions on the disc jzj < 1 , therefore, from any infinite sequence from this family, we can extract a subsequence which converges uniformly on the disc jzj < 1 . Let f (z) be the limit function of such a sequence, f (z) = lim fn (z); n!+1

lim n = 0:

n!+1

The convergence being uniform, by (2), we have: Z Z f (z)dz = lim fn (z)dz = 0; n!+1

C

C

therefore, f (z) is analytic and uniform, for jzj < 1 . On the other hand, f (0) = 0 , f (1) = 1 , and when z describes the circle jzj = 1 in the positive sense, w = f (z) describes the circle jwj = 1 in the same sense, therefore f (z)  z . Consequently, lim f (z) = z;  !0

and the functions f (z) , 0 <  < 1 being equally continuous, the convergence is uniform, which proves the proposition. From the lemma proved, we deduce the following proposition: Theorem 1. Let D be a simply connected domain containing the point w = 0 , and let w = f (z; ) be an almost analytic function of class C 1 realising a homeomorphic representation of the disc jzj < 1 onto D and such that:

19 On a class of continuous representations

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1. f (0; ) = 0 ; 2. f (1; ) = w1 where w1 is a “point” 6 of the frontier of D ; and 3. the characteristic function p(z) of f satisfies the condition 1  p(z)  1 + . Let us denote by w = F (z) , F (0) = 0 , F (1) = w1 the function which realises the conformal representation of the disc jzj < 1 onto D . Then, if  approaches 0 , f (z; ) converges uniformly to F (z) on the disc jzj < 1 . If the frontier of D is a Jordan curve, then the convergence is uniform on the closed disc jzj  1 . In the following, we need to make this theorem more precise. Theorem 1 0 . Under the conditions of Theorem 1, we have, for jzj  r , jf (z; )

F (z)j < 1 (; r; R)

where 1 (; r; R) depends only on  , r and R and does not depend on D . Indeed, let z = Φ(w) be the inverse function of F . Let us consider the function ! = Φ[f (z; )] : this function satisfies the condition of Lemma 3, therefore Φ[f (z; )] = z + ˛(z)

where j˛(z)j < () , lim !0 () = 0 , consequently f (z) = F [z + ˛(z)]

which proves the proposition. Lemma 4. If w = f (z) , f (0) = 0 is an almost analytic function of class C 1 and realises a homeomorphic representation of the disc jzj < 1 onto the disc jwj < 1 and if 1  p(z)  1 +  , then for any point z0 , jz0 j < 1 , for any number  , 0 <  < 1 jz0 j , and for any two points z1 and z2 of the crown (1 1 ) < jz z0 j <  , we have: ˇ ˇ ˇ f (z2 ) f (z0 ) ˇ ˇ ˇ < 1 + (; 1 ); 1 (; 1 ) < ˇ (3) f (z1 ) f (z0 ) ˇ ˇ ˇ ˇ ˇ ˇ arg f (z2 ) f (z0 ) arg z2 ˇ < (; 1 ) (4) ˇ f (z1 ) f (z0 ) z1 ˇ where lim1 !0 (; 1 ) = 0 and  depends only on  and 1 , and not on f .  !0

6 If w1 is attained by a simple arc which is contained in D , then we understand the word “point” in the usual sense; otherwise, point is in the sense of simple target of Mr. Carathéodory.

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Let us prove the existence of  in the inequality (3). By extending the homeomorphic correspondance w = f (z) between the discs jzj  1 , jwj  1 to the discs jzj  2 and jwj  2 and performing the transformation (1), we can reduce the proof of inequalities (3) and (4) to the proof of those inequalities7 for z0 = 0 . Let us assume, for contradiction, that the first part of the lemma for z0 = 0 is not true. Let us construct in the z -plane the crown (1 1 ) < jzj <  < 1 and let us denote by L the image of this crown in the w -plane. From the hypothesis done, there exists a number k > 1 such that however small are the numbers  and 1 , there exists a function w = f (z) satisfying the condition of the lemma and a domain L ,  = (; 1 ) containing two points w1 and w2 such that: ˇ ˇ ˇ w2 ˇ ˇ ˇ > k: ˇw ˇ 1 From Lemma 3, we see that lim1 !0 (; 1 ) = 0 .  !0

Let us construct in the z -plane, the disc jzj = n = 1 < 1 , where n , n > 1 , is a large enough positive number that we shall define later. Let us denote by D1 the image of this disc in the w -plane and let r be the distance between the origin w = 0 and the frontier8 of D1 . This being set, let us consider the function 1 f (1 ; z) = F (z): r Such a function F has the following properties: w=

1. it realises a homeomorphic representation of the disc jzj < 1 onto a domain ∆; 2. the domain ∆ contains the disc jwj < 1 and the frontier of ∆ contains a point at distance 1 from the origin w = 0 ; 3. the characteristic function p1 (z) of F satisfies the condition 1  p1 (z)  1 + ;

4. the image of the crown (1 1 ) n1 < jzj < points w10 = wr1 and w20 = wr2 , ˇ 0ˇ ˇ w2 ˇ ˇ ˇ > k: ˇ w0 ˇ 1

1 n

, let it be, , contains the

(5)

Let us denote by w = Φ(z) , Φ(0) = 0 , Φ(1) = F (1) , the function which realises the conformal representation of the disc jzj < 1 onto the domain ∆ . Let

0 be the image of the crown (1 1 ) n1 < jzj < n1 . From the known properties of univalent analytic functions, there exists a number n0 , n0 > 0 , which depends only 7 See the proof of Lemma 2. 8 The domain D contains the disc jwj < r and does not contain the disc jwj < r1 for r1 < r . 1

19 On a class of continuous representations

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on k and a number 10 such that, for any two points !1 and !2 of 0 , we have for n  n0 , 1  10 ˇ ˇ ˇ !2 ˇ 1 + k ˇ ˇ< < k: (6) ˇ! ˇ 2 1 On the other hand, 1 1 1 1 j!1 j  ; j!2 j  : (7) n n The number n being fixed, let us take the numbers  , 1 small enough so that: 1. n < 1 ; this is possible since lim1 !0  = 0 ;  !0

1 k 1 2. the distance between the domains and 0 is smaller than 2n ; this k +1 0 is possible by Theorem 1 . From (6) and (7), for this value of  , for any points !10 and !20 of 0 , we have !10 w0 < k ; in particular w20 < k which is impossible, by (5). Thus, the existence of !20 1 a function (; 1 ) for inequality (3) is proved. The proof of the second part of the lemma is completely analogous. For the proof of the fundamental existence theorem, we need a proposition on the conformal representation of the complementary domains.

Theorem 2. For any real analytic function x 0 = '(x) of the real variable x , '( 1) = 1 , '(1) = 1 , ' 0 (x) > 0 for jxj  1 , we can construct two analytic functions f1 (z; h) and f2 (z; h) , z = x + iy , h = const such that: a. f1 (z; h) is regular and univalent on jxj < 1 , h < y  0 ; f2 (z; h) is regular and univalent on jxj < 1 ; 0  y < h , where h is some fixed positive number; b. f1 (x; h) = f2 ['(x); h] , 1 < x < 1 ; c. if 1 < x1 < 1 , h < y1 < 0 and 1 < x2 < 1 , 0 < y2 < h , then f1 (x1 + iy1 ) ¤ f2 (x2 + iy2 ):

Let us prove the proposition for small enough values of h . Let us set f1 (z; h) = '(z) , f2 (z; h) = z . The functions f1 and f2 satisfy condition (a), and by the properties of the function ' , for h small enough, f1 and f2 satisfy also conditions (b) and (c). Let us assume that the proposition is true for a value h = h0 , and let us prove in this case that it remains true for h = 2h0 . Indeed, let D1 and D2 be the images of the rectangles 1 < x < 1 , h0 < y < 0 and 1 < x < 1 , 0 < y < h0 in the representations z = f1 (z; h0 ) and z = f2 (z; h0 ) , and let Γ be the image of the segment 1 < x < 1 of the real axis. Let us denote by D the domain formed by the union of D1 , D2 and Γ . Let us also set z1 = f1 (1; h0 ) , z2 = f1 ( 1; h0 ) , z3 = f2 ( 1; h0 ) , z4 = f2 (1; h0 ) . Let us realise a conformal representation w = g(z) of the domain D onto the rectangle 1 < u < 1 , h0 < v < h0 (w = u + iv) . We take such a conformal representation in such a way that to the points z1 , z2 , z3 ,

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and z4 correspond the vertices of the rectangle.9 Let us denote by D10 and D20 the images of the domains D1 and D2 by the transformation w = g(z) (Figure 1). Let us consider the functions w = g[f1 (z; h0 )] = f1 (z; 2h0 ); w = g[f2 (z; h0 )] = f2 (z; 2h0 ):

ℎ0

1

ℎ0

Figure 1

It is easy to see that the functions f1 and f2 constructed are the functions sought for. Indeed, the function f1 (z; 2h0 ) realises the conformal representation of the rectangle 1 < x < 1 , h0 < y < 0 onto the domain D10 . In this representation, to the segment parallel to the real axis y = h0 , 1 < x < 1 corresponds the segment v = k , 1 < u < 1 parallel to the real axis, and the domain D10 is situated above this segment. Therefore, by the Schwarz principle, the function is regular and univalent in the rectangle 1 < x < 1 , 0 < y < 2h0 . By the same 9 g (z1 ) = 1

i k , g (z2 ) =

1

ik .

19 On a class of continuous representations

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reasonings, we see that the function f2 is regular and univalent in the rectangle 1 < x < 1 , 0 < y < 2h0 . Thus, property (a) is proved. Properties (b), and (c) are straightforward consequences of the properties of f1 (z; h0 ) , f2 (z; h0 ) , and of the construction of f1 (z; 2h0 ) , f2 (z; 2h0 ) .

2. Existence theorems It is easy to deduce from Lemma 4 and Theorem 2 the fundamental theorem on the existence of an almost analytic function, having the given characteristics p(z) and  (z) . Theorem 3. For any functions p(z) , p(z)  1 , and (z) defined for jzj  1 , p(z) continuous for jzj  1 and (z) continuous if p(z) ¤ 1 , we can construct an almost analytic function w = f (z) , f (0) = 0 , f (1) = 1 that realises the homeomorphic representation of the disc jzj  1 onto the disc jwj  1 and which has given characteristics10 p(z) and (z) . Let us introduce a special class of homeomorphic representations of the disc jzj  1 onto the disc jwj  1 . We say that an almost analytic function w = f (z) of class C 1 belongs to the class P in a domain ∆ if it possesses the following properties: 1. it realises a homeomorphic representation of the domain ∆ ; 2. let z0 be an arbitrary point of the domain ∆ , let us construct the ellipse Ez0 in such a way that z0 is the center of Ez0 , the angle between the major axis and the real axis is equal to (z) and, a and b being the two axes of Ez0 we have: 1  ab = p(z) . This being set, we have ˇ ˇ ˇ f (z ) f (z ) ˇ ˇ 1 0 ˇ lim ˇ ˇ=1 a !0 ˇ f (z2 ) f (z0 ) ˇ where z1 and z2 are the points of Ez0 for which the expression jf (z) f (z0 )j attains respectively its maximum and its minimum. The functions p and  and the number  ,  > 0 , being given, let us prove that there exists a function w = f (z; ) , f (0; ) = 0 , f (1; ) = 1 of class P which realises a homeomorphic representation from the disc jzj  1 onto the disc jwj  1 . Let z0 be a point of the disc jzj < 1 ; let us denote by ı the square having its center at the point z0 and let h be the length of the sides of ı . Let us construct the linear transformation z = ˛z + ˇ¯z which transforms the ellipse Ez0 into a disc of the z -plane. By the properties of the functions p(z) and  (z) , there exists 10 Instead of assuming that p and  are continuous, one can impose that they satisfy the conditions introduced in the definition of the almost analytic function and that p (z ) < M = const .

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a number h , depending only on  , such that the function z = ˛z + ˇ¯z is of class P in the closed square ı . Let us now assume that we have in the disc jzj < 1 , two simply connected domains D1 and D2 with no common points and an analytic arc belonging to the frontier of D1 and the frontier of D2 . Let us assume furthermore that there exist two functions z = F1 (z) and z = F2 (z) that are of class P in the domains D1 and D2 , respectively. This being set, let us prove that there exists a function of class P in the domain D formed by the union of the points of D1 , D2 , and . Let ∆1 and ∆2 be the images of D1 and D2 by F1 and F2 respectively, and let ˛1 and ˛2 be the arcs in the frontiers of ∆1 and ∆2 which correspond to . Let us denote by w = g1 (z) ( w = g2 (z) ) the function that realises the conformal representation of the domain ∆1 ( ∆2 ) onto the rectangle 1 < u < 1; 1 < v < 0 (resp. 1 < u < 1; 0 < v < 1 ) in such a way that to the arc corresponds the segment 1 < u < 1 of the real axis. Let u be a point of the segment 1  u  1 , let A be the point of which corresponds to u in the representation g1 [F1 (z)] , and let u0 be the point of the segment 1  u  1 which corresponds to the point A in the representation g2 [F2 (z)] . Thus, u0 is a function of u : u0 = '(u) . The function ' satisfies the conditions of Theorem 2,11 therefore we can construct two functions f1 and f2 of Theorem 2. It follows from the properties of the functions f1 and f2 that the function w = F (z) equal to f2 [g1 ¹F1 (z)º] for z contained in D1 and equal to f2 [g2 ¹F2 (z)º] for z contained in D2 is the desired function. It follows from the previous arguments that there exists a function z = (z) of class P in the disc jzj  1 . Let us denote by ∆ the image of the disc jzj < 1 . Let us realise the conformal representation w = g(z) of the domain ∆ onto the disc jwj < 1 in such a way that to the point z0 = (0) corresponds the point w = 0 and to the point z1 = (1) corresponds the point w = 1 . Then, the function w = g[ (z)] = f (z; ) is a function of class P in the disc jzj < 1 ; besides, this function realises a homeomorphic representation of the disc jzj < 1 onto the disc jwj < 1 , f (0; ) = 0 , f (1; ) = 1 . From Lemma 2, the family of functions f (z; ) , 0 <  < 1 as well as the family of inverse functions, are families of equally continuous functions on the discs jzj < 1 and jwj < 1 respectively. Therefore, there exists a sequence of functions f (z; 1 );

f (z; 2 );

:::;

1 >  2 >


>  n >


;

f (z; n );

lim n = 0

:::;

(8)

n!1

converging uniformly to a function w = f (z) , f (0) = 0 , f (1) = 1 , which realises a homeomorphic representation of the disc jzj < 1 onto the disc jwj < 1 . By applying Lemma 4, it is easy to prove that the function w = f (z) is the function sought of by theorem. 11 If ' (u) is not analytic, then we can always render it analytic by changing, as small as we wish, the functions F1 and F2 . It is obvious that we can do this change in such a way that the functions keep these indicated properties.

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Indeed, let  be an arbitrary positive number and z0 a point of the disc jzj < 1 . We denote by k , a number such that12 (2n ; n ) < 

for each value of n , n  k . From the definition of the functions f (z; ) , there exists a number a0 such that for any value of a , a  a0 we have ˇ ˇ ˇ f (z ;  ) f (z ;  ) ˇ ˇ 1 k 0 k ˇ 1 k < ˇ (9) ˇ < 1 + k ˇ f (z2 ; k ) f (z0 ; k ) ˇ where z1 and z2 are arbitrary points of the ellipse13 Ez0 = Ez0 (a) . Let us set z = f (z; k ); w = f (z; n );

n > k:

(10)

Equations (10) define a homeomorphic correspondance w = (z) , (0) = 0 , between the discs jzj < 1 and jwj < 1 . From the definition of f (z; ) , it follows that the characteristic function p(z) of the almost analytic function (z) satisfies the inequality 1  p(z)  1 + 2k : Thus, by Lemma 4, we have 1

ˇ ˇ (z1 ) (2k ; k ) < ˇˇ (z2 )

ˇ (z0 ) ˇˇ < 1 + (2k ; k ) (z0 ) ˇ

(11)

where z0 = f (z0 ; k ) and z1 and z2 are two arbitrary points of the annulus (1

k ) < jz

z0 j < :

Consequently, for each value of n , n > k , and for each value of a , a < a0 , we have ˇ ˇ ˇ f (z ;  ) f (z ;  ) ˇ ˇ 1 n 0 n ˇ 1 

1 2 dr 1  1+ dr 2 dr L > 2  = ; 2 r 2 r 2 r

Therefore,

 0 > 0:

1 0 1  > log r; C = const 0  2 which is impossible since when r ! 1 the left hand side stays bounded whereas the right hand side tends to infinity. C

19 On a class of continuous representations

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For the theorem to be meaningful, we still have to prove the existence of surfaces that satisfy the conditions of the theorem. We shall give an example of such a surface. Example. We define the surface S sought for in the cylindrical coordinates r; '; t , x = r cos ' , y = r sin ' in the following manner:

2

Figure 2

Let n be an integer n  2 . For r = n +  , 0   < 1 , the curve t = F (r; ') , 2 r = const of the ('; t ) -plane is a periodic curve whose period is 23 n . Let 2 us define F (r; ') , r = n +  for 0  ' < 23 n . The curve t = F (r; ') is 2 a polygonal curve with 2
32n+1 1 vertices joining the points (0; 0) and (23 n ; 0) (Figure 2). We set cos(˛(r)) = 2 2

r2

;

r = n + ;

yr = (1 ) n3 n tan ˛(n) + (n + 1)3 1 xr = yr cot ˛(r); r

(n+1)2

2

tan ˛(n + 1);

1 23 n 2xr ; yr0 = yr rır tan ˛(r): 2n+1 2 3 1 Then the coordinates of the odd vertices of F are (xr + 2kır ; yr ) , k = 0; 1; : : : ; 32n+1 1 , the coordinates of the even vertices are (xr + (2k + 1)ır ; yr0 ) , k = 0; 1; : : : ; 32n+1 2 . For r  1 , we set F (r; ')  0 and for 1 < r  2 we set ır =

F (r; ') = (r

1)F (2; '):

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Mikhaïl Lavrentieff

From the definition of F , we have ˇ ˇ ˇ = r tan ˛(r) ; 1. for each value of r , r  2 , we have ˇ @F @'

! 2. for r  2 , the angle formed by grad F (r; ') and the vector 0; re i' is greater than a positive number; 3. limr !1 F (r; ') = 0 .

