Conformally Invariant Metrics and Quasiconformal Mappings [1 ed.] 3030320677, 9783030320676

Table of contents :
Preface
Notation and Terminology
Contents
List of Figures
Part I Introduction and Review
1 Introduction
2 A Survey of Quasiregular Mappings
Part II Conformal Geometry
3 Möbius Transformations
3.1 Poincaré Extension
3.1.1 Stereographic Projection
3.1.2 Balls in the Chordal Metric
3.1.3 Absolute Ratio
3.2 Automorphisms of the Unit Ball
3.2.1 The Lipschitz Constant of Ta|Bn
3.2.2 The Ahlfors Bracket
3.3 Chordal Isometries
4 Hyperbolic Geometry
4.1 The Poincaré Half Space
4.2 The Poincaré Unit Ball
4.3 Bounds for Hyperbolic Distance
5 Generalized Hyperbolic Geometries
5.1 The Quasihyperbolic Metric
5.2 Möbius Invariant Metrics
5.2.1 The Point-Pair Invariant mG
5.2.2 The Symmetric Ratio
5.2.3 The Generalized Hyperbolic Metric ρG
5.3 Properties of Generalized Hyperbolic Metrics
6 Metrics and Geometry
6.1 Uniform Domains and Generalizations
6.1.1 Ptolemy's Theorem
6.2 Whitney Squares and (a,b,s)-Admissible Families
6.2.1 Whitney Decomposition
6.2.2 (a,b,s)-Admissible Families
6.3 Harnack Functions
Part III Modulus and Capacity
7 The Modulus of a Curve Family
7.1 Basic Properties of the Modulus
7.1.1 The Cylinder
7.1.2 The Spherical Ring
7.1.3 Constants and Dimension
7.2 Comparison Principle for the Modulus
7.2.1 The Modulus of a Ring
7.3 Grötzsch and Teichmüller Rings
7.4 Hypergeometric Functions and Elliptic Integrals
7.4.1 Elliptic Integrals and γ2(s)
7.4.2 The Gaussian Hypergeometric Function
7.4.3 Landen Transformation
7.4.4 Arithmetic-Geometric Mean
7.4.5 Modular Equations
7.4.6 Jacobi's Infinite Products
7.4.7 Ramanujan's Approximation of μ(r) [29, 5.51], [64, p. 91, (2.4)]
8 The Modulus as a Set Function
8.1 The Construction of c(E)
8.1.1 Proof of Theorem 8.1
8.2 Metric Concentration of Sets
8.2.1 Thickness and Capacity
8.2.2 Conformal Invariants in the Plane
8.2.3 Reduced Modulus
8.3 Tubular Neighborhoods
9 The Capacity of a Condenser
9.1 Spherical Symmetrization
9.2 Grötzsch and Teichmüller Condensers
9.3 Hyperbolic Metric and Capacity
9.4 Dimension Cancellation and Special Functions
10 Conformal Invariants
10.1 Two Conformal Metrics
10.2 Ferrand's Modulus Metric
10.3 Teichmüller's Function
10.4 QED Domains
10.5 Capacitary Geometry
Part IV Intrinsic Geometry
11 Hyperbolic Type Metrics
11.1 Metrics Determined by One Boundary Point
11.2 Metrics Determined by Two Boundary Points
11.2.1 Special Values of sBn
11.2.2 Ptolemy-Alhazen Problem and s-Metric
12 Comparison of Metrics
12.1 The Unit Ball
12.2 The Upper Half Space
12.3 General Domains
13 Local Convexity of Balls
13.1 Distance Ratio Metric
13.2 Quasihyperbolic Balls
13.3 Apollonian Metric
13.4 Seittenranta Metric
14 Inclusion Results for Balls
14.1 The Punctured Space
14.2 The Upper Half-Space
Part V Quasiregular Mappings
15 Basic Properties of Quasiregular Mappings
15.1 Topological Properties of Discrete Open Mappings
15.1.1 An Open Problem
15.2 Path Lifting
15.3 Analytic Properties of Quasiregular Mappings
15.4 Curve Families and Quasiconformal Mappings
15.4.1 Ferrand's Problem
15.4.2 Open Problem
16 Distortion Theory
16.1 The Schwarz Lemma and Quasiregular Maps
16.2 Bounds for Moduli of Continuity
16.3 The Schwarz Lemma in the Planar Case
16.3.1 Summary of Main Ideas
16.4 Further Results
16.4.1 An Open Problem
17 Dimension-Free Theory
17.1 Quasiregular Mappings and Harnack Functions
17.2 Quasihyperbolic Metric and Quasiregular Mapping
18 Metrics and Maps
18.1 An Open Problem on Quasicircles
19 Teichmüller's Displacement Problem
19.1 On Krzyż's Constant
Part VI Solutions
20 Solutions to Exercises
20.1 Solution to Exercises in Part II
20.2 Solution to Exercises in Part III
20.3 Solution to Exercises in Part IV
20.4 Solution to Exercises in Part V
A Some Open Problems
Problems from [553, p. 193]
New Problems
B Formulary
Hyperbolic Functions and Their Inverses
Some Biographical Data
References
Index
Author Index

Citation preview

Springer Monographs in Mathematics

Parisa Hariri Riku Klén Matti Vuorinen

Conformally Invariant Metrics and Quasiconformal Mappings

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Parisa Hariri • Riku Klén • Matti Vuorinen

Conformally Invariant Metrics and Quasiconformal Mappings

123

Parisa Hariri Department of Mathematics and Statistics University of Turku Turku, Finland

Riku Klén Turku PET Centre University of Turku Turku, Finland

Matti Vuorinen Department of Mathematics and Statistics University of Turku Turku, Finland

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-32067-6 ISBN 978-3-030-32068-3 (eBook) https://doi.org/10.1007/978-3-030-32068-3 Mathematics Subject Classification (2010): 32-02, 30C62 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Classical function theory (CFT) has a long history. Its roots lie in the mathematics of the eighteenth and nineteenth centuries, in the works of L. Euler, A.L. Cauchy, C.F. Gauss, C.G.J. Jacobi, B. Riemann, H.A. Schwarz, F. Klein, and many others. Some of the cornerstones of CFT are geometric and algebraic properties of complex numbers, power series expansions, explicit constructions of conformal mappings, and linear structure of the space of analytic functions, all of which are restricted to the two-dimensional case and have no counterpart in higher dimensional vector spaces. Extending CFT to euclidean spaces Rn , n ≥ 3, seems therefore to be a nontrivial task. We may nevertheless ask whether there exists an n-dimensional function theory that generalizes some geometric, topological, or analytic parts of CFT to dimensions n ≥ 3. The theory of quasiconformal and quasiregular mappings, developed during the past century, provides an affirmative answer to this question. As the reader will see, these mappings generalize conformal mappings and analytic functions, respectively, to the higher dimensional case. The present book is an introduction to this n-dimensional geometric function theory. As the literature shows, there are many ways to develop the theory of quasiconformal and quasiregular mappings. Our approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role throughout the whole book. The intended readership consists of mathematicians from beginning graduate students to researchers. The prerequisite requirements are modest: only some familiarity with basic ideas of real and complex analysis is expected. Our aim is to bring under one roof results previously scattered in many research papers published during the past 50 years, since the origin of the theory of quasiconformal and quasiregular mappings in dimensions n ≥ 3. This research area is indebted to its pioneers F.W. Gehring, O. Martio, Yu.G. Reshetnyak, S. Rickman, and J. Väisälä for their groundbreaking contributions, which have stimulated mathematicians of several student generations, led to hundreds of research papers, and pointed out

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Preface

directions for future research. Prior to their work, only the two-dimensional theory had been studied. About one half of the book is adopted in updated form from the book Conformal Geometry and Quasiregular Mappings, 1988 [555] of the third author. Already in that book the key idea was to use, in the spirit of F. Klein’s Erlangen Program, the notion of invariant geometry as a leading principle. Here we develop these ideas much further. The new material, published in research journals mainly during the past decade, deals with what we call hyperbolic type metrics and their applications to quasiconformal mapping theory. The study of these new types of metrics, for instance the comparison of the geometries generated by these metrics to other geometries, offers a large spectrum of open problems for future research. The text is enriched by a large set of exercises, most of them with complete solutions or hints for solutions. We hope that the exercises and the open research problems collected at the end of the book will offer stimulation and challenges for all readers from graduate students to researchers. Finally, we wish to express our thanks to all who have helped us in our work: our friends, colleagues, fellow researchers and especially our family members. Thanks are also due to Springer Verlag for their cooperation in publishing the book. Turku, Finland Turku, Finland Turku, Finland

Parisa Hariri Riku Klén Matti Vuorinen

Notation and Terminology

The standard unit vectors in euclidean space Rn , n ≥ 2, are denoted by e1 , . . . , en . A point x in Rn can be represented as a vector (x1 , . . . , xn ) or as a sum of vectors x= x1 e1 + · · · + xn en . For x, y ∈ Rn the inner product is defined by x · y = n n 1/2 . The ball centered at i=1 xi yi . The length (norm) of x ∈ R is |x| = (x · x) n n n x ∈ R with radius r > 0 is B (x, r) = { y ∈ R : |x − y| < r } , and the sphere with the same center and radius is S n−1 (x, r) = { y ∈ Rn : |x − y| = r }. We employ the abbreviations B n (r) = B n (0, r) ,

Bn = B n (1) ,

S n−1 (r) = S n−1 (0, r) , S n−1 = S n−1 (1) . n

The Möbius space R = Rn ∪ {∞} is the one-point compactification of Rn . Let X be a nonempty set and let d : X × X → [0, ∞) be a function satisfying (1) d(x, y) = d(y, x) , for all x, y ∈ X, (2) d(x, y) ≤ d(x, z) + d(z, y) , for all x, y, z ∈ X, (3) d(x, y) ≥ 0 and d(x, y) = 0 ⇐⇒ x = y . If (1)–(3) hold, then (X, d) is called a metric space. If d(x, x) = 0 for all x ∈ X and if (1) and (2) hold, then (X, d) is a pseudometric space. The notion of a metric space was introduced by M. Fréchet in 1906 [111, p. xv], [419, p.38]. For a metric space (X, d) let BX (y, r) = { x ∈ X : d(x, y) < r }. If we want to emphasize the role of the metric d rather than that of the space X , we write Bd (y, r) instead of BX (y, r) . We say that the metric space (X, d) is proper if for all x ∈ X and all r > 0 the closure of the ball BX (y, r) is compact. If A, B ⊂ X are nonempty let d(A, B) = inf{ d(x, y) : x ∈ A, y ∈ B } and d(A) = sup{ d(x, y) : x, y ∈ A }. For x ∈ X set d(x, A) = d({x}, A). For nonempty A ⊂ X and r > 0 we denote the tubular neighborhood of A by A(r) = {z ∈ X : d(z, A) < r} = ∪y∈A BX (y, r) .

vii

viii

Notation and Terminology

For two nonempty sets A, B ⊂ X the Hausdorff distance is H (A, B) = inf {A ⊂ B(r) and B ⊂ A(r)} . r>0

If (X, dX ), (Y, dY ) are metric or pseudometric spaces and f : (X, dX ) → (Y, dY ) is continuous, then ωf (t) = sup{ dY (f (x), f (y)) : dX (x, y) ≤ t ,

x, y ∈ X} , t > 0 ,

(*)

is called the modulus of continuity of f . This definition clearly depends on the metrics dX and dY . If confusion seems possible we shall specify the metrics. It is clear that ωf : (0, ∞) → (0, ∞] is increasing and that ωf (t) → 0 as t → 0 iff f : (X, dX ) → (Y, dY ) is uniformly continuous; in this case we set ωf (0) = 0 . A theorem that yields an upper bound for the modulus of continuity is often called a distortion theorem. If there are α ∈ (0, 1], C > 0, t0 > 0 such that ωf (t) ≤ Ct α for all t ∈ (0, t0 ) , then f is called Hölder continuous. In the special case α = 1 , f is Lipschitz continuous. For x, y ∈ Rn let [x, y] = { (1 − t)x + ty : 0 ≤ t ≤ 1 } and for x ∈ Rn \ {0} let n [x, ∞] = { sx : s ≥ 1 } ∪ {∞}. The Möbius space R = Rn ∪ {∞}, equipped with the chordal (spherical) distance q, is a metric space. In addition to (Rn , | |) and n (R , q) we shall require some other metric spaces such as the hyperbolic spaces (Bn , ρBn ) and (Hn , ρHn ), Hn = {x ∈ Rn : xn > 0}, as well as (G, kG ) where G ⊂ Rn is a domain and kG is the quasihyperbolic metric on G. The set of natural numbers 0, 1, 2, . . . is denoted by N and the set of all integers by Z. The set of complex numbers is denoted by C. We often identify C = R2 . n For a set A in Rn or R the topological operations A (closure), ∂A (boundary), n n and R \ A (complement) are always taken with respect to R . Thus the domain Rn \ {0} has two boundary points, 0 and ∞, and the half-space Hn = { x ∈ Rn : xn > 0 } has ∞ as a boundary point. A domain is an open connected nonempty set. A neighborhood of a point is a domain containing it. The notation f : D → D  n usually includes the assumption that D and D  are domains in R . A domain D in n R is said to be simply connected if every curve C in D , homeomorphic to S 1 , can 2 be contracted to a point in D by means of a continuous mapping f : B → D with 1 fS = C . n We say that a domain G in R is a Jordan domain if ∂G is homeomorphic to n−1 S . For instance, bounded convex domains are well-known examples of Jordan domains [341, Theorem 3.9]. If n = 2 the boundary of a Jordan domain G divides 2 the plane into two domains, G and D = R \ G , whose common boundary ∂G is, and G is homeomorphic to the unit disk and D is homeomorphic to its complementary domain. This is the content of the Jordan curve theorem in the plane. In the case n = 3 there are Jordan domains G homeomorphic to the unit 3 ball B3 such that their complementary domain R \ G is not homeomorphic to B3 . A famous example is Alexander’s horned sphere. Other examples illustrating exotic Jordan domains in R3 can be constructed using the Fox-Artin arc; see [161, p. 310

Notation and Terminology

ix

Thm 7.2.1]. A domain G in Rn is said to be locally connected at x ∈ ∂G , if there are arbitrarily small neighborhoods U of x such that U ∩ G is a connected set. It is a well-known (but nontrivial) fact that Jordan domains are locally connected at every boundary point. Let G be an open set in Rn . A mapping f : G → Rm is differentiable at x ∈ G if there exists a linear mapping f  (x) : Rn → Rm , called the derivative of f at x, such that f (x + h) = f (x) + f  (x)h + |h|(x, h) where (x, h) → 0 as h → 0. The Jacobian determinant of f at x is denoted by Jf (x). Assume next that n = m and that all the partial derivatives exist at x ∈ G (thus f need not be differentiable at x). In this case one defines the formal derivative of f = (f1 , . . . , fn ) at x as the linear map defined by f  (x)ei =

 ∂f

1

∂xi

(x), . . . ,

∂fn T (x) , ∂xi

i = 1, . . . , n .

For an open set D ⊂ Rn and for k ∈ N, C k (D) denotes the set of all those continuous real-valued functions of D whose partial derivatives of order p ≤ k exist and are continuous. The n-dimensional volume of the unit ball mn (Bn ) is denoted by n and the (n − 1)-dimensional surface area of S n−1 by ωn−1 . Then ωn−1 = nn and n =

π n/2 (1 +

1 2 n)

 ,

∞

(s) =

x s−1 e−x dx

0

for all n = 2, 3, . . . where stands for Euler’s gamma function. For k = 1, 2, . . . we have by the well-known properties of the gamma function [AS, 6.1] ω2k−1 =

2π k ; (k − 1)!

ω2k =

2k+1 π k . 1 · 3 · · · (2k − 1)

Algorithms suitable for numerical computation of (s) and other special functions are given in [1] and in [58, 440]. We next give a list of the additional notation used. Hn = Rn+ P (a, t) GM O(n) M  x,  f π(x), π2 (x) q(x, y)

the Poincaré half-space, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 an (n − 1)-dimensional hyperplane . . . . . . . . . . . . . . . . . . . . . 25 the group of Möbius transformations . . . . . . . . . . . . . . . . . . . . 27 the group of orthogonal mappings . . . . . . . . . . . . . . . . . . . . . . 27 the group of sense-preserving Möbius transformations . . . . 27 a generic point of { x ∈ Rn+1 : xn+1 = 0 } . . . . . . . . . . . . . . 27 the stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . 28, 30 the chordal (spherical) distance between x and y . . . . 28, 29

x

 x Bq (x, r) | a, b, c, d | a ∗ = a/|a|2 Ta Lip(f ) A[x,y] tx ρ(x, y) JD [x, y] x∗ , y∗ Bρ (x, M) jD (x, y) L(x, y) kD (x, y) BkG (x, M) σG (x, y) δG (x, y) sG (x, y) ρG (x, y) pX (A, t) |γ |

(γ ) Mp ( ), M( ) (E, F ; G) (E, F ) cn RG,n (s) RT ,n (s) γn (s) = γ (s) τn (s) = τ (s) μ(r) K(r) , E(r) F (a, b; c; r) ϕK,n (r) c(E) p−cap E, cap E α (F ) log n (s) log n (s) λn

Notation and Terminology

the antipodal (diametrically opposite) point . . . . . . . . . . . . . . 28 the chordal ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 the absolute (cross) ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 the image of a point a under an inversion in S n−1 . . . . . . 35 a hyperbolic isometry with Ta (a) = 0 . . . . . . . . . . . . . . . . . . 35 the Lipschitz constant of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 the Ahlfors bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 a spherical isometry with tx (x) = 0 . . . . . . . . . . . . . . . . . . . . 40 the hyperbolic distance between x and y . . . . . . . . . . . 50, 55 the geodesic segment joining x and y in D ∈ {Hn , Bn } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 the endpoints of the hyperbolic geodesic through x and y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 56 the hyperbolic ball with center x and radius M . . . . . .52, 56 the distance ratio metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 the line through x in the direction of y . . . . . . . . . . . . . . . . 65 the quasihyperbolic distance between x and y . . . . . . . . . . 68 the quasihyperbolic ball with center x and radius M . . . . 84 Ferrand’s metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Möbius metric/Seittenranta’s metric . . . . . . . . . . . . . . . . . . . . 75 a point-pair invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 the generalized hyperbolic metric . . . . . . . . . . . . . . . . . . . . . . . 76 the number of balls in a covering of the set A . . . . . . . . . . . 98 the locus of a path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 the length of a curve γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 the ( p -)modulus of a curve family . . . . . . . . . . . . . . . . . 103 the family of all closed nonconstant curves joining E and F in G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 (E, F ; Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 the constant in the spherical cap inequality . . . . . . . . . . . . . 114 the Grötzsch ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 the Teichmüller ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 the capacity of RG,n (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 the capacity of RT ,n (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 the modulus of the Grötzsch ring RG,2 (s) . . . . . . . . . . . . . 122 the complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . 122, 313 the Gaussian hypergeometric function . . . . . . . . . . . . . . . . . 123 a special function related to the Schwarz lemma . . . . 122, 167 a set function related to the modulus . . . . . . . . . . . . . . . . . . . 136 the ( p -)capacity of a condenser . . . . . . . . . . . . . . . . . . . . . . . 150 the α -dimensional Hausdorff measure of F . . . . . . . . . . . 155 the modulus of the Grötzsch ring . . . . . . . . . . . . . . . . . . . . . . 157 the modulus of the Teichmüller ring . . . . . . . . . . . . . . . . . . . 157 the Grötzsch ring constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Notation and Terminology

rG (x, y) λG (x, y) μG (x, y) p(x) mG (x, y) sG (x, y) cG (x, y) vG (x, y) pG (x, y) ∗ (x, y) jG αG (x, y) δG (x, y) μ(y, f, D) μ(f, D) Bf dim E J (G) i(x, f ) U (x, f, r) N(f, A) K(f ) KO (f ) KI (f ) H (x, f ) λ(K)

xi

a point-pair function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 a conformal invariant (by J. Ferrand) . . . . . . . . . . . . . . 173, 187 the modulus (conformal) metric . . . . . . . . . . . . . . . . . . . . . . . 173 the Teichmüller function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178 a point-pair invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 the triangular ratio metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 the Cassinian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 the visual angle metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 a point pair invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 th(jG (x, y)/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80, 200 the Apollonian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 the Seittenranta metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 the topological degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 the topological degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 the branch set of a mapping f . . . . . . . . . . . . . . . . . . . . . . . . 282 the topological dimension of a set E . . . . . . . . . . . . . . . . . . 282 a collection of subdomains of a domain G . . . . . . . . . . . . . 282 the local (topological) index of f at x . . . . . . . . . . . . . . . . 282 a normal neighborhood of x . . . . . . . . . . . . . . . . . . . . . . . . . .283 the maximal multiplicity of f in A . . . . . . . . . . . . . . . . . . 284 the maximal dilatation of f . . . . . . . . . . . . . . . . . . . . . . . . . . 288 the outer dilatation of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 the inner dilatation of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 the linear dilatation of a mapping f at x . . . . . . . . . . . . . . 295 a special function related to the linear dilatation . . . . . . . . . 298

Contents

Part I

Introduction and Review

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3

2

A Survey of Quasiregular Mappings . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7

Part II

Conformal Geometry

3

Möbius Transformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Poincaré Extension .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Stereographic Projection . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Balls in the Chordal Metric . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Absolute Ratio . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Automorphisms of the Unit Ball . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 The Lipschitz Constant of Ta |Bn . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 The Ahlfors Bracket .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Chordal Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 28 28 31 33 35 36 38 40

4

Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Poincaré Half Space .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Poincaré Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Bounds for Hyperbolic Distance . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

49 50 55 59

5

Generalized Hyperbolic Geometries.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 The Quasihyperbolic Metric . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Möbius Invariant Metrics . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 The Point-Pair Invariant mG . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 The Symmetric Ratio . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 The Generalized Hyperbolic Metric ρG . . . . . . . . . . . . . . . . . . 5.3 Properties of Generalized Hyperbolic Metrics . .. . . . . . . . . . . . . . . . . . . .

67 67 73 74 75 76 78

6

Metrics and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Uniform Domains and Generalizations . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 Ptolemy’s Theorem .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83 84 86 xiii

xiv

Contents

6.2

6.3 Part III 7

Whitney Squares and (a, b, s)-Admissible Families . . . . . . . . . . . . . . . 6.2.1 Whitney Decomposition.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 (a, b, s)-Admissible Families . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Harnack Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

92 92 93 96

Modulus and Capacity

The Modulus of a Curve Family . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Basic Properties of the Modulus .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.1 The Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.2 The Spherical Ring . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.3 Constants and Dimension . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Comparison Principle for the Modulus.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 The Modulus of a Ring . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Grötzsch and Teichmüller Rings . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Hypergeometric Functions and Elliptic Integrals .. . . . . . . . . . . . . . . . . . 7.4.1 Elliptic Integrals and γ2 (s) . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.2 The Gaussian Hypergeometric Function . . . . . . . . . . . . . . . . . . 7.4.3 Landen Transformation.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.4 Arithmetic-Geometric Mean . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.5 Modular Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.6 Jacobi’s Infinite Products .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.7 Ramanujan’s Approximation of μ(r) [29, 5.51], [64, p. 91, (2.4)] . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 106 106 107 114 115 120 120 122 122 123 124 124 125 126

8

The Modulus as a Set Function . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 The Construction of c(E) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Proof of Theorem 8.1 .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Metric Concentration of Sets . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Thickness and Capacity . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 Conformal Invariants in the Plane . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.3 Reduced Modulus . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Tubular Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

133 136 141 141 141 143 144 144

9

The Capacity of a Condenser. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Spherical Symmetrization.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Grötzsch and Teichmüller Condensers . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Hyperbolic Metric and Capacity .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Dimension Cancellation and Special Functions . . . . . . . . . . . . . . . . . . . .