We shall prove that there exists a constant K such that for any domain D of S containing the origin, we have  < KL

where  is the area of D and L is the length of the frontier of D . For that purpose, let us denote by D 0 the projection of D onto the (r; ') -plane. Without loss of generality, we shall assume that D 0 contains the disc r < 2 and that any straight line starting from the origin cuts the frontier of D 0 at a finite number of points. From the properties on F , we have “ 1  < K1 rdrd'; cos ˛(r) D0

Z L > K2

rd' cos ˛(r)

Γ

where K1 and K2 are two absolute constants and Γ is the frontier of D 0 . 2 Substituting cos 1˛(r) = 2r , we obtain: Z Z K1 2 2 < 2r d'; L > K2 r2r d'; r  2: 2 log 2 Γ

Γ

Hence, 
0:

(1)

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This problem, also called the conformal representation problem, asks fo a generalization of a result of Gauss, who introduced the notion of isothermal coordinates and showed their existence for surfaces embedded in the Euclidean space of dimension three (see [6]). We recall that the expression “isothermal coordinates” refers to local coordinates in which the Riemannian metric becomes conformally equivalent to the Euclidean metric. Classically, the quest for such coordinates amounts to solving a first-order partial differential equation which in complex notation is the so-called Beltrami equation. More precisely, by setting z = x + iy , equation (1) can be rewritten as ds 2 = e f (z) jdz + (z) d z¯j2 ;

(2)

where  is a complex-valued function depending on E , F , and G and satisfying jj < 1 , and finding a solution of (2) amounts to finding a solution of the Beltrami equation @f @f = (z) ; (3) @¯ z @z where, with the standard notation, @ 1 @ @  @ 1 @ @  = +i and = i : @¯ z 2 @x @y @z 2 @x @y A function f which is a solution of (3) is usually called  -conformal. The function  is not necessarily differentiable. Laventieff’s pair of characteristic functions associated with an almost analytic function are analogues of the complexvalued function  . In fact, one may express  in terms of p and  (see [19], p. 18): p(z) 1 2i(z) (z) = e : p(z) + 1 A first version of the existence of a solution of equation (3) was obtained by Korn in his 1914 paper Zwei Anwendungen der Methode der sukzessiven Annäherungen [12] and Lichtenstein in his 1916 paper Zur Theorie der konformen Abbildungen [18]. In these papers, the authors essentially prove the existence result under the hypothesis that  is Hölder continuous. In particular, the metric is not required to be Riemannian. Lavrentieff, in his paper, mentions the work of Lichtenstein. His result amounts to saying that it suffices to assume  to be continuous. About the time Lavrentieff published his paper, the condition on the Hölder continuity of  was relaxed to measurability by Morrey in [21], using methods of partial differential equations. Lavrentieff’s solution of the above problem is a version of what became known later as the measurable Riemann mapping theorem, or the existence theorem for  -conformal functions. Later, the theory was generalized by Ahlfors in his paper Conformality with respect to Riemannian metrics (1955) [1] and by Ahlfors and Bers in their paper Riemann’s Mapping Theorem for variable metrics (1960) [2].

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Lavrentieff had already addressed these questions in his 1928 ICM talk in Bologna [14], where his aim was to obtain the solution of the Beltrami equation as the limit of a sequence of explicit mappings which are solutions of partial differential equations, using a minimization principle. An early trace of the notion of almost analytic functions is found in the same paper. In his 1935 paper [16], the question is studied in more detail. We note incidentally that a use of quasiconformal mappings in this kind of problems is mentioned by Teichmüller in the last part of his 1939 paper which is translated in Volume V of the present series [24]. Unlike the theories that were developed by Tissot and Grötzsch before him, and Teichmüller after him, Lavrentieff makes no attempt to study the notion of an “extremal” almost analytic function. In the rest of this essay, we review in more detail Lavrentieff’s paper.

2. Almost analytic functions We now recall more precisely Lavrentieff’s definition of an almost analytic function f defined on a domain D of the complex plane, along with its associated characteristics. 1. f is continuous. 2. In the complement of a countable closed subset of D , f is orientationpreserving and a local homeomorphism. 3. There exist two functions, p(z)  1 and  (z) , called the characteristic functions of f ( and which Lavrentieff, sometimes, simply calls characteristics) defined on D and satisfying the following: • There is subset E of D consisting of a finite number of analytic arcs such that p is continuous and  is continuous at any point z satisfying p(z) 6= 1 . • On every domain ∆  D which does not contain any point of E and whose frontier is a simple analytic curve, p is uniformly continuous. Furthermore, if such a domain ∆ and its frontier do not contain any point satisfying p(z) = 1 , then  is also uniformly continuous on ∆ . • For any point z0 in D which is not in E , let E be the infinitesimal ellipse centered at z0 which is the image of an infinitesimal circle in the domain of f . Then (z) is the angle between its great axis and the real axis of the complex plane, and p(z0 ) = ab  1 the ratio of the great axis a to the small axis b of E . Let z1 and z2 be two points on the ellipse E at which the expression jf (z) f (z0 )j attains its maximum and minimum respectively. Then, ˇ ˇ ˇ f (z ) f (z ) ˇ ˇ 1 0 ˇ lim ˇ ˇ=1 a !0 ˇ f (z2 ) f (z0 ) ˇ

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Finding a solution of (3) is equivalent to the problem of constructing an almost analytic function whose characteristic functions are the given functions p and  . Lavrentieff’s characteristics remind us of Tissot’s indicatrix (see [26] in the present volume). Lavrentieff notes in his paper [16] that if we assume that the function p is bounded, then we recover a class of functions that are analogous to the ones considered by Grötzsch in [8], that is, quasiconformal mappings. Grötzsch, in his 1930 article [9], in which he deals with a class of mappings with bounded indicatrix, already noted that Lavrentieff’s second characteristic function is the Tissot indicatrix. Soon after Lavrentieff wrote his Comptes Rendus note in which he introduced the definition of an almost analytic function, another note was published by Stoïlov [23], showing that the first two conditions in Lavrentieff’s definition imply that the function f is an interior transformation (“transformation intérieure”). Such a transformation is a topological map from a surface into the Riemann sphere which is continuous, open and such that the pre-images of points are zero-dimensional. Stoïlov’s remark on Lavrentieff’s paper amounts to the fact that an almost analytic function is topologically equivalent to an analytic functions as a mapping from a surface (a twodimensional topological manifold) into the Riemann sphere. Stoïlov’s investigation of this subject was motivated by the so-called Brouwer problem, that is, the problem of topologically characterizing analytic functions. He obtained such a characterization in his paper [22]. His addition to Lavrentieff’s result says that for any almost analytic function w = f (z) in the sense of Lavrentieff, there exists a topological transformation of the domain of the variable, z = t (z 0 ) , such that the composed map f (t (z 0 )) is analytic. Stoïlov adds in [23] that all the topological properties of almost analytic functions are the same as those of analytic functions. This is an empirical fact which also holds for quasiconformal mappings in the sense of Grötzsch and Teichmüller, and it reminds us of the so-called “second Bloch principle” also called the principle of topological continuity (“principe de continuité topologique”) which says that in general, properties of holomorphic functions, or holomorphic families, should remain true if one modifies the topological data. The principle was vaguely stated by André Bloch in his 1926 paper La conception actuelle de la théorie des fonctions entières et méromorphes (The current conception of the theory of entire and meromorphic functions) [5], and it had a certain influence on mathematics, especially the theory of functions which, at that time, was at the highest peak. Bloch writes, after his statement: “It would be premature to try to give this statement a precise form; we shall restrict ourselves to indicate in some particular cases what significance it may have and what use can be done of it” ([5], p. 93). Besides the fact that individual almost analytic functions share properties that are common to analytic functions, Lavrentieff obtains in his paper [16], p. 417, a compactness result for families of almost analytic functions with uniformly bounded characteristics which is a generalization of a property that holds for analytic functions.

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The following result of Lavrentieff says that one can recover an almost analytic function f from its characteristic functions p and  (Theorem 1 in [15], p. 1011, and Theorem 3 in [13], p. 414): Theorem 2.1. Let p(z)  1 and (z) be two functions defined on the closed unit disc such that p(z) is continuous and  is continuous at any point z satisfying p(z) 6= 1 . Then, there exists an almost analytic function w = f (z) satisfying f (0) = 0 , f (1) = 1 , sending homeomorphically the closed unit disc jzj  1 to the closed disc jwj  1 , and having the functions p and z as characteristics functions. Lavrentieff says in a footnote that instead of assuming that p and  are continuous, one may suppose that they satisfy the conditions requested in the definition of an almost analytic function with p(z) bounded by a constant M . He also obtains the following result which says that an almost analytic function is uniquely determined by its characteristic functions ([15], p. 1012]): Theorem 2.2. If two almost analytic functions defined on some domain D have the same characteristic functions p and  and coincide on a subset which has a limit point in D , then they coincide on the whole set D . The two preceding results are versions of the integrability of almost-complex structures in dimension 2. They are also geometric versions of the existence and uniqueness results for quasiconformal mappings with a given dilatation which were later obtained by Teichmüller and others. There is an enormous literature dedicated to precise analytic versions of these results, with a minimum amount of continuity assumptions on the dilatation. These results culminated in the so-called Ahlfors–Bers measurable Riemann mapping theorem and its refinements (see [1] and [2]). The following result of Lavrentieff gives an existence property of an almost analytic function that realizes a homeomorphic representation of the unit disc jzj  1 onto the unit disc jwj  1 , provided the maximal value of the characteristic function p satisfies some boundedness condition expressed by the divergence of some integral (see Theorem 2 in [15], p. 1011, and Theorem 7 in [16], p. 418). Let p(z)  1 and  (z) be two functions defined on the closed unit disc such that p is continuous and  is continuous at every point satisfying p(z) 6= 1 . For every r < 1 let q(r) = max p(z): (4) jzj=r

R 1 dr Theorem 2.3. If the integral 0 rq(r) is divergent, then one can construct an almost analytic function w = f (z) satisfying f (0) = 0 , f (1) = 1 , whose characteristic functions are p and  and which sends homeomorphically the closed unit disc onto itself.

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3. Applications In this section, we review a few of applications of almost analytic functions that Lavrentieff gives in his paper [16]. They concern a Picard-type theorem, a compactness theorem, and the type problem. The paper contains more. The following is a generalization of the Picard big theorem. Theorem 3.1. Let f be an almost analytic function satisfying the hypothesis of R 1 dr Theorem 2.3, that is, the integral 0 rq(r) is divergent, and assume that for z ! 0 , the function F (z) has neither finite nor infinite limit. Then, for any complex number a except possibly for a single value a , the equation f (z) = a admits an infinite number of solutions in the neighborhood of the point z = 0 . In a footnote, Lavrentieff says that Grötzsch in [8] proved the same theorem under the condition that the function p is bounded (that is, f satisfies the strong form of quasiconformality). The following is a compactness theorem for almost analytic functions that have convergent distortion characteristics, which is close to the Montel theorem for holomorphic functions. Theorem 3.2. Let (fn ) , n  1 be a sequence of almost analytic functions satisfying the following three properties: 1. for each n  1 , fn (0) = 0 , fn (1) = 1 ; 2. for each n  1 , fn is a homemorphism of the unit disc; 3. for each n  1 , the sequence of characteristic functions pn (z) of fn (z) converges uniformly to a functions p(z) , and the sequence of characteristic functions n (z) converges uniformly in the interior of the domain jzj < 1 , p(z) ¤ 1 to a function (z) . Then the sequence (fn ) converges uniformly to an almost analytic function f satisfying f (0) = 0 and f (1) = 1 and which is a homeomorphism of the disc, and the characteristic functions of f (z) are the limits as n ! 1 of those of fn . The next two theorems (proved in §4 of Lavrentieff’s paper) concern at the same time the conformal representation problem and the type problem. Theorem 3.3. Let R be a Riemannian metric on the unit disc in the complex plane defined by the differential quadratic form ds 2 = E dx 2 + 2F dx dy + G dy 2 ;

where E , F , and G are continuous and satisfy EG F 2 > 0 for all (x; y) . Then there exists a conformal representation of R onto the unit disc equipped with the standard complex structure.

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After stating this theorem, Lavrentieff raises the question of the Lipschitz behavior of such a conformal representation. More precisely, he asks the following: If A and A0 are two points on the surface, and B and B 0 their images by the 0) conformal representation onto the unit disc, to know whether the ratios (A;A (B;B 0 ) 0

(B;B ) and (A;A 0 ) are bounded or not, where  and  denote the distance functions on the surface and on the unit disc respectively. To the best of our knowledge, this kind of question was not addressed by the other founders of the theory of quasiconformal mappings. Lavrentieff gives an example where the second ratio is unbounded. For the next theorem, we consider a differentiable surface S in the 3-dimensional space equipped with a coordinate system (x; y; t ) , defined by an equation of the form t = t (x; y) , where (x; y) varies in the unit disc. Let q(r) be now the maximum of 1 + jgrad t (x; y)j on the circle x 2 + y 2 = r 2 . Lavrentieff proves the following:

Theorem 3.4. If

R1 1

dr rq(r)

diverges, then S is of hyperbolic type.

It is conceivable that this was the first time where quasiconformal mappings are used in the study of the type problem. Later on, Teichmüller and Ahlfors also used quasiconformal mappings in the type problem. See the survey [3] in the present volume. The divergence condition that appears in this parabolicity criterion is comparable to a divergence condition given by Teichmüller in his later paper [25], where he considers the type problem from a completely different point of view. It is interesting to note that Milnor, several years later, studied the type problem in a similar setting; cf. [20]. Lavrentieff also gives the following criterion for the existence of surfaces of hyperbolic type, using the same notation for the surface S . The notation is the same as in Theorem 3.4. Theorem 3.5. The surface S is of hyperbolic type if and only if for any  > 0 and for any domain D contained in S and containing the point (0; 0; t (0; 0)) , the area of D is smaller than l 2  where l is the length of the frontier of D . In a short section that concludes his paper, Lavrentieff proposes to extend the study of the class of almost analytic functions to a larger class of functions on the unit disc which satisfy conditions (2) and (3) in the definition of an almost analytic function together with a metric property of annuli concerning annuli on the surface, saying that there exists a constant M such that for any jzj < 1 nd for every 0   < 2 and 0  < 2 , we have jf (z + re i  ) r !0 jf (z + re i )

(z) = lim

f (z)j  M; f (z)j

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This definition is at the basis of the notion of quasiconformality which was developed several years later by Gehring and others, for the general setting of mappings between metric spaces, see, e.g., [7], [17], and [11]. Lavrentieff states a Montel-type result about families of functions belonging to this class.

References [1] L. V. Ahlfors, Conformality with respect to Riemannian metrics. Ann. Acad. Sci. Fenn. Ser. A. I 1955, no. 206, 22 pp. MR 0074855 Zbl 0067.30702 R 443, 446 [2] L. V. Ahlfors and L. Bers, Riemann’s Mapping Theorem for variable metrics. Ann. of Math. (2) 72 (1960), 385–404. MR 0115006 Zbl 0104.29902 R 443, 446 [3] V. Alberge, M. Brakalova-Trevithick, and A. Papadopoulos, Teichmüller’s work on the type problem. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 24, 543–560. R 448 [4] V. Alberge and A. Papadopoulos, On five papers by Herbert Grötzsch. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 18, 393–415. R 442 [5] A. Bloch, La conception actuelle de la théorie des fonctions entières et méromorphes. Enseignement Math. 24 (1926), 83–103. JFM 52.0315.03 R 445 [6] C. F. Gauss, Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daßdie Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird. In H. C. Schumacher (ed.), Astronomische Abhandlungen. Drittes Heft, Altona, 1825, 1–30. Reprint in Carl Friedrich Gauss Werke. Vierter band, Göttingen 1880, 189–216. English translation by H. P. Evans, On conformal representation. In D. E. Smith (ed.), A source book in mathematics. Dover Publication, New York, 1959, 463–475. R 443 [7] F. W. Gehring, The definitions and exceptional sets for quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I 281 (1960), 28 pp. MR 0124488 Zbl 0090.05303 R 449 [8] H. Grötzsch, Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes. Leipz. Ber. 80 (1928), 503–507. English translation by M. Brakalova-Trevithick, On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 15, 371–374. JFM 54.0378.02 R 442, 445, 447 [9] H. Grötzsch, Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche. Leipz. Ber. 82 (1930), 69–80. English translation by M. Karbe, On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 16, 375–385. JFM 56.0298.02 R 442, 445 [10] H. Grötzsch, Über möglichst konforme Abbildungen von schlichten Bereichen. Leipz. Ber. 84 (1932), 114–120. English translation by M. Brakalova-Trevithick, On closest-to-conformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA

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[11] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), no. 1, 1–61. MR 1654771 Zbl 0915.30018 R 449 [12] A. Korn, Zwei Anwendungen der Methode der sukzessiven Annäherungen. In C. Carathédory, G. Hessenberg, E. Landau, and L. Lichtenstein (eds.), Mathematische Abhandlungen Hermann Amandus Schwarz zu seinem fünfzigjärigen Doktorjubiläum. Springer-Verlag, Berlin 1914, 215–229. JFM 45.0568.01 R 443 [13] M. A. Lavrentieff, Sur un problème de M. P. Montel. Comptes Rendus Acad. Sc. Paris 184 (1927), 1634–1635. JFM 53.0282.02 R 446 [14] M. A. Lavrentieff, Sur une méthode geométrique dans la représentation conforme. In Atti del Congresso Internazionale dei Matematici (Bologna 3–10 settembre 1928). Tomo III. Bologna 1930, 241–242. JFM 56.0295.03 R 444 [15] M. A. Lavrentieff, Sur une classe de représentations continues. Comptes Rendus Acad. Sci. Paris. 200 (1935), 1010–1013. JFM 61.0300.02 Zbl 0012.21501 R 441, 446 [16] M. A. Lavrentieff, Sur une classe de représentations continues. Mat. Sbornik 42 (1935), 407–424. English translation by V. Alberge and A. Papadopoulos, this volume, Chaprer 19. JFM 61.1131.04 Zbl 0014.31905 R 441, 444, 445, 446, 447 [17] O. Lehto and L. Virtanen, Quasiconformal mappings in the plane. Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, 126. Springer-Verlag, Berlin etc., 1973. Original German edition, Quasikonforme Abbildungen. Springer-Verlag, Berlin etc., 1965. MR 0344463 MR 0188434 (original) Zbl 0267.30016 Zbl 0138.30301 (original) R 449 [18] L. Lichtenstein, Zur Theorie der konformen Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete. Bull. Int. Acad. Sci. Cracovie, Cl. Sci. Math. Ser. A. (1916), 192–217. JFM 46.0547.01 R 443 [19] G. S. Migirenko, Mihail Alekseevič Lavrentiev. Amer. Math. Soc. Transl. (2) 104 (1976) 1–39. Zbl 0329.01014 R 443 [20] J. Milnor, On deciding whether a surface is parabolic or hyperbolic. Amer. Math. Monthly 84 (1977), no. 1, 43–46. MR 0428232 Zbl 0356.53002 R 448 [21] C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. MR 1501936 JFM 62.0565.02 Zbl 0018.40501 R 443 [22] S. Stoïlov, Sur les transformations continues et la topologie des fonctions analytiques. Ann. Sci. École Norm. Sup. (3) 45 (1928), 347–382. MR 1509289 JFM 54.0607.02 R 445 [23] S. Stoïlov, Remarques sur la définition des fonctions presque analytiques de M. Lavrentieff. C. R. Acad. Sci. Paris 200 (1935), 1520–1521. JFM 61.0300.03 Zbl 0012.21502 R 445 [24] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 22 (1939), 1–197. Reprint in Gesammelte Abhandlungen. L. V. Ahlfors and F. W. Gehring, eds. Springer-Verlag, Berlin etc., 1982, 335–531. English translation by G. Théret, Extremal quasiconformal mappings and quadratic differentials. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. V. IRMA Lectures in Mathematics and Theoretical Physics, 26. European Mathematical Society (EMS), Zürich, 2016, 321–483. MR 0003242 JFM 66.1252.01 Zbl 0024.33304 Zbl 1344.30044 (translation) R 444

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[25] O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung. Deutsche Math. 3 (1938), 621–678. Reprint in Gesammelte Abhandlungen (L. V. Ahlfors and F. W. Gehring, eds.), Springer-Verlag, Berlin etce, 1982, 205–262. English translation by M. Brakalova-Trevithick and M. Weiss, Investigations on conformal and quasiconformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 22, 463–531. JFM 64.0313.06 Zbl 0020.23801 R 448 [26] A. Papadopoulos, A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2019, Chapter 9, 289–299. R 442, 445

Chapter 21

An application of quasiconformal mappings to the type problem Oswald Teichmüller (translated from the German by Melkana Brakalova-Trevithick)

Contents 1 2 3 4 5 6

The problem . . . . . . . . . . . . . The dilatation quotient . . . . . . . . Quasiconformal mappings . . . . . . Surfaces with the same line complex Invariance of the type . . . . . . . . Control of L on the characteristic . .

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453 454 456 458 459 461

1. The problem Simply connected Riemann surfaces W , branched over only finitely many points w = a1 ; w = a2 ; : : : ; w = aq in the w -plane, have been frequently treated in connection with the new value distribution theory. In order to describe such surfaces, one constructs through the points a1 ; a2 ; : : : ; aq (in the given order) on the w -sphere a continuous simple closed curve L; which separates the sphere into a positively circulated region I and a negatively circulated region A . L is separated into q arcs s1 ; s2 ; : : : ; sq by the points a1 ; a2 ; : : : ; aq : One can then build a Riemann surface with finitely or infinitely many copies of the “half-planes” I and A by gluing to each copy of I a certain copy of A; along a certain arc s ; in this case, each boundary arc s should be applied to each I and to each A once. Different surfaces, branched over the same points a1 ; a2 ; : : : ; aq ; differ from each other only by the gluing process if the separating curve L is fixed. As it is known, one represents the latter graphically through a line complex. A Riemann Original title: Ëine Anwendung quasikonformer Abbildungen auf das Typenproblem. Deutsche Math. 2 (1937), 321–327.

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surface is uniquely determined by the points a1 ; a2 ; : : : ; aq ; the curve L; and its line complex. Recently, the problem of determining the properties of the schlicht mapping of W from its line complex has been frequently studied. So far in the foreground has been the type problem: How can one determine, from the given line complex, if the corresponding surface W can be mapped one-to-one and conformally onto the whole plane, the punctured plane, or the unit disk? One is still very far from finding sufficient and necessary conditions. Here we would like to answer the preliminary question of whether such a type criterion is at all possible. The surface W does not only depend on the line complex but also on the points a1 ; : : : ; aq ; and L . Can an alteration of the last data change the type of W? This appears quite unlikely as, until now, there are no known examples of whether these data have any known effect on the schlicht mapping of W: (Such an example will be reported later on.) We will actually prove that two Riemann surfaces with the same line complex can always be mapped simultaneously either onto the plane, or onto the punctured plane, or onto the unit disk. For this proof we leave pure function theory behind and apply mappings that are not conformal but quasiconformal. Moreover, the proof will be pretty simple. The method should, however, when expanded accordingly, prove itself useful in more difficult questions. Our result is so far of a practical importance, as in some type problem investigations, the general requirements on L disrupt noticeably the calculations. If one wants to determine the type of W and not necessarily the more precise 2 behavior of the mapping, one can always assume that a = e q and that L is the unit circle.

2. The dilatation quotient Let a mapping of a region in the z -plane onto a region in the w -plane be one-toone and, in both directions, continuously differentiable. We set z = x + iy and w = u + iv: Then, at every point, to every differential dz = dx + i dy corresponds through du = ux dx + uy dy; dv = vx dx + vy dy;

a complex differential dw = du + i dv: The quotient dw remains unchanged if dz one multiplies dz by a real number. On the other hand, in general, it depends essentially (as a linear fraction), ˇ ˇon the ratio dx : dy; thus on the direction of ˇ will assume a maximum and a minimum value, the dz: The variable quantity ˇ dw dz

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as dz takes successively all possible directions. We set ˇ ˇ ˇ Max ˇ dw dz ˇ dw ˇ Dz/w = D = Min ˇ dz ˇ and call D the dilatation quotient of our mapping z ! w: By assumption, ux vy vx uy ¤ 0; and therefore D is a finite number: 1  D < 1: In order to compute D; we reduce the quadratic form jdwj2 = (u2x + vx2 ) dx 2 + 2(ux uy + vx vy )dx dy + (uy2 + vy2 ) dy 2 ˇ ˇ ˇ; and the to a diagonal form: the larger of the resulting eigenvalues is Max ˇ dw dz ˇ dw ˇ ˇ ˇ smaller is Min dz . In this way one obtains:   u2 + u2 + v 2 + v 2 2  x y x y D4 + 2 D 2 + 1 = 0; ux vy vx uy 1 u2x + uy2 + vx2 + vy2 : 2 ux vy vx uy We always have D  1; and D = 1 if and only if the mapping z ! w is either conformal or a conjugation of a conformal mapping at the considered point. Our requirements are symmetric with respect to z and w ; therefore one can determine Dz/w by knowing Dw/z . We have ˇ ˇ ˇ ˇ ˇ dz ˇ ˇ dz ˇ 1 1 ˇ ˇ ˇ ˇ= ˇ ˇ ; Min ˇ ˇ ˇ; Max ˇ = ˇ ˇ ˇ dw ˇ Min ˇ dw dw Max ˇ dw dz dz D = jKj +

p K2

1

with K =

therefore

ˇ dz ˇ ˇ ˇ ˇ ˇ Max ˇ dw Max ˇ dw dz ˇ dz ˇ = ˇ dw ˇ = Dz/w : Dw/z = Min ˇ dw ˇ Min ˇ dz ˇ Moreover, let a region in the s -plane be mapped one-to-one and continuouslydifferentiably in both directions, onto a z -region. Then it holds that: ˇ ˇ ˇ ˇ ˇ ˇ ˇ dw ˇ ˇ ˇ ˇ ˇ ˇ  Max ˇ dz ˇ
Max ˇ dw ˇ ; Max ˇˇ ˇ ˇ ˇ ˇ ds ds dz ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ dw ˇ ˇ dz ˇ ˇ dw ˇ ˇ ˇ ˇ ˇ ˇ ˇ; Min ˇ  Min ˇ ˇ
Min ˇ ds ˇ ds dz ˇ

and therefore Ds/w

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Max ˇ dw ˇ Max ˇ dw Max ˇ dz ds ds dz ˇ dw ˇ  ˇ dz ˇ
ˇ dw ˇ = Ds/z
Dz/w : = ˇ ˇ ˇ ˇ ˇ Min ds Min ds Min dz ˇ

From these calculations follows that Dz/w does not change if one subjects the z -plane to a conformal mapping. As a matter of fact, if the mapping s $ z is conformal, we have Ds/w  Ds/z
Dz/w = Dz/w  Dz/s
Ds/w = Ds/w ;

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thus Ds/w = Dz/w : In the same manner Dz/w remains invariant under a conformal mapping of the w -region. To an infinitesimally small circle jdzj2 = const: corresponds, under ˇ ˇ the ˇ and mapping z ! w; an infinitesimally small ellipse with half-axes jdzj
Max ˇ dw dz ˇ dw ˇ 2 jdzj
Min ˇ dz ˇ: The circle with area jdzj is transformed into an ellipse with area ˇ ˇ ˇ ˇ ˇ dw ˇ ˇ ˇ ˇ
Min ˇ dw ˇ : jdzj2
Max ˇˇ ˇ ˇ dz dz ˇ If we denote the surface element by dz = dx dy;

dw = du dv;

we obtain jux vy

vx uy j =

dw dz

ˇ ˇ ˇ ˇ ˇ dw ˇ ˇ dw ˇ ˇ ˇ ˇ ˇ: = Max ˇ
Min ˇ dz ˇ dz ˇ

From here follows ˇ ˇ ˇ ˇ ˇ ˇ2 ˇ dw ˇ2 ˇ dw ˇ2 ˇ ˇ dw 1 dw ˇ ˇ ˇ  Max ˇ dw ˇ = D = Min ˇˇ ; ˇ ˇ ˇ ˇ ˇ D dz dz dz dz dz ˇ ˇ ˇ: for all of possible values of ˇ dw dz After the above justification of the theory of dilatation quotients, in the intended function-theoretic applications, the complex notation will be placed in the foreground as opposed to the real components.

3. Quasiconformal mappings If a region in the z -plane is mapped onto a region in the w -plane and the mapping is one-to-one and continuously-differentiable in both directions, then it is called quasiconformal if the dilatation quotient Dz/w is bounded. This is obviously only a restriction concerning the boundary behavior of D: A slight generalization is advantageous for the applications. We want to allow that the continuous-differentiability breaks off at individual isolated points, while, naturally, the mapping remains one-to-one and continuous. In a neighborhood of discontinuity of the partial derivatives, D must remain bounded. From the rules Dw/z = Dz/w ; Ds/w  Ds/z
Dz/w ;

follows that the inverse of a quasiconformal mapping is also quasiconformal, as is the composition of two quasiconformal mappings. Every conformal mapping is quasiconformal.