149 155 157 162 167

10 Conformal Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Two Conformal Metrics . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Ferrand’s Modulus Metric . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Teichmüller’s Function .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 QED Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Capacitary Geometry.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

173 173 174 178 182 184

127

Contents

Part IV

xv

Intrinsic Geometry

11 Hyperbolic Type Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Metrics Determined by One Boundary Point . . .. . . . . . . . . . . . . . . . . . . . 11.2 Metrics Determined by Two Boundary Points . .. . . . . . . . . . . . . . . . . . . . 11.2.1 Special Values of sBn . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.2 Ptolemy-Alhazen Problem and s-Metric.. . . . . . . . . . . . . . . . .

191 193 198 205 206

12 Comparison of Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 The Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 The Upper Half Space.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 General Domains .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

209 210 219 223

13 Local Convexity of Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 Distance Ratio Metric .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 Quasihyperbolic Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Apollonian Metric .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 Seittenranta Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

239 240 246 252 255

14 Inclusion Results for Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 261 14.1 The Punctured Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 265 14.2 The Upper Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271 Part V

Quasiregular Mappings

15 Basic Properties of Quasiregular Mappings . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 Topological Properties of Discrete Open Mappings .. . . . . . . . . . . . . . . 15.1.1 An Open Problem.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Path Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Analytic Properties of Quasiregular Mappings .. . . . . . . . . . . . . . . . . . . . 15.4 Curve Families and Quasiconformal Mappings . . . . . . . . . . . . . . . . . . . . 15.4.1 Ferrand’s Problem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4.2 Open Problem.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

281 281 285 285 288 289 294 298

16 Distortion Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.1 The Schwarz Lemma and Quasiregular Maps . .. . . . . . . . . . . . . . . . . . . . 16.2 Bounds for Moduli of Continuity.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 The Schwarz Lemma in the Planar Case . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3.1 Summary of Main Ideas . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.4 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.4.1 An Open Problem.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

299 299 307 313 316 317 319

17 Dimension-Free Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 321 17.1 Quasiregular Mappings and Harnack Functions .. . . . . . . . . . . . . . . . . . . 322 17.2 Quasihyperbolic Metric and Quasiregular Mapping .. . . . . . . . . . . . . . . 328 18 Metrics and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 333 18.1 An Open Problem on Quasicircles . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 344

xvi

Contents

19 Teichmüller’s Displacement Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 347 19.1 On Krzy˙z’s Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 350 Part VI

Solutions

20 Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.1 Solution to Exercises in Part II . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.2 Solution to Exercises in Part III . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.3 Solution to Exercises in Part IV . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.4 Solution to Exercises in Part V . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

359 359 396 423 435

A

Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 453

B

Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 457

Some Biographical Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 467 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 469 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 493 Author Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 497

List of Figures

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9

Stereographic projection (3.4) and chordal metric (3.5) visualized.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Inversion π2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Chordal ball (3.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Inversion in a sphere .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Möbius center in Exercise 3.21 (2) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Lemma 3.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Lemma 3.27 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Left: Proof of Lemma 3.29 (1). Right: Proof of Lemma 3.29 (4) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2 onto B2 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Möbius map of S+

29 31 31 36 38 42 43 44 46 51 51 52 53

Fig. 4.6

Hyperbolic geodesics in H2 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Hyperbolic geodesic in the upper half plane as in (4.7). . . . . . . . . . . . . Hyperbolic distance and absolute ratio (4.9) . . . .. . . . . . . . . . . . . . . . . . . . The hyperbolic disk Bρ (te2 , M) in H2 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Left: Hyperbolic geodesics in B2 . Right: The hyperbolic disk Bρ (z, M) in B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Möbius transformation Tx . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. 5.1

Proof of Lemma 5.7(1) . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

70

Fig. 6.1 Fig. 6.2

Whitney squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Admissible family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

92 94

Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4

Left: A cylinder (7.1.1). Right: Remark 7.9 . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Lemma 7.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Lemma 7.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Left: Grötzsch ring, cap RG,n = M( s ) = γn (s). Right: Teichmüller ring, cap RT ,n = M(s ) = τn (s) .. . . . . . . . . . . . . . . . . . . . The functions μ(r) , 0 < r ≤ 1, and μ(1/r), r > 1 . . . . . . . . . . . . .

107 113 117

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5

Fig. 7.5

55 57

121 123

xvii

xviii

List of Figures

Fig. 7.6

Ring domains in Exercise 7.45 . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130

Fig. 8.1

Function c(E) defined in (8.7) . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137

Fig. 9.1 Fig. 9.2 Fig. 9.3

Spherical symmetrization of a set E . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 156 Spherical symmetrization in Theorem 9.14 . . . . .. . . . . . . . . . . . . . . . . . . . 156 The graph of γ3 (1/r) , 0 < r < 1, lies in the shaded region. Lower bounds are f (r) = 4c3 log 1+r 1−r (dotted 4π (solid line). Note that line) and g(r) = 2 log (9.9002/r) λ3 < 9.9002 and c3 = 4b3 ≈ 0.59907,  b3 is as in  where 1−r (7.8). Upper bounds are F (r) = 4c3 μ 1+r (dotted line)

Fig. 9.4

4π and G(r) = μ(r) 2 (solid line) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 161 Spherical symmetrization in Lemma 9.22 .. . . . . .. . . . . . . . . . . . . . . . . . . . 163

Fig. 10.1 Conformal invariants λG (left) and μG (right) . . . . . . . . . . . . . . . . . . . . 174 Fig. 11.1 From left to right: visualization of the extremal point z ∈ ∂G for metrics sG , cG and vG . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 11.2 Distance vH2 (x, y) = ω in two different cases .. . . . . . . . . . . . . . . . . . . . r Fig. 11.3 Left: Apollonian metric αG . Right: Apollonian circles Sx,y ...... Fig. 11.4 The s-metric disks Bs (z, t) are euclidean starlike with respect to the center z . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.1 Fig. 12.2 Fig. 12.3 Fig. 12.4 Fig. 12.5 Fig. 12.6

Distance vB2 (x, y) in general case . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The 3 cases in the proof of Lemma 12.4 .. . . . . . . .. . . . . . . . . . . . . . . . . . . . The 3 cases in the proof of Lemma 12.13 . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Lemma 12.25, vD (x, y) = θ = (x, z, y) . . . . . . . . . . . . . . . Proof of Theorem 12.29 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Condition H (δ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

194 196 199 206 210 212 219 233 234 237

Fig. 13.1 Line l, euclidean balls B1 = B n (y1 , c|x − y1 |), B2 = B n (y2 , c|x − y2 |) and B3 = B n (z, c|x − z|) and points y1 , y2 and z . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . .. . . . . . . . . . . . . . . . . 243 Fig. 13.2 Points y1 and y2 and circles B n y1 , d(y1 ) , B n z, d(z) and B n y2 , d(y2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 244 Fig. 14.1 Left: Inclusions Bj (e1 , m) ⊂ Bk (e1 , r) ⊂ Bj (e1 , r) from Proposition 14.1 and Theorem 14.7. Middle: Inclusions Bj (e1 , m) ⊂ Bq (e1 , r) ⊂ Bj (e1 , M) from Theorem 14.12. Right: Inclusions Bk (e1 , m) ⊂ Bq (e1 , r) ⊂ Bk (e1 , M) from Theorem 14.14 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 262

List of Figures

xix

Fig. 14.2 Left: Inclusions Bj (3e2 /2, m) ⊂ Bk (3e2 /2, r) ⊂ Bj (3e2 /2, r) from Proposition 14.1 and Theorem 14.18. Middle: Inclusions Bj (3e2 /2, m) ⊂ Bq (3e2 /2, r) ⊂ Bj (3e2 /2, M) from Theorem 14.24. Right: Inclusions Bk (3e2 /2, m) ⊂ Bq (3e2 /2, r) ⊂ Bk (3e2 /2, M) from Theorem 14.27 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 262 Fig. 15.1 Open mapping f in Exercise 15.11 .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 284 Fig. 15.2 Proof of Theorem 15.36 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 Fig. 20.1 Fig. 20.2 Fig. 20.3 Fig. 20.4 Fig. 20.5 Fig. 20.6

The unit circle and orthogonal circles Ya , Yb . . .. . . . . . . . . . . . . . . . . . . . The unit circle and orthogonal circle Yb . . . . . . . .. . . . . . . . . . . . . . . . . . . . The unit circle and the orthogonal circle Ya . . . .. . . . . . . . . . . . . . . . . . . . Visualization of the solution of Exercise 6.33 .. .. . . . . . . . . . . . . . . . . . . . Selection of ak and bk in Exercise 10.28 . . . . . .. . . . . . . . . . . . . . . . . . . . Exercise 15.37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. B.1

Stewart’s theorem: man + dad = bmb + cnc . . . . . . . . . . . . . . . . . . . . . 458

365 366 367 394 421 438

Part I

Introduction and Review

Chapter 1

Introduction

About a hundred years ago, in the beginning of the twentieth century, many new ideas gained footing in mathematical literature, significantly enriching later research. The objects of mathematical research, such as curves, surfaces, functions, and geometric bodies, had been traditionally assumed to be smooth or at least piecewise smooth. Well-known examples show that if one takes a countable sequence of such objects converging to a limit object, the limiting object need no longer be piecewise smooth. A famous example of such limiting behavior is K. Weierstrass’ construction of a uniformly continuous function f : R → R as a countable sum of trigonometric cosine functions, hence all partial sums are infinitely many times differentiable [331, Thm 1.12, p. 9]. This function is α-Hölder continuous for every α ∈ (0, 1) but nowhere differentiable. Another example of surprising behavior is Cantor’s middle-third function c : [0, 1] → [0, 1] [116, 122], which is a Hölder continuous function with c(0) = 0, c(1) = 1 , and c (x) = 0 a.e. and for which the fundamental result of calculus fails, because 

1

c(1) − c(0) = 1 =

c (x) dx .

0

These and other similar constructions showed that highly “nonsmooth” mathematical objects, functions, and sets, are more than just a curiosity. The classical differential and integral calculus was not adequate to study such nonsmooth objects. New tools were provided by careful analysis of the notion of convergence, exact formulations of mathematical proofs, and H. Lebesgue’s new theory of measure and integration. These tools also strongly influenced real, complex, and functional analysis, and differential equations and prepared the road for modern analysis. In modern analysis smooth functions were replaced by functions from appropriate function spaces such as Sobolev spaces or ACL functions, i.e. functions that are absolutely continuous on lines.

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_1

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1 Introduction

Quasiconformal and quasiregular mappings in Rn are natural generalizations of conformal and analytic functions of one complex variable, respectively. Nonsmooth mathematical objects are recurrent in this context and hence we see that modern analysis is a condition “sine qua non” for the theory of these mappings. In the two-dimensional case quasiconformal mappings were introduced by H. Grötzsch [179] in 1928. L.V. Ahlfors coined the term “quasiconformal” in his famous 1935 paper [7]. The higher-dimensional case was first studied by M.A. Lavrent’ev [321] in 1938. O. Teichmüller [510, 511] extended the theory to Riemann surfaces in a series of papers 1938–1943. World War II interrupted this progress. After the war, the research of quasiconformal maps was revitalized due to the work of L.V. Ahlfors [8], L. Bers, I.N. Vekua, and their students. The systematic study of quasiconformal mappings in Rn was begun by F. W. Gehring [150] and J. Väisälä [519] in 1961, and the study of quasiregular mappings by Yu. G. Reshetnyak in 1966 [453]. In a highly significant series of papers published in 1966–69 Reshetnyak proved the fundamental properties of quasiregular mappings by exploiting modern tools from differential geometry, nonlinear PDE theory, and the theory of Sobolev spaces. In 1969–72 O. Martio, S. Rickman and J. Väisälä [363–365, 525] gave a second approach to the theory of quasiregular mappings that was based on some results of Reshetnyak, most notably on the fact that a non-constant quasiregular mapping is discrete and open. On the other hand, their approach made use of tools from the theory of quasiconformal mappings, such as curve families and moduli of curve families. The extremal length and modulus of a curve family in the plane were introduced by L. V. Ahlfors and A. Beurling in their landmark paper [16] on conformal invariants in 1950. B. Fuglede [145] extended these notions to euclidean spaces of dimension n ≥ 3 in 1957. A third approach was suggested by B. Bojarski and T. Iwaniec [82] in 1983. Their methods were based on modern real and harmonic analysis. Their work was to a large extent independent of Reshetnyak’s work. As the authors said in the introduction, they planned to publish a monograph based on the ideas reported in this huge paper, which had many new results. In our book the central idea is to make systematic use of quantities defined on a given domain in Rn that are either conformally invariant or Möbius-invariant or at least quasi-invariant under some large class of mappings of the domain. For instance, quasi-invariance with respect to the class of L-bilipschitz mappings is often a very useful property. The fundamental invariant of the theory of quasiconformal mappings, the modulus of a curve family, offers a good starting point for our work, too. The modulus provides for us a link between conformal invariants and geometry. The Möbius-invariant hyperbolic geometry and its cousins, which we call hyperbolic type geometries, provide us a geometric view of the subject. Hyperbolic type metrics share some but not all properties of the hyperbolic metric. It is one of the key problems of the theory to prove results which are sharp or asymptotically sharp when the maximal dilatation K → 1 . In the limiting case K = 1 the mappings are conformal. Therefore we make an effort, whenever possible, to prove sharp or asymptotically sharp results.

1 Introduction

5

The seminal ingredient in our approach is to define and apply two conformally invariant metrics μG (x, y) and λG (x, y)1/(1−n) in a proper subdomain G of Rn . These metrics had been introduced by I.S. Gál [147] and J. Ferrand [135, 136, 328], respectively. As we shall see below, they are solutions to extremal problems of conformal geometry: for particular domains they reduce to classical problems of H. Grötzsch and O. Teichmüller, respectively. Since both metrics can be expressed in terms of moduli of curve families and since quasiconformal mappings are bilipschitz mappings in both metrics, many results follow now easily. For example, the quasiconformal version of the Schwarz lemma in a sharp form follows immediately. This same method also applies to quasiregular mappings, but now there are a few caveats because these mappings may be non-injective. In the case of the unit ball G = Bn both μG (x, y) and λG (x, y)1/(1−n) can be expressed in terms of the hyperbolic metric ρBn (x, y). In the planar case n = 2 both of these expressions involve classical special functions, complete elliptic integrals, which enables us to give accurate upper and lower bounds. For n = 2 in the case G = R2 \ {0} , the Ferrand metric λG (x, y)−1 is essentially the same thing as the solution of a classical extremal problem of geometric function theory, the modulus problem of O. Teichmüller, and an explicit formula in terms of elliptic integrals can be found in G.V. Kuzmina’s book [317, Ch. V]. These facts show that three things are tightly bound together: geometry of domains, extremal problems of conformal geometry, and special functions. Indeed, special functions provide the bridge that connects the geometry of domains with function-theoretic extremal problems. Then, for a wide class of domains, we find upper and lower bounds for these metrics. We begin with a survey of the theory of quasiconformal maps, covering topics that are either of general interest or close to our own research. As the list of monographs at the end of our survey shows, the theory has blossomed during the three decades since the publication of [555]. Because of lack of competence and of space, we cannot much discuss these developments in our survey but give telegram style pointers to the literature. The main content of the book is divided into six parts. Part I is comprised of the introduction and the survey. Part II deals with geometric preliminaries, including a discussion of Möbius transformations and hyperbolic geometry. In Part III we study conformal invariants; in particular, the modulus of a curve family is in our focus. Part IV studies metrics of hyperbolic type and the geometries defined by these metrics. In Part V we study quasiregular maps and their distortion in terms of hyperbolic type metrics. Part VI provides hints and solutions to exercises. Finally, the book ends with a list of open problems and a formulary, given in Appendices A and B. We have made an effort to make this book accessible for a wide readership. Therefore the prerequisites required of readers are modest: basic knowledge of real and complex analysis and of metric spaces is enough. An excellent reference is Väisälä’s book [524, pp. 1–50]. When results from [524] or from other books or from papers are cited, precise citations are given. Detailed solutions are provided to almost all exercises at the end of the book.

Chapter 2

A Survey of Quasiregular Mappings

The goal of this survey is to give the reader a brief overview of the theory of quasiconformal (qc) and quasiregular (qr) mappings and of some related topics. We shall also indicate the many ways in which the classical function theory of one complex variable (CFT) is related to quasiregular mapping theory (QRT) in Rn , as well as to point out some differences between CFT and QRT. At the end of this survey we give a list of books for further reading on the subject. Best starting points for learning QRT are the surveys of L. V. Ahlfors [11], L. Bers [65], F.W. Gehring [156, 158], O. Lehto [324], and J. Väisälä [527]. These papers already give a wide overview of QRT. One of the corner stones of the higher dimensional QRT is the book of Väisälä [524], which we strongly recommend. 1. Foundations In his pioneering papers [453–461], in which were laid the foundations of QRT, Yu. G. Reshetnyak successfully combined the powerful analytic machinery of PDE’s in the sense of Sobolev with some geometric ideas from CFT. Reshetnyak showed that the basic properties of qr mappings can be derived from the properties of the function uf (x) = log |f (x)|, where f is qr. He proved that uf satisfies a nonlinear elliptic PDE which for n = 2 is linear and coincides with the Laplace equation. It follows from the work of J. Moser [401], F. John–L. Nirenberg [272], and J. Serrin [487] that the solutions of this equation satisfy the Harnack inequality in { z : uf (z) > 0 }. Note that if f is analytic, then log |f (z)| has a similar role in CFT. Some of the main results of Reshetnyak are given in his books [463–466]. Obviously only a part of CFT can be carried over to QRT: for instance, power series expansions and the Riemann mapping theorem have no n-dimensional counterpart. 2. Quasiconformal Balls By Riemann’s mapping theorem a simply-connected plane domain with more than one boundary point can be mapped conformally onto the unit disk B2 . Liouville’s theorem says that the only conformal mappings defined on subdomains of Rn , n ≥ 3 , are the Möbius transformations. Thus © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_2

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Riemann’s mapping theorem has no counterpart in Rn when n ≥ 3 : since Möbius transformations preserve spheres, the unit ball Bn in Rn can be mapped conformally only onto another ball or a half-space. A quasiconformal counterpart of the Riemann mapping theorem is also false: for n ≥ 3 there are Jordan domains in Rn homeomorphic to Bn which cannot be mapped quasiconformally onto Bn although their complements can be so mapped. For a concrete example, see [158, Ex. 5.2, p. 9]. Also, the unit ball Bn , n ≥ 3 , can be mapped quasiconformally onto a domain with non-accessible boundary points, as shown by Gehring and Väisälä in [166]. A similar construction also shows that for each n ≥ 3 , there exists a domain D ⊂ Rn that can be mapped quasiconformally onto Bn such that D does not have a tangent plane at any point of ∂D [154, Example 1]. These facts show that for each n ≥ 3 the quasiconformal mappings in Rn constitute a class of mappings substantially larger than the class of Möbius transformations. Väisälä [520] established the following version of the Carathéodory extension theorem for quasiconformal mappings in space. If f : B3 → G is a quasiconformal mapping onto a domain G and if G is locally connected at each point of its boundary ∂G (relative to the Möbius space), then f can be extended to a 3  homeomorphism of B onto G . This result also has a counterpart for a subclass of qr maps, so called proper qr maps, which have the property that boundary of the domain of definition is mapped onto the boundary of the image domain, see M. Vuorinen [547, Thm 4.7]. In fact, many properties of qc maps can be extended for this subclass of qr maps. R. Näkki [404, 406] built a systematic theory of boundary behavior of qc maps. For instance, he discussed domains of [166] having inward- or outward-directed “wedges with zero angle” on the boundary (see the figures on pp. xxii-xxiii of [29]). For n ≥ 3 a Jordan domain homeomorphic to the unit ball is not necessarily quasiconformally homeomorphic to it by [166]. T. Kuusalo [316] constructed an example of Jordan domains D in Rn , n ≥ 3 , and a quasiconformal mapping f : D → D for which there is no homeomorphic extension f such that f : D → D and f |D = f . 3. Topological and Analytic Properties A basic fact from CFT is that a nonconstant analytic function is discrete (i.e. point-inverses f −1 (y) are discrete sets if f is analytic) and open (i.e. f A is open whenever f is analytic and A is open). By Reshetnyak’s fundamental work a similar result holds in QRT. Next, let Bf denote the set of all points where f fails to be a local homeomorphism. In CFT it is a basic fact that the branch set Bf is a discrete set if f is non-constant and analytic. A topological difference between the cases n = 2 and n ≥ 3 is that Bf is never discrete if f is qr in Rn , n ≥ 3 , and Bf = ∅ . By a result of A. V. Chernavski˘ı if G is a domain in Rn and if f : G → Rn is discrete and open, then dim Bf = dim f Bf ≤ n − 2 [98, 99, 523]. Also the metric properties of the branch set are different: if n = 2 and f is analytic, then cap Bf = 0 , while if n ≥ 3 and f is qr in Rn , then either Bf = ∅ or cap Bf > 0 (for the definition of the capacity see Chap. 9; see also [362, 461, 480]).