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We can transfer this concept to Riemann surfaces right away. We call now a oneto-one and continuous mapping between two surfaces quasiconformal if everywhere, except at isolated points, it is continuously-differentiable and its dilatation quotient is bounded. The continuous differentiability is to be proved through transition to local uniformizing parameters; the points with non-trivial local parameter (at infinity or at a branch points) are isolated and one should simply ignore them. It happens very often that one wishes to put together a quasiconformal or, in general, continuously-differentiable mapping. Let G1 ; G2 be two regions in the z -plane, separated by a differentiable curve C , continuous with respect to the arc length; the regions G1 and G2 are mapped continuously-differentiably onto corresponding regions in the w -plane. Each of these mappings is continuouslydifferentiable on C: When does one have a continuously-differentiable mapping of the region which results in gluing together G1 ; G2 and C ?—First, one needs to assign to each point on C the same image point for both mappings. Moreover, the limit values of the partial derivatives from both regions have to approach the same value on C; and this condition could lead to further requirements. However, this last condition is automatically fulfilled if both given mappings are conformal at C . For the proof, we could clearly assume that C and its w -image have horizontal tangents at the relevant points. Therefore vx  0; whether one approaches our C -point from G1 or from G2 : Further, because both mappings exert the same effect on C , ux has the same limit when approaching from G1 and from G2 : Now, when we approach our point from both domains, the Cauchy–Riemann equations uy =

vx ;

vy = ux :

should also hold; this gives the continuity of all four partial derivatives. Now one still has to solve an important mapping problem. The periphery of the unit disk in the  -plane is mapped one-to-one, continuously-differentiably, and orientation-preserving onto the periphery of the unit disk in the ! -plane. This mapping has to be extended to a quasiconformal mapping from jj  1 to j!j  1 so that the mapping is conformal at the peripheral points. We set z = x + iy = (1

i ) log ;

w = u + iv = (1

i) log !:

From here, the punctured disk 0 < jj < 1 is mapped onto the half-plane x < y: Equal values of  correspond to z -values that differ by a multiple of 2 + 2 i; similar for ! ! w . By the given boundary correspondence, let the boundary point x = y be mapped to the boundary point u = v = f (x) = f (y): Here f is continuous, differentiable, f 0 (x) > 0 and f (x + 2) = f (x) + 2 . Then we map the half-plane x < y onto the half-plane u < v using u = f (x);

v = f (y):

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This mapping is one-to-one and continuously-differentiable (also at the boundary). Because of ˇ ˇ 0 2 2 0 2 2 ˇ dw ˇ2 ˇ ˇ = f (x) dx + f (y) dy ; ˇ dz ˇ dx 2 + dy 2 the dilatation quotient is ° f 0 (x) f 0 (y) ± Max¹f 0 (x); f 0 (y)º Dz/w = = Max ; : Min¹f 0 (x); f 0 (y)º f 0 (y) f 0 (x) It actually approaches 1 at each finite boundary point x = y; and since f 0 (x + 2) = f 0 (x) , it is bounded. If one increases both x and y by 2 , the same happens to u and v ; therefore through  !z !w !!

the disk jj  1 is mapped onto the disk j!j  1 one-to-one, quasiconformally, and conformally at the boundary. The only point of discontinuity for the partial derivatives that comes into question is  = 0 $ ! = 0:

4. Surfaces with the same line complex After these preparations, let us turn back to our problem. We were given a surface W branched only over a1 ; : : : ; aq ; which is separated into half-sheets I and A by means of a closed simple continuous curve L passing through a1 ; : : : ; aq ; the line complex gives the rule of gluing the half-sheets. Next, we want to map two different such surfaces W; with the same line complex, quasiconformally onto each other. Obviously, the line complex of W does not change if one substitutes L by a sufficiently close regular curve passing through the points a1 ; : : : ; aq . Therefore, we want to assume right away that e.g. the curve L is a circular polygonal region with everywhere continuous tangents. Now we map A conformally onto the exterior of the unit disk; the mapping is differentiable at the boundary (to prove this, one takes e.g. a boundary point and its image to infinity and reflects.) Again, we map the periphery of the unit disk continuously-differentiably onto itself so that the images of a are transformed to 2 i

e q : This mapping is extended to a quasiconformal mapping of the outside of the unit disk. Altogether, A is mapped quasiconformally on the outside of the unit disk 2 i

and conformally at the boundary in such a manner that a is carried to e q : Now we map I conformally onto a new unit disk; the mapping is still continuously-differentiable at the boundary. The periphery of the new disk is mapped onto L; and L is mapped continuously-differentiably onto the periphery of the previous unit disk. This mapping of the periphery onto itself is extended quasiconformally to the unit disk so that it is conformal at the boundary. Now, altogether, we have mapped quasiconformally I and A onto the interior and exterior of the unit disk in a w 0 -plane, so that it is conformal at the boundary. The points

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on L are mapped by both mappings to the same peripheral points, a drops to 2 i

e q : Because of the continuity and boundary conformality at L; we are dealing with a quasiconformal mapping of the w -plane onto the w 0 -plane. The surface W was built from half-sheets I and A , following a rule determined by the line complex. Following the same rule, we can construct over the w 0 -plane a surface W0 from the interior I0 and exterior A0 of the unit disk. By mapping now each I onto the corresponding I0 and each A onto the corresponding A0 by the earlier constructed mapping w ! w 0 ; we obtain a one-to-one continuous and (apart from isolated points) continuously differentiable mapping of W onto W0 : This mapping is quasiconformal, as the dilatation is nowhere bigger than the dilatation of the mapping from the w -plane to the w 0 -plane. Every other surface W with a topologically equivalent line complex can be mapped quasiconformally onto W0 ; as well. We see that one can always map two surfaces with equal line complexes quasiconformally onto each other.

5. Invariance of the type It is generally known that one can map conformally and one-to-one a surface W onto the entire z -sphere if and only if it is made of finitely many half-sheets and also when the line complex is finite. The difficulty lies in the distinction between the limit-point and the limit-circle cases for an infinite line complex. We have just mapped surfaces with the same line complex quasiconformally onto each other. Our assertion will be proved when it is shown that the type of a simply connected open Riemann surfaces is invariant under a quasiconformal mapping, that is, if two surfaces are mapped quasiconformally onto each another, one cannot be mapped onto the unit disk and the other onto the punctured plane one-to-one and conformally. Otherwise, one can map one-to-one, first the punctured plane conformally onto one of the surfaces, this quasiconformally onto the other surface, and the last conformally onto the unit disk, which gives a quasiconformal one-to-one mapping of the punctured z -plane onto the unit disk jwj < 1: We can obviously assume that z = 0 is mapped onto w = 0: From here we will derive a contradiction. Let z = re i' ;

 = log z = log r + i';

log w = !: The ! -image of the segment log r = const , 0  '  2 has length at least 2 , because its endpoints are at a distance 2 : ˇ Z2ˇ ˇ d! ˇ ˇ d': 2  ˇˇ d ˇ 0

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Oswald Teichmüller

(Here for d! we use one of its values that corresponds to arg d  = d Schwarz inequality, it follows that

 :) 2

By the

ˇ Z2 Z2ˇ ˇ d! ˇ2 ˇ d'; 4  d'
ˇˇ d ˇ 2

0

0

ˇ Z2ˇ ˇ d! ˇ2 ˇ ˇ d': 2  ˇ d ˇ 0

The dilatation quotient Dz/w should be bounded, and because the mappings z $  and w $ ! are conformal, also D /! = Dz/w  K:

According to an inequality derived earlier, ˇ ˇ ˇ d! ˇ2 d! d! ˇ ˇ K ; ˇ d  ˇ  D /! d d if one takes this and integrates with respect to Zr1

dr r

from r0 to r1 ; one obtains

Zr1 Z2 d! 2 K d'd log r; d r0 r0 r0 “ 2(log r1 log r0 )  K d! : dr  r

On the right appears the area of the ! -image of the rectangle log r0 < log r < log r1 ; 0 < ' < 2: That image lies, however, between the lines R ! = h and R ! = 0 ( h = Min R !) and contains for each line R ! = const a segment R = log r0

of total length at most 2; since otherwise it must contain two points at a difference 2n i: Therefore “ d!  2h; log r1  log r0 + Kh: This is a contradiction as r1 is arbitrarily large. Amendment during publication. H. Wittich alerted me to a publication by Grötzsch: Über die Verzerrung bei schlichten nicht konformen Abbildungen und über eine damit zussamenhängende Erweiterung des Pikardschen Satzes, Leipz. Ver. 80 (1928), whose final part, except for the notation, is closely related to the proof of the invariance of the type given here.

21 An application of quasiconformal mappings to the type problem

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6. Control of L on the characteristic Again, we have just learned of a result that the position of a1 ; : : : ; aq and L have hardly any influence on the schlicht mapping of M: An example of the opposite will be now briefly presented on. One can interpret the real axis and a sequence of circles jzj = n 12 as a topological image of a line complex with q = 4: Each accompanying surface M can be mapped onto the punctured plane, so w will be a meromorphic function of z: Its characteristic is T (r) = A(log r)2 + B(r) log r; B is bounded. a1 ; : : : ; a4 have each index 1/2; defects are missing. But the real constant A > 0 depends essentially on the cross-ratio of a1 ; a2 ; a3 ; a4 ; as well as on the choice of the separating curve L: The proof is not difficult but it should be carried out in a different context.

Chapter 22

Investigations on conformal and quasiconformal mappings Oswald Teichmüller (translated from the German by Melkana Brakalova-Trevithick and Matthias Weiss)

Introduction In the summer of 1936, while H. Wittich and I were reflecting on a class of Riemann surfaces introduced by E. Ullrich,1 I thought that a proof of the following lemma seemed desirable: Let the punctured z -plane be mapped one-to-one and quasiconformally onto the punctured w -plane. Let the dilatation quotient satisfy D  C (jzj) where C (r) converges to 1 , as r ! 1 , so fast that Z1 dr (C (r) 1) r converges. Then, approaching infinity, we have jwj  const
jzj:

At first, the problem seemed accessible through the analytic methods with which Ahlfors2 investigated schlicht mappings of surfaces treated by Nevanlinna. But the matter was not that simple. In the above mentioned work, Ahlfors shows that the schlicht conformal image of certain simply connected surfaces cannot be bounded; from here, with the help of a differential inequality, it is concluded that this schlicht image is the punctured plane.The latter is self-evident: otherwise it would be mapped conformally onto a bounded region. This simple observation led me to the idea that it would be probably more appropriate to employ Ahlfors’ estimates only after applying suitable auxiliary conformal mappings. Since in our problem above, one has to consider Original title: Untersuchungen über konforme und quasikonforme Abbildung. Deutsche Math. 3 (1938), 621–678. 1 E. Ullrich, Zum Umkehrproblem der Wertverteilungslehre (preliminary notification), Gött. Nachr. N. F. 1. 2 L. Ahlfors, Über eine in der neueren Wertverteilungslehre betrachtete Klasse transzendenter Funktionen. Acta math. 58.

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only neighborhoods of z = 1; w = 1; but exclude these points, one comes close to work with annuli or general ring domains and their conformal mapppings. The following question also leads to ring domains: One considers a region bounded by n analytic curves with h interior and k boundary marked points, which one maps conformally. In what cases does there exist exactly a one-parameter family of conformally non-equivalent configurations? Answer: I. II. III. IV. V.

One One One Two Two

boundary boundary boundary boundary boundary

curve, two interior points. curve, one interior point, two boundary points. curve, four boundary points. curves. curves, one boundary point.

The typical conformal invariants in cases I and II are Green’s function and the harmonic measure, respectively, while V: brings nothing new to IV . Thus we will look for the conformal invariant and its properties in cases III and IV . At the same time case IV , the ring domain, is more prominent in this work because it is often possible to go from III to IV but not vice versa. Initially I investigated distortion theorems which started as an analogue to a proof of the distortion theorem3 presented by E. Schmidt at a mathematical seminar in Berlin. It turned out that the above mentioned theorem can be proved in a much simpler and a completely new way. The decisive lemma contains a sufficient condition for a set lying between two domains on the sphere to be close to circular. In these investigations arose the opportunity to prove some of Ahlfors’ distortion theorems, in even sharper form, without using differential inequalities. Further, one obtains almost automatically a new sufficient condition for a Riemann surface to be of a hyperbolic type. As is known, so far, there are sufficient conditions for hyperbolic type only in qualitative forms, such as the disk theorems4 and their special cases. On the other hand, sufficient quantitative criteria, such as, for example, the one that measures ramification, exist only for a parabolic type.5 We start out with ring domains and quadrilaterals and their conformal invariants and follow with the investigations of Grötzsch6 on extremal problems of the conformal mappping, but always keep an eye on the goal to obtain limit results on the asymptotic behavior of conformal and quasiconformal mappings through a limit process; this will become more meaningful by the end of the work. One can actually talk about a combination of Grötzsch’s and Ahlfors’ methods. 3 R. Nevanlinna, Eindeutige Analytische Funktionen, Berlin 1936, Ch. IV, §3. 4 L. Ahlfors, Zur Theorie der Überlagerungsflächen, Acta math. 65. 5 According to a review in Zbl. Math. Grenzgebiete, Ch. Kakutani has written in Japan J. Math 13, a paper : Applications of the theory of pseudo-regular functions to the type-problem of Riemann surfaces, in which he proves, with the help of quasiconformal mapppings, a necessary criteria for hyperbolic type. This work is not yet accessible to me. (Cf. remark 30a.) 6 Leipz. Ber. 80–84.

22 Investigations on conformal and quasiconformal mappings

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Due to the size of this work, we also indicate short paths to some key results. In order to get acquainted with the new proof of Ahlfors’ distortion theorem,7 one needs to study only §1, §2.2, 6, 7 and §3.4–7. The amendment of one part of the proof of the work on surfaces with finitely many logarithmic ends2 follows from §1, §2.5, §4.1–3, and §5.1–3. In order to prove the quasiconformal mapping result, mentioned at the beginning, one has to accept only §6.1, 3, 6. The hyperbolic criterion, which will be highlighted in §7 relies only on §1.1–4 and §6.1, 3, 5. It has not emphasized yet that §1, §2.1–5 and §3.1, 2 include some known facts in new notation and arrangements. I have provided this at the beginning partly for the uniformity and clarity of the entire work. I would like to thank Mr. A. Dinghas for the effort to produce, based on my design, the print-ready drawings.

Contents 1 2 3 4 5 6 7

The module of a ring domain Distortion theorems . . . . . . The quadrilateral . . . . . . . The Modulsatz . . . . . . . . Taking the limit . . . . . . . . Quasiconformal mapppings . . A type criterion . . . . . . . .

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465 474 484 494 508 514 521

1. The module of a ring domain 1. The notion of a ring domain. We first consider a schlicht doubly connected domain G in the z -plane. Its boundary, as is well known, consists of two closed connected sets R1 and R2 , separated by the domain G ; Ri is either a point or a continuum. The set of all points in the z -plane that do not belong to G is separated by G into two closed connected sets K1 and K2 ; Ri is then the boundary of Ki ; Ki is a point if and only if Ri is a point, otherwise it is a continuum. We say that G separates the points (or continua) P1 and P2 if P1 is in K1 and P2 is in K2 . The nature of the boundary of G requires a multiple case distinctions: I. K1 and K2 are points. II. K1 is a continuum and K2 is a point. III. K1 and K2 are continua. 7 L. Ahlfors, Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta. Soc. Sci. Fenn. N. S. 1.

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These three types are invariant under schlicht conformal mapppings, since, by the so-called Removable Singularities Theorem, under a schlicht conformal mapping, to an isolated boundary point always corresponds an isolated boundary point, not a continuum. In case I, one can normalize G so that the points K1 and K2 are mapped through a fractional linear transformation to 0 and 1 , resp. In case II, there exists a linear fractional transformation that brings K2 to 1: Now, if one adds the point at 1 to the domain G , it will become simply connected. Since it still has a boundary K1 that is a continuum, one can map G conformally8 to the exterior of the unit circle; here 1 could be mapped onto 1: Again, by removing the point at 1 one obtains a conformal mapping from G onto the domain 1 < jwj < 1: (Naturally, one can consider the domain 0 < jwj < 1 as an alternative choice for normalization.) For case III, we will be interested in the following. By a ring domain, we understand a doubly connected domain bounded by two continua. K1 and K2 are named complementary continua and R1 and R2 are named boundary components of G . 2. Mapping onto a circular ring: the module. Applying known methods from Uniformization theory9 we prove that Every ring domain can be mapped conformally onto a concentric circular ring r < jwj < R . Proof. We may assume that z = 0 belongs to one and z = 1 to the other complementary continuum of G . Using  = log z , we map conformally the relatively unramified simply connected covering surface of G onto a simply connected domain Γ of the  -plane. To each simple closed curve Cz in G separating 0 from 1 corresponds a 2 i -periodic curve C whose ends, for I ! ˙1; define two accessible boundary points10 at  = 1 ; these points are different from each other as a sufficiently large circle in the  -plane contains parts of the images of K1 and K2 and Γ is simply connected. Now we map Γ conformally onto j!j < 1: The translation  !  + 2 i which transforms Γ onto itself, corresponds to a fractional linear transformation S which takes the unit circle onto itself and has no fixed points in the interior of j!j < 1 . If !0 lies on the image of the curve C , then for n ! 1; resp. n ! 1; S n !0 ; converges, on the one hand, to the fixed points of S on the unit circle and, on the other hand, to 8 Under a conformal mapping we always understand a mapping that is one-to-one. 9 P. Koebe, Abhandlungen zur Theorie der konformen Abbildung I. Die Kreisabbildung des

allgemeinsten einfach und zweifach zusammenhängenden schlichten Bereichs und die Ränderzuordung bei Konformer Abbildung. J. reine und angew. Math. 145. 10 Cf. e.g. L. Bieberbach, Lehrbuch der Funktionentheorie II, Berlin 1927, 1931, Kap. I, §5.

22 Investigations on conformal and quasiconformal mappings

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the ! -image of the accessible points on the boundary, at  = 1; defined through the ends of C : The latter are, however, different from one another and therefore S has two different fixed points on j!j = 1; and is therefore hyperbolic. Now we map j!j < 1 conformally onto a parallel strip in some F -plane so that the two fixed points of S are mapped onto the ends of the strip (lying at F = 1 .) Applying, if needed, a similarity transformation, one can make sure that S corresponds exactly to the translation F = F + 2 i: Then, this is some strip t1 < RF < t2 which is uniquely determined, up to a translation, by S , so that t2 t1 is uniquely specified by the domain G . (Because even if one substitutes S by S 1 ; the positive number t2 t1 does not change.) A single revolution in the positive direction around z = 0 in G corresponds to a shift of F by 2 i; under the mapping w = e F ; the F -strip goes to a concentric circular ring with center 0; and the mapping z ! w is one-to-one. Thus G is mapped conformally onto a normalized domain 0 < r < jwj < R < 1: As far as the quotient Rr is concerned, it is uniquely determined by G; as one can see immediately from the proof. We assign to the logarithm of this ratio of radii, as well as to the quantity t2 t1 , from the previous proof, the notion of module of G : M = log R log r: If one maps conformally the ring domain G onto a concentric circular ring r < jwj < R; then one calls M = log R log r the module of G: It is a conformal invariant of G . We will get yet another proof of the module invariance. 3. Estimates of the module in terms of logarithmic area. We begin now by establishing a connection between the module of a ring domain and its geometric properties. Here it is appropriate to introduce the logarithmic ’metric in the’ z -plane, punctured at 0 and 1 : the logarithmic length of a curve is jd log zj = jdzj ; the jzj ’ ’ dz logarithmic area is d log z = . In the last formulas we denote by dz jzj2 and d log z the area element in the z -, resp., log z -plane: dz = dx dy = r dr d' (z = x + iy = re i' ) . This can also be interpreted as meaning that lengths and areas are first transferred to the log z -plane, before being measured, where, naturally, one is to take11 log z mod 2 i: For example, the logarithmic length of the circle jzj = r is 2; the logarithmic area of r < jzj < R is 2(log R log r) . 11 The logarithmic metric could be interpreted, after Koebe, as the usual metric of one infinitely long straight circular cylinder with radius 1 , onto which one maps the plane punctured at 0 and 1.

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Let ring domain G separates 0 from 1 . Let F be its logarithmic area and M be its module. Then 2M  F . Equality holds only for a circular ring r < jwj < R: Proof. According to §1.2, one can map the domain G conformally onto r < jwj < R , where log R log r = M: We set w = %e i : The circle jwj = % corresponds to a simple closed curve in G which separates the two boundary components. Therefore log z must change after one revolution around it by 2 i; and it follows that ˇ Z2ˇ ˇ d log z ˇ ˇ % d: 2  ˇˇ dw ˇ 0

By the Schwarz inequality, it follows that ˇ Z2 Z2 ˇ ˇ d log z ˇ2 2 ˇ ˇ % d ; 4  d ˇ dw ˇ 2

0

2  %

0

ˇ Z2 ˇ ˇ d log z ˇ2 ˇ ˇ ˇ dw ˇ % d: 0

Integrating with respect to % from r to R leads to ZR 2 r

d%  %

ˇ “ ZR Z2ˇ ˇ d log z ˇ2 ˇ ˇ % d% d = ˇ dw ˇ r

0

d log z ;

G

2M = 2(log R

log r)  F:

If there is an equality in the above estimate, there are equalities in all steps of the derivation. Therefore, when we first integrate over the z -image of the circle R log z jwj = % = const; we always have jd log zj = 2; i.e. arg dd log = 0 or ; thus w this image becomes a circleˇ jzj =ˇ const : Second, if there is only an equality in the ˇ ˇ Schwarz inequality, then ˇ d log z ˇ% = ˇ d log z ˇ is a function of % only. One must dw

also have

d log z d log w

d log w

= ˙1; z = const
w or z =

const w

.

The above was a simple example of a method often applied by Grötzsch and Ahlfors, which we will use further. At the same time, the proof shows that one cannot map conformally a ring domain onto one that has a smaller radii ratio. We have thus a new proof of the module invariance.

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4. Monotonicity of the module. From the important inequality proved above we immediately derive two corollaries, which will be used again and again. Other corollaries will be derived in §2. If a ring domain G0 with module M 0 is a subset of another ring domain G with module M such that it separates its two complementary continua, one from another, then M 0  M:

Equality holds only if G0 = G: The topological assumption in the statement is not superfluous, since in every disk of every ring domain G; one can fit in annuli of arbitrary large moduli. The latter, however, do not separate the boundary points of the ring domain G . Proof. We map G onto r < jwj < R : log R log r = M: Then G0 is transformed into a ring domain, which is part of this circular ring, whose both boundaries separate the boundary components jwj = r and jwj = R , and thus, indeed, separate w = 0 and w = 1: If F 0 is its logarithmic area, then by §1.3 2M 0  F 0  2(log R

log r) = 2M:

Equality can hold only if the w -image of G0 is a circular ring with center 0 and logarithmic area 2(log R log r); which is therefore identical to the circular ring r < jzj < R; therefore G = G0 : 5. A ring domain containing two disjoint ring domains If a ring domain with module M contains two disjoint ring domains with moduli M 0 ; M 00 so that each separates the complementary continua of G , then12 M 0 + M 00  M:

Proof. Again, we can immediately assume that G is the domain r < jzj < R: If F 0 and F 00 are the logarithmic areas of G0 and G00 , then by §1.3 holds 2(M 0 + M 00 )  F 0 + F 00  2(log R

log r) = 2M:

Under the assumption that G : r < jzj < R , equality can obviously hold only when G0 : r < jzj < % and G00 : % < jzj < R; and vice versa. 12 H. Grötzsch, Über einige Extremalprobleme der konformen Abbildung, Leipz. Ver. 80, note on p. 370. [Editor’s note: English translation by A’Campo-Neuen, On some extremal problems of the conformal mapping. In A. Papadopoulos (ed.), Handbook of Teichmüller theory. Vol. VII. IRMA Lectures in Mathematics and Theoretical Physics, 30. European Mathematical Society (EMS), Zürich, 2020, Chapter 13, 355–363.]

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6. The reduced module. Let G be a simply connected domain containing 0 but not 1 . For sufficiently small %; the circle jzj = % lies entirely in G ; the intersection of G with jzj > % is a ring domain G% ; unless G is the punctured plane; we exclude this case. For future investigations, we need to have certain knowledge of the behavior of the module M% of G% as % ! 0: We mapˇ G ˇ conformally onto jwj < 1; so that z = 0 corresponds to w = 0; dz ˇ and we set ˇ dw = R . R is the radius of the circle centered at 0 onto which G 0 is mapped conformally so that 0 corresponds to 0 and the derivative at the origin is 1 . One calls R the mapping radius of G . The image of jzj = % under the mapping of G onto jwj < 1 is a simple closed curve which, by the distortion theorem, lies between jwj = %1 and jwj = %2 ; where %1 and %2 are determined by the equalities %1 %2 R =R = %: (1 %1 )2 (1 + %2 )2 The second equality has a solution in the interval 0 < %2 < 1 only for % < R4 , and for %  R4 one should set %2 = 1 . For % < R4 one obtains: r R R 4% % %2 %1 = 1 + 1+ > 2 2; 2% 2% R R R r R R 4% % %2 %2 = 1 1 < +2 : 2% 2% R R R(R 4%) (Perhaps the simplest way to obtain the estimates is to expand in powers of R% .) We see that %1 and %2 are in first approximation equal to R% : G% is thus mapped conformally onto a ring domain contained in the circular ring %1 < jwj < 1; and itself containing the circular ring %2 < jwj < 1: By §1.4 holds 1 1 log < M% < log %2 %1 or 2% 1 % % < log < log < M% + log 2% R 4% R%2 R 1 + R 4% < log

% 1 2% 2% < log < < 2% R%1 R 2% R 4% 1 R

or jM% + log %

log Rj 

R

2% : 4%

We derive our first corollary: As % ! 0; M% + log % converges to log R: This result justifies the following definition: z The logarithm of the mapping radius of G is called the reduced module M of G .

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z = log R = lim (M% + log %) . Naturally, one cannot talk of invariance Thus M %!0

of the reduced module though it will show to be a valuable tool. Only later, e.g. toward the end of §4, we will see why it is desirable to define the logarithm of the radii quotient and the mapping radius, resp., as module and reduced module; here, it suffices to hint on the link with the logarithmic metric from §1.3. As % decreases, M% + log % increases. This is because for %0 < % the circle jzj = % cuts the domain G%0 with module M%0 in two parts: the circular ring %0 < jzj < % with module log % log %0 and the ring domain G% with module M% . By §1.5, (log % log %0 ) + M%  M%0 ; M% + log %  M%0 + log %0 :

From here, together with the estimates derived above, follows: z , and the part of G in jzj > % has module If G has reduced module M z M M% ; then for % < e4 holds z M

2% e Mz

4%

z < M% + log %  M:

Note that, by a theorem of Koebe, the circle jzj = % lies entirely in G only if z % < R4 ; which explains the appearance in the denominator of R 4% = e M 4%: Presumably there is a much better estimate. Using the transformation z ! z1 ; one obtains easily: Let G be a simply connected domain that contains 1 but not 0 (which is not the plane punctured at the origin). The reciprocal of the mapping radius R of G is the radius of the circle centered at 0; onto the exterior of which one maps conformally G; so that 1 goes to 1 and the derivative at infinity is equal to 1 . We call z = log R M the reduced module of G: If for P > 4R; MP is the module of the intersection of G with jzj < P (which is a ring domain), then, as P ! 1; MP log P converges z ; more precisely, in a monotone and increasing way to M z M

2

Pe Mz

4

< MP

z log P  M:

7. The reduced module and the reduced logarithmic area. Let G be again a simply connected domain that contains 0 and not 1; and which is different from the punctured plane at 1 . Since we have introduced the reduced module z = lim (M% + log %) , we should also define the reduced logarithmic area and M %!0

then apply §1.3 over.

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Let % be so small that the circle jzj = % lies entirely in G , let G% be the part of G in jzj > %; and let F% be the logarithmic area of G% (0 < F%  1) . Then F% + 2 log %

is independent of %; because for %0 < %; obviously, F%0 = 2(log %

log %0 ) + F% :

We call this quantity independent of % the reduced logarithmic area Fz of G : Fz = F% + 2 log %: z and the reduced There is an inequality between the reduced module M logarithmic area Fz z  F: z 2 M

Equality holds only if G is the disk jzj < R . We denote again the module of G% by M% and in that case, by §1.3, it holds that 2M%  F% ; 2(M% + log %)  F% + 2 log %:

The limit % ! 0 leads to the asserted inequality. This proof provides no clue of when to expect an equality sign. However, because of applications in §4, we need to consider it and we prove the result one more time, following directly the method used in §1.3. We map G conformally onto jwj < 1 so that 0 corresponds to 0 ; at the same ˇ dz ˇ ˇ = e Mz : Set w = %e i : The logarithmic length of the z -image of the time ˇ dw 0 circle jwj = % = const is at least 2 and is strictly greater than 2 when the z -image of jwj = % is not a circle centered at 0 : ˇ Z2ˇ ˇ d log z ˇ ˇ ˇ d: 2  ˇ d log w ˇ 0

log z If for some % -interval holds an equality, then dd log is real and hence a constant w which is 1 , and then z = const
w: —From the Schwarz inequality follows

ˇ Z2 Z2ˇ ˇ d log z ˇ2 ˇ ˇ d; 4  d
ˇ d log w ˇ 2

0

0

ˇ Z2ˇ ˇ d log z ˇ2 ˇ d: 2  ˇˇ d log w ˇ 0

22 Investigations on conformal and quasiconformal mappings

Equality holds in the same way as above. Now ˇ ³ “ Z1 ² Z2ˇ ˇ d log z ˇ2 ˇ ˇ d 2 d% = ˇ d log w ˇ % %1

0

473

d log z + 2 log %1 :

jwj>%1

In the right-hand side appears the area of the subset of G that is separated from 0 by the z -image of jwj = %1 : The last curve, however, differs in the logarithmic metric, for sufficiently small %1 ; by an arbitrary small amount from the circle ˇ ˇ z ˇ = e Mz ). The subset jzj > %1 e Mz of G has logarithmic jzj = %1 e M (since ˇ dw dz 0 area z z 2 log %1 : F%1 eMz = Fz 2 log(%1 e M ) = Fz 2 M ’ As a result jwj>%1 d log z changes little as %1 ! 0; i.e. our right-hand side z as %1 ! 0 : converges to Fz 2 M; ˇ Z1 ² Z2ˇ ˇ d log z ˇ2 ˇ ˇ ˇ d log w ˇ d 0

³ 2

d% = Fz %

z 2 M:

0

The integrand on the left-hand side is  0 and it is 0 only when z = const
w , thus G is a disk jzj < R: Bieberbach’s method is much simpler here. One sets z = we P(w)

and calculates ˇ ˇ ˇ ˇ z = log ˇ dz ˇ M ˇ dw ˇ 0

and Fz = lim

I

log jzjd arg z

%!1 jwj=%

through the coefficients of the power series P(w): Then one can see right away that z  Fz and that equality holds only when P(w) = const . This is, of course, 2 M long known, but since in this entire work we are applying only geometric methods, I prefer the Grötzsch–Ahlfors method.13 8. Conclusions. Here we put together some simple conclusions that correspond to §1.4 and 1.5. By analogy with §1.4 we obtain the result that the reduced module (also the mapping radius) of a subdomain G0 of a domain G; which contains 0 , is smaller than that of G . This theorem is so well known that it is worthwhile to pay close attention to it. 13 By yet another but related method, our result was proved by H. Grunsky, Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusamenhängender Bereiche. Schriften des Mathematischen Instituts und des Institutes für angewandte Mathematik der Universität Berlin, vol. 1, issue 3 (1932).

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If a simply connected domain G , different from the punctured plane and containing 0 but not 1 , with reduced module M; contains a simply connected domain G0 containing 0 with reduced module M 0 as well as a disjoint domain G00 with module M 00 which separates G0 from the boundary of G , then M 0 + M 00  M:

In the proof, w.l.o.g. we assume that G is the disk log jzj < M: It has reduced logarithmic area 2M: If F 0 is the reduced logarithmic area of G0 ; and F 00 is the reduced logarithmic area of G00 ; it obviously holds 2(M 0 + M 00 )  F 0 + F 00  2M:

Equal sign holds (under the assumption that G is the disk jzj < e M ) only if G0 is the disk log jzj < M 0 and G00 is the circular ring M 0 < log jzj < M: The transition to the case when the domain contains 1 but not 0 is self-evident. Especially important is the following theorem to which we will return again in §4. Let G0 and G00 be two disjoint simply connected domains; G0 containing z = 0 and G00 containing z = 1 , and let M 0 and M 00 be the reduced moduli of G0 and G00 . Then M 0 + M 00  0: Equality holds only when G0 W jzj < R and G00 W jzj > R . Proof. Let jzj = % lie entirely in G0 and jzj = P in G00 . Let F%0 be the logarithmic area of the subset jzj > % of G0 and FP00 of the subset jzj < P of G00 . Then F%0 + 2 log % and FP00 2 log P are the reduced logarithmic areas of G0 and G00 . From §1.7 follows ∗ 2M 0  F%0 + 2 log % 2M 00  FP00 2 log P 0 00 2(M + M )  F%0 + FP00 2(log P log %)  0 because F%0 and FP00 are the logarithmic areas of disjoint subsets of the circular ring % < jzj < P: If we have an equality sign, first, there should be an equality sign in 2M 0  F%0 + 2 log %; which means that G0 must be some disk jzj < R0 ; second G00 must be some disk jzj > R00 ; and third there should be an equality sign in the inequality F%0 + FP00  2(log P log %); which means that R0 = R00 .

2. Distortion theorems 1. Grötzsch’s extremal region. First of all we will prove some properties of ring domains whose one boundary component is a circle. A fractional linear ∗ Translator’s note. There is a typo in the original paper; in the last inequality, the term 2 is missing.

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transformation maps this circle to the unit circle, and a point in the complementary continuum of the other boundary component to 1 . After this preliminary normalization, the complements of G will consist of jzj  1 as well as a disjoint simply connected set containing 1 . Now we follow Grötzsch.14 Instead of Koebe’s extremal region, we consider GP , the exterior of the unit disk cut along the real axis from z = P > 1 to z = 1 . Let the module of this ring domain be log Φ(P) (1 < Φ(P) < 1) . For P < P0 , GP is a ring domain properly contained in GP0 , therefore by §1.4, Φ(P) < Φ(P0 ) : Φ is a strictly monotone increasing function. GP contains the circular ring 1 < jzj < P; so §1.4 yields the rather crude estimate Φ(P) > P. To show that Φ(P) is a continuous function, it suffices to show that an arbitrary circular ring 1 < jwj < R can be mapped conformally to a suitable GP ; then the monotone function Φ(P) takes on all values between 1 and 1 and hence must be continuous. Indeed, it is possible to map the circular semiring 1 < jwj < R; =w > 0 conformally onto jzj > 1; =z > 0 , sending the boundary points w = R; 1; +1 to z = 1; 1; +1 , resp. Should w = +R be mapped onto z = P, a reflection in w = R


1; 1


R and z = 1


1; 1


P on the real axis will yield the desired map of 1 < jwj < R onto GP (cf. figure 1). Explicitly calculating this requires the introduction of elliptic functions and one can observe a connection between the function Φ(P) and the values of the elliptic modular function for purely imaginary periodic ratios. In the interest of preserving the purity of the method used and to avoid tedious calculations, we will not make use of this connection at all; rather, we derive everything we wish to know about the function Φ(P) from its geometric definition. For P ! 1 we have Φ(P)  4P: This formula can be obtained from a wellknown series expansion of the modular function, but we will prove it as follows: log Φ(P) is the module of the subset jzj > P1 of the z -plane cut along the real axis from z = 1 to 1 . According to §1.6, as P ! 1 , log Φ(P) + log P1 converges to the reduced module of the z -plane cut along the real axis from z = 1 to 1 . This reduced module, however, equals log 4 , because this region is generally ˇ dz ˇknown as the image 4w ˇ ˇ = 4 . Therefore, of the unit disk jwj < 1 under the function z = (1+w) 2 : dw 0 lim log Φ(P)

P !1

lim

P !1

log P = log 4;

Φ(P) = 4: P

The proof in §1.6 even shows that  2  log 4 log 1 + < log Φ(P) 4P 4 14 Compare with the work cited in footnote 12.

log(P) < log 4

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Oswald Teichmüller

Figure 1

or 4P 1+

We only take note that

Φ(P)

P

1 2(P 1)

< Φ(P) < 4P:

converges to 4 in an increasing manner.