2 A Survey of Quasiregular Mappings

9

V. A. Zorich [579, 581] proved a surprising injectivity result, conjectured by M.A. Lavrent’ev: a locally homeomorphic K -qr map of the whole space Rn , n ≥ 3 , is a homeomorphism. The exponential function shows that a similar claim for n = 2 is false. O. Martio, S. Rickman, and J. Väisälä [365] generalized this result: they proved that for n ≥ 3, K ≥ 1 there exists a constant z(n, K) ∈ (0, 1) such that a K -qr local homeomorphism f : Bn → Rn is injective in B n (0, z(n, K)) . It seems to be an open problem whether we can replace z(n, K) by a constant independent of the dimension n . Martio and Srebro [372] extended this result to the case of quasimeromorphic mappings. Zorich also proved another version of his result where quasiregularity (qrty) is replaced by the requirement that the maximal dilatation have a controlled growth [580, 581]. S. P. Novikov then raised the question [581, p. 140] about a manifold version of the Zorich theorem [579], and this was partially answered by M. Gromov [581, p. 140], [177, 178]. For further results, see I. Holopainen and P. Pankka [238]. By a result of S. Stoilow, a qr mapping f of B2 onto a domain D can be factorized as f = g ◦ h , where h is a qc mapping of B2 onto itself and g is an analytic function [326, p. 247],[340]. Thus, by this Stoilow’s factorization theorem, the powerful two-dimensional arsenal of CFT is applicable to the “analytic part” of f , greatly facilitating the study of two-dimensional qr mappings. No such result is known for the multidimensional case. Topological aspects of CFT are studied by M. Cristea [106], see also G. Whyburn [569, 570]. Another result that is known only for dimension n = 2 is the powerful existence theorem for plane qc mappings (cf. [14, 15, 36, 80, 326]), based on C. Morrey’s [400] existence results (1938) for the solutions of Beltrami equations. This result, which is the analytic definition of qc mappings, is sometimes called the measurable mapping theorem. It is in fact the end result of the work of several generations of mathematicians starting from the time of C.F. Gauss. In the multidimensional case there is no general existence theorem for qc mappings, and all examples of qc and qr mappings known to the authors are based on direct constructions. In the qc case several examples are given in [166] and [524, Ch 16]. In the qr case a basic mapping is the winding mapping, given in cylindrical coordinates (r, ϕ, z) by (r, ϕ, z) → (r, kϕ, z) , k a positive integer [363]. An important example of a qr mapping is the so-called Zorich mapping [370, 579] and its various generalizations due to Rickman (cf. e.g. [472]). Additional examples are given in [463, pp. 27– 32], [369, 371], and [222]. B. N. Apanasov [32] has constructed highly nontrivial examples of locally homeomorphic qr mappings for dimension n = 3 . One can also construct new qc (qr) mappings by composing qc (qr) mappings. Let Hn = {x ∈ Rn : xn > 0} . By Poincaré’s extension theorem a Möbius transformation f : ∂Hn → f (∂Hn ) = ∂Hn , n ≥ 2 , can be extended to a Möbius transformation  f : Hn → Hn with  f |∂Hn = f [51, 347]. In their famous paper A. Beurling and L.V. Ahlfors [71] proved an extension theorem for qc mappings when n = 2 . Their work also shows that one-dimensional qc mappings, the boundary values of a qc homeomorphism of H2 onto itself, need not be absolutely continuous. Ahlfors [9] then went on and generalized this extension theorem for

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2 A Survey of Quasiregular Mappings

n = 3 , and L. Carleson [96] for n = 4 . Finally, P. Tukia and J. Väisälä [518] proved the general case n ≥ 3 of the extension theorem. The case n ≥ 3 differs from the case n = 2 , because now boundary values are absolutely continuous. Another extension method for qc maps in the plane, different from that in [71], is due to A. Douady and C. Earle [114], [245, pp. 184–194]. 4. Quasiconformality Versus Lipschitz and Hölder Maps A homeomorphism f : G → f G , G ⊂ Rn , is said to be K -qc if M( )/K ≤ M(f ) ≤ K M( )

(*)

for all curve families in G , where M( ) is the modulus of (see Chap. 7 below). This definition is somewhat implicit because the concrete value of the modulus is known only for very few curve families. To clarify the geometric consequences of (∗) let us point out that the linear dilatation is finite

|f (x) − f (z)| H (x, f ) = lim sup : |z − x| = r = |y − x| ≤ d(n, K) |f (x) − f (y)| r→0 for all x ∈ G , where d(n, K) < ∞ depends only on n and K . A well-known property of conformal mappings can be expressed by stating that H (x, f ) = 1 for K = 1 . Note that here the best constant d(n, K) → 1 when K → 1 , n ≥ 3 ; see Remark 15.47. A homeomorphism f : G → f G satisfying |x − y|/L ≤ |f (x) − f (y)| ≤ L|x − y| for all x, y ∈ G , is called L-bilipschitz. It is easy to show that L-bilipschitz maps are L2(n−1) -qc. But the converse is false. The standard counterexample is the qc radial stretching [524, 16.2] x → |x|α−1 x , x ∈ Bn , α ∈ (0, 1) , which is not bilipschitz. For many topological facts about Lipschitz maps, see J. Luukkainen and J. Väisälä [341]. All qc mappings are, however, locally Hölder continuous; e.g., if f : Bn → Bn is K -qc, then for |x|, |y| ≤ 12 , |f (x) − f (y)| ≤ A(n, K) |x − y|α , α = K 1/(1−n) , where A(n, K) depends only on n and K . For details see Chap. 16 below. Let B , QC , and H denote the classes of all bilipschitz, qc, and locally Hölder continuous mappings, respectively. By what was said above, the inclusions B ⊂ QC ⊂ H hold, where the first inclusion is strict. Simple examples can be constructed to show that also the second inclusion is strict. Many fundamental features of qc mappings are related to the strictness of the inclusion B ⊂ QC . For instance, one can construct qc mappings such that the image of a segment is not even locally rectifiable and such that the Hausdorff dimension of a set is different from the Hausdorff dimension of its image [167].

2 A Survey of Quasiregular Mappings

11

The Hölder continuity of qc mappings on the boundary of the domain of definition has been studied by many authors, see e.g. O. Martio and R. Näkki [360] and R. Näkki and B. Palka [407]. 5. Lp -Integrability From the analytic definition of qcty [524, 34.6] it can be deduced [524, 32.4] that the partial derivatives of a K -qc mapping are locally Lp -integrable. It is a deep result that these partial derivatives are even locally Lp integrable for some p = p(n, K) > n . This was proved by B. Bojarski [79] for n = 2 as an application of some results of A. Calderon and A. Zygmund [92]. The multidimensional case was established by F. W. Gehring in a highly influential paper [152]. The method of proof in [152], which makes use of so-called reverse Hölder inequalities, has found several applications to the calculus of variations and to PDE theory [168, 256, 257, 499, 500]. Some estimates dealing with the case K → 1 were given by Yu. G. Reshetnyak in [464] (see also [181]). The sharp integrability exponent is not known for n ≥ 3 . For n = 2 it was found by K. Astala in his important paper [35]. One of the main themes of the monumental book Astala– Iwaniec–Martin [36] is the study of applications of Astala’s higher integrability result. An alternative proof for Astala’s result was established by A. Eremenko and D.H. Hamilton [130, 189]. In connection with qr mappings the integrability has also been studied by B. Bojarski and T. Iwaniec [82], Iwaniec [254, 255], and O. Martio [353]. The paper by S. Donaldson and D. Sullivan [112] led T. Iwaniec and G. J. Martin [256, 258] to deeply study the analytic properties of qr mappings and differential structures on manifolds in terms of differential forms. A survey of this work is [255]. One of the highlights of the book [259] is a treatment of these issues, another highlight being the study of mappings with finite distortion. 6. Stability Theory The stability theory of K -qc and K -qr mappings in Rn in the sense of this book deals with the quantitative description of the behavior of these mappings when K → 1 . This terminology is due to Yu. G. Reshetnyak [458]. Roughly speaking, the expectation is that the mapping should become more or less like a conformal mapping under this passage to the limit. By Liouville’s classical theorem the two cases n ≥ 3 and n ≥ 2 are substantially different, and we shall therefore consider them separately. Case A. n ≥ 3 Liouville’s classical theorem, which was mentioned above in connection with quasiconformal balls, requires that the mappings be sufficiently smooth ( C 3 is enough). By deep results of F. W. Gehring [151] and Yu. G. Reshetnyak [454, 464, 466] the differentiability assumption can be replaced by the requirement that the mapping be 1 -qc or even 1 -qr. Another proof was given by B. Bojarski and T. Iwaniec [81]. Next, as shown by Reshetnyak [454, 462, 464, 466], one can show that as K → 1 a K -qr mapping must approach a Möbius transformation. For the exact statement of these results the reader is referred to [464, 466]. The methods of [464, 466] involve normal family arguments. Unfortunately the “speed” with which the convergence to Möbius transformations takes place as K → 1 is usually only qualitatively

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defined and no quantitative estimate for the “speed” in terms of K and n are known. Additional results have been proved by A. P. Kopylov [301], J. Sarvas [481], V. I. Semenov [484, 485], D. A. Trotsenko [514], and others. In 2005 K. Rajala [446] made significant progress in stability theory by showing that for each n ≥ 3 there exists a constant K1 with an explicit formula such that every K -qr mapping with K ∈ [1, K1 ) is locally homeomorphic. Case B. n ≥ 2 There are relatively few theorems for K -qc or K -qr mappings in Rn , n ≥ 2 , that are asymptotically sharp as K → 1 and that provide quantitative distortion estimates. This state of affairs is partly due to the fact that the proof of such results seems, in many cases, to depend on sharp estimates for certain little-known special functions. Many results of qc mapping theory depend more or less directly on the dimension n or on the maximal dilatation K or on both. It is a natural question raised by J. Väisälä [536] to study this dependence more closely. G. Anderson, M. Vamanamurthy, and M. Vuorinen studied this topic in a series of papers [24–28], deriving numerous new results. Some of their main discoveries, summarized in [29], were: (1) The distortion function ϕK,n (r) of the Schwarz lemma for K -qc mappings admits an explicit dimension-free upper bound and, moreover, the function ϕK,n (r) → r when K → 1 [24]. (2) The linear dilatation of a K -qc map of Rn has an explicit dimension-free upper bound, tending to 1 when K → 1 [557]. (3) Special functions have a crucial role in proving the sharp bounds in (1) and (2) [27]. In particular, complete elliptic integrals and elliptic functions are needed. The branch of special functions mentioned in item (3), designed for the needs of geometric function theory, was later significantly developed by many authors; see [49] and the bibliography of [30]. What remains open on the basis of these discoveries is whether there exists a new kind of stability property in high dimensions. For instance, if K > 1 is fixed, is it true that ϕK,n (r) → r when n → ∞ ? Or does the aforementioned upper bound in (2) for the linear dilatation have a similar convergence behavior? See [557, (4.3)] and Open problem 15.4.2 below. It seems that the solution of these questions would require new information about special functions such as the capacity of the Grötzsch ring in Rn , n ≥ 3 , which is not currently available. Many open problems of this kind were formulated in [27]. For n = 3 numerical computations for the capacity of the Grötzsch ring were reported in [478], which was part of Samuelsson’s PhD thesis [477]. 7. Quasiconformal Images of Spheres and Circles Qc maps are differentiable at almost every point with non-zero Jacobian, and thus their local behavior is well understood at almost every point. Let E stand for the exceptional set of measure zero where a qc map f : G → f G in Rn , n ≥ 2, is not differentiable or where the Jacobian vanishes. Close to this set E the local behavior of f can be wild. For the purpose of analyzing the local behavior of a qc or qr map at a given point z ∈ E ,

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13

C.J. Bishop, V. Ya. Gutlyanski˘ı, O. Martio, and M. Vuorinen [75] applied the notion of an infinitesimal space introduced in [183]. The question about the differentiability of a qc map at a given point has been studied by many authors; see the surveys [117] and [489] for the case n = 2 . For n ≥ 3 see Yu. G. Reshetnyak [464, 466] and N.A. Kudryavtseva [308]. Theorems giving sufficient conditions for the differentiability of a qc map at a given point, are often called Teichmüller-Wittich-Belinskii type theorems; these are studied in detail in [80]. A. Golberg [170] gave several explicit examples illustrating the curiosities of the pointwise behavior of plane qc maps; see also A. Fletcher and B. Wallis [141] who analyzed the structure of the infinitesimal space at a point of nondifferentiability. The bilipschitz images of circles may exhibit highly nonsmooth local behavior. All bilipschitz images of circles are rectifiable, but they may have points where no tangent exists because of “spiraling character” of the image curve at a point. See [37, 80]. Recall that bilipschitz maps are differentiable only a.e. by the Rademacher–Stepanov theorem [524, p. 97], [344]. The image of the unit sphere S n−1 under a K -qc map f : Rn → Rn is called a K -quasisphere or, for n = 2 , a K -quasicircle. F. W. Gehring and J. Väisälä proved [167] that the Hausdorff dimension of a quasisphere is always less than n and it can be any number in (n − 1, n) ; Ch. Pommerenke and J. Becker [57] found a nearly sharp upper bound for n = 2 . The sharp upper bound for the Hausdorff dimension of a quasicircle is due to S. Smirnov [491]. See also S. Smirnov and I. Prause [438] and K. Astala, T. Iwaniec, and G. Martin [36, p.336]. A bounded Jordan curve γ ⊂ R2 is said to satisfy the three-point condition if there exists a constant d ≥ 1 such that for all a, b ∈ γ we have |a − c| + |b − c| ≤ d|a − b| for all c in the smaller subarc of γ \ {a, b} . This three-point condition, due to Ahlfors [159, Thm 2.2.5, p.23], [167], gives a purely geometric characterization for quasicircles, with no reference to qc maps needed. There is no counterpart of Ahlfors’ three-point condition for quasispheres, n ≥ 3 . The well-known von Koch snowflake curve, which has Hausdorff dimension > 1 , is a quasicircle (see the figure [29, p.xxii]). This snowflake curve, which is the limit of a countable sequence of polygonal curves, shows that quasicircles need not have tangents at any point and that they may be locally non-rectifiable. A counterpart of the snowflake curve in R3 was constructed by D. Meyer [384]. He constructed a quasisphere having Hausdorff dimension > 2 . In 1990 P. Mattila and M. Vuorinen [379] studied the local behavior of a K -quasisphere S ⊂ Bn and showed that there exists δ = δ(n, K) ∈ (0, 1) with δ → 0 when K → 1 with the following property: For each x ∈ S and all small enough r > 0 there exists an (n − 1) -dimensional hyperplane V , x ∈ V , such that S ∩ B n (x, r) lies in a set V + B n (δr) (see the figure [29, p. xxv]). In the plane case, this linear approximation property says that a K -quasicircle can be locally approximated by narrow rectangles (the ratio of side lengths ≈ δ ), with better accuracy when K is near 1 . This result was, in part, based on asymptotically sharp bounds for the linear dilatation of a K -qc mapping

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of M. Vuorinen [557]. Further results were proved by I. Prause [435, 436] and by M. Badger, J. Gill, S. Rohde and T. Toro [40]. In his important paper on the travelling salesman problem (TSP), P. W. Jones [275] used the idea of covering planar sets with families of narrow rectangles and discovered a necessary and sufficient condition for the rectifiability of planar curves. Jones’ work has found many applications, see e.g. H. Pajot [415], and it was generalized to the n -dimensional case; see G. David and S. Semmes [107]. The papers [275] and [379] were independently published at the same time, in 1990. In their book C. Bishop and Y. Peres [77] study fractal sets of analysis and probability theory. One chapter of their book is devoted to TSP and they also investigate the Hausdorff dimension of quasicircles in that chapter. See also J.B. Garnett and D.E. Marshall [149, Ch X] and V. Vellis and J.-M. Wu [544]. Further results about TSP appear in [41]. Let f : Rn → Rn be qc and let K(t) be the maximal dilatation of the restriction of the mapping to an annular t - neighborhood {x ∈ Rn : d(x, S n−1 ) < t} of the unit sphere S n−1 . If the convergence K(t) → 1 as t → 0 is fast enough, then f S n−1 is rectifiable, i.e. it has finite (n − 1) -dimensional Hausdorff measure [379]. Inspired by this result, Yu. G. Reshetnyak proved a stronger conclusion [466, pp. 378]. Further results appear in [182]. P.W. Jones and C. Bishop [76] studied a conformal map f of the unit disk onto a bounded quasidisk  and gave a necessary and sufficient condition for the rectifiability of ∂ in terms of the Schwarzian derivative of f [149, Thm 6.2, p. 394]. An explicit univalent function of the unit disk mapping the unit circle onto a quasicircle is given in [427, pp. 304–305]—this quasicircle does not have a tangent at any point. The rectifiability problems of sets have been studied also in the recent papers [39, 439]. 8. Nonlinear Potential Theory Let G be a domain in R2 and v : G → R harmonic. For a domain D ⊂ G with D ⊂ G let Fv (D) = { u : G → R : u|∂D = v|∂D , u ∈ C 2 (G) } . A well-known extremal property of the class of harmonic functions, the Dirichlet principle, states that they minimize the Dirichlet integral [516, pp. 9–14]. In the above notation this means that   |∇v|2 dm = inf |∇u|2 dm . D

u∈Fv (D) D

Analogous Dirichlet integral minimizing properties hold as well for the solutions of the nonlinear elliptic PDE’s which arise in connection with qr mappings. This important fact was proved by Yu. G. Reshetnyak [457]. These solutions are called n -harmonic functions. The resulting potential theory is the subject of the monograph of Gol’dshte˘ın and Reshetnyak [171]. Inspired by Reshentyak’s work, V. M. Miklyukov [386] studied subsolutions of these PDE’s.

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15

In a series of papers S. Granlund, P. Lindqvist, and O. Martio have considerably extended these results [174, 175, 336, 337, 356]. They also found a unified approach to some some function-theoretic parts of QRT including, in particular, the n harmonic measure, which is a generalization of the classical notion. Martio’s surveys [356, 357] discuss these issues. The monograph of Heinonen, Kilpeläinen and Martio [219] extends these results to more general A -harmonic equations. This book builds a potential theory of the solutions of these nonlinear equations, which has many features in common with the potential theory of classical harmonic functions. This highly cited book is considered to be one of the cornerstones of nonlinear potential theory. The book of J. Malý and W. P. Ziemer [345] studies similar topics from a bit different point of view. This nonlinear potential theory was extended by A. and J. Björn [78] to the case of metric spaces with a doubling measure. 9. Value Distribution Theory In 1967 V. A. Zorich [579] asked whether Picard’s theorem holds for spatial qr mappings and whether the value distribution theory of Nevanlinna [410] has a counterpart in this context. These questions have been answered by S. Rickman in a series of papers [468–472], the main results being reviewed in [470] and [471, 473]. Additional results appear in [378] as well as in [423]. A culmination of Rickman’s deep work was the proof of Picard’s theorem for qr mappings [469]. One of the methods used in [469] is a two-constants theorem for qr mappings (analogous to the two-constants theorem of CFT [410]), which Rickman derived from an estimate for the solutions of certain nonlinear elliptic PDE’s due to V. G. Maz’ya [380]. An alternative proof which only makes use of curve family methods is given in [471, 473]. Rickman’s results related to the Picard theorem and his ingenious methods of constructing qr mappings are some of the most profound results of qr theory. Alternative proofs for the Picard theorem were also given by A. Eremenko and J. Lewis [131], [219, p. 279], Lewis [333], and M. Bonk and P. Poggi-Corradini [84]. See also the paper of I. Holopainen and P. Pankka [239]. Rickman’s book [473] gives a carefully crafted introduction to qr mappings with emphasis on value distribution theory. A recent paper of D. Drasin and P. Pankka [118] demonstrates the sharpness of some of Rickman’s value distribution results. 10. Special Classes of Domains The standard domain, on which most of the CFT is developed, is the unit disk. During the past three or four decades many authors have studied CFT on domains more general than the unit disk. In the early 1960s two highly important classes of domains were introduced by L. V. Ahlfors and F. John, respectively. Ahlfors studied domains bounded by quasicircles, i. e. images of the unit circle under a qc mapping of R2 , and found remarkable properties of these domains. In a paper related to elasticity properties of materials, John introduced a class of domains, nowadays known as John domains [271]. See the survey [408]. In connection with his studies of injectivity of qr mappings, Yu. G. Reshetnyak [458, 462] applied John domains. John domains found applications in the work of O. Martio and J. Sarvas [368], who also introduced the important class of uniform domains. Since their introduction uniform domains have become very popular

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in CFT and related parts of analysis. P. W. Jones [273, 274] studied extension operators of function spaces and proved, for instance, that BMO-functions of a plane Jordan domain can be extended to the whole plane if and only if its boundary is a quasicircle. Plane Jordan domains bounded by quasicircles are uniform domains. Jones’ work generalizes earlier work of H.M. Reimann and T. Rychener [451] concerning the case of BMO-functions of the unit disk; see also the discussion of Reimann’s work in [36, pp. 343–344]. M. Zinsmeister [578] studied harmonic analysis on so-called Lavrentiev domains. Applications to function spaces and their extension to a larger domain have also been proved by S. K. Vodop’yanov, V. M. Gol  dshte˘ın, and Yu. G. Reshetnyak in [546]. See also [171, Chapter 6], [163, 514, 529], [158, pp. 14–15] [159, p.139], [530, 533]. One of the particular topics of interest of F. W. Gehring was the study of quasidisks. In his surveys [155, 157] he presented numerous characterizations of these domains and underlined their key role in geometric function theory and related areas. The richly illustrated monograph Gehring–Hag [159], a jewel of mathematical literature, makes stimulating reading of the instances where quasidisks occur. 11. Related Classes of Mappings Some mathematical research topics, originally studied in CFT, have been extended in several directions, of which QRT is one. We list here a few topics, where either research themes or methods are similar to those of QRT. •

•

• •

•

Maps qc in the mean were mentioned by Ahlfors [8]. These maps and some other classes were studied in the monograph of Martio, Ryazanov, Srebro, and Yakubov [366]. Maps with finite distortion form an important generalization of qr maps. See the monographs of Iwaniec–Martin [259] and Hencl–Koskela [224]. The boundary behavior of maps with controlled growth of dilatation were studied by Zorich [580, 582]. Almost quasiconformal maps and almost solutions of A -harmonic equations were studied by V.M. Miklyukov [387].  Maps with finite Dirichlet integral |∇u(x)|p w(x) dx < ∞ , sometimes called BLD or Beppo Levi functions, have been studied with various positive weights w and exponents p > 0 . See J. Lelong-Ferrand [328], G.D. Suvorov [508], V.M. Miklyukov [385, 388], P. Koskela–J. Manfredi–E. Villamor [303], Y. Mizuta [392], M. Ohtsuka [413], E. Stein [498]. Homeomorphisms between plane domains are called harmonic maps if both coordinate functions are harmonic; see the survey of D. Bshouty and W. Hengartner [89] and the monograph of P. Duren [123]. Harmonic maps form a large chapter in the voluminous scientific work of S. Ponnusamy; see e.g. [101, 334, 431]. M. Dorff and J. Rolf [113] have authored a richly illustrated introduction to mapping problems of univalent harmonic functions with supporting Java Applets. The Schwarz lemma has a counterpart for harmonic maps; see e.g. P. Duren [123] and D. Kalaj and M. Vuorinen [279]. T. Adamowicz [2] investigated maps whose coordinate functions are p -harmonic.

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•

17

Maps which are qc and harmonic were perhaps first introduced by Martio in 1968 [352]. Thirty years later many authors began to study these maps, and the intensive research continues until the present time [101, 277, 278, 375, 421]. The famous Schoen conjecture asks whether a quasisymmetric map (i.e. a 1 dimensional qc map) of the unit circle onto itself, can be always extended to a harmonic map of the unit disk onto itself. This conjecture was recently solved by V. Markovic [348]. M. Jian- L. Liu- H. Yao [269] showed that the DouadyEarle extension is not always harmonic. Note that in the Beurling-Ahlfors [71] extension result the extended map is qc. The survey [422] compares various extension methods; see also [430, pp.112–120] and [240].