2. The closest boundary point problem. In analogy with a well-known theorem of Koebe, the following theorem of Grötzsch holds here. Let G be a ring domain which separates its boundary component jzj = 1 from z = 1 . Suppose that there exists a point which does not belong to G , at a distance P > 1 from z = 0: Then the module M of G satisfies the estimate M  log Φ(P): Equality holds only if G can be obtained from GP by a rotation about z = 0 . Proof. We may assume that z = P itself does not belong to G . We then map the extremal region GP conformally onto 1 < jwj < R = Φ(P) (see figure 1). Through a reflection in the slit z = P


1 and a corresponding one in the circle jwj = R; one obtains a conformal mapping from GP onto the circular ring R < jwj < R2 , which is an analytic continuation of the original map. By gluing the two copies of GP criss-cross along the slit P


1 , we obtain the Riemann surface FP with quadratic branch points at z = P and z = 1 . FP is now mapped conformally onto 1 < jwj < R2 , with the twice run slit z = P


1 corresponding to the circle jwj = R . The map w becomes a two-to-one function of z for jzj > 1 ; however, it does not change its value after a complete revolution around the circle jzj = 1 + " , where " < P 1 . According to the monodromy theorem, each branch of w is single valued in our region G : the two branches of w(z) map G onto two disjoint ring domains in 1 < jwj < R2 , with boundary components jzj = 1 and jzj = R2 ,

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resp. Both have module M , and §1.5 tells us that the sum 2M of their moduli is at most equal to the module 2 log Φ(P) of the circular ring 1 < jwj < R2 , and equality holds only if the w -images of G are annuli with module log Φ(P) ; that is, G = GP , as we wanted to show. In a similar way (even easier, in many of the details,) one can prove Koebe’s theorem with the exact constant 14 : start with a simply connected domain G containing neither P nor 1 but 0; compare with Koebe’s extremal region, and at the end of the proof, instead of using the result in §1.5, use the final result in §1.8 on reduced moduli. The latter follows also from §1.7; one proves the theorem in §1.7 without any reference to moduli, and one gets an easy approach to the distortion theorem following the method outlined at the end. 3. A different proof. The proof that M  Φ(P); carried out above, uses in an essential way a type of geometric monodromy: A closed path in G can never go from one sheet of FP to another; instead, it is closed on FP ; as well. Therefore one can fit two disjoint copies of G on the surface so that they are mapped under z ! w into the above mentioned ring domains with module M contained in 1 < jwj < R2 . The conclusion is without doubt sharp, but the application of this method, in more difficult problems, can cause a break up; as for example in §4 or the problem of the maximal reduced module for simply connected regions containing 0 but not containing a given set of points, which was investigated by Grötzsch.15 We therefore give a second proof which uses the same basic idea (of estimating M using the logarithmic area in the w -plane), which is carried out without the Riemann surface FP and operates only with mappings of schlicht regions. Let G be again a ring domain with module M , separating its boundary component jzj = 1 from z = 1 and not containing z = P. Just as before, we map GP conformally into 1 < jwj < R = Φ(P) , so that z = P is carried to w = R . GP is symmetric about the real axis, the w -image of 1 appears to be mapped one-to-one onto 1 < jwj  R , except for the segment P < z < 1 , each point of which always corresponds to two points on jwj = R , symmetric with respect to the real axis. 15 H. Grötzsch, Über ein Variationsproblem der konformen Abbildung. Leipz. Ver. 82. 16 This can be proved by considering the conformal mapppings of a circular ring onto itself, as in §1.3; see op. cit. in footnote 9.

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Figure 2

Under this map, the image of G in 1 < jwj  R is not necessarily connected unless one identifies the complex conjugate points Re i' and Re i' ; 0 < ' <  (see figure 2). This point set is guaranteed to have logarithmic area  2 log Φ(P) . In a similar way, as in §1.3, we will conclude from here that M  log Φ(P) . We map G conformally onto a circular ring 1 < j!j < e M and set ! = %e i . The circle j!j = % = const : corresponds to a certain curve C% in the w -plane, which is not necessarily closed unless we identify Re i' and Re i' . Following §1.3, we will first show that the logarithmic length of C% is at least 2 . Let n(') be the number of intersections of the segment 1 < jwj < R , arg w = ' with C% . The two segments 1 < jwj < R;

arg w = ˙'

(0 < ' < );

correspond to a continuous curve in the z -plane, which begins and ends at jzj = 1 , but crosses P < z < 1 and, together with the unit circle, separates the points

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z = P and z = 1 . Such a curve cannot lie in the ring domain G , because P and 1 belong to the same complementary continuum of G . Hence, it must leave somewhere j!j < % in G and enter it again elsewhere, and thus have at least two intersections with the z -image of jwj = % . It follows that n(') + n( ')  2

(0 < ' < ):

Integrating from 0 to  yields Z+ Z n(')d' = ¹n(') + n( ')ºd'  2: 

0

However, I

jd log wj 

j!j=%

I

jd arg wj =

Z n(')d'  2: 

j!j=%

Equality holds only when C% is some circle jwj = const : After we have shown, in a topological way, that C% has length at least 2 , the formalism already applied in §1.3 comes into effect: ˇ Z2ˇ I ˇ d log w ˇ ˇ ˇ d = jd log wj  2; ˇ d log ! ˇ 0

j!j=%

ˇ Z2 Z2ˇ ˇ d log w ˇ2 ˇ ˇ d; 4  d
ˇ d log ! ˇ 2

0

0

ˇ Z2ˇ ˇ d log w ˇ2 ˇ ˇ d ; 2  ˇ d log ! ˇ 0 eM

Z 2M = 2 1

d%  %

eM

Z 1

ˇ “ Z2ˇ ˇ d log w ˇ2 ˇ ˇ d d% = ˇ d log ! ˇ %

d log w  2 log Φ(P);

0

M  log Φ(P): Equality holds only in the case w = const
! ; and then G = GP .

4. Refinement. Let G be again a ring domain with module M separating its boundary component jzj = 1 from z = 1 . If the boundary point, not on jzj = 1 and closest to z = 0 , has absolute value P1 , then M  log Φ(P1 ) ; if, however, the farthest boundary point has distance P2 from 0 , then G lies in the circular ring 1 < jzj < P2 , and according to §1.4 we have M  log P2 . But what can be said if one has estimates for the closest and farthest boundary points? One can guess right away the extremal region: it is the circular ring 1 < jzj < P2 cut along the segment from z = P1 to z = P2 . We need to calculate its module

480

Oswald Teichmüller

X (P1 ; P2 ) as a function of P1 and P2 to obtain an upper bound for the moduli of all ring domains that separate their boundary component jzj = 1 from jzj  P2 and do not contain P1 . Just as before, this can be proved exactly. The calculation of X (P1 ; P2 ) appears likely to be easiest, if one maps 1 < jzj < P2 onto a GP0 so that that z = P2 is carried to the end of the cut P0 : our extremal region will then be carried into GP00 with module that one can consider as known. This auxiliary mapping, by the way, reduces our problem back to the one solved earlier. According to §1.4, log P1  X (P1 ; P2 )  log P2 , and if we keep P2 fixed, X will be a strictly monotone increasing function of P1 . From this last property follows: For every " > 0 and for every fixed P2 , there exists ı > 0 with the property that every ring domain separating its jzj = 1 boundary component from jzj  P2 and with module at least log P2 ı contains the circular ring 0 < log jzj < log P2 " entirely. Simply, one needs to set X(P2 e " ; P2 ) = log P2 ı .

5. A normal family. We consider schlicht conformal mapppings f from a fixed, for the time being, circular ring 1 < jwj < R; onto a ring domain in the z -plane which has jzj = 1 as a boundary component and separates it from z = 1 : z = f (w) . Here the boundary components jwj = 1 and jzj = 1 correspond to each other. By the reflection principle, f has an analytic continuation to a schlicht mapping from R1 < jwj < R onto an easily recongizable ring domain in the z -plane, so that f is also regular on jwj = 1 . We could now explore, in analogy with the well-known theory of schlicht mappings on the unit disc, some properties of these mappings, and in the process the mapping from §2.1 will often take on the role of an extremal mapping. However, since we do not want to get lost in details, we would rather refer to the papers by Grötzsch.6 We only wish to only briefly mention here that our functions f (w) are a normal family on 1 < jwj < R . This is clear because jf j > 1 . This normal family is closed in a sense that the limit function f (w) of any uniformly convergent sequence fn of functions of this family, on the whole interior of 1 < jwj < R , (i.e. on every closed subset of this domain) also belongs to this family. First of all, every fn (w) assumes somewhere on every circle jwj = r (1 < r < R) , an absolute value of the amount r ; otherwise the assigned image of 1 < jwj < r under z = fn (w) would be either properly contained in 1 < jzj < r , or would properly contain the circular ring 1 < jzj < r as a subset; in both cases, according to §1.4, the module would not be log r . Therefore the limit function f (w) must assume, somewhere on every circle jwj = r , a value of the amount jf (w)j = r . Therefore f is not a constant and is schlicht as a limit of schlicht functions. In addition, like all fn , it omits the value of 1 . This can be deduced, in the usual way, from Rouché’s theorem. What remains is to study f in a neighborhood of the unit circle.

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p The sequence fn (w) converges uniformly on jwj = R and is therefore bounded on this circle. By reflecting in the unit circle, one obtains a sequence of p 1 1 regular functions fn on R < jwj < R which is bounded on R < jwj < R and p converges to f on 1 < jwj < R .pBy Vitali’s theorem, fn converges uniformly to f in the interior of R1 < jwj < R ; in particular, the schlicht regular function f (w) possesses analytic boundary values on the unit circle of absolute value 1 and thus belongs again to our normal family. Let z = fn (w) map 1 < jwj < Rn conformally onto a ring domain separating its boundary component jzj = 1 from 1; so that w = 1 is carried to z = 1: Let Rn ! 1: Then fn (w) ! w uniformly on every circular ring 1  jwj  R:

Proof. By the diagonalization method, there is a subsequence of fn which converges uniformly in every circular ring 1 < jwj < R . The limit of each such subsequence belongs in each 1 < jwj < R to the above discussed normal family and thus maps 1 < jwj < 1 conformally onto a doubly connected region separating its jzj = 1 boundary from z = 1 so that w = 1 goes to z = 1 . This limit function must be w . Because jfn j > 1 , and w is the limit of every subsequence of fn which converges uniformly on every 1 < jwj < R , fn itself must converge uniformly to w on 1 < jwj < 1 . This theorem should also follow from Grötzsch’s distortion theorems.12 We will use it only once in §5.2. 6. The problem of the closest and farthest boundary points. We now leave the investigation of ring domains with one circular boundary component behind and turn towards a problem, which we will later connect to Ahlfors’ proof of Denjoy’s conjecture: a transfer of results from §2.2 to an arbitrary ring domain. We consider a ring domain G which separates the pair of points z = 0 and z = 1 ; the farthest boundary point in the part of the complementary continuum of G containing 0 has distance % from the origin, and the closest boundary point in the 1 -containing complementary continuum of G is at a distance P from the origin. For a given P% the module M of G will be estimated from above. With this goal in mind, we temporarily consider the union of G with the complementary continuum containing 1 : this results in a simply connected region, namely the complement set of the complementary continuum of G containing z = 0 . We map this simply connected region onto jsj > 1 , so that 1 is carried to 1 . The image of G will then separate the boundary jsj = 1 from s = 1 ; this ring domain with a circular boundary component can, in turn, be mapped conformally onto the circular ring 0 < log jwj < M . In this way the transformation of G onto a circular ring is a composition of two maps about which something is already known.

482

Oswald Teichmüller

ˇ ds ˇ ˇ = p . By Koebe’s theorem, one can estimate p from above in terms Let ˇ dz 1 of a known17 % , the bound will only be attained if the complementary continuum of G containing 0 is the segment from 0 to %e i' . By the distortion theorem, one can further find an upper bound18 for the s -image of the closest boundary point z = Pe i of the complementary continuum of G containing 1 ; this bound is worst for p largest, as in the extremal case just discussed, and it is attained only if we have, in addition, e i = e i' . Now by §2.2 one can find an upper bound for the module M of G by a transfer to the s -plane; this bound is worst when the distance between the s -image of Pe i and s = 0 is as large as possible; that is, in the same extremal case as mentioned before, and, moreover, it is attained only if the s -image of G is the disk jsj > 1 , cut radially from the s -image of Pe i to s = 1 . All in all, we see that the module M of G , for given % and P, will be the largest when and only when G can be rotated into the z -plane cut along the real axis from % to 0 and from P to 1 .

If the two complementary continua of a ring domain contain 0 and %e i' , on the one hand, and 1 and Pe i , on the  other hand, then the module is P P at most equal to log Ψ % : Here, log Ψ % is the module of the z -plane cut along the real axis from % to 0 and from P to 1: Equality holds only in the case e i' = e i , when the region is the plane cut along the segments %e i'


0 and Pe i


1 . Finding the extremum for fixed %; P; ' , and  would be a nice exercise. It seems as though this will require elliptic functions whose period ratio is no longer purely imaginary. However, we will not be addressing this as well as many other questions from conformal geometry and thus are completing this section with a few remarks on the inequality just proved.
7. Calculating Ψ P% . We can easily express Ψ in terms of the function Φ introduced in §2.1. We only need to follow the reasoning of the proof we just completed: the z -plane cut along the segment from % to 0 can be mapped conformally onto jsj > 1 through r   %(s 1)2 2z % z= ; s =1+ 1+ 1+ ; 4s % z

so that 1 is carried to 1 ; z = P will now get mapped to r   2P % s =1+ 1+ 1+ % P 17 p 

4 %

.

18 The bound is 1 +

Pp 2

  q 1 + 1 + P4p .

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483

and G to the exterior of the unit circle cut along the real axis from there to 1 . Hence r    P 2P % Ψ =Φ 1+ 1+ 1+ : % % P
There is another way to calculate Ψ P% : one shifts the extremal region via the translation z + % to the plane cut along the real axis from 0 to % and from % + P to 1p . It will then be symmetric with respect to the circle with center 0 and radius %(% + P) , and hence can be mapped to a circular ring with module q 
%+ P %+ P 2 log Φ p%(%+ = 2 log Φ so that % P) s  P %+P 2 Ψ =Φ : % % Since both expressions for Ψ must be equal to each other, this yields an already known formula for calculating the elliptic modular function under a doubling of the period ratio. By §2.1 Φ(P)  4P ; both our representations for Ψ yield P P P Ψ  16 as ! 1: % % % One would also like to obtain an upper estimate for Ψ : Φ(PP) tends to 4 in an increasing manner, thus Φ(P) < 4P; from the first formula for Ψ we obtain a good estimate r   P 8P % P Ψ 0 , with vertices going to e  ; 1; +1; +e  . Reflection in the real axis yields a conformal transformation of the circular ring 1 < jwj < e  onto the plane with cuts on the real axis from 1 to 0 and from +1 to 1 , which then has also module  . † Translator’s note. There is a typo in the original version where in the last inequality, it is 1 and not 12 .

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In §2.6, % indicated the maximum distance of the points in the complementary continuum of G containing 0; and P the minimum distance to 0 from the points in its componentary continuum containing 1; where G separates 0 and 1 . Obviously, the assumption that P%  1 means precisely that there is no circle jzj = const : which runs entirely inside G . Since Ψ is monotone, from this assumption for the module M of G follows the inequality P M  log Ψ  Ψ(1) = : % Conversely, one obtains: The interior of a schlicht conformal image of a circular ring separating 0 and 1 with radii quotient > e  always contains a circle with center 0: The transcendental constant e  cannot be replaced here by any smaller one.