12. Qc Mappings on Manifolds and Metric Spaces The study of qc mappings on Riemann surfaces has a long history going back to L. Ahlfors [7] and O. Teichmüller [511]. Teichmüller’s work led to the birth of the theory of Teichmüller spaces, nowadays a very extensive theory [148, 253, 325, 342, 486], which has a close connection with low dimensional topology [245, 347] and applications to theoretical physics [418]. For a survey of Teichmüller spaces, with emphasis on their connection with univalent functions, see T. Sugawa [502]. In the 1970s, J. Ferrand [135, 136, 328] applied the curve family method of Ahlfors and Beurling to study qc maps on manifolds of dimension n ≥ 2 . M. Gromov [177, 178] studied metric spaces and maps between such spaces and generalized the notion of curvature to this context. Hyperbolic spaces in the sense of Gromov have become an active area of research during the past forty years [281]. See [88, 119] and P. Pansu [416], M. Bonk, J. Heinonen, P. Koskela [83], and the survey of J. Väisälä [538]. Qc mappings have also found applications in computer graphics. For instance, visualization methods have been developed for medical measurement data, enabling researchers to produce pictures of the human brain on a two-dimensional plane which are useful for the planning of medical treatments. Qc mappings are applied here to keep the distortion of the image under control, see [180, 270]. Since the 1980s qc mapping theory has been studied in metric spaces augmented with some additional structure. Väisälä developed his theory of qcty in the context of metric, Banach, and Hilbert spaces in a series of papers [531, 532, 534, 535, 537]. As opposed to the theory of qc maps in euclidean spaces, this theory is independent of the dimension, and Väisälä calls these maps freely qc maps. His approach is based on the use of several metrics, including, in particular, the quasihyperbolic and distance ratio metrics. There is a list of open problems in [537, pp. 115–116]. Recently this “theory of the free world” has been studied by many authors, e.g. by M. Huang, Y. Li, S. Ponnusamy, A. Rasila, X. Wang [241, 242]. Differential forms had an important role in the work of Yu. G. Reshetnyak [460, 463] and also in the later works of V. M. Miklyukov [386] and of B. Bojarski and T. Iwaniec [81]. The important research of S. Donaldson and D. Sullivan [112] used, among other things, multilinear algebra and the language of Hodge theory in their study of manifolds. Then Iwaniec and Martin [255, 256, 258] applied these methods to study singularities of qr mappings. Miklyukov’s book [387] and the

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papers [143, 358, 359] established a theory of qr maps between two manifolds of the same dimension, also using the methods of multilinear algebra and Hodge formalism. For instance, Phragmén-Lindelöf type results can be proved in this setup. See also the book R. Agarwal–S. Ding–G. Nolder [5]. Another approach to qc maps in metric spaces is based on real analysis on spaces with a doubling measure. The paper of J. Heinonen and P. Koskela [220] was one of the first pioneering works in this connection, and nowadays many people are working on this area; see the survey of D. Herron [225]. Here methods and ideas come from measure theory, real and functional analysis, and metric geometry of manifolds. These topics are discussed in the books of J. Heinonen [217] and of J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson [221]. Qcty on Carnot and Heisenberg groups has been studied e.g. in [46, 218, 302, 416, 417, 545]. 13. Dynamics and Qr Maps Dynamical systems, based on iterations of the form ¯ →C ¯ is a complex function, have been studied fn+1 (z) = f (fn (z)) , where f : C by numerous authors. The book [87] of B. Branner and N. Fagella provides a clear and richly illustrated introduction to this subject; see also Branner’s article [173, pp. 179–196]. R. Stankewitz and J. Rolf [496] have written a short introduction to complex dynamics emphasizing computational aspects and supporting computer experiments with Java Applets. For instance in the case of quadratic polynomials f (z) the iteration leads to a rich theory where exotic measure-theoretic fractals such as Mandelbrot sets are objects of study [52, 97, 383], [159, 506, 507, p. 16]. Quasidisks are also recurrent in such studies [87, 159, 383, 430]. Part of this theory has its counterpart in higher dimensions n ≥ 3 . Qc and qr mappings have a crucial role here. See W. Bergweiler-D. Drasin-A. Fletcher [62] and Iwaniec-Martin [259]. See also [62, 63, 97, 138, 139, 235, 236, 350]. 14. Intrinsic Geometry of Domains The famous German mathematician F. Klein had a vision for how the different research areas of geometry—euclidean, hyperbolic, and spherical—could be unified. These ideas became known under the name Klein’s Erlangen Program (1872); see J. Gray’s article on geometry in [173, pp. 83–95], in particular, p. 93. In a nutshell, let G be one of these three geometric spaces and consider a group T of automorphisms of G with the property that given x, y ∈ G there exists a group element h ∈ T with h(x) = y . Two configurations, e.g. triangles, are considered equivalent if one of them can be mapped onto the other one by an element of the group. The groups considered are subgroups of the Möbius automorphism group of the space. The space G is equipped with a metric which is invariant under the group operation. Thus the euclidean distances are invariant under translations and rotations, the hyperbolic distances under all Möbius automorphisms of the space, and the chordal (spherical) distances under rotations of the Riemann sphere. The notion of invariance was therefore an inherent feature of Klein’s Erlangen program. This influential program also became popular in geometric function theory. In particular, conformal invariance and Möbius invariance are the most important

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19

notions of invariance, and a key notion is the conformally invariant hyperbolic metric. In the higher dimensional case n ≥ 3 , by the Liouville theorem conformal maps are restrictions of Möbius transformations, and hence conformal invariance is less useful than in the plane case. In the case n = 2 , by virtue of Riemann’s mapping theorem one can map a simply-connected domain with non-degenerate boundary onto the unit disk and use this mapping to define the hyperbolic metric in the given domain. This procedure yields a well-defined metric in the given domain, independent of the conformal map [54, 283]. In the higher dimensional case there is no metric nearly as good for the study of the intrinsic geometry of a domain as the hyperbolic metric is in the case n = 2 . But some substitutes for it have been found for a general domain G ⊂ Rn , n ≥ 3, that share some but not all properties of the hyperbolic metric. These metrics take into account not only the distance between the points x, y ∈ G but also the position of the points relative to the boundary. We call these hyperbolic type metrics or relative metrics. These metrics are important tools for our work. An important step in the development of hyperbolic type metrics was the introduction of the quasihyperbolic and distance ratio metrics by F.W. Gehring and B.P. Palka in 1976 [165]. These two metrics became soon very useful tools in the theory of multidimensional qc maps, and for instance J. Väisälä’s theory of qc maps in Banach spaces [537] makes systematic use of these two metrics. During the past four decades, the intrinsic geometry generated by hyperbolic type metrics has developed fast; some results are discussed in the surveys of P. Hästö [204] and of M. Vuorinen [559, 560]. Also in the context of hyperbolic type metrics the ideas derived from Klein’s Erlangen Program have turned out to be very fruitful. During the past twenty years numerous authors have contributed to this field, leading to a plenitude of open problems. Some of these problems have been solved in PhD theses, or parts thereof, [73, 191, 199, 215, 227, 248, 287, 290, 291, 396, 482, 483, 561, 562, 575]. It seems natural to expect that the fruits of all this work will become visible in the coming years. The role of hyperbolic type metrics is crucial for many parts of CFT. A. Papadapoulos lists in [419, pp. 42–48] twelve metrics which occur in CFT. Various topics where metrics have a role, are discussed in the following monographs [51, 54, 262, 283, 299, 300]. 15. Numerical Approximation of Conformal Invariants Conformal invariants, for example the harmonic measure, the hyperbolic distance between points in a plane domain, and the extremal length and modulus of a curve family, are some of fundamental tools in geometric function theory. This is, in part, due to the fact that the class of meromorphic functions is invariant under pre- and postcomposition with conformal maps. Conformal invariants are often directly or indirectly connected with extremal problems and bounds for these invariants can be obtained by symmetrization methods, see A. Baernstein [43], V.N. Dubinin [121], G.V. Kuz’mina [317], and A. Yu. Solynin [493]. However, the analytic or numerical values of conformal invariants are only known in relatively few special cases.

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There are a number of numerical approximation methods applicable to numerical approximation of moduli of curve families connecting the two plates of condensers or, which is the same thing, capacities of the corresponding condensers. Nevertheless, there are very few results on this topic. Neither systematic compilation of the literature reporting computed values of condenser capacities, nor listings of numerical values of capacities of condensers of specific type, nor published computer codes for this purpose exist. Some computational results and pointers to the literature may be found in the bibliographies of the papers [70, 72, 185–187]. Apparently, a numerical approach to potential theory, efficient algorithms for harmonic measures, capacities of condensers, and for evaluation of the hyperbolic distance between two points in a simply connected domain could open new gateways for research: testing the accuracy of theoretical results might lead to new discoveries. In a recent paper M.M.S. Nasser and M.K. Vuorinen [409] gave algorithms for computing several conformal invariants in simply-connected domains. 16. Books, Collections of Surveys We indicate here some information about the literature on QRT. First, extensive bibliographies appear in C. Andreian-Cazacu [31], P. Caraman [94], and R. Kühnau [312]. Second, the bibliograhies of the books listed below give many further pointers to the most recent results. For a survey on the history of qc maps, see A. Papadopoulos [420]. Books Before 2000 For the case n = 2 the collected papers of L.V. Ahlfors [13] and O. Teichmüller [511] document the original publications of pioneers. The papers in these volumes have had far-reaching consequences for this research field. The monographs L.V. Ahlfors [14], P.P. Belinskii [59], S.L. Krushkal’–R. Kühnau [305], R. Kühnau [309], O. Lehto–K.I. Virtanen [326], and I.N. Vekua [543] are well-known reference books of the planar QRT. In the higher dimensional case n ≥ 3, the books of J. Väisälä [524], P. Caraman [94], V. M. Gol  dshte˘ın and Yu. G. Reshetnyak [171], G.D. Anderson–M.K. Vamanamurthy–M.K. Vuorinen [29] discuss qc maps, and those of Yu.G. Reshetnyak [463–466], M. Vuorinen [555], J. Heinonen–T. Kilpeläinen–O. Martio [219], S. Rickman [473] deal with qr mappings. Books After 2000 Since the year 2000 many books have been published about QRT: K.Astala–T.Iwaniec–G.Martin [36], T.Iwaniec–G.Martin [259], B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov [80], O. Martio, V. Ryazanov, U. Srebro, E. Yakubov [366], F.W. Gehring–K. Hag [159], F.W. Gehring–G. Martin–B. Palka [161], S. Hencl–P. Koskela [224], A. Fletcher–V. Markovic [140], M. Mateljevi´c [375], A. Vasil´ev [542]. During the past few decades, QRT was developed by numerous authors in the setup of metric spaces and manifolds. See for instance M. Gromov [178], J. Väisälä [537] and J. Heinonen–P.Koskela–N. Shanmugalingam–J. Tyson [221]. The handbook edited by R. Kühnau [311, 312] contains many surveys, with good bibliographies, on quasiconformal mappings. The multivolume handbook on Teichmüller spaces edited by A. Papadopoulos [418] contains numerous papers with many pointers to the literature. See also the following collections of surveys:

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21

[124, 184, 432, 558]. The initial chapter of T. Iwaniec–G. Martin [259] provides a nice survey of the higher dimensional QRT highlighting some of the main results of QRT. For information about euclidean and hyperbolic geometry we recommend [13, 51, 54, 86, 197, 283, 450]. Moduli of curve families and other conformal invariants have many applications to CFT. For various applications of these methods, see e.g. B. Branner and N. Fagella [87], V. N. Dubinin [121], M. Ohtsuka [412, 413], J. B. Garnett and D.E. Marshall [149], Ch. Pommerenke [427, 430].

Part II

Conformal Geometry

The first two chapters are devoted to a study of some geometric quantities that remain invariant under the action of the group of Möbius transformations or under one of its subgroups. Examples of such subgroups are (1) translations, (2) orthogonal maps, (3) self-maps of Rn+ = {x ∈ Rn : xn > 0}, and (4) spherical isometries. The Möbius invariance of the absolute (cross) ratio is of fundamental importance in such studies. The following three metric spaces will be central to our discussions: (a) the euclidean space Rn , (b) the Poincaré half-space Rn+ = Hn , and (c) the Möbius n space R = Rn ∪ {∞}. Each of these metric spaces is endowed with its own natural metric that is invariant under rigid motions of the space. In later chapters it will become clear that these metrics are often more natural than euclidean metric in geometric function theory. Our main sources are A. F. Beardon’s book [51] and L. V. Ahlfors’ lecture notes [12]. The German mathematician F. Klein had a vision of studying the above three geometries in a unified way using groups of isometries associated with the metrics of each space. These ideas became known as Klein’s Erlangen Program, which had a great influence not only on geometry but also on geometric function theory. In their book D. Mumford, C. Series, and D. Wright [403] apply this vision to study discrete groups, combining the beauty of computer graphics with deep theory. The Erlangen Program was an important step in the progress which lead to the present central role of conformal invariants (such as hyperbolic metric, harmonic measure, extremal length of curve family) in geometric function theory.

Chapter 3

Möbius Transformations

For x ∈ Rn and r > 0 let B n (x, r) = { z ∈ Rn : |x − z| < r }, S n−1 (x, r) = { z ∈ Rn : |x − z| = r }, denote the ball and sphere, respectively, centered at x with radius r. The abbreviations B n (r) = B n (0, r), S n−1 (r) = S n−1 (0, r), Bn = B n (1), S n−1 = S n−1 (1) will be used frequently. For t ∈ R and a ∈ Rn \ {0} we denote P (a, t) = { x ∈ Rn : x · a = t } ∪ {∞}. n

Then P (a, t) is a hyperplane in R = Rn ∪ {∞} perpendicular to the vector a, at distance t/|a| from the origin. Definition 3.1 Let D and D  be domains in Rn and let f : D → D  be a homeomorphism. f conformal if (1) f ∈ C 1 , (2) Jf (x) = 0 for all x ∈ D,  We call  and (3) f (x)h = f (x) |h| for all x ∈ D and all h ∈ Rn . If D and D  are n

domains in R , we call a homeomorphism f : D → D  conformal if the restriction of f to D \ {∞, f −1 (∞)} is conformal.

Examples 3.2 Some basic examples of conformal mappings are the following elementary transformations. (1) A reflection in P (a, t): f1 (x) = x − 2(x · a − t)

a , f1 (∞) = ∞ . |a|2

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_3

25

26

3 Möbius Transformations

(2) An inversion (reflection) in S n−1 (a, r): f2 (x) = a +

r 2 (x − a) , f2 (a) = ∞ , f2 (∞) = a . |x − a|2

(3) A translation f3 (x) = x + a , a ∈ Rn , f3 (∞) = ∞. (4) A stretching by a factor k > 0: f4 (x) = kx , f4 (∞) = ∞. (5) An orthogonal mapping, i.e. a linear map f5 with |f5 (x)| = |x| , f5 (∞) = ∞ . Then f1 and f2 are involutions, i.e. mappings that are their own inverses. Remark 3.3 The translation x → x + a can be written as a composition of reflections in P (a, 0) and P (a, 12 |a|2). The stretching x → kx, k > 0, can be √ written as a composition of inversions in S n−1 (0, 1) and S n−1 (0, k ). It can be proved that an orthogonal mapping can be composed of at most n + 1 reflections in planes (see [51, p. 23, Theorem 5.1.3]). Exercise 3.4 (1) Let f be an inversion in S n−1 (a, r) as defined in 3.2(2). Show that f −1 = f and that |x − a||f (x) − a| = r 2 for all x ∈ Rn \ {a}. By considering similar triangles show that the following identity holds for x, y ∈ Rn \ {a} : |f (x) − f (y)| =

r 2 |x − y| . |x − a||y − a|

(3.1)

(2) Using complex numbers one can write the inversion in S 1 (c, r) as z → c + r 2 /(z − c) . Show that if S 1 (c, r) is orthogonal to the unit circle S 1 , this formula can be written as z →

cz − 1 . z−c

(3) Let S 1 (c, r) be orthogonal to the unit circle. Show that a, b ∈ S 1 (c, r) iff c=

u ; v

u = a(1 + |b|2) − b(1 + |a|2) , v = ab − ab .

Use the formula in (2) to find the image of a point z under the reflection (inversion) in S 1 (c, r) in the case when c has this value. (4) Let S 1 (c, r) be orthogonal to the unit circle. Find the points of intersection of these two circles.

3 Möbius Transformations

27

Exercise 3.5 (1) For x, y ∈ Rn \ {0} let g(x, y) = |x − y|2 /(|x||y|). Applying (3.1) show that g(x, y) = g(f (x), f (y)) if f is a stretching or an inversion in S n−1 (r), r > 0. (2) For x, y ∈ Rn \ {0} let ϕ ∈ [0, π] be the angle between the segments [0, x] and [0, y] and denote s = sin(ϕ/2) . Show that |x − y| ≤ ||x| − |y|| + 2s|y|

and |x − y| ≥ s||x| + |y|| .

(3) Let x, y ∈ Rn \ {0} and |y| ≥ |x|. Show that d(y, [0, x]) ≥ |x−y| 2 . 1 (4) Let a ∈ (0, 1) and w = 2 (a + 1/a) + it, t ∈ R . Show that S 1 (w, |w − a|) is perpendicular to the unit circle S 1 . n

n

Definition 3.6 A homeomorphism f : R → R is called a Möbius transformation if f = g1 ◦· · ·◦gp where each gj is one of the elementary transformations in 3.2(1)– (5) and p is a positive integer. Equivalently (see 3.3), f is a Möbius transformation if f = h1 ◦ · · · ◦ hm where each hj is a reflection in a sphere or in a hyperplane and m is a positive integer. It follows from the inverse function theorem and the chain rule that the set of all n conformal mappings of R is a group. It is left as an easy exercise for the reader to show that the set of all Möbius transformations constitutes a subgroup of the group n of conformal mappings, and we denote it by GM(R ) or GM. Further, we shall write n

GM(D) = { f ∈ GM(R ) : f D = D } n

for D ⊂ R . We denote by O(n) the set of all orthogonal maps in Rn . A map f in GM with f (∞) = ∞ is called a similarity transformation if |f (x) − f (y)| = c|x − y| for all x, y ∈ Rn where c is a positive number. A similarity transformation can be written in the form x → cU (x) + a , where c > 0 , U ∈ O(n) , and a ∈ Rn . n

Definition 3.7 Let D and D  be domains in R . We call a C 1 -homeomorphism f : D → D  sense-preserving (orientation-preserving) if Jf (x) > 0 for all x ∈ D \ {∞, f −1 (∞)} . If Jf (x) < 0 for all x ∈ D \ {∞, f −1 (∞)} then we call f sense-reversing (orientation-reversing). One can show that reflection in a hyperplane or in a sphere is sense-reversing and hence so is the composition of an odd number of reflections. The composition of an even number of reflections is sense-preserving. For these results the reader is referred to [445, pp. 137–145]. The set of all sense-preserving Möbius transforman tions is denoted by M(R ) or M. Also we let M(D) = { f ∈ M : f D = D } if n D⊂R . Remark 3.8 One can extend Definition 3.7 so as to make it applicable to a wider class of mappings (including quasiregular mappings). This extended definition makes use of the topological degree of a mapping, which will be briefly discussed in 15.1.

28

3 Möbius Transformations

3.1 Poincaré Extension n

It will be convenient to identify R with the subset { x ∈ Rn+1 : xn+1 = 0 } ∪ {∞} n+1 of R . The identification is given by the embedding x →  x = (x1 , . . . , xn , 0) ;

x = (x1 , . . . , xn ) ∈ Rn .

(3.2)

We are now going to describe a natural two-step way of extending a Möbius n n+1 n transformation of R to a Möbius transformation of R . First, if a ∈ R and f in n , t) GM(R ) is a reflection in P (a, t) or in S n−1 (a, r), let  f be a reflection in P (a n n  or S (a , r), respectively. Then if x ∈ R and y = f (x), by 3.2(1)–(2) we get  (x) . f (x1 , . . . , xn , 0) = (y1 , . . . , yn , 0) = f

(3.3)

f preserves the plane By (3.3) we may regard  f as an extension of f . Note that  xn+1 = 0 and each of the half-spaces xn+1 > 0 and xn+1 < 0. These facts follow n from the formulae 3.2(1)–(2). Second, if f is an arbitrary mapping in GM(R ) it has a representation f = f1 ◦ · · · ◦ fm where each fj is a reflection in a plane or a m is the extension of f , and it preserves the half-spaces f 1 ◦· · ·◦f sphere. Then  f = n xn+1 > 0, xn+1 < 0, and the plane xn+1 = 0. In conclusion, every f in GM(R ) n+1 has an extension  f in GM(R ). It follows from [51, p. 31, Theorem 3.2.4] that f is called the Poincaré extension such an extension  f of f is unique. The mapping  of f . In the sequel we shall write x, f instead of  x,  f , respectively. Many properties of plane Möbius transformations hold for n-dimensional n Möbius transformations as well. The fundamental property that spheres of R (which are spheres or planes in Rn , see Exercise 3.15 below) are preserved under Möbius transformations is proved in [51, p. 28, Theorem 5.2.1].

3.1.1 Stereographic Projection The stereographic projection π(x) = en+1 +

n

π : R → S n ( 12 en+1 , 12 ) is defined by

x − en+1 , x ∈ Rn ; π(∞) = en+1 . |x − en+1 |2 n

(3.4)

Then π is the restriction to R of the inversion in S n (en+1 , 1) . In fact, we can identify π with this inversion. Because f −1 = f for every inversion f , it follows n that π maps the “Riemann sphere” S n ( 12 en+1 , 12 ) onto R .

3.1 Poincaré Extension

29

Fig. 3.1 Stereographic projection (3.4) and chordal metric (3.5) visualized

n

The spherical (chordal) metric q in R is defined by n

q(x, y) = |π(x) − π(y)| ; x, y ∈ R ,

(3.5)

where π is the stereographic projection (3.4). From the definition (3.4) and by (3.1) we obtain ⎧ |x − y| ⎪ ⎪  ; x = ∞ = y , ⎨q(x, y) =  1 + |x|2 1 + |y|2 (3.6) 1 ⎪ ⎪ . ⎩q(x, ∞) =  1 + |x|2 For x ∈ Rn \ {0} the antipodal (diametrically opposite) point  x is defined by  x=−

x |x|2

(3.7)

and we set ∞  = 0,  0 = ∞ . Then, by (3.6), q(x,  x ) = 1 and hence π(x) , π( x) are indeed diametrically opposite points on the Riemann sphere (Fig. 3.1). Exercise 3.9 Note that 

|x − y| (1 + |x||y|)2

+ (|x| −

|y|)2

= q(x, y) ≤

|x − y| . |x| + |y|

Applying (3.1) show that q(x, y) = q

 x y  |x − y| , 2 ≤ 2 |x| |y| |x||y|

30

3 Möbius Transformations

holds for x, y ∈ Rn \ {0} . Show also that |x − y| 2|x − y| ≤ q(x, y) ≤ (1 + |x|)(1 + |y|) (1 + |x|)(1 + |y|) n

for all x, y ∈ Rn . Show that q(x, y) ≥ 12 |x −y| for x, y ∈ B and that q(x, y) = 1 n 2 |x − y| for x, y ∈ ∂B . Exercise 3.10

√ (1) For 0 < t < 1 let w(t) = t/ 1 − t 2 . Show that q(0, w(t)e1 ) = t and that t w(t) 2t < < s w(s) s √ for 0 < s < t < 12 3 . (2) Let x, y ∈ Bn , n = 2 , with s = q(0, x) , t = q(0, y) . Show that   q(x, y) ≤ s 1 − t 2 + t 1 − s 2 ≤ t + s . (3) Let x, y ∈ Rn \ {0} , n = 2 , with q(0, x) > q(0, y) . Show that the strict inequality q(x, y) > q(0, x) − q(0, y) holds. (4) Show that for x, y, z ∈ Bn , x = z , √ |x − y| q(x, y) 1 |x − y| ≤ ≤ 2 . √ q(x, z) |x − z| 2 |x − z| Definition 3.11 Let (X1 , d1 ) and (X2 , d2 ) be metric spaces and let f : X1 → X2 be a homeomorphism. We call f an isometry if d2 (f (x), f (y)) = d1 (x, y) for n all x, y ∈ X1 . A map f in GM(R ) is called a spherical (chordal) isometry n if q(f (x), f (y)) = q(x, y) for all x, y ∈ R . A similarity transformation f is called a euclidean isometry if |f (x) − f (y)| = |x − y| for all x, y ∈ Rn . Orthogonal mappings and the inversion in S n−1 are examples of spherical isometries, while the translation x → x + e1 and the stretching x → 12 x are not spherical isometries (see 3.29). Reflection in a hyperplane and translations are examples of euclidean isometries. Remark 3.12 The inversion π2 (x) = en+1 +

2(x − en+1 ) , x ∈ Rn ; π2 (∞) = en+1 , |x − en+1 |2 n

is also sometimes called the stereographic projection. It maps R onto S n so that π2 (0) = −en+1 , π2 (∞) = en+1 and π2 (S n−1 ) = S n−1 . From (3.1) it follows that the spherical metric q in (3.5) can be defined in terms of π2 as

3.1 Poincaré Extension

31

Fig. 3.2 Inversion π2

Fig. 3.3 Chordal ball (3.9)

n

q(x, y) = 12 |π2 (x) − π2(y)| , x, y ∈ R . We can identify π2 with the inversion in √ S n (en+1 , 2 ) . We see that π2 maps the half-space xn+1 < 0 onto Bn+1 in such a way that π2 (−en+1 ) = 0 , π2 (0) = −en+1 (Fig. 3.2).

3.1.2 Balls in the Chordal Metric n

For x ∈ R and r ∈ (0, 1) we define the chordal ball n

Bq (x, r) = { z ∈ R : q(x, z) < r } .

(3.8)

We sometimes also use the notation Q(z, r) for Bq (z, r) . Its boundary sphere is denoted by ∂Bq (x, r) . From the Pythagorean theorem it follows that (cf. (3.7))    n x, 1 − r 2 . Bq (x, r) = R \ Bq 

(3.9)

To gain insight into the geometry of chordal balls Bq (x, r) it is convenient to study the image πBq (x, r) under the stereographic projection π (see the Fig. 3.3). Indeed, by definition (3.5) we see that   n 1 S ( 2 en+1 , 12 ). πBq (x, r) = B n+1 π(x), r

32

3 Möbius Transformations

Either by this formula or, more directly, by the definition of the spherical metric (plus the fact that Möbius transformations preserve spheres) we see that in the euclidean geometry, Bq (x, r) is a point set of one of the following three kinds (a) an open ball B n (u, s) , n n (b) the complement of B (v, t) in R , n (c) a half-space of R . n Clearly, ∂Bq (x, r) is either a sphere √ or a hyperplane of R . Formula (3.9) shows, in particular, that πBq (x, 1/ 2 ) is a half-sphere of the Riemann sphere S n ( 12 en+1 , 12 ) . The next lemma gives a quantitative form of the above observations.

Lemma 3.13 For b ∈ Rn and s ∈ (0, 1] , the following relations hold: (1) If 0 < s < (1 + |b|2)−1/2 , then Bq (b, s) = B n (a, r) , where r=

ss  (1 + |b|2 ) , |1 − s 2 (1 + |b|2)|

s =



1 − s2, n

a=

b . 1 − s 2 (1 + |b|2)

n

(2) If s > (1 + |b|2 )−1/2 , then Bq (b, s) = R \ B (a, r) , where r and a are as in (1) , (3) If b = 0 and s = (1 + |b|2 )−1/2 = 1 , then Bq (b, s) = Rn . (4) If b = 0 and s = (1 + |b|2)−1/2 , then Bq (b, s) is the open half space b · x > (|b|2 − 1)/2 .  

Proof See [29, Lemma 7.16].

Remark 3.14 In complex analysis, the spherical metric is often defined in the following way. If γ is a rectifiable curve in Rn set  |dx| σ (γ ) = 2 γ 1 + |x| and for x, y ∈ Rn define σ (x, y) = inf σ (γ ) γ

where γ runs through the collection of all rectifiable curves γ with x ∈ γ and n y ∈ γ . In a natural way this definition is extended then to all x, y ∈ R . It is n easy to show that σ (x, y) is a metric on R . Making use of (3.6) one can show that σ (x, y) = σ h(x), h(y) if h is a spherical isometry. This spherical metric is equivalent to the metric q . In fact, the two relationships σ (x, y) = 2 arcsin q(x, y) , n

hold for all distinct x, y ∈ R .

1≤

σ (x, y) ≤π, q(x, y)

3.1 Poincaré Extension

33

Exercise 3.15

√ √ (1) Show that { x ∈ Rn : xn > 0 } = Bq (en , 1/ 2 ) , Bn = Bq (0, 1/ 2 ) , √ √ n n R \ B √= Bq (∞, 1/ 2 ) , Bq (0, r) = B n (r/ 1 − r 2 ) , and B n (t) = Bq (0, t/ 1 + t 2 ) . If t ∈ (0, 1) , x = te1 , y = − 1−t 1+t e1 , show that √ x,  x , e2 , −e2 ∈ ∂Bq (y,√1/ 2 ) . (2) Show that Bq (te1 , 1/ 2 ) = B n (ue1 , r) , where u = 2t/(1 − t 2 ) , r = (1 + t 2 )/(1 − t 2 ) , provided t ∈ (0, 1) . Discuss also the case t ∈ (1, ∞) . (3) Find q(Bq (x, r)) and q(∂Bq (x, r)) , the spherical diameters of Bq (x, r) and n ∂Bq (x, r) , respectively, for x ∈ R , r ∈ (0, 1) . [Hint: Consider first the case √ r < 1/ 2 .] (4) Show that √ if z ∈ Rn , then there exists e√∈ S n−1 such that e and −e are in ∂Bq (z, 1/ 2 ) . Conclusion: If  Bq (z, 1/ 2 ) = B n (a, r) , then r 2 = 1+|a|2 .   n Show conversely that q B (b, 1 + |b|2 ) = 1 for all b ∈ Rn .

3.1.3 Absolute Ratio n

For an ordered quadruple a, b, c, d of distinct points in R we define the absolute (cross) ratio by | a, b, c, d | =

q(a, c) q(b, d) . q(a, b) q(c, d)

(3.10)

It follows from (3.6) that for distinct a, b, c, d in Rn | a, b, c, d | =

|a − c| |b − d| . |a − b| |c − d|

One of the most important properties of Möbius transformations is that they preserve absolute ratios, i.e. if f ∈ GM , then | f (a), f (b), f (c), f (d) | = | a, b, c, d |

(3.11)

n

for all distinct a, b, c, d in R . To verify this Möbius invariant property of the absolute ratio it is enough to check that (3.11) holds for each of the elementary transformations in (3.2). For inversions use (3.1). As a matter of fact, the preservation of absolute ratios is a characteristic property of Möbius transformations. It is n n proved in [51, p. 72, Theorem 5.2.7] that a mapping f : R → R is a Möbius transformation if and only if f preserves all absolute ratios. It follows from (3.11) that | a, b, c, d | = λ if and only if there exists an f in GM with f (a) = 0 , f (b) = e1 , f (d) = ∞ , |f (c)| = λ .

(3.12)

34

3 Möbius Transformations n

By property (3.11), the absolute ratio is GM(R )-invariant. Besides the absolute n ratio we shall consider later some other GM(R )-or GM(D)-invariant quantities. For instance, |a − b|/|a − c| and d(E)/d(0, E) , E ⊂ Rn \ {0} , are GM(Rn )invariant quantities. Remark 3.16 The absolute ratio depends on the order of the points. In fact, | 0, e1 , x, ∞ | = |x| =

1 , | 0, x, e1, ∞ |

| 0, e1 , ∞, x | = |x − e1 | = | 0, ∞, x, e1 | =

1 , | 0, ∞, e1 , x |

1 |x| = . |x − e1 | | 0, x, ∞, e1 |

If f ∈ GM maps a, b, c, d → 0, e1 , x, ∞ , then |x| = |a, b, c, d| and |x −e1 | = |a, b, d, c| . A thorough discussion of the complex cross-ratio can be found in [51, pp. 75–78] and [60, pp. 126–127, 6.3]. n

Lemma 3.17 Let f be a Möbius transformation of R that maps a quadruple 0, e1 , x , ∞ of distinct points onto the quadruple −e1 , y , −y , e1 , respectively, such that |y| ≤ 1 . Then |y| =

|x − e1 | 1 + |x| + t

and |y + e1 |2 =

4 |y − e1 |2 = , |x| 1 + |x| + t

 1/2 where t = (1 + |x|)2 − |x − e1 |2 . Proof By the Möbius invariance (3.11), the identities |0, e1 , x, ∞| = | − e1 , y, −y, e1 |

and |0, e1 , ∞, x| = | − e1 , y, e1 , −y|

hold, which yield the following equations |y − e1 |2 = |x||y + e1 |2

and 4|y| = |x − e1 ||y + e1 |2 .

Simplifying these equations gives 1 − |x| 4|y| − |y|2 − 1 = 2y · e1 = (1 + |y|2 ) . |x − e1 | 1 + |x| Solving this quadratic in |y| , we obtain |y| and substituting into the first equation gives |y + e1 |2 .  

3.2 Automorphisms of the Unit Ball

35 n

Exercise 3.18 For distinct points a, b, c, d ∈ R let q(a, d)2 q(b, c)2 , q(a, b) q(b, d) q(a, c) q(c, d)   sf (a, b, c, d) = s f (a), f (b), f (c), f (d) . s(a, b, c, d) =

Then s is symmetric: s(a, b, c, d) = s(d, b, c, a) = s(b, a, d, c) = s(a, c, b, d) . n Show that s is GM(R )-invariant, i.e. sf (a, b, c, d) = s(a, b, c, d)

(3.13)

n

whenever a, b, c, d ∈ R and f ∈ GM . Applying (3.6) show that s(0, x, y, ∞) = |x − y|2/(|x||y|) . It should be noted that the invariance property in Exercise 3.5 is a special case of (3.13). [Hint: Show that s(a, b, c, d) is the product of two absolute ratios.]

3.2 Automorphisms of the Unit Ball We shall give a canonical representation for the maps in M(Bn ) . Assume that f is in M(Bn ) and that f (a) = 0 for some a ∈ Bn . We denote a∗ =

a , a ∈ Rn \ {0} , |a|2

(3.14)

and 0∗ = ∞ , ∞∗ = 0 . Fix a ∈ Bn \ {0} . Let σa (x) = a ∗ + r 2 (x − a ∗ )∗ , r 2 = |a|−2 − 1

(3.15)

be the inversion in the sphere S n−1 (a ∗ , r) orthogonal to S n−1 . Then σa (a) = 0 , σa (a ∗ ) = ∞ (Fig. 3.4). Let pa denote the reflection in the (n − 1)-dimensional plane P (a, 0) through the origin and orthogonal to a and define a sense-preserving Möbius transformation by Ta = pa ◦ σa . Then, by (3.15), Ta Bn = Bn , Ta (a) = 0 , and with ea = a/|a| we have Ta (ea ) = ea , Ta (−ea ) = −ea . For a = 0 we set T0 = id , where id stands for the identity map. The proof of the following fundamental fact can be found in [12, p. 21], [51, p. 40, Theorem 5.5.1]. Lemma 3.19 If g ∈ GM(Bn ) , then there is k ∈ O(n) such that g = k ◦ Ta where a = g −1 (0) .

36

3 Möbius Transformations

Fig. 3.4 Inversion in a sphere

Definition 3.20 Let (X1 , d1 ) , (X2 , d2 ) be metric spaces, let f : X1 → X2 be continuous and L ≥ 1 , and let ω : [0, ∞) → [0, ∞) be a continuous strictly increasing function with ω(0) = 0 . We say that f is uniformly continuous with modulus of continuity ω if d2 (f (x), f (y)) ≤ ω(d1 (x, y)) for all x, y ∈ X1 , and L–Lipschitz if   d2 f (x), f (y) ≤ L d1 (x, y) for all x, y ∈ X1 . The least constant L with this property is denoted by Lip(f ) . If, in addition, f is a homeomorphism and   d1 (x, y)/L ≤ d2 f (x), f (y) ≤ L d1 (x, y) for all x, y ∈ X1 , we say that f is L–bilipschitz or that f is an L–quasiisometry. We call f a Lipschitz (bilipschitz) mapping if it is L–Lipschitz (resp. L– bilipschitz) for some L ≥ 1 . n

If h ∈ GM and x ∈ R we sometimes write hx instead of h(x) .

3.2.1 The Lipschitz Constant of Ta |Bn Let Ta = pa ◦ σa be as in 3.2, a ∈ Bn \ {0} . Since pa is a reflection in a plane and hence preserves euclidean distances, it follows that |Ta x − Ta y| = |σa x − σa y| . As

3.2 Automorphisms of the Unit Ball

37 n

|a|−1 − 1 ≤ |z − a ∗ | ≤ |a|−1 + 1 for all z ∈ B , using (3.1) we get  |a| 2   1 + |a| |x − y| , |a|−2 − 1 |x − y| = 1 − |a| 1 − |a|  |a| 2   1 − |a| |Ta x − Ta y| ≥ |x − y| |a|−2 − 1 |x − y| = 1 + |a| 1 + |a|

|Ta x − Ta y| ≤

for all x, y ∈ Bn . Hence Lip(Ta |Bn ) = sup

|T x − T y|

1 + |a| a a : x, y ∈ Bn , x = y ≤ . |x − y| 1 − |a|

In fact, Lip(Ta |Bn ) = Lip(Ta |S n−1 ) =

1 + |a| 1 − |a|

(3.16)

as we see by applying (3.1) to a pair of points x, y ∈ S n−1 with |x −a ∗| = |y −a ∗| and then letting |x − a ∗ | → |a|−1 − 1 . As Ta−1 = T−a , it follows from (3.16) that Ta |Bn is bilipschitz with the constant (1 + |a|)/(1 − |a|) . Exercise 3.21 (1) Show that |Tx y| =

|x − y| |x||y − x ∗ |

for all x, y ∈ Bn . [Hint: Apply (3.1).] (2) Let r ∈ (0, 1) . Find the “Möbius center” of the segment [0, re1 ] , i.e. the point a ∈ (0, re1 ) such that Ta (0) = −Ta (re1 ) where Ta is as in 3.2.1. [Hint: First observe that |Ta (a) − Ta (0)| = |Ta (re1 ) − Ta (a)| and hence, by the definition of Ta , a similar equality holds √ with σa in place of Ta . Next apply (1.5) to σa to obtain |a| = r/(1 + 1 − r 2 ) . Note that the point a can be found by a geometric construction, as in Fig. 3.5. See also 4.11.] Exercise 3.22 (1) Show that if ϕ ∈ (0, 12 π) , xϕ = (cos ϕ, sin ϕ) , yϕ = (cos ϕ, − sin ϕ) then there exists a Möbius transformation Ta : B2 → B2 with Ta e1 = e1 , 2 Ta (−e1 ) = −e1 , Ta (xϕ ) = e2 = −Ta (yϕ ) , and Lip(Ta | B ) = cot 12 ϕ . 1 [Hint: By 3.2.1 we see that a ∗ = |a| e1 and e2 , xϕ , a must be collinear. Now 1 |a|

= tan( 14 π + 12 ϕ) ,

1+|a| 1−|a|

= cot 12 ϕ , and the result follows from (3.16).]

38

3 Möbius Transformations

Fig. 3.5 Möbius center in Exercise 3.21 (2)

(2) Let ϕ ∈ (0, 12 π) , Fϕ = { x ∈ S n−1 : x1 = cos ϕ } , and let Ta ∈ M(Bn ) with Ta Fϕ = Fϕ . Show that a = (cot 12 ϕ)2 e1 . Assume next that 0 < α ≤ β < 12 π and Tb Fα = Fβ . Find b . Exercise 3.23 (1) Let 0 < s < 1 . Applying (3.1) as in 3.2.1 show that 1 − s2 1 |x − y| ≤ |Ta x − Ta y| ≤ |x − y| 2 2 (1 + s ) 1 − s2 n

for all a, x, y ∈ B (s) . (2) For a, x ∈ Bn with a = 0 and ea = a/|a| show that |Ta x| ≤ |Ta (−|x|ea )| . [Hint: See (3.18).]

3.2.2 The Ahlfors Bracket Vector algebra gives, with x ∗ = x/|x|2, |x|4 |y − x ∗ |2 = (y|x|2 − x) · (y|x|2 − x) = |x|2 (1 − 2x · y + |x|2 |y|2)

3.2 Automorphisms of the Unit Ball

39

and hence |x|2 |y − x ∗ |2 = 1 − 2x · y + |x|2 |y|2 . The Ahlfors bracket A[x, y] is defined by A[x, y]2 =1 − 2x · y + |x|2|y|2 = |x|2 |y − x ∗ |2 = (1 − |x|2 )(1 − |y|2 ) + |x − y|2 . (3.17) Because pa is a euclidean isometry and Ta = pa ◦ σa , we have |Ta x| = |Ta x − Ta a| = |σa x − σa a| =

|x − a| |x − a| r 2 |x − a| = = ∗ ∗ ∗ |x − a ||a − a | |x − a ||a| A[x, a] (3.18)

where we used r 2 = |a|−2 (1 − |a|2 ), |a − a ∗ | = (1 − |a|2)/|a| . By (3.17), (3.18) we obtain 1 − |Ta x|2 = 1 −

|x − a|2 (1 − |x|2 )(1 − |a|2 ) = . A[x, a]2 A[x, a]2

(3.19)

x − a∗ . A[x, a]2

(3.20)

Now we can also write σa x = a ∗ + r 2 (x − a ∗ )∗ = a ∗ + (1 − |a|2 )

The Ahlfors bracket is a handy notation for calculations involving Möbius transformations and hyperbolic geometry, see [29, Chapter 7], [12]. Lemma 3.24 For a ∈ Bn \ {0} , x ∈ Rn , A = A[x, a] we have Ta x =

(1 − |a|2)(x − a) − |x − a|2 a . A2

(3.21)

Proof Because pa z = z − 2(a · z)a ∗ and Ta = pa ◦ σa , we get by (3.20) A2 Ta x = A2 pa ◦ σa x = a ∗ A2 + (1 − |a|2)(x − a ∗ ) − 2(A2 + (1 − |a|2)(a · x − 1))a ∗ where we used a · a ∗ = 1 . Further, A2 Ta x = a ∗ (−A2 − 2(1 − |a|2)(a · x − 1)) + (1 − |a|2)(x − a ∗ ) = a ∗ (−|x|2 |a|2 + 2a · x − 1 − 2(1 − |a|2 )(a · x − 1) − (1 − |a|2 )) + (1 − |a|2 )x = a ∗ (−|x|2 |a|2 + 2|a|2a · x − |a|2 ) + (1 − |a|2 )x .

40

3 Möbius Transformations

Thus Ta x = −

a (1 − |a|2 )x (1 − |a|2 )(x − a) − |x − a|2a 2 (|x| − 2a · x + 1) + = . A2 A2 A2  

3.3 Chordal Isometries We are now going to study the action of chordal isometries, proving a representation for them similar to that of GM(Bn ) in 3.19. By (3.16) and 3.19 we see that g ∈ GM(Bn+1 ) is a euclidean isometry iff g(0) = 0 . Next we we shall reformulate n this fact for maps in GM(R ) . Let√ p be the reflection in the hyperplane xn+1 = 0 n n+1 and f1 the inversion in S (en+1 , 2 ) , and set f = f1 ◦ p . Then f Rn+1 + =B n and f (en+1 ) = 0 , q(x, y) = 12 |f (x) − f (y)| for all x, y ∈ R . Assume now n n+1 that h ∈ GM(R ) is given and that  h ∈ GM(R ) is its Poincaré extension. We see that ϕ = f ◦  h ◦ f −1 ∈ GM(Bn+1 ) is a euclidean isometry if and only if  . One can show that h is a spherical isometry iff  h(e ) = e h(e ) = e n+1

n+1

n+1

n+1

(see [51, p. 42, Theorem 3.6.1]). In particular, the inversion in S n−1 (a, r) ⊂ Rn is a spherical isometry iff en+1 ∈ S n (a , r) ⊂ Rn+1 , i.e. iff r 2 = 1 + |a|2 .

(3.22)

√ We recall (see (3.9), 3.15) that by virtue of (3.22) B n (a, r) = Bq (z, 1/ 2 ) for some z ∈ Rn . n n We define a spherical isometry tz in M(R ) which maps a given point z ∈ R to 0 as follows. For z = 0 let tz = id and for z = ∞ let tz = p ◦ f , where f is the inversion in S n−1 and p is the reflection in the (n − 1)-dimensional plane x1 =0 . For z ∈ Rn \ {0} let sz be the inversion in S n−1 (−z/|z|2 , r) , where r = 1 + |z|−2 . According to the criterion (3.22), the inversion sz is a spherical isometry and it is easy to show that sz (z) = 0 . Let pz be the reflection in the plane P (z, 0) . Defining tz = pz ◦ sz

(3.23)

n

we see that tz ∈ M(R ) is a spherical isometry with tz (z) = 0 . Hence    tz (Bq (z, r)) = Bq (0, r) = B n r/ 1 − r 2 , |tz (x)|2 = n

for all x, z ∈ R , r ∈ (0, 1) .

q(x, z)2 1 − q(x, z)2

(3.24)

3.3 Chordal Isometries

41

In the above discussion we showed that the inversion sz is a spherical isometry by exploiting a result from [51]. Next we shall show this by a direct computation. n Lemma 3.25 For  a point z ∈ R \ {0} let sz be the inversion in the sphere n−1 2 S (−z/|z| , 1 + |z|−2 ) . Then sz (z) = 0 and sz is a chordal isometry.