3. The quadrilateral 1. The module of a quadrilateral. In conformal geometry, regions that can be mapped conformally onto each other are not considered essentially different. But we define: A quadrilateral is a simply connected region with four chosen distinct accessible boundary points, called vertices. The vertices partition the boundary into four subsets, called sides. Instead of an “accessible boundary point” one should really say a “prime end” in order to obtain complete conformal invariance. However, we will not use this meaning. A quadrilateral can be mapped conformally onto a circle, and the images of the vertices will be four points on the circumference, whose cross-ratio is the only conformal invariant of the quadrilateral and can hence be called its module. Paatero19 works with this module. If we connect the four vertices of the circle diagonally with two arcs orthogonal to it, they will intersect at an angle 2˛ . A linear fractional transformation maps the circle to the unit circle, the point of intersection to 0 , and the vertices to ei ˛ ;

e i(

˛)

;

e i ( +˛) ;

e

i˛

:

2˛ 

We can now take as the module of the quadrilateral. This definition is clearly consistent with the following one of R. Nevanlinna’s: One determines, on each fixed continuous path l; connecting two opposite sides of a quadrilateral V , a and c; the minimum of the harmonic measure of the boundary set a + c; measured in the quadrilateral V ; take as module the supremum of these minima over all curves l . This construction is conformally invariant and yields, under the above unit circle normalization, the value 2˛ .  19 V. Paatero. Zur Theorie der beschränkten Funktionen. Ann. Acad. Sci. Fenn. 46.

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The functional connection with the originally mentioned module is elementary. In this fashion, Paatero’s theorem is subordinate to Nevanlinna’s principle of harmonic measure growth. There is a third normalization, which is most useful to us: map the circle to a rectangle using an elliptic integral of the first kind, so that the points on the circumference we already called vertices are mapped to the actual vertices of the rectangle. The ratio of the sides of this rectangle can be taken as the module of the quadrilateral. Because every rectangle can be mapped conformally onto a circle, there is a one-to-one correspondence between this module and the two previous ones. This correspondence is continuous and monotone, each of these properties is implied by the other. For the proof, we could appeal to the continuity of the elliptic modular function. One could, on the other hand, take two rectangles of different side length ratio and map one, scaled if necessary, onto the other. Then map both rectangles onto circles and apply Paatero’s or Nevanlinna’s theorem to the resulting map of one circle onto part of another. Finally, using obvious transformations and reflections as in §2.8, reduce a circle-rectangle transformation to a twice-cut plane-circular ring transformation, in which case we already know about the continuity of the module. 2. Estimating the module using geometric mesurements. Now we arrive at a simple inequality that corresponds to §1.3, is actually the basis of Grötzsch– Ahlfors’ method, but explicitly occurs first in Rengel.20 Let V , a bounded quadrilateral with sides a; b; c; d; be mapped conformally to the rectangle 0 < u < a , 0 < v < b in the w = u + iv -plane, where a corresponds to the side v = 0; b to u = a; c to v = b; and d to u = 0: Let ˇ be the infimum of the length of all curves in V connecting a and c; and let F be the area of V . Then a F  2: b ˇ Equality holds only when V is a rectangle with sides a; b; c; d . Proof. For example, the length of the image of the segment u = const : in the z -plane containing V is at least ˇ : ˇ Zb ˇ ˇ dz ˇ ˇ ˇ dv: ˇ ˇ dw ˇ 0

20 E. Rengel. Über einige schlitztheoreme der Konformen Abbildung. Schriften des Mathematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin, Vol. 1, issue 4 (1933).

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From the Schwarz inequality, it follows that ˇ Zb ˇ ˇ dz ˇ2 ˇ ˇ dv: ˇ b ˇ dw ˇ 2

0

Integrating with respect to u from 0 to a gives ˇ “ Za Zb ˇ ˇ dz ˇ2 ˇ ˇ dvdu = b ˇ ab ˇ dw ˇ 2

0

0

dz = bF:

V

Equality can ˇonlyˇ occur if the curves u = const : in V ˇare ˇ line segments, and dz ˇ dz ˇ if moreover ˇ dw is a function of u only. But then ˇ dw = const : and the transformation is linear. 3. Remarks I. If ˛ denotes the lower bound of the lengths of all curves joining b and d in V , then we have next a F  2 b ˇ

and b F  2: a ˛

Multiplying yields 1

F2 ; ˛2ˇ2

˛ˇ  F:

Equality holds only when V is a rectangle with sides a; b; c; d . This is a rather curious extremal property of the rectangle! II. The same inequality remains still valid if the quadrilateral V is a subset of a differentiable surface with analytic metric. In that case, one can introduce isothermal parameters in the neighborhood of any location, and conformally map it onto a piece of the plane; thus the surface is a Riemannian manifold and, by the general uniformization theorem, there exists a conformal mapping from V to a planar rectangle, and then one can draw the same conclusions as above. Equality is attained only if the curves u = const : are geodesics and when, moreover, on each of these curves the scale factor from the w -plane to the surface is a constant; that is, a function of u only. Moreover, u and v must be isothermal parameters. Thus, by a theorem from differential geometry, the map is (up to a multiplicative constant) an isometry.

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Let V be a quadrilateral with sides a; b; c; d on a surface with analytic metric. Let F be the area of V; ˛ a lower bound of the lengths of all curves joining b and d; and ˇ a lower bound of the lengths of all curves joining a and c in V: Then ˛
ˇ  F: Equality holds only when V can be mapped isometrically to a planar rectangle, preserving the vertices. It can be observed that the inequality Min(˛; ˇ)2  F holds, where equality indicates isometric images of squares. Here Min(˛; ˇ) is the lower bound of the lengths of all curves joining opposite sides of a quadrilateral. If the Gaussian curvature does not vanish in at least one point of V , we must have ˛
ˇ < F . It seems plausible that the upper bound of the inequality could be improved by taking into account an integral involving the Gaussian curvature which would have to vanish in the case ˛
ˇ = F . III. Let us recall a connection between the question we study and a problem of Beurling.21 For simply connected domains, Beurling considers the lower bound of the lengths of all curves that join two fixed interior points or an interior point and a boundary arc, divides by the square root of the area and asks for the maximum of this quantity after subjecting this configuration (domain with two interior points or one interior and a boundary arc, resp.) to a conformal transformation. If one replaces the pair of points, the interior point and the boundary arc, resp., with a pair of boundary arcs, one obtains the problem solved in §3.2, because, there, we could also write ˇ b p p : F ab The extremal domain is thus a rectangle. 4. Strip domains. In order to prove the asymptotic value theorem, Ahlfors7 investigates the so-called strip domains. According to Nevanlinna,22 a strip domain is defined as a simply connected region, with two designated accessible boundary points r1 and r2 with 0 and ı(") be determined as in §4.2. Let the circular ring r < jzj < R with module M = log R log r contain two ring domains G0 , G00 with moduli M 0 , M 00 , positioned so that G0 separates 0 from G00 and 1 , and G00 separates 1 from G0 and 0 . Let M 0 + M 00  log R

log r

ı("):

From here we want to conclude that all points separated by G0 from 0 and by G00 from 1 belong to the circular ring log r + M 0

"  log jzj  log R

M 00 + ":

z 0 be the region obtained by adding to G0 its complementary continuum Let G z 00 be the region obtained from G00 by adjoining to containing 0 ; likewise, let G z 0 contains G0 with module M 0 it its complementary continuum containing 1 . G

22 Investigations on conformal and quasiconformal mappings

497

and the disjoint disk jzj < r with reduced module log r ; according to §1.8, for the z 0 we have, z 0 of G reduced module M z 0  M 0 + log r: M z 00 contain G00 and the disjoint disk jzj > R with reduced module Let G z 00 z 00 of G according to §1.8 we have for the reduced module M z 00  M 00 M

log R ;

log R:

It follows that z0 +M z 00  M 0 + M 00 M

(log R

log r)  ı("):

By the special Modulsatz (4.2), the points which are separated by G0 from 0 , and z 0 nor in G z 00 ; lie in the region formed by the by G00 from 1; but are neither in G circular ring z 0 "  log jzj  M z 00 + ": M However, this is contained in the circular ring M 0 + log r

"  log jzj 

M 00 + log R + ":

4. The extremal regions of the special Modulsatz. The proof just demonstrated is purely existential: for every " , there exists ı , but we have learned nothing about its magnitude. Now we want to determine the best (largest) ı for every " , and for this we prove one more time the special Modulsatz in a different way; in the process we get to know the extremal regions. It was through this different approach that I first discovered the Modulsatz. Let G0 and G00 be disjoint, simply connected regions, of which G0 contains 0 and G00 contains 1 . Let their reduced moduli be M 0 and M 00 , and let z0 be a point which belongs neither to G0 nor to G00 . The special Modulsatz restricts the location of z0 for given M 0 and M 00 ; for an arbitrary complex number a ¤ 0 , one can apply the transformation z ! az and replace M 0 by M 0 + log jaj , M 00 by M 00 log jaj , and z0 by az0 . In this way we normalize M 00 = 0 . On the other hand, we assume that z0 = 1 . Thus we ask what can be said about M 0 and M 00 if one assumes that neither G0 nor G00 contains z = 1 . From §1.8 we already know that M 0 + M 00  0 ; and Koebe’s theorem implies further that M 0  log 4; M 00  log 4 . In addition, it should now be determined exactly which pairs (M 0 ; M 00 ) are appropriate. We first construct a family of extremal regions, prove a number of inequalities about M 0 and M 00 ; demonstrating the extremal properties of each of the regions, from there we determine the set of appropriate (M 0 ; M 00 ) and then calculate the best possible ı(") .

498

Oswald Teichmüller

Figure 6. In reality, the circles have the same radius 1

In order to construct the extremal regions, we first consider the interior I and exterior A of the unit circle in the w -plane. We want to glue these regions to obtain an ideal closed surface, by using nontrivial boundary correspondence. Let q > 1 be a parameter. Using   = q' j'j  ; q we map the boundary arc w = e i' , j'j  q of I onto the boundary w = e i , j j   of A . We glue the leftover boundary piece of I : w = e i' , q  j'j   so that for q  j'j  ; e i' and e i' always correspond to each other. The boundary point w = 1 of I corresponds to itself, whereas the three boundary  points, w = e ˙i q of I and w = 1 of A correspond to each other (see figure 6). In this way we obtain a closed surface of genus 0 . We can now easily construct a local uniformizing parameter in a neighborhood of each point. For w0 in I , one can take w w0 , and for w0 in A , choose e.g. w1 w10 . If w0 lies on the unit
circle, then first w0 can be a point on w = e i'0 of I j'0 j  q , corresponding to w0 = e iq'0 in A . In that case, the local uniformization is given by the function ´ q(log w i'0 ) in I; log w iq'0 in A:

22 Investigations on conformal and quasiconformal mappings

499


Secondly, w0 can be a result of identifying e i'0 and e i'0 q < '0 <  , in which case ´ log w i'0 in the neighborhood of w = e i'0 ; jwj  1; log w i'0 in the neighborhood of w = e i'0 ; jwj  1 is a local uniformization. Still left are both singularities. In a neighborhood of w0 = 1 of I , a local uniformization is (log( w))2 : 

Finally in a neighborhood of the boundary points w0 = e ˙i q of I and w0 = 1 of A , 

qi log

w  23 i q

e  w  23 iq log i  e q 2

(log( w)) 3



in the neighborhood of w = e i q ; jwj  1; in the neighborhood of w = e

i q

; jwj  1;

in the neighborhood of w = 1; jwj  1

is a local uniformization.—We will not use these details further; we simply notice that a schlicht mapping of neighborhoods of the two singularities, (log w log w0 )2 2 resp. (log w log w0 ) 3 , will be a one-to-one first order vanishing schlicht function, and that otherwise everything else is regular. According to the uniformization theory one can map our ideal closed surface conformally onto the entire z -plane; here w = 0 of I is carried to z = 0 , the point w = 1 of the domain A to z = 1 , and the boundary point w = 1 of I to z = 1 . Because of symmetry, the real axis in the w -plane and the boundary arc w = e ˙i' ; q  '   of I are carried to the real axis in the z -plane; the 

boundary point w = e ˙i q of I and w = 1 of A correspond to a point z = b on the real axis (see figure 6). Later we will prove that b = q 2 . Then I is carried to a domain G0q , and A to a domain G00q . G0q and G00q are separated from each other by the z -image of the boundary parts w = e i' , j'j  q of I ; w = e i , j j   of A . The boundary part w = e i' , q  j'j   corresponds to a cut in G0q along the segment b


1 on the real axis. Let G0q and G00q have respectively moduli Mq0 and23 Mq00 . 5. Proof of the extremal property. Now we want to show how, and to what extent, our extremal problem is solved through these one-parameter families of extremal domains G0q and G00q . We prove this by the method from §2.3: 23 One could also map the circular triangle bounded by 1  w  +1 , and jwj = 1;  arg w   , and arg w =  1; 0; 1 q ; 1  jwj  1 to the upper z -half-plane so that correspond to 1; 0; 1 , then replace w by w q in jwj > 1 and reflect, which results in the same map of the slit w -plane onto the z -plane.  q

500

Oswald Teichmüller

Let G0 and G00 be two disjoint simply connected domains containing 0; resp. 1; with reduced moduli M 0 ; M 00 ; both not containing z = 1: Then for all q > 1 q 2 M 0 + M 00  q 2 Mq0 + Mq00 : Equality holds only when G0 = G0q and G00 = G00q . Proof. Based on the mapping just constructed of G0q ; resp. G00q onto the interior I; resp. the exterior A of the unit circle in the w -plane, we set ´ ! = q log w w in I; z in G0q ; ! = log w w in A; z in G00q : We primarily use the differential d! which does not depend on the specific branch of the logarithm; we may, e.g. take the main branch in the plane cut along the negative real axis. Now we map conformally G0 onto jj < 1 and G00 onto jj > 1 so that 0 goes to 0; resp. 1 to 1; and set  = %e i :

According to §1.6, ˇ ˇ ˇ dz ˇ M = log ˇˇ ˇˇ ; d 0 0

00

M =

ˇ ˇ ˇ dz ˇ log ˇˇ ˇˇ : d 1

Now we fix a curve in the z -plane corresponding to jj = % = const : , let ni (') be the number of its intersections with the z -image of the segment jwj  1; arg w = ' and let na ( ) be the number of its intersections with the z -image of jwj  1; arg w = . For ' < q , the union of the z -images of jwj  1; arg w = '

and

jwj  1; arg w = q'

is a continuous curve from z = 0 to z = 1 . On the other hand, the z -image of jj = % is a closed curve, separating z = 0 from z = 1 . Therefore it must have at least one point of intersection with our curve:   ni (') + na (q')  1; j'j < : q For

 q

< ' <  , the union of the z -images of jwj  1;

arg w = ˙'

is a closed curve which starts and ends at z = 0 , but separates z = 1 from z = 1 . Because of the last property, it cannot belong only to G0 , but must instead leave somewhere the part of G0 corresponding to jj  % for 0 < % < 1 , and enter

22 Investigations on conformal and quasiconformal mappings

501

it elsewhere again, thus it has at least two points of intersection with the z -image of jj = % :   ni (') + ni ( ')  2; 0; K = 0:

We only have to follow the course of the proof above: First was M (~; )

M (; )  2'(~) + '() < 3 C e

M (~; )

c~

:

In order to arrive at §5.2, we set 3 Ce

c~

= ı(") 

"2 ; log 1"

thus ~ becomes a function of " . After defining  as a function of " through 1  =  + log ; " ˇ ˇ ˇ ˇ then ˇM (; ) log 1" ˇ  C e c yields 1 + e M (~ ;) ı(") "  4": 1 e M (~ ;)+" When one, however, expresses " as a function of  , one obtains !()  ı(") + 2" + log

"  const
e

c log  c+2

:

Hence, for large  , !() < const
e

c log  c+2

:

From j(log r1 ()

)

(log r1 ()

)j  2'(~) + 2 Max¹!(~); !(); !()º;

assuming only  > ~;  > ~ (this ~ has nothing to do with the previous one), taking the limit  ! 1 yields the final formulas j log r1 ()



˛j  const
e

c log  c+2

;

accordingly c log 

j log r2 ()  ˛j  const
e c+2 : ˇ ˇ In Ahlfors2 (see §5.1) ˇM (%1 ; %2 ) P2 (log %2 log %1 )ˇ  const ; in order to be able % to apply §5.3, we have to set 2  = log %; P so that c = P2 . We therefore obtain j log ri ()



˛j  const
e

P 2 log  P+4

(i = 1; 2):

Ahlfors obtained with differential inequalities merely const j log ri ()  ˛j  p (i = 1; 2): 3 

514

Oswald Teichmüller

I suspect that my estimates may not yet be the best possible. These questions could gain more importance in the investigation of Riemann surfaces generated by meromorphic functions of infinite order.