Proof It is easy to show that sz (z) = 0 . By (3.1) we obtain for x, y ∈ Rn (1 + |z|−2 )|x − y| (1 + |z|−2 )|x − y| = , |sz (x) − sz (y)| = x + z/|z|2 y + z/|z|2 |x − z||y − z| where  z = −z/|z|2 as in (3.7). Further by (3.1) |sz (x)| = |sz (x) − sz (z)| = |sz (y)| =

|x − z| (1 + |z|−2 )|x − z| = , |x − z||z − z| |z||x − z|

|y − z| . |z||y − z|

Substituting these identities into (3.6) we obtain ⎧   |sz (x) − sz (y)| ⎪ ⎪ q sz (x), sz (y) =   ⎪ ⎪ ⎪ 1 + |sz (x)|2 1 + |sz (y)|2 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩= 

(3.25) (1 + |z|2 )|x − y|  . |z|2 |x − z|2 + |x − z|2 |z|2 |y − z|2 + |y − z|2

z )|2 = 1 or, By the Pythagorean theorem |π(x) − π(z)|2 + |π(x) − π( 2 2 equivalently, q(x, z) + q(x, z ) = 1 or |x − z|2 |x − z|2 + =1. (1 + |x|2 )(1 + |z|2 ) (1 + |x|2 )(1 + |z|−2 ) This relation implies ⎧ z|2 |x − z|2 |x − z|2 |x − z|2 |z|2 |x − ⎪ 2 ⎪ = + = + , 1 + |x| ⎪ ⎨ 1 + |z|2 1 + |z|−2 1 + |z|2 1 + |z|2 ⎪ ⎪ z|2 |y − x|2 |z|2 |y − ⎪ ⎩ 1 + |y|2 = + . 1 + |z|2 1 + |z|2 By substituting (3.26) into (3.25) we obtain   q sz (x), sz (y) = q(x, y) ,

(3.26)

42

3 Möbius Transformations

Fig. 3.6 Proof of Lemma 3.25

showing that sz preserves the spherical distance between points x, y in Rn . It is left as an exercise for the reader to prove the case when x or y equals ∞ (Fig. 3.6).   Lemma 3.26 A Möbius transformation h is a chordal isometry if and only if th(0) ◦ h ∈ O(n) . Proof Assume that h ∈ GM is a chordal isometry. Then f = th(0) ◦ h ∈ GM(Bn ) with f (0) = 0 , and hence f ∈ O(n) by 3.19. The converse implication is   trivial. In some questions it is useful to apply the following isometric decomposition of an inversion, which follows from [51, p. 31, Theorem 3.2.4]. Lemma 3.27 Let a ∈ Rn , r > 0 , and let b ∈ Rn , u > 0 , be such that B n (a, r) = Bq (b, u) . If f is the inversion in S n−1 (a, r) , then f = tb−1 ◦ f1 ◦ tb , where tb√is the chordal isometry defined in (3.23) and f1 is the inversion in S n−1 (u/ 1 − u2 ) = ∂Bq (0, u) . Exercise 3.28 Show that B n (a, r) and B n (v) , where r 2 < 1 + |a|2 , v= 

2r (1 + (|a| +

r)2 )(1 + (|a| −

r)2 ) + 1 + |a|2 − r 2

3.3 Chordal Isometries

43

Fig. 3.7 Lemma 3.27

have equal spherical diameters. Note that v < 1 . Conclusion: The inversion f1 in 3.27 is in fact the inversion in a euclidean sphere with radius v and center 0 (Fig. 3.7). Lemma 3.29 Each of the following Möbius transformations is a bilipschitz mapping in the spherical metric with the given constant: (1) f (x) = kx , k ≥ 1 : Lip(f ) = k . (2) The inversion in S n−1 (t) , t ∈ (0, 1) : Lip(f ) = t −2 . (3) The inversion in S n−1 (a, r) , r 2 < 1 + |a|2 :  Lip(f ) =

(1 + (|a| + r)2 )(1 + (|a| − r)2 ) + 1 + |a|2 − r 2 2r

2 .

   (4) f (x) = x + b : Lip(f ) = 1 + 12 |b| |b| + 4 + |b|2 . Proof (1) Clearly f B n ( 1k ) = Bn . If π2 is the map in 3.12, then √  π2−1 B n ( 1k ) = S n B n (−en+1 , 2/ 1 + k 2 ) = A n = { x ∈ Sn : x and π2 Bn = S− n+1 < 0 } . −1 n n Hence π2 ◦ f ◦ π2 : S → S n maps A onto S− . Let α be the angle 1 between [0, en+1 ] and [en+1 , k e1 ] . Obviously tan α = k1 and the Lipschitz constant of π2 ◦ f ◦ π2−1 in the euclidean metric of Rn+1 (restricted to S n ) is the same as the Lipschitz constant of f in the spherical metric, Lip(f ) . It follows from 3.22(1) that Lip(f ) = k (Fig. 3.8). (2) Since the proof is similar to the above proof, we indicate only the changes. First n f maps S n−1 (t 2 ) onto S n−1 (and B n (t 2 ) onto R \ Bn ). As above in the proof of part (1) we see that Lip(f ) = t −2 . (3) The proof follows from 3.27, 3.28, and part (2).

44

3 Möbius Transformations

Fig. 3.8 Left: Proof of Lemma 3.29 (1). Right: Proof of Lemma 3.29 (4)

(4) Again the proof is similar to the one in (1). Observe first that g = π2 ◦ f ◦ π2−1 preserves the 2 -dimensional plane containing en+1 , −en+1 and −b , and that g(en+1 ) = en+1 , g(π2 (−b)) = −en+1 . By 3.19 we see that g = k◦ Ta , k ∈ O(n + 1) , Ta ∈ GM(Bn+1 ) . By elementary geometry |a| = 1/ 1 + 4|b|−2 , and hence (3.16) yields     4 + |b|2 + |b| 1 + |a| = = 1+ 12 |b| |b|+ 4 + |b|2 , Lip(f ) = Lip(g) = 2 1 − |a| 4 + |b| − |b|  

which completes the proof.

The important basic formula (3.1) about euclidean distances under inversion has a counterpart for chordal distances given in the next theorem. Theorem 3.30 Let B n (a, t) = Bq (b, r) , and let g be the inversion in S n−1 (a, t) . Then q(g(x), g(y)) =

(rr  )2 q(x, y) , H (x)H (y)

r =



1 − r2 ,

where H (x) =



r  4 q(x, b)2 + r 4 q(x,  b)2

and  b = −|b|−2b .   √ n−1 Corollary 3.31 Let u ∈ (0, 1/ 2 ] , let f be the inversion in S (a, r) , and let n S n−1 (a, r) = ∂Bq (b, u) for some b ∈ R . Then Lip(f ) = u−2 − 1 . Proof See [28, Theorem 7.29].

3.3 Chordal Isometries

45

Proof By 3.27 f and tb ◦ f ◦ tb−1 = g have equal Lipschitz constants. By 3.27 g √ is the inversion in S n−1 (u/ 1 − u2 ) = ∂Bq (0, u) . Hence by 3.29(2) and 3.15(1), Lip(f ) = u−2 − 1 .   Exercise 3.32 Let x, y, w be three points in Rn . Show that q(x, y)/c ≤ q(x − w, y − w) ≤ c q(x, y) where c = Lip(h) and h(x) = x − w . [Hint: 3.29(4).] Exercise 3.33 (Continuation to 3.10(4) and 3.28) Assume that x, y, z ∈ B n (a, r) with x = z . Show that q(z, y) |z − y| 1 |z − y| ≤ ≤c , c |z − x| q(z, x) |z − x| n

where c depends only on q(B (a, r)) .

√ Exercise 3.34 Let 0 < a < 1, b = a/(1 + 1 − a 2 ) and for 0 < t < 1 let Yt be the circle centered at (t + 1/t)/2 and the segment [t, 1/t] as its diameter. Show that Yt is orthogonal √ to the unit circle and that Yb intersects the unit circle at the point p = (a, 1 − a 2 ). Show that the  point (b, 0) is on the segment

[p, s], s = (0, −1). Find the point q = (q1 , 1 − q12 ) in the upper half plane where Ya intersects the unit circle and show that (0, 1), q, (1/a, 0) are collinear. Exercise 3.35 Find a Möbius transformation z →

az + b , cz + d

ad − bc = 0,

which maps H2 = {(x, y) ∈ R2 : y > 0} onto B2 = {(x, y) ∈ R2 : x 2 + y 2 < 1} such that (−1, 0, 1) → (1, i, −1) . Exercise 3.36 Prove the following facts. (1) Distinct points z1 , z2 , z3 ∈ C are collinear iff (z1 − z3 )/(z2 − z3 ) ∈ R. (2) Given three points z1 , z2 , z3 ∈ C, find a formula for the area of the positively oriented triangle [z1 , z2 , z3 ] . Hint: One version of the formula is 1 |z1 (z2 − z3 ) + z2 (z3 − z1 ) + z3 (z1 − z2 )|. 4 Another variant is: 1 Im(z1 z2 + z2 z3 + z3 z1 ) . 2

46

3 Möbius Transformations

(3) Let a, b ∈ C be distinct points and w the image of a point z under reflection in the line through the points a and b . Show that w=

b−a b−a

z+

ab − ba b−a

.

Exercise 3.37 (1) Show that if x ∈ Bn , a ∈ Bn \ {0} , then σa (x) ∈ Bn , where σa (x) is defined in (3.15). (2) Let h(w) = r 2 w/|w|2 , r > 0, w ∈ Rn \{0} . Show that if x, y ∈ Rn \{0}, |x| ≤ |y|, λ = (|x| + |x − y|)/|x|, z = λx, then |h(x) − h(z)| ≤ |h(x) − h(y)| ≤ 3|h(x) − h(z)| . √ Exercise 3.38 Show that the inversion in S n−1 (−en , 2) maps W = {z ∈ Rn : |z| < 1, zn = 0} onto S n−1 ∩ Hn , where Hn = {z ∈ Rn : zn > 0}. For B, C ∈ W \{0} consider the circular arc U ⊂ W with B, C ∈ U and perpendicular to ∂W at the points A and D ( A, B, C, D are on U in this order). Let B1 , C1 ∈ S n−1 be the image points of B, C under the inversion. Let B2 be the projection of B1 onto the segment [A, D]. By looking at Fig. 3.9, show that (1) 0, B, B2 are collinear. (2) |A, B, C, D| = |A, B1 , C1 , D| . Exercise 3.39 Let h(x) = x/|x|2. Show that h maps the sphere S n−1 (be1 , s) (assume that b > 1 + s ) onto a sphere. (Hint: Write u = (b − s)e1 , v = (b + s)e1 . If the image is a sphere S n−1 (c, t) , then clearly c = (h(u) + h(v))/2 and t = |h(u) − h(v)|/2. Hence it remains to show that |z − be1 | = s implies |h(z) − c| = t . )

2 Fig. 3.9 Möbius map of S+ 2 onto B

1

C

1

0.5 D

B

1

0

C B

0.5

A

1 1 0 1

1

0.5

0

0.5

1

3.3 Chordal Isometries

47

Exercise 3.40 The sets [−e1 , 0] and [ae1, ∞] , a > 0, can be mapped by a Möbius transformation onto [−e1 , e1 ] and [be1, ∞] ∪ [−be1, ∞] , resp., for some b > 1 . Express b in terms of a . Notice that [x, ∞] = {xt : t ≥ 1} , if x ∈ Rn \ {0} . Exercise 3.41 Show that for all a, x, y ∈ Bn |Ta x − Ta y|2 |x − y|2 = . (1 − |Ta x|2 )(1 − |Ta y|2 ) (1 − |x|2 )(1 − |y|2 ) (Hint: Ahlfors bracket.) Exercise 3.42 For ϕ ∈ (0, 12 π) define the points xϕ = (cos ϕ, sin ϕ) and yϕ = (cos ϕ, − sin ϕ). Then there exists a Möbius transformation Ta : B2 → B2 with Ta e1 = e1 , Ta (−e1 ) = −e1 , Ta (xϕ ) = e2 = −Ta (yϕ ). Find |a|. Exercise 3.43 (1) Show that the inversion f in S n−1 (a, r) , preserves the upper half-space f (Hn ) = Hn , if an = 0 . (2) Show that the expression |x − y|2 , 2xn yn where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) , is invariant under the inversion in S n−1 (a, r) when an = 0 . Remark 3.44 As shown e.g. by the Riemann mapping theorem, the class of conformal mappings in the plane is extremely large. According to a deep classical theorem of Liouville the multidimensional case is radically different: If D is a domain in Rn , n ≥ 3 , and f : D → f D ⊂ Rn is conformal, then f can n be written in the form f = g|D where g ∈ GM(R ) . This result was proved by Liouville for C 3 -mappings, and a similar result under weaker hypotheses was obtained by F. W. Gehring in 1962 [151, Theorem 16] and by Yu. G. Reshetnyak in 1967 (see [464] for more details). Another proof was given by B. Bojarski and T. Iwaniec [81]. Further generalizations of these results were obtained by Yu. G. Reshetnyak (see [464, 466] and the references therein). For additional references and further historical details, see [161, pp. 64–65], [450, p. 142], [571, p. 437]. Notes 3.45 The significance of F. Klein’s Erlangen program is analysed in [403, 434], [450, pp. 32–34]. Good references for the theory of Möbius transformations n in R are A. F. Beardon’s book [51] and L. V. Ahlfors’ lecture notes [12]. The book of J.G. Ratcliffe [450] provides a reader-friendly and detailed introduction, with many historical remarks, to the topics of this and the next chapter [450, pp.98– 99, 142–143]. For instance, according to [450, p.142], the stereographic projection was already known to Ptolemy and the cross ratio was introduced by Möbius. The comprehensive geometry book of M. Berger [60], [61] contains numerous

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3 Möbius Transformations

excellent illustrations related to the topic of this chapter. For two-dimensional Möbius transformations, see D. Hilbert and S. Cohn–Vossen [234], C. Carathéodory [95], L. R. Ford [142]. For the geometry of complex numbers see [109, 394, 399].

Chapter 4

Hyperbolic Geometry

The parallel postulate of the euclidean geometry says that given a line and a point not on it, there exists only one line through this point and not intersecting the given line. Since the days of Euclid, it had been an open problem studied by many generations of mathematicians, whether the parallel postulate follows from the other axioms of geometry. The discovery of noneuclidean geometry by N.I. Lobachevsky in 1829 solved this two millennia old question in the negative [338]. He showed that without the parallel axiom one can build a noneuclidean geometry, which we shall refer to as hyperbolic geometry, where many results of the euclidean geometry have their counterparts but which is fundamentally different from euclidean geometry. A few years later, in 1832, J. Bolyai independently made some similar discoveries. For these and other historical remarks about the origin of hyperbolic and noneuclidean geometries, see [450, pp. 32–34, 98–99] and [389]. In his survey [173, pp. 89–93] on the history of geometry, J. Gray mentions Alhazen (also known as Ibn al-Haytham) and Omar Khayyam among the many mathematicians who studied the role of the parallel postulate in euclidean geometry. Hyperbolic geometry can be developed in the context of two spaces or, as they are sometimes called, models. These two models of the hyperbolic space are the unit ball Bn and the Poincaré half-space Hn = Rn+ = { (x1 , . . . , xn ) ∈ Rn : xn > 0 } . In the planar case n = 2 these models of hyperbolic geometries were studied, in particular, by H. Poincaré. Each model is equipped with a metric, called the hyperbolic metric ρ , also known as the Poincaré metric. It is unique up to a multiplicative constant in each model. These two constants are chosen in such a

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_4

49

50

4 Hyperbolic Geometry

way that for all x, y ∈ Bn   ρHn h(x), h(y) = ρBn (x, y) whenever h ∈ GM and hBn = Hn . Therefore both models are conformally compatible in the sense that the two metric spaces (Bn , ρ) and (Hn , ρ) can be identified. In what follows we shall use the symbols Rn+ and Hn interchangeably.

4.1 The Poincaré Half Space For A ⊂ Rn let A+ = { x ∈ A : xn > 0 }. We define a weight function w : Rn+ → R+ = { x ∈ R : x > 0 } by w(x) =

1 , x = (x1 , . . . , xn ) ∈ Rn+ . xn

(4.1)

If γ : [0, 1) → Rn+ is a continuous mapping such that γ [0, 1) is a rectifiable curve with length s = (γ ), then γ has a normal representation γ 0 : [0, s) → Rn+ parametrized by arc length (see J. Väisälä [524, p. 5]). Then |γ 0 (r)−γ 0 (t)| ≤ |r −t| for all r, t ∈ [0, s]. The hyperbolic length of γ [0, 1) is defined by 

s

h (γ [0, 1)) = 0

 |dx| 0  0 w(γ (γ ) (t) (t))dt = . γ xn

(4.2)

If A ⊂ Rn+ is a (Lebesgue) measurable set we define the hyperbolic volume of A by  mh (A) =

w(x)n dm(x) ,

(4.3)

A

where m stands for the n-dimensional Lebesgue measure and w is as in (4.1). If a, b ∈ Rn+ , then the hyperbolic distance between a and b is defined by  ρ(a, b) = inf h (α) = inf α∈ ab

α∈ ab α

|dx| , xn

(4.4)

where ab stands for the collection of all rectifiable curves in Rn+ joining a and b . Sometimes the more complete notation ρRn (a, b) or ρHn (a, b) will be + employed. The infimum in (4.4) is in fact attained: for given a, b ∈ Rn+ there exists a circular arc L perpendicular to ∂Rn+ such that the closed subarc J [a, b] of L with end points a and b satisfies  ρ(a, b) = h (J [a, b]) = J [a,b]

|dx| . xn

(4.5)

4.1 The Poincaré Half Space

51

Fig. 4.1 Hyperbolic geodesics in H2

If a and b are located on a normal of ∂Rn+ , then J [a, b] = [a, b] = { (1 − t)a + tb : 0 ≤ t ≤ 1 } (cf. [51, p. 134]). Because of the (hyperbolic) length-minimizing property (4.5), the arc J [a, b] will be called the geodesic segment joining a and b (Fig. 4.1). Knowing the geodesics, we calculate the hyperbolic distance in two special cases. First, for r, s > 0 we obtain  ρ(ren , sen ) =

r s

dt r = log . t s

(4.6)

Second, if ϕ ∈ (0, 12 π) we denote uϕ = (cos ϕ)e1 + (sin ϕ)en and calculate (Fig. 4.2)  ρ(en , uϕ ) = J [uϕ ,en ]

dα = sin α

π/2

dα = log cot 12 ϕ . sin α

(4.7)

ϕ

We shall often make use of the hyperbolic functions sh x = sinh x , ch x = cosh x , th x = tanh x , cth x = coth x and their inverse functions, listed below

Fig. 4.2 Hyperbolic geodesic in the upper half plane as in (4.7)

52

4 Hyperbolic Geometry

Fig. 4.3 Hyperbolic distance and absolute ratio (4.9)

in 4.1. The above formulae (4.6) and (4.7) are special cases of the general formula (see [51, p. 35]) ch ρ(x, y) = 1 +

|x − y|2 , x, y ∈ Hn = Rn+ . 2xn yn

(4.8)

Note that by this formula the hyperbolic distance ρ(x, y) is completely determined once the euclidean distances xn = d(x, ∂Hn ) , yn = d(y, ∂Hn ) , and |x − y| are known. For another formulation of (4.7) let z, w ∈ Hn , let L be an arc of a circle perpendicular to ∂Hn with z, w ∈ L and let {z∗ , w∗ } = L∩∂Hn , the points being labelled so that z∗ , z, w, w∗ occur in this order on L (Fig. 4.3). Then (cf. [51, p. 133, (7.26.)]) ρ(z, w) = log | z∗ , z, w, w∗ | .

(4.9)

Note that (4.6) is a special case of (4.9) when z∗ = 0 and w∗ = ∞ because | 0, z, w, ∞ | = |w|/|z| for z, w ∈ Hn . The invariance of ρ is apparent by (4.9) and (3.11): Given f in GM(Hn ) and x, y ∈ Hn , then ρ(x, y) = ρ(f (x), f (y)) .

(4.10)

For a ∈ Hn and M > 0 the hyperbolic ball { x ∈ Hn : ρ(a, x) < M } is denoted by Bρ (a, M) . It is well known that Bρ (a, M) = B n (z, r) for some z and r (this also follows from (4.10)! ). This fact together with the observation that λten , (t/λ)en ∈ ∂Bρ (ten , M) , λ = eM (cf. (4.6)), yields (Fig. 4.4) ⎧   n ⎪ ⎪ ⎨Bρ (ten , M) = B (t ch M)en , t sh M ,

B n (ten , rt) ⊂ Bρ (ten , M) ⊂ B n (ten , Rt) , ⎪ ⎪ ⎩r = 1 − e−M , R = eM − 1 .

(4.11)

Remark 4.1 The hyperbolic functions sh x , ch x , th x , cth x and their inverse functions arsh x , arch x , arth x , arcth x (denoted by some authors as sinh−1 x ,

4.1 The Poincaré Half Space

53

Fig. 4.4 The hyperbolic disk Bρ (te2 , M) in H2

cosh−1 x etc.) occur often in what follows. Recall that ⎧ √ ⎪ arsh x = log(x + x2 + 1 ) , x ≥ 0 , ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨arch x = log(x + x 2 − 1 ) , x ≥ 1 , x ⎪ arth x = 12 log 11 + ⎪ −x , 0≤x 1. For easy reference we record the following simple formulae: ⎧ √ ⎪ log(1 + x) ≤ arsh x ≤ 2 log(1 + x) , x ≥ 0 , ⎪ ⎪ ⎪  ⎪  ⎨     2 log 1 + 12 (x − 1) ≤ arch x ≤ 2 log 1 + 2(x − 1) , x ≥ 1 , ⎪   ⎪ ⎪ ⎪ ⎪ ⎩arch x = 2 log( x + 1 + x − 1 ) , x ≥ 1 . 2 2 (4.12) Exercise 4.2 Verify the following elementary relations. (1) 1 − e−s ≤ th s ≤ 1 − e−2s ≤ (2) If s ≥ 0 , then

2s for s ≥ 0 . 1+s

th s =

th 2s  . 1 + 1 − th2 2s

54

4 Hyperbolic Geometry

Further, if u ∈ [0, 1] and 2s = arth u , then th s =

 1 u u3 1 ≤ 2 (u + u3 ) . ≤ u+ √ √ 2 2 2 1+ 1−u 1+ 1−u

(3) log th s = −2 arth(e−2s ) , s > 0 . x ≤ 1 − e−x ≤ x , for x > −1 [1, 4.2.32]. (4) 1+x (5) Show that  1 + th x p 1 + th px = 1 − th px 1 − th x for p = 1, 2, . . . and x > 0 . 2t . Show that g  (t) > 0 for t > −1 and hence (6) Let g(t) = log(1 + t) − 2+t √t g(t) > 0 for t > 0 . Note that with u = this last inequality also 1+ 1+t implies for t > 0 log(1 + t) = 2 log(1 + u) > 2

4t 2u 2t = , > √ 2 2+u 2+t (1 + 1 + t)

which is a better lower bound for log(1+t) than g(t) > 0 . (Because g(t) > 0 for t > 0 it also follows that v(t) = (1/t) log(1 + t) is decreasing for t > 0 . This monotonicity property implies Bernoulli’s inequality (5.6).) t (7) Let h(t) = √ − log(1 + t) . Show that h (t) > 0 for t > −1 and hence 1+t h(t) > 0 for t > 0 . As in (6) we have for t > 0 log(1 + t) = 2 log(1 + u) < √

2u 1+u

=

2t √ √ (1 + 1 + t) 4 1 + t

which yields a better upper bound for log(1 + t) than h(t) > 0 . (8) Let 0 < a ≤ 1 . Then, the function f1 (t) =

log(1 + t) log(1 + t a )

is increasing on (0, ∞) with range (0, 1/a) . (9) Let 0 < a ≤ 1 . Then, the function f2 (t) =

log(1 + t a ) . loga (1 + t)

is decreasing on (0, 1) and increasing on (1, ∞) .

4.2 The Poincaré Unit Ball

55

4.2 The Poincaré Unit Ball So far we have discussed only the hyperbolic geometry of Hn = Rn+ . Now we are going to give the corresponding formulae for Bn . The weight function w : Bn → R+ is now defined by w(x) =

2 , x ∈ Bn , 1 − |x|2

(4.13)

(cf. (4.1)). The hyperbolic distance between a and b in Bn , denoted by ρBn (a, b) = ρ(a, b) , is defined by a formula analogous to (4.5); the same is true about the hyperbolic volume of a measurable set A ⊂ Bn . For a, b ∈ Bn the geodesic segment J [a, b] joining a to b is an arc of a circle orthogonal to S n−1 . In a limiting case the points a and b are located on a euclidean line through 0 (Fig. 4.5). In particular, J [0, te1 ] = [0, te1 ] for 0 < t < 1 and we have  ρ(0, te1 ) = [0,t e1 ]

2|dx| = 1 − |x|2

t 0

2ds 1+t = 2 arth t . = log 2 1−t 1−s

(4.14)

It follows from (4.14) that for s ∈ (−t, t) 1 + t 1 − s  ρ(se1 , te1 ) = log · . 1−t 1+s

(4.15)

A counterpart of (4.8) for Bn is sh2

 2 ρ(x, y) =

1

|x − y|2 , x, y ∈ Bn , (1 − |x|2 )(1 − |y|2 )

Fig. 4.5 Left: Hyperbolic geodesics in B2 . Right: The hyperbolic disk Bρ (z, M) in B2

(4.16)

56

4 Hyperbolic Geometry

(cf. [51, p. 40]). As in the case of Hn , we see by (4.16) that the hyperbolic distance ρ(x, y) between x and y is completely determined by the euclidean quantities |x − y| , d(x, ∂Bn ) , d(y, ∂Bn ) . Finally, by [53, Lemma 3.1] we also have ρ(x, y) = log | x∗ , x, y, y∗ | = sup{log |u, x, y, v| : u, v ∈ ∂Bn } ,

(4.17)

where x∗ , y∗ are defined as in (4.9): If L is the circle orthogonal to S n−1 with x, y ∈ L , then {x∗ , y∗ } = L ∩ S n−1 , the points being labelled so that x∗ , x, y, y∗ occur in this order on L . It follows from (4.17) and (3.11) that ρ(x, y) = ρ(h(x), h(y))

(4.18)

for all x, y ∈ Bn whenever h is in GM(Bn ) . Finally, in view of (3.11), (4.9), and (4.17) we have ρBn (x, y) = ρHn (g(x), g(y)) , x, y ∈ Bn ,

(4.19)

whenever g is a Möbius transformation with gBn = Hn . Thus we see that the obvious counterpart of (4.17) also holds for Hn . It is well known that the balls Bρ (z, M) of (Bn , ρ) are balls in the euclidean geometry as well, i.e. Bρ (z, M) = B n (y, r) for some y ∈ Bn and r > 0 . Making use of this fact, we shall find y and r . Let Lz be a euclidean line through 0 and z and {z1 , z2 } = Lz ∩ ∂Bρ (z, M) , |z1 | ≤ |z2 | . We may assume that z = 0 since with obvious changes the following argument works for z = 0 as well. Let e = z/|z| and z1 = se , z2 = ue , u ∈ (0, 1) , s ∈ (−u, u) . It follows from (4.15) that  1 + |z| 1 − s  ρ(z1 , z) = log · =M, 1 − |z| 1 + s  1 + u 1 − |z|  · =M. ρ(z2 , z) = log 1 − u 1 + |z| Solving these equations for s and u and using the fact that Bρ (z, M) =   + z2 ), 12 |u − s| one obtains the following formulae (Exercise: Verify the computation.): B n 12 (z1

⎧ n ⎪ ⎨Bρ (x, M) = B (y, r) , (1 − |x|2 )t x(1 − t 2 ) ⎪ ⎩y = , r = , t = th 12 M , 1 − |x|2t 2 1 − |x|2 t 2

(4.20)

and ⎧     n n ⎪ ⎨B x, a(1 − |x|) ⊂ Bρ (x, M) ⊂ B x, A(1 − |x|) , t (1 + |x|) t (1 + |x|) ⎪ ⎩a = , A= , t = th 12 M . 1 + |x|t 1 − |x|t

(4.21)

4.2 The Poincaré Unit Ball

57

Fig. 4.6 Möbius transformation Tx

We shall often need a special case of (4.20): Bρ (0, M) = B n (th 12 M) .

(4.22)

A standard application of formula (4.22) is the following observation. Let Tx be in M(Bn ) as defined in 3.2 with Tx (x) = 0 . Fix x, y ∈ Bn and z ∈ J [x, y] with ρ(z, x) = ρ(z, y) = 12 ρ(x, y) (Fig. 4.6). Then Tz (x) = −Tz (y) and (4.22) yields  |Tx (y)| = th 12 ρ(x, y) , |Tz (x)| = th 14 ρ(x, y) .

(4.23)

We next derive a few inequalities from (4.21). By studying the expression for the radius r in (4.21) we see that d( Bρ (z, M) ) ≤ d( Bρ (0, M) ) = 2 th 12 M

(4.24)

for all z ∈ Bn and all M > 0 . This yields a sharp inequality between the euclidean and hyperbolic distances as follows. For given x, y ∈ Bn choose z ∈ J [x, y] with ρ(x, z) = ρ(z, y) = 12 ρ(x, y) . Then with M = 12 ρ(x, y) (4.24) yields the useful inequality |x − y| ≤ 2 th 14 ρ(x, y) .

(4.25)

Equality holds here if x = −y . Because th A ≤ A for A ≥ 0 , (4.25) also yields the crude estimate |x − y| ≤ 12 ρ(x, y) for x, y ∈ Bn . Exercise 4.3 (1) Let a, b, c, d ∈ C be four distinct points and let L(a, b) and L(c, d) be the lines through a, b and c, d , resp. Show that if the lines L(a, b) and L(c, d)

58

4 Hyperbolic Geometry

have exactly one point of intersection w , then w=

(ab − ab)(c − d) − (a − b)(cd − cd) (a − b)(c − d) − (a − b)(c − d)

.

(2) Let a, b, c ∈ C be three distinct noncollinear points. Show that the center m of the circle through these three points is m=

|a|2 (b − c) + |b|2(c − a) + |c|2 (a − b) a(b − c) + b(c − a) + c(a − b)

.

(3) [307, Problem 1.1.27] Suppose that z1 and z2 are points in the unit disk with |z1 | = 0, 1 and 0, z1 , z2 are not collinear. Show that the circle through z1 , z2 , z¯ 1−1 has the center a and radius r , with a=

z1 (1 + |z2 |2 ) − z2 (1 + |z1 |2 ) , z1 z¯ 2 − z¯ 1 z2

r=

|z1 − z2 ||1 − z1 z¯ 2 | . |z1 z¯ 2 − z¯ 1 z2 |

Find also the points of intersection of the circle with the unit circle. Exercise 4.4 Applying (4.21) show that if Bρ (x, M) = B n (y, r) , then r admits an estimate (1 − |y|) b ≤ r ≤ (1 − |y|) B where b and B depend only on M . Show also that the numbers a and A in (4.21) have lower and upper bounds depending only on M . In particular, A/a has an upper bound depending only on M . Exercise 4.5 Let x0 ∈ Bn , M > 0 and v = min{ |z − x0 | : ρ(x0 , z) = M } , V = max{ |z − x0 | : ρ(x0 , z) = M } . Find an upper bound for V /v by applying 3.23(1). Given distinct points x and y in Bn or Hn one can express the Poincaré distance ρ(x, y) in terms of the absolute ratio | x∗ , x, y, y∗ | by virtue of the formulae (4.9) and (4.17) where x∗ and y∗ are the “end-points” of a geodesic line containing x and y . Sometimes it will be convenient to express ρ(x, y) in a different way without referring to the points x∗ and y∗ at all. For instance, one can express ρ(x, y) in terms of the supremum of |x, a, y, b| , a, b ∈ ∂Bn , see Theorem 5.16. Such an expression can be used to define hyperbolic type metrics in a proper subdomain G ⊂ Rn ; see Theorems 5.16 and 5.18 and Lemma 11.9.

4.3 Bounds for Hyperbolic Distance

59

4.3 Bounds for Hyperbolic Distance The formulae (4.8) and (4.16), which give explicit expressions for ρHn (x, y) and ρBn (x, y) , respectively, are of fundamental importance for hyperbolic geometry. As a matter of fact, many formulae of this chapter can be derived directly from these formulae. For many applications it would be formally adequate to define the hyperbolic distance in terms of (4.8) and (4.16) without any reference to the geometric interpretation in terms of the length-minimizing property of geodesics. These geometric notions and their invariance properties are, however, the reason why the hyperbolic metric is so useful and natural in many applications. The reader may show as an exercise that (4.21) follows from (4.16). The explicit expressions (4.8) and (4.16) for ρ(x, y) are somewhat complicated. Often it will be sufficient to give bounds for ρ(x, y) in terms of simple comparison functions. We now introduce such a function. For an open set D in Rn , D = Rn , define d(z) = d(z, ∂D) for z ∈ D and  jD (x, y) = log 1 +

 |x − y| min{d(x), d(y)}

(4.27)

for x, y ∈ D [552, (2.26)]. If A ⊂ D is non-empty define   jD (A) = sup jD (x, y) : x, y ∈ A .

(4.28)

The next lemma shows that jD (x, y) is a metric on D , and we call it the distance ratio metric. Lemma 4.6 The function jD is a metric. The following inequalities d(x) (1) jD (x, y) ≥ log , d(y)  d(x) |x − y|  ≤ 2 jD (x, y) , (2) jD (x, y) ≤ log + log 1 + d(y) d(x) hold for all x, y ∈ D . Proof ([483], [29, Thm 7.44]) To prove that jD is a metric, we need only to verify the triangle inequality jD (x, y) + jD (y, z) ≥ jD (x, z)

for all x, y, z ∈ D.

Let dx = inf{d(x, z) : z ∈ ∂D} . By symmetry, we may assume that dx ≤ dz . Then we have to prove that d(x, y) d(y, z) d(x, y)d(y, z) d(x, z) + + ≥ . min{dx, dy} min{dy, dz} min{dx, dy} min{dy, dz} dx

(4.29)

60

4 Hyperbolic Geometry

If dy ≤ dx , then (4.29) follows immediately. On the other hand, if dx < dy , then the left side of (4.29) is equal to 1 dx

   dx + d(x, y) d(x, y) + d(y, z) , min{dy, dz}

which is greater than or equal to d(x, z)/dx. (1) The proof follows because d(y) ≤ d(x) + |x − y| . (2) If d(x) ≤ d(y) , the proof is obvious in view of (4.27). If d(x) > d(y) ,  d(x) d(x) |x − y|   |x − y|  jD (x, y) = log 1 + ≤ log + d(y) d(y) d(y) d(y)  d(x) |x − y|  = log + log 1 + ≤ 2 jD (x, y) , d(y) d(y) where in the last step the inequality in part (1) was applied.

 

Exercise 4.7 (1) Let G and G be proper subdomains of Rn with G ⊂ G . Show that jG (x, y) ≤ jG (x, y) for all x, y ∈ G . For w ∈ Rn set Rw = Rn \ {w} and define hG (x, y) = sup{ jRw (x, y) : w ∈ ∂G } for all x, y ∈ G . Show that if w ∈ ∂G |y − w| jG (x, y) = hG (x, y) ≥ log ; x, y ∈ G . |x − w|

(4.30)

Moreover, if d(x) ≤ d(y) and z ∈ ∂G with |x − z| = d(x) prove that   |y − z| ≥ 12 exp jG (x, y) − 1 . |x − z| P. Hästö and H. Lindén [210] studied the metric hG and coined the term “halfApollonian metric” for it. (2) Let G be a domain in Rn and s ∈ (0, 1/4] . Show that there exist numbers s1 , s2 > 0 such that for all z ∈ G , x, y ∈ B n (z, sd(z)) , w ∈ ∂G s1 ≤

|x − w| ≤ s2 . |y − w|

4.3 Bounds for Hyperbolic Distance

61

Exercise 4.8 (1) Let B = S n−1 (en , 1) ∩ { x ∈ Hn : xn ≥ 1 } . Find max{ ρHn (en , x) : x ∈ B} and min{ ρHn (en , x) : x ∈ B} . (2) For an open set D ⊂ Rn , D = Rn , define [164, p. 51]  |x − y|  |x − y|   1+ . jD (x, y) = log 1 + d(x) d(y) By [159, p. 37, Ex. 3.3.7], ρD (x, y) ≤  jD (x, y) for all x, y ∈ D when D ∈ {B2 , H2 } . Show that for all x, y ∈ D jD (x, y) ≤  jD (x, y) ≤ 2 jD (x, y) . In the next lemma we show that jD yields simple two-sided estimates for ρD both when D = Bn and when D = Hn . Lemma 4.9 (1) jBn (x, y) ≤ ρBn (x, y) ≤ 2 jBn (x, y) for x, y ∈ Bn . (2) jHn (x, y) ≤ ρHn (x, y) ≤ 2 jHn (x, y) for x, y ∈ Hn . Proof (1) First, let y = −x . Then   2|x| 1 + |x| jBn (x, −x) = log 1 + , = log 1 − |x| 1 − |x| and hence ρ(x, −x) = 2jBn (x, −x) . Next, if x = y , the lemma is trivial. By symmetry we may assume that |y| ≤ |x|, x = y. It easily follows from (3.17) and (4.16) that th

ρBn (x, y) |x − y| = 2 A[x, y]

and hence jBn (x, y) ≤ ρ(x, y) iff 1 +

iff

|x − y| A[x, y] + |x − y| ≤ 1 − |x| A[x, y] − |x − y|

2|x − y| |x − y| ≤ 1 − |x| A[x, y] − |x − y|

iff A[x, y] ≤ |x − y| + 2(1 − |x|), which follows from (3.2.2) and the triangle inequality.

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4 Hyperbolic Geometry

In the same way, ρBn (x, y) ≤ 2 jBn (x, y) iff



A[x, y] + |x − y| (1 − |x|2 )(1 − |y|2)

≤1+

|x − y| . 1 − |x|

Since |y| ≤ |x| , this is true if A[x, y] + |x − y| ≤ 1 − |x|2 + (1 + |x|)|x − y|, iff A[x, y] ≤ 1 − |x|2 + |x||x − y|, which follows from the triangle inequality and (3.2.2). (2) Denote u = 1 + |x − y|2 /(2xn yn ) . By (4.8) and (4.12) we get ρHn (x, y) ≤ 2 log(1 +



 |x − y|  ≤ 2jHn (x, y) . 2(u − 1) ) = 2 log 1 + √ xn yn

For the proof of the lower bound we may assume that xn ≤ yn and x = xn en . Let y  = (xn + |x − y|)en . Because (y  )n ≥ yn it follows from (4.8) and (4.6) that  |x − y|  ρHn (x, y) ≥ ρHn (x, y  ) ≥ log 1 + = jHn (x, y) xn  

and the assertion follows.

Exercise 4.10 For an open set D ⊂ Rn with D = Rn and for a non-empty set A in D with d(A, ∂D) > 0 put rD (A) =

d(A) . d(A, ∂D)

Show that 1 2

      log 1 + rD (A) ≤ log 1 + 12 rD (A) ≤ jD (A) ≤ log 1 + rD (A) .

Show also that for a compact set A ⊂ Bn , ρBn (A) ≥ log(1 + c rBn (A)) with c = 1/2 . (We do not know the best constant c in this inequality.) Exercise 4.11 Solve 3.21(2) with the help of the hyperbolic metric. [Hint: Because 2  = 1+r of (4.15) the requirement that ρ(0, a) = 12 ρ(0, re1 ) leads to 1+|a| 1−|a| 1−r , √ 2 i.e. |a| = r/(1 + 1 − r ) .] There are several geometric methods for finding the hyperbolic midpoint of a geodesic segment, see [561].

4.3 Bounds for Hyperbolic Distance

63

Exercise 4.12 For an open set D in Rn , D = Rn , let   |x − y| |x − y|2

, ϕD (x, y) = log 1 + max √ ; x, y ∈ D . d(x)d(y) d(x)d(y) Show that jD (x, y)/2 ≤ ϕD (x, y) ≤ 2 jD (x, y) . Remark 4.13 Let D ⊂ Rn be a domain and  |x − y|  hD,c (x, y) = log 1 + c √ , x, y ∈ D , c > 0 . d(x)d(y) It was shown in [115] that hD,c is a metric iff c ≥ 2 . This metric was also studied by P. Hästö [198]. Recently, this metric found some applications [17, 411]. Exercise 4.14 (1) Observe first that, for t ∈ (0, 1) ,   ρHn (ten , en ) = ρHn ten , S n−1 ( 12 en , 12 ) (cf. (4.8)). Making use of this observation and (4.11) show that B n ( 12 en , 12 ) =



t ∈(0,1) Bρ (ten ,

log 1t ) .

  (2) For p > 0 and t > 0 let A(t) = ρHn (0, t p ) , (t, t p ) . Find the limits limt →0 A(t) and limt →∞ A(t) in the three cases p < 1 , p = 1 , and p > 1 . Exercise 4.15 The stereographic projection π2 (see 3.12) provides a connection between the hyperbolic geometries (Bn , ρ) and (Rn+1 , ρ− ) and the spherical  −  n geometry of (R , q) . Verify that ρ(0, ae1 ) = ρ− π2 (0), π2 (ae1 ) , a ∈ (0, 1) , by computing the absolute ratios | − e1 , 0, ae1 , e1 | and | π2 (−e1 ), π2 (0), π2 (ae1 ), π2 (e1 ) | (see (4.9) and (4.18)). Note that 2q(0, ae1) = |π2 (0) − π2 (ae1 )| . Let be1 be the orthogonal projection of π2 (ae1 ) onto the x1 -axis. Show that ρ(0, be1) = 2ρ(0, ae1) . [Hint: See Fig. 3.5 in 3.21(2).] Exercise 4.16 (Continuation of 4.15) Show that π2 (ae1) ∈ S n ∩ S n (x, r) where S n (x, r) is a sphere orthogonal to S n with ae1 ∈ S n (x, r) . Find x and r .

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4 Hyperbolic Geometry

Exercise 4.17 Let x, y ∈ Bn and let Tx ∈ M(Bn ) be as defined in 3.2. Show that |x − y| s |Tx y| =  , = √ 2 2 2 1 + s2 |x − y| + (1 − |x| )(1 − |y| ) where s 2 = |x − y|2 /((1 − |x|2)(1 − |y|2)) . [Hint: By (4.23) and (4.17) |Tx y|2 = th2

1

 2 ρ(x, y) = 1 −

1 ch2 ( 12 ρ(x, y))

=

s2 .] 1 + s2

Next let z ∈ J [x, y] be the hyperbolic midpoint of J [x, y] as in (4.23). Show that |Tz x| = |Tz y| = = 

s √ 1 + 1 − s2 |x − y|

|x − y|2 + (1 − |x|2 )(1 − |y|2 ) +



(1 − |x|2 )(1 − |y|2 )

,

√ where s is as above. [Hint: Because th A = t/(1 + 1 − t 2 ) , t = th 2A , one can apply (4.23) and the above computation.] Moral: Instead of using these lengthy expressions for |Tx y| and |Tz y| involving euclidean distances it will often be more convenient to use the equivalent formula (4.23) involving the hyperbolic distance ρ(x, y) . Exercise 4.18 Let x, y ∈ Rn and let tx be a spherical isometry as defined in (3.23). Show that |x − y| . |tx y| =  (1 + |x|2)(1 + |y|2 ) − |x − y|2 [Hint: This follows immediately from (3.24) and (3.6).] Let α ∈ [0, 12 π] be such that sin α = q(x, y) . Then α is the angle between the segments [en+1 , tx x] = [en+1 , 0] and [en+1 , tx y] at en+1 (see (3.4).) Show that the above formula can be rewritten as |tx y| = tan α . Note the analogy with (4.23). Exercise 4.19 Show that |x − y|2 |f (x) − f (y)|2 = (1 − |f (x)|2 ) (1 − |f (y)|2 ) (1 − |x|2) (1 − |y|2 ) for all f in GM(Bn ) and all x, y ∈ Bn . [Hint: Apply (4.16) and (4.18).]

4.3 Bounds for Hyperbolic Distance

65

Exercise 4.20 (Contributed by M. K. Vamanamurthy) Starting with the identity (cf. 4.17) th 12 ρ(x, y) = 

|x − y| |x

− y|2

+ (1 − |x|2)(1 − |y|2 )

for x, y ∈ Bn verify the following inequalities (1)

|x − y| |x − y| ≤ th 12 ρ(x, y) ≤ , 1 + |x||y| 1 − |x||y|

(2)

|x| − |y| |x| + |y| ≤ th 12 ρ(x, y) ≤ , 1 − |x||y| 1 + |x||y|

(3)

1 2 |x

≤

− y| ≤

|x − y| ≤ th 14 ρ(x, y) 1 + |x||y| + |x| |y|

|x| + |y| , 1 − |x||y| + |x| |y|

 where |x| = 1 − |x|2 . Can you find similar inequalities for the spherical chordal metric? [Hint: 4.18.] Notation for Exercise: 4.21–4.23 Let L(x, y) be the line through the points x and y as in Exercise 4.3. Let ∂H2 be the real axis. A complex number x is denoted by x = x1 + ix2 or x = |x|eiα . Let x = x1 + ix2 , y = y1 + iy2 ∈ B2 \ {0} such that 0, x, y are noncollinear, and x ∗ = x/|x|2 , y ∗ = y/|y|2 . Let |x − y| x|y|2 − y y(1 + |x|2 ) − x(1 + |y|2 ) and ra = . a=i 2(x2y1 − x1 y2 ) 2|y||x1y2 − x2 y1 |

(4.31)

It follows from 4.3(3) that the four points x, y, x ∗ , y ∗ are in S 1 (a, ra ) . Moreover, the circle S 1 (a, ra ) is orthogonal to S 1 . Exercise 4.21 Let x, y ∈ B2 \ {0} be points such that 0, x, y are noncollinear and |x| = |y| . Let x ∗ = x/|x|2 , y ∗ = y/|y|2 and w = L(x, y) ∩ L(x ∗ , y ∗ ) . Construct the circle S 1 (w, rw ) , which is orthogonal to the circle S 1 (a, ra ) , where a, ra are as in (4.31). Show that the circle S 1 (w, rw ) is orthogonal to the circle S 1 and  |x − y| (1 − |x|2 )(1 − |y|2 ) y(1 − |x|2) − x(1 − |y|2) and rw = w= . |y|2 − |x|2 |y|2 − |x|2

66

4 Hyperbolic Geometry

Let x = eiα , y = eiβ with α, β ∈ (0, π) and α = β . The midpoint z of the hyperbolic segment J [x, y] in H2 is obtained from (4.7)  z = e , δ = arc cos iδ

cos β+α 2 cos β−α 2

 .

(4.32)

Exercise 4.22 Let x, y ∈ S 1 ∩ H2 , {x∗ , y∗ } = S 1 ∩ ∂H2 , let x∗ , x, y, y∗ occur in this order on S 1 . Let a, ra be as in (4.31) and z be as in (4.32). Furthermore, 2 let w = L(x, y) ∩ ∂H2 , v = L(x, x∗ ) ∩ L(y, y∗ ) , n = L(x, y) ∩ S 1 ( w2 , |w| 2 )∩H . Then (1) (2) (3) (4) (5) (6) (7)

the line L(a, z) is orthogonal to ∂H2 . the line L(w, z) is tangent to the circle S 1 . the circle S 1 (a, ra ) is orthogonal to the circle S 1 ( w2 , |w| 2 ). 2 the line L(v, z) is orthogonal to ∂H . the point v is on the circle S 1 (a, ra ) . the point n is the midpoint of the euclidean segment [x, y] . (y1 , z1 , y) = (x1 , z1 , x) .

(Here the notation (x, z, y) means the angle in the range [0, π] between the segments [x, z] and [y, z] . ) √ √ Exercise 4.23 Fix 0 < t < 1 , let x∗ = t + i 1 − t 2 and y∗ = t − i 1 − t 2 . Construct the orthogonal circle S 1 (a, ra ) intersecting the unit circle at the points x∗ , y∗ . Let {x, y} ∈ S 1 (a, ra ) ∩ B2 , be such that x∗ , x, y, y∗ occur in this order on S 1 (a, ra ) and let x ∗ = x/|x|2 , y ∗ = y/|y|2 , and u = L(x, y ∗ ) ∩ L(y, x ∗ ) . (1) Then u=

y(1 − |x|2 ) + x(1 − |y|2 ) , 1 − |x|2 |y|2

and Re u = Re w , where w = L(x, y) ∩ L(x ∗ , y ∗ ) . (2) The three points w, x∗ , y∗ are collinear and hence Re u = t . Notes 4.24 The main sources for this chapter are A.F. Beardon [51] and the other references given at the end of Chap. 3. The book [197] provides a richly illustrated approach to euclidean and noneuclidean geometry. See also [19, 86], [516, pp. 508– 514], [434] and [283]. Some applications and refinements of the sharp inequality (4.25) were given by C.J. Earle and L.A. Harris [127] and by R. Klén and M. Vuorinen [294].

Chapter 5

Generalized Hyperbolic Geometries

In geometric function theory, invariance properties of metrics are important. In our work below, two notions of invariance are most important; invariance with respect to the group of Möbius transformations and invariance with respect to the group of similarity transformations. Examples of metrics invariant with respect to the first and second transformation groups are the hyperbolic and distance ratio metrics, resp., studied in Chap. 4. In Chap. 4 we also saw that the hyperbolic metric can be defined in (at least) three different ways: (1) as a weighted metric, (2) in terms of an explicit formula, (3) in terms of the absolute ratio. We will use these ideas also here and begin with the most important metric, the similarity invariant quasihyperbolic metric introduced by F.W. Gehring and B.P. Palka [165] in 1976. The basic properties of the quasihyperbolic metric will be reviewed; these will have many applications later. Thereafter we discuss briefly four Möbius invariant metrics defined on proper subdomains of the Möbius space. These invariant metrics will be studied in Part IV in more detail. Two of these four metrics were already studied by the third author in 1988 [555], but the systematic study of these two metrics begun after P. Seittenranta [483] and P.A. Hästö [203] had proved the triangle inequalities for these metrics in general domains as well as many fundamental properties. The other two Möbius invariant metrics are J. Ferrand’s version of the quasihyperbolic metric 1988 [135] and A.F. Beardon’s Apollonian metric 1998 [53]. For further remarks on the above metrics see Notes 5.27.

5.1 The Quasihyperbolic Metric In an arbitrary proper subdomain D of Rn one can define a metric, the quasihyperbolic metric of D, which shares some properties of the hyperbolic metric of Bn or Hn . We shall now give the definition of the quasihyperbolic metric and state without © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_5

67

68

5 Generalized Hyperbolic Geometries

proof some of its basic properties which we require later on. The quasihyperbolic metric was introduced and systematically developed and applied by F. W. Gehring and his collaborators. During the past four decades that have elapsed since its introduction, it has become a standard tool of quasiconformal mapping theory both in the euclidean space and metric space contexts. Throughout this chapter D will denote a proper subdomain of Rn . In D we define a weight function w : D → R+ by w(x) =

1 ; x∈D. d(x, ∂D)

(5.1)

Using this weight function one defines the quasihyperbolic length q (γ ) = D q (γ ) of a rectifiable curve γ by a formula similar to (4.2). The quasihyperbolic distance between x and y in D is defined by  kD (x, y) = inf

α∈ xy

D q (α)

= inf

α∈ xy α

w(x)|dx| ,

(5.2)

where xy is a collection of rectifiable curves in D joining x and y. This metric was introduced in [165]. It is clear that kD is a metric on D. It follows from (5.2) that kD is invariant under similarity transformations. Lemma 5.1 ([165]) For given points x, y ∈ D , there exists a geodesic segment JD [x, y] of the metric kD , a curve α joining x and y , for which the infimum in (5.2) is attained. However, very little is known about the structure of such geodesic segments JD [x, y] when D is given. Some regularity properties of geodesic segments have been obtained by G. Martin [349]. Definition 5.2 We say that an arc C in D is c-convex with respect to balls, or simply c-convex, if for each x in D and r ≤ c dist(x, ∂D), the set C∩ B¯ n (x, r) is connected. Theorem 5.3 ([349]) Each quasihyperbolic geodesic of D is 1-convex with respect to balls in D. D. Herron [226] has also studied convexity properties of several metrics. As in (4.3) one can define the quasihyperbolic volume of a (Lebesgue) measurable set A ⊂ D [575]  dm mh (A) = . (5.3) n A d(x) Remark 5.4 Clearly, kHn = ρHn , and we see easily that ρBn ≤ 2 kBn ≤ 2 ρBn (cf. (5.1),(4.13)). Hence, the geodesics of (Hn , kHn ) are those of (Hn , ρHn ), but it is a difficult task to find the geodesics of kD when D is given. The following monotone

5.1 The Quasihyperbolic Metric

69

property of kD is clear: if D and D  are domains with D  ⊂ D and x, y ∈ D  , then kD  (x, y) ≥ kD (x, y). In order to find some estimates for kD (x, y) we shall employ, as in the case of Hn and Bn , the metric jD defined in (4.27). The metric jD is indeed a natural choice for such a comparison function since both kD and jD are invariant under similarity transformations. Lemma 5.5 ([531]) Let x, y ∈ D and γ be a curve joining them in D. Then  

(γ )

k (γ ) ≥ log 1 + . d(x) Proof Let γ be a rectifiable arc joining x and y in D. Set λ = (γ ), and let α : [0, λ] → γ be the arc-length parametrization of γ . Since d(α(t)) ≤ d(x) + t, we obtain    λ 

(γ ) |dx| dt ≥ = log 1 + .

k (γ ) = d(x)   0 d(x) + t γ d(x) Corollary 5.6 Let x, y ∈ D. Then  kD (x, y) ≥ log 1 +

|x − y| min{d(x), d(y)}

 = jD (x, y) .

(5.4)

In combination with 4.6, (5.4) yields d(x) kD (x, y) ≥ log , d(z) = d(z, ∂D) . d(y)

(5.5)

For easy reference we record Bernoulli’s inequality log(1 + as) ≤ a log(1 + s) ; a ≥ 1 , s > 0 . Lemma 5.7 ([549, Lemma 2.11]) (1) If x ∈ D, y ∈ Bx = B n (x, d(x)), then  kD (x, y) ≤ log 1 +

 |x − y| . d(x) − |x − y|

(2) If s ∈ (0, 1) and |x − y| ≤ s d(x), then kD (x, y) ≤

1 j (x, y) . 1−s D

(5.6)

70

5 Generalized Hyperbolic Geometries

Fig. 5.1 Proof of Lemma 5.7(1)

z y x

Bx

D

Proof (1) Select z ∈ ∂Bx such that y ∈ [x, z] (Fig. 5.1). Because [x, y] ∈ xy , from 5.4 we obtain  kD (x, y) ≤ kBx (x, y) ≤ [x,y]

|dw| ≤ d(w)

 [x,y]

|dw| = |w − z|

d(x) 

dt t

d(x)−|x−y|

  |x − y| d(x) = log 1 + d(x) − |x − y| d(x) − |x − y|   = jRn \{z} (x, y) . = log

(2) For the proof of (2) we apply part (1), Bernoulli’s inequality (5.6), and the definition of jD to obtain |x − y|  (1 − s)d(x)  |x − y|  1 1 log 1 + ≤ j (x, y) ≤ 1−s d(x) 1−s D

 kD (x, y) ≤ log 1 +

 

as desired.

Lemma 5.8 ([160, Lemma 2.1]) Suppose that x, y are points in D and that d1 = d(x, ∂D), d2 = d(y, ∂D), t = |x − y| . If t < d1 + d2 , then kD (x, y) ≤ log

d1 + d2 + t . d1 + d2 − t

(5.7)

This bound is sharp. If t < d2 , then kD (x, y) ≤ log(1 +

2t ). d1

(5.8)

5.1 The Quasihyperbolic Metric

71

Proof Let α = [x, y] and B1 = B n (x, d1 ) , B2 = B n (y, d2 ) . The triangle inequality implies that d1 ≤ d2 + t and d2 ≤ d1 + t . Then by making a preliminary change of variables, we may assume that 0, x, y lie on a line λ and that d12 − |x|2 = d22 − |y|2 = d 2 .

(5.9)

d(z, ∂D)2 ≥ d(z, ∂(B1 ∪ B2 ))2 = d 2 + |x|2

(5.10)

Since B1 ∪ B2 ⊂ D

for z ∈ α . Suppose that λ is parametrized with respect to arclength s with λ(0) = 0 , λ(s1 ) = x , λ(s2 ) = y and s2 > 0 ; by relabeling we may assume that s1 < s2 . Then t = s2 − s1 and we obtain  ds d2 + s2 d1 + d2 + t (5.11)  = log = log kD (x, y) ≤ 2 2 d + s d1 + d2 − t 1 1 d + |z| from integration and (5.9). Next if D = B1 ∪ B2 and γ is any arc joining x and y in D , then d(z, ∂D)2 ≤ d 2 + |z|2 for z ∈ γ and we obtain equality in (5.7). Finally, (5.7) implies (5.8) whenever   t ≤ d2 . Lemma 5.9 ([160, Lemma 2.6]) Suppose that γ is an arc which joins points x, y in D and that d1 = d(x, ∂D), d2 = d(y, ∂D), t = (γ ) . Then kD (x, y) ≥ log

(d1 + d2 + t)2 . 4d1d2

(5.12)

This bound is sharp. In particular, kD (x, y) ≥ log(1 +

t ). d1

(5.13)

Proof If γ is parametrised by arc length with γ (0) = x , then d(z, ∂D) ≤ d1 + s ,

d(x, ∂D) ≤ d1 + t − s ,

for z ∈ γ . Hence r = 12 (t + d2 − d1 ) ∈ [0, t] and we obtain (5.12) from  kD (γ ) = γ

= log

ds = d(z, ∂D)



r 0

(d1 + d2 + t)2 . 4d1 d2

ds + d1 + s



t r

ds d1 + t − s

(5.14)

72

5 Generalized Hyperbolic Geometries

Equality holds if x, y are on an open subinterval β of a line λ and D = (Rn \ λ) ∪ β . Finally, (5.13) follows from (5.12) and the fact that d2 ≤ d1 + t .   Next we shall study the quasihyperbolic balls BkG (x, M) = { z ∈ G : kG (x, z) < M } when x ∈ G and M > 0. It follows from (5.5) that e−M d(x) ≤ d(z) ≤ eM d(x)   holds for z ∈ B kG (x, M). Next, for z ∈ B n x, (1−e−M )d(x) we deduce by 5.7(1)   that kG (x, z) < M and for z ∈ Rn \ B n x, (eM − 1)d(x) we find by (5.4) and (5.5) that kG (x, z) > M. In conclusion, we have proved that ⎧ ⎪ B n (x, rd(x) ) ⊂ BkG (x, M) ⊂ B n (x, Rd(x) ) , ⎪ ⎨ B kG (x, M) ⊂ { z ∈ G : e−M d(x) ≤ d(z) ≤ eM d(x) } , ⎪ ⎪ ⎩ r = 1 − e−M , R = eM − 1 .

(5.15)

For G = Hn one can show that the numbers r and R are the best possible (see (4.11)). We shall write Bk (x, M) for BkG (x, M) if there is no danger of confusion. Exercise 5.10 ([483, Theorem 3.8]) Show that in (5.15) we can replace BkG (x, M) balls by BjG (x, M) balls. Exercise 5.11 Fix x ∈ G and t ∈ (0, 1). Show that      BkG x, log(1 + t) ⊂ B n x, t d(x) ⊂ BkG x, log

1  1−t

and that there exist numbers c1 , c2 > 0 such that for all z1 , z2 ∈ B n (x, t d(x)) c1 |z1 − z2 | ≤ d(x) kG (z1 , z2 ) ≤ c2 |z1 − z2 | . [Hint: Apply (5.15).] Möbius transformations are hyperbolic isometries. That is, if each one of the domains G, G is a ball or a half-space in Rn and if f is a Möbius transformation mapping G onto G , then   ρG (x, y) = ρG f (x), f (y) for x, y ∈ G (cf. (4.9), (4.17), (4.19)). Although the metric kG does not have this invariance property it is not changed by more than the factor 2 under Möbius transformations. P. Hästö [204, 205] has proved that, under some general hypotheses, isometries of the quasihyperbolic metric are Möbius transformations.

5.2 Möbius Invariant Metrics

73

Lemma 5.12 ([165]) If G and G are proper subdomains of Rn and if f is a Möbius transformation of G onto G , then 1 2 kG (x, y)

  ≤ kG f (x), f (y) ≤ 2 kG (x, y)

for all x, y ∈ G.

5.2 Möbius Invariant Metrics J. Ferrand [135] introduced a Möbius invariant version of the quasihyperbolic metric. This interesting metric has largely gone unnoticed, but some of its basic properties were found by P. Seittenranta [483] and very recently by D. Herron and P. K. Julian in their paper [227]. See also D. Herron [226]. n

n

Definition 5.13 Let G ⊂ R be a domain with card(R \ G) ≥ 2 and |a − b| , x ∈ G \ {∞} . |x − a||x − b| a,b∈∂G

wG (x) = sup Define a metric σG in G

 σG (x, y) = inf

γ ∈ γ

wG (t)|dt| .

This Ferrand’s Möbius invariant metric has the following properties: Theorem 5.14 ([135, p.122]) The function σG is a Möbius invariant metric: if G ⊂ n n R is a domain such that its complement R \ G contains at least two points and n h ∈ GM(R ) , then for all x, y ∈ G σhG (h(x), h(y)) = σG (x, y) . Moreover, σG = ρG for G ∈ {Bn , Hn } . Furthermore, if G is a proper subdomain of Rn , then kG ≤ σG ≤ 2kG . Note that another metric named after J. Ferrand will be introduced and studied below in Chap. 10. We now proceed to define another Möbius invariant metric. For distinct a, b, c, d n in R let m(a, b, c, d) = max{ | a, b, d, c | , | a, c, d, b | } .

(5.16)

74

5 Generalized Hyperbolic Geometries n

n

If G ⊂ R is a domain with card(R \ G) ≥ 2 then let mG (b, c) = sup{ m(a, b, c, d) : a, d ∈ ∂G } .

(5.17)

It is clear that m is symmetric, that is, m(a, b, c, d) = m(a, c, b, d) = m(b, a, d, c)

(5.18)

n

and also GM(R )-invariant, that is, mf (a, b, c, d) = m(f a, f b, f c, f d) = m(a, b, c, d)

(5.19)

n

for all f ∈ GM(R ) (cf. (3.11)). For x, y ∈ Rn \ {a} ( a ∈ Rn ) m(a, x, y, ∞) =

|x − y| . min{ |x − a|, |y − a| }

(5.20)

It follows from (5.20) that jG (x, y) = log(1 + mG (x, y)) , G = Rn \ {a} ,

(5.21)

for all x, y ∈ G where jG is as in (4.27).

5.2.1 The Point-Pair Invariant mG Next we record some properties of the Möbius invariant symmetric function mG n n for an arbitrary domain G ⊂ R with card(R \ G) ≥ 2 . The following facts are immediate: (1) G1 ⊂ G2 and x, y ∈ G1 ⇒ mG1 (x, y) ≥ mG2 (x, y) . (2) For a fixed y ∈ G , mG (x, y) → 0 iff x → y and mG (x, y) → ∞ iff x → ∂G . (3) mG (x, y) ≥ q(∂G) q(x, y) . (4) mG (x, y) ≤ q(∂G) q(x, y)/q({x, y}, ∂G)2 . For x, y ∈ G , define δG (x, y) = log(1 + mG (x, y)) . Lemma 5.15 ([555, 8.39]) For G ∈ {Bn , Hn } and b, c ∈ G δG (b, c) = ρG (b, c) .

(5.22)

5.2 Möbius Invariant Metrics

75

Proof Without loss of generality we may assume that G = Bn . By GM(Bn )invariance we may assume b = −re1 = −c . Then ρ(b, c) = 2 log[(1+r)/(1−r)] or, equivalently, r = th 14 ρ(b, c) . For all a, d ∈ ∂Bn m(a, b, c, d) ≤

2|b − c| 4r = = m(−e1 , −re1 , re1 , e1 ) 2 (1 − r) (1 − r)2

and hence mBn (b, c) =

4 th 14 ρ(b, c) (1 − th

1 2 4 ρ(b, c))

= eρ(b,c) − 1  

and the assertion follows.

The metric δG is called the Möbius metric or the Seittenranta metric; he proved in his PhD thesis the triangle inequality for δG and applied it to study quasiconformal mappings. Seittenranta [483] adopted the term Möbius metric for this purpose. Theorem 5.16 ([483, Thm 3.4, 3.12]) The function δG is a Möbius invariant n n metric: if G ⊂ R is a domain such that its complement R \ G contains at n least two points and h ∈ GM(R ) , then for all x, y ∈ G δhG (h(x), h(y)) = δG (x, y) ≤ σG (x, y) . Moreover, δG = ρG for G ∈ {Bn , Hn } . Furthermore, if G is a proper subdomain of Rn , then jG ≤ δG ≤ 2jG . Finally, if G = Rn \ {0} , then δG = jG . Proof See [483] and Lemma 5.15. The triangle inequality is also proven in [28,   Theorem 7.47]. As the similarities between Theorems 5.14 and 5.16 show, the pair of two Möbius invariant metrics σG , δG may be regarded as a Möbius invariant counterpart of the pair kG , jG , where both metrics are invariant under similarity transformations. Next we will define a third Möbius invariant metric, the generalized hyperbolic metric.

5.2.2 The Symmetric Ratio n

For distinct points a, b, c, d in R define the symmetric ratio by s(a, b, c, d) = | a, b, d, c || a, c, d, b | .

(5.23)

76

5 Generalized Hyperbolic Geometries

Then by (3.10) s(a, b, c, d) =

q(a, d)2 q(b, c)2 , q(a, b) q(b, d) q(a, c) q(c, d)

(5.24)

which we recognize as the expression studied in 3.18. We recall that s is symmetric, i.e. s(a, b, c, d) = s(a, c, b, d) = s(d, b, c, a) = s(b, a, d, c) , n

and also GM(R )-invariant in the sense that sf (a, b, c, d) = s(f a, f b, f c, f d) = s(a, b, c, d) n

n

(5.25) n

for all f in GM(R ) . Let D ⊂ R be an open set with card(R \ D) ≥ 2 and define   sD (b, c) = sup 12 s(a, b, c, d) : a, d ∈ ∂D .

(5.26)

It follows from (5.24) and (3.6) that s(a, x, y, ∞) =

|x − y|2 . |a − x||y − a|

(5.27)

5.2.3 The Generalized Hyperbolic Metric ρG We next list some immediate properties of the point-pair invariant sG (b, c) when n G⊂R . (1) (2) (3) (4) (5)

sG (x, y) = sG (y, x) ,   n sf G f (x), f (y) = sG (x, y) for f ∈ GM(R ) and x, y ∈ G , G ⊂ G and x, y ∈ G imply sG (x, y) ≥ sG (x, y) , for fixed y ∈ G , sG (x, y) → 0 iff x → y and sG (x, y) → ∞ iff x → ∂G , sG (x, y) ≥ (q(∂G) q(x, y))2 .

Lemma 5.17 ([555, 3.26]) For G ∈ {Bn , Hn } and b, c ∈ G sG (b, c) = ch ρG (b, c) − 1 . Proof Without loss of generality we may assume that G = Bn . Because this equality is GM(Bn )-invariant, we may assume that b = −re1 = −c , r ∈ (0, 1) .

5.2 Möbius Invariant Metrics

77

Then r = th 14 ρ(x, y) by (4.23). It follows from (5.23) that for a, d ∈ S n−1 we obtain s(a, b, c, d) =

4r 2 |a − d|2 4r 2 |a − d|2 = . |a − b||b − d||a − c||c − d| |a − b||a − c||d − b||d − c|

It is left as an exercise for the reader to show that min{ |a − b||a − c| : a ∈ S n−1 } = 1 − r 2 , and similarly for |d − b||d − c| . Thus s = sup{ s(a, b, c, d) : a, d ∈ S n−1 } ≤

 4r 2 4r 2 22 = . 2 2 (1 − r ) 1 − r2

This upper estimate is in fact attained if a = −e1 = −d . Hence s = 16 sh2



1 4

     ρ(b, c) ch2 14 ρ(b, c) = 4 sh2 12 ρ(b, c) = 2 (ch ρ(b, c) − 1)

and sBn (b, c) = ch ρ(b, c) − 1 as desired.

 

Note also that Lemma 5.17 yields a formula for ρ(b, c) involving the absolute ratio ch ρ(b, c) = 1 + sup

1

2 | a, b, d, c || a, c, d, b |

: a, d ∈ S n−1 } .

(5.28)

Recall that another formula for the hyperbolic metric ρ in terms of absolute ratios was given in (4.17). An advantage of (5.28) over (4.17) is that it generalizes to every n n domain G in R with card(R \ G) ≥ 2 . For such a domain G define a function ρG by ch ρG (b, c) = 1 + sG (b, c) ,

(5.29)

when b, c ∈ G . This function defines a metric as shown by P. Hästö [201]. He termed it the generalized hyperbolic metric and compared it to the Seittenranta metric δG and to some other metrics [203]. Theorem 5.18 ([201, Thm 5.1], [203, Thm 1]) The function ρG is a Möbius n n invariant metric: if G ⊂ R is a domain such that its complement R \ G contains n at least two points and h ∈ GM(R ) , then for all x, y ∈ G ρhG (h(x), h(y)) = ρG (x, y)

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5 Generalized Hyperbolic Geometries

and (1) δG ≤ ρG ≤ cδG , c = (arch(3))/ log 3 . Moreover, if G ⊂ Rn is a proper subdomain, then (2) jG ≤ ρG ≤ djG , d = (arch(3))/ log 2 . Both inequalities in (1) and the first inequality in (2) are sharp. Finally, for G ∈ {Bn , Hn } the metric (5.29) agrees with the hyperbolic metric. As this theorem of Hästö shows, both of the metrics δG and ρG are very similar, they are even comparable in all domains. Yet another Möbius invariant metric, the Apollonian metric, was introduced by A. F. Beardon [53], and we compare it to δG in Chap