6. Quasiconformal mapppings 1. The dilatation quotient. In an earlier short work25 I explained the concept of dilatation quotient and its main properties. It concerns the one-to-one mapping of a z = x + iy domain onto a w = u + iv domain. Let u and v be differentiable functions of x and y; and x , y differentiable functions of u; v . Let us determine that the mapping is such that not only one can pass from z to w; but also from z + dz to w + dw; which assigns to each dz = dx + i dy; in a linear and homogeneous manner, dw = du + i dv: For a conformal mappping, dw is dz dw independent of dz but in the general case dz will be a function of arg dz: The dilatation quotient is defined through ˇ ˇ ˇ ˇ Max ˇ dw dz ˇ ˇ ˇ: Dzjw = D = ˇ ˇ Min ˇ dw dz ˇ An infinitesimally small circle in the z -plane is transformed into an infinitesimally small ellipse in the w -plane, and D is the quotient of its axes. If D = 1; then the mapping at the corresponding point is infinitesimally a stretch and, therefore, conformal or conformal after a reflection; otherwise D > 1: If one maps the z -domain conformally onto some  -domain, and the w -domain conformally onto some ! -domain, then jdzj = p jd j;

jd!j = q jdwj

and, therefore, the dilatation quotient of the composition  ! z ! w ! ! ˇ ˇˇ ˇˇ ˇ ˇ ˇ ˇ d! ˇ ˇ dw ˇ ˇ dz ˇ ˇ ˇ Max ˇ dw p q Max ˇ dw ˇ ˇ dz ˇ ˇ d  ˇ dz ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ = Dzjw Dj! = = ˇ d! ˇ ˇ dw ˇ ˇ dz ˇ ˇ ˇ Min ˇ dw p q Min ˇ dw ˇ ˇ dz ˇ ˇ d  ˇ dz ˇ is the same as the one for the original mapping z ! w: The dilatation quotient of the mapping is a conformal differential invariant of first order, moreover the only one, up to a sign of the functional determinant. In place of the equality valid for conformal mapppings ˇ ˇ ˇ dw ˇ2 dw ˇ ˇ ; ˇ dz ˇ = dz 25 Teichmüller, Eine Anwendung quasikonformer Abbildungen auf das Typenproblem. Deutsche Mathematik 2.

22 Investigations on conformal and quasiconformal mappings

515

where, as earlier, we denote the area elements by dz = dx dy; and dw = dudv , the corresponding important inequality ˇ ˇ ˇ dw ˇ2 dw ˇ ˇ ; ˇ dz ˇ  D dz is valid for each value of the varying differential quotient dw : dz A mapping has been called quasiconformal26 when the dilatation quotient is bounded. However, one does not necessarily get at the intended function-theoretic applications with this notion. We define, in general: A quasiconformal mappping is a differentiable mapping whose dilatation quotient, in one of the two planes, is estimated from above. For example, one could prescribe that the dilatation quotient does not grow too fast toward the boundary or that it converges to 1 sufficiently fast. Whether a mapping is quasiconformal depends on how it is applied and if one can arrive or not at conclusions based on estimates of the dilatation quotient. Observe that we will derive conclusions about the behavior of the mapping from an estimate of D in the z -plane, and not in the w -plane. We should make our differentiability assumptions more precise. Of course we consider only one-to-one mappings that are continuous in both directions. Further, the partial derivatives ux ; uy ; vx ; vy are continuous and the functional determinant ux vy vx uy should be ¤ 0; so that the partial derivatives of x; y with respect to u; v are also continuous. Isolated exceptional points are, however, permitted. Finally, we also allow that the partial derivatives have discontinuities over analytic curves that do not accumulate inside the considered domain. 2. Examples. The simplest examples of quasiconformal mapppings are naturally the conformal mapppings. We discuss now an important class of informative examples which map the z = re i' -plane onto the w = %e i -plane as follows: % = %(r);

 = ':

Here %(r) is a monotone, continuous and piecewise continuously differentiable and the same holds for the inverse r(%): Naturally one could map in this manner the unit disk onto the punctured plane. The bestˇ way ˇ to compute the dilatation quotient is through transition to wˇ d log % logarithms: ˇ dd log log z oscillates between 1 and d log r ; so ° d log % d log r ± Dzjw = Dlog zj log w = Max ; : d log r d log % From here follows 1 d log %  D D d log r 26 Op. cit., footnote 25 also op. cit., footnote 5, and R. Nevanlinna, Eindeutige analytishe Funktionen, Berlin 1936, Chapter XIII, §8.

516

Oswald Teichmüller

and after integration Z r2 r1

1 dr  log %2 D r

Z r2 dr log %1  D : r r1

If this mapping maps the punctured plane z ¤ 1( 1  log r < 1 ) onto the unit disk jwj < 1 ( 1  log % < 0); it follows that Z r2 1 dr 0  log %2  log %1 + ; D r r1

R1 1

dr therefore, must be convergent. If from an upper estimate on D it follows D r that the integral is divergent, then it is not possible to construct a mapping of the punctured z -plane onto the unit disk.—When, conversely, jzj < 1 is to be mapped onto w ¤ 1; it follows from Z r2 dr log %2  log %1 + D ; r r1

R 1 dr that D r is divergent which is impossible when D increases sufficiently slowly. Finally, if z ¤ 1 is mapped onto w ¤ 1 , then one can write Z r2 (D r1

dr 1)  r

Z r2 1 r1

 log %2 Z r2  (D r1

R1 In case (D

1) dr r

1  dr D r

log r2 1)

log %1 + log r1

dr : r

converges, then lim (log %

r !1

log r) = ˛;

also converges and, as r ! 1 , %  e ˛ r:

All this holds for our example, however, we will prove similar theorems for arbitrary quasiconformal mapppings. 3. The behavior of module under quasiconformal mapppings A circular ring r1 < jzj < r2 is mapped quasiconformally onto a circular ring %1 < jwj < %2 such that the dilatation quotient satisfies D  C (jzj):

22 Investigations on conformal and quasiconformal mappings

517

Then Z r2 r1

1 dr  log %2 C (r) r

Z r2 dr log %1  C (r) : r r1

In the limiting case C = 1; namely of conformal mapppings, this theorem turns into an earlier result, from S1, about the invariance of the module. The proof will reveal that essentially the bounds are obtained only by the examples from §6.2. Proof. Let z = re i' . To each circle jzj = r = const , there corresponds a curve in the w -plane which separates jwj = %1 from jwj = %2 ; and which also has logarithmic length of at least 2 : ˇ Z2ˇ ˇ d log w ˇ ˇ ˇ d': 2  ˇ d log z ˇ 0 w For dd log log z we choose the value that corresponds to a purely imaginary d log z: The Schwarz inequality implies

ˇ Z2ˇ Z2 d log w ˇ d log w ˇ2 ˇ d'  2 C (r) 4  2 ˇˇ d'; d log z ˇ d log z 2

0

0

the last follows from the inequality already mentioned in §6.1, ˇ ˇ d log w ˇ d log w ˇ2 ˇ ˇ ; ˇ d log z ˇ  D d log z

and D  C (r): Division by 2 C (r) throughout, multiplication by integration yield Z r2 2 r1

1 dr  C (r) r

Z r2Z2

=

d log z

r1 0

“

d log w

d log w

= 2(log %2 Z r2 r1

d'

1 dr  (log %2 C (r) r

log %1 );

log %1 ):

dr r

dr ; r

and

518

Oswald Teichmüller

The segment r1 < jzj < r2 ; arg z = ' = const , corresponds to a curve in the w -plane, which connects jwj = %1 to jwj = %2 , with logarithmic length at least log %2 log %1 : v ˇ Z r2 ˇ Z r2u u d log w dr ˇ d log w ˇ dr u ˇ ˇ t log %2 log %1   C (r) : ˇ d log z ˇ r r d log z r1

r1

w For dd log log z , one sets the value that corresponds to a real d log z: From the Schwarz inequality, it follows Z r2 Z r2 d log w dr dr 2 (log %2 log %1 )  C (r)
: r d log z r r1

r1

Integration with respect to ' gives 2(log %2

log %1 )2 

“ Z r2 dr C (r)
d log w r r1

Z r2 dr = C (r)
2(log %2 r

log %1 );

r1

log %2

Z r2 dr log %1  C (r) : r r1

It would be easy to derive the corresponding formulas for the general case when D is bounded, from above, by a function depending not only on r but also on ': Based on the properties of the module already established and the inequalities just proved, we arrive at conclusions about the behavior of quasiconformal mapppings without having to carry out again similar integral estimates. 4. The case of the punctured plane z ¤ 1 onto the unit disk If the punctured plane z ¤ 1 is mapped quasiconformally onto the unit R 1 1 dr disk jwj < 1 and if D  C (jzj); then converges. C (r) r Proof. The module of the Rw -image of the circular ring r1 < jzj < r2 is, according 1 1 dr to §6.3, at least equal to . On the other hand, by §1.4, it is at most C (r) r equal to the module of the domain bounded by the w -image R r of jzj = r1 as well as jwj = 1 , and therefore bounded as r2 ! 1 . Therefore r12 C 1(r) dr is also bounded r for r2 ! 1 .

22 Investigations on conformal and quasiconformal mappings

519

§6.2 indicates that one cannot conclude more about C (r): The theorem is quite trivial.27 5. The case of the unit disk onto the punctured plane w ¤ 1 If the unit disk jzj < 1 is mapped Rquasiconformally onto the punctured 1 plane w ¤ 1 and if D  C (jzj); then C (r) dr diverges. r Proof. Let r1 > 0 be large enough so that the w -image of jzj  r1 contains the point w = 0: Let % = Max jwj: For every X , there exists an r2 with jzj=r1

Min jwj  %e X :

jzj=r2

According to §1.4, the module of the ring domain with boundaries the w -image of jzj = r1 and jzj = r2 ; which contains the ring % < jwj < %e X ; is at R r2 circular dr least X: By §6.3 this module is at most r1 C (r) r : For every X , there exists Rr some r2 ; such that r12 C (r) dr  X: r Again, §6.2 indicates that the theorem cannot be really sharpened. We have not applied directly to the given mapping z ! w the method from §6.3, involving integral estimates. Rather, we first map conformally the w -image of the circular ring r1 < jzj < r2 on another circular ring. Else, we would have obtained simply Rr 2d 2  r12 C (r) dr ; where d is the logarithmic distance between the w -images F r of jzj = r1 and jzj = r2 ; and F is the logarithmic area of the w -image of 2 r1 < jzj < r2 : But lim dF = 1 is not obvious. Perhaps, it is possible to conclude r1 !1 Rr this directly from the convergence of r12 C 1(r) dr but such a proof would be rather r complicated. I have not examined if the differential inequality method is applicable. In my opinion, the proof given here, based on §1.4 and §6.3, is the simplest. 6. The case of the punctured plane z ¤ 1 onto the punctured plane w ¤ 1 Let the punctured plane zR ¤ 1 be mapped onto the punctured plane 1 w ¤ 1: If D  C (jzj) and (C (r) 1) dr converges, then there exists r a constant > 0 with jwj  jzj: R1 One observes that in the claim the condition (C (r) 1) dr cannot be r replaced by a weaker one for mappings that are conformal in a neighborhood of 1; by the theorem on removable singularities, and as also shown by the example in §6.2. 27 This can be found in the work of M. Lavrentieff, Sur une classe de représentations continues. C. R. 200, Rec. Math. Moscou 42.

520

Oswald Teichmüller

Proof. Let C be the w -image of jzj = e  ; for sufficiently large ; C separates w = 0 from w = 1: The assumptions from §5.1 are all fulfilled. M (; ) denotes, in the same manner as there, the module of the ring domain bounded by C and C : By §6.3, for  <  , Ze

Ze dr C (r) ; r

e

e

1 dr  M (; )  C (r) r

Ze (C (r)

Ze

dr 1)  r

e

1

1  dr C (r) r

e

 M (; )

(

)

e

Z 

(C (r)

1)

dr ; r

e

jM (; )

(

)j  '();

Z1 where '() = (C (r)

1)

dr : r

e

Here, lim '() = 0: By §5.3 there exists a real constant ˛ such that !1

j log jwj



˛j < "

for all w on C ;

where " ! 0 as  ! 1: This means that ˇ w ˇ ˇ ˇ log ˇ ˛ ˇ ! 0 for jzj = e  ! 1; e z or if one sets e ˛ = ; jwj  jzj:

Wittich and I have attempted in vain to prove this theorem using the differential inequality method. That only succeeded if we made already some assumptions about C : It seems, however, that one could avoid the use of the Modulsatz if one generalizes the distortion theorem from §2. But we did not pull through only with Ahlfors’2 analytic methods without applying any auxiliary conformal mapppings to the w -plane. À propos, it is possible to substitute the assumption for D  C (jzj) , R1 ’ (C (r) 1) dr < 1 by the weaker one 1) d log z < 1 , by jzj>M (D r using more accurate estimates in §6.3.

22 Investigations on conformal and quasiconformal mappings

If one assumes that C (r)  1 + const (c > 0) , then clearly rc converges. Then Z1 dr '() = (C (r) 1) < const
e c : r

R1 (C (r)

521

1) dr r

e

By §5.4 we have jlog jwj

log jzj

˛j 

const


p log jzj

c+2 c

jzj c+2

:

7. Applications to quasiconformal mappings of strips. In these investigations, one can naturally substitute ring domains by quadrilaterals. The results can be derived from the theorems already proved. For example we will transfer the result from §6.6. Let the strip 0  =  b be mapped quasiconformally (and continuously on the boundary) on an equally wide strip 0  =!  b so that  = +1 corresponds to R 1! = +1 and  = 1 corresponds to ! = 1: In addition, D  G( 1 –halfsheet which is glued to the previous half-sheet along the arc a a +1 of the unit circle. Then comes a node which is connected to its predecessor via q 1 edges, which corresponds to a jwj < 1 –half-sheet, and which is glued to the previous jwj > 1 –half-sheet along the arcs a +1 a +2 ; a +2 a +3 ; : : : ; a 1 a , where the points a +2 ; : : : ; a 1 do not represent any problems (indices mod q ). Then, again a jwj > 1 –half-sheet is glued along a a +1 to the previous jwj < 1 –halfsheet, and attached to it in turn, along a +1 a +2 ; a +2 a +3 ; : : : ; a 1 a ; is another jwj < 1 –half-sheet and so on. Finally, a jwj < 1 is glued once along a a +1 to a jwj < 1 –half-sheet, which corresponds to a ramification node, whose vertices a are all logarithmic branch points of W . Now, we construct the straight lines 0 a and 0 a +1 on the initial half-sheet, and on the terminal half-sheet the radial straight lines a 1 and a +1 1 , since the ramification nodes belong only to the q -th component of our chain. These four segments bound a subregion of the Riemann surface which winds finitely many times around a and a +1 , and the function w a +1 z = log +i c w a maps it to a simply connected domain G in a z = x + iy -plane. If we carry out the same construction on W0 , we only need to observe that here l = 1 ; we simply obtain a sector 2 < arg w 0 < 2( q+1) bounded by the lines 0 a 1 and q 0 a +1 1 which is likewise mapped by z 0 = log

w 0 a +1 + i c0 w 0 a

onto a region G0 of the z 0 -plane. The behavior of both maps is illustrated in figure 9. If we choose the still undetermined additive constant in such a way that the real axis becomes an asymptote for the lower boundaries of G and G0 , then we can characterize G and G0 by the following inequalities: GW

f (x) < y < l + f (x);

G0 W

f (x 0 ) < y 0 <  + f (x 0 ):

Now W is obviously made up of parts that correspond uniquely to the individual chains and that have been mapped onto one of the infinitely many regions G; the ultimate collection of which is prescribed by the line complex. Likewise, W0

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consists of parts whose images are called G0 . We obtain a quasiconformal mapping from W to W0 by mapping the individual G quasiconformally onto the G0 . We must, however, ensure that this piecewise map from W to W0 remains continuous along the boundaries of the various parts of the surface. This can be achieved by prescribing that the boundaries of G and G0 , resp., are mapped onto each other via S , the disk jj < 1 Sk ( Rk ) contains only parts of the  -images of those half-sheets of W0 that correspond to nodes V1 ; V2 ; : : : ; Vk 1 . However, the dilatation quotient here is at most (k) , because in the just mentioned part of W0 protrude only images of G that correspond to chains of the line complex, which are contained in the finite part of the line complex of W bounded by Vk . If we let S S C (r) = (k) for 1 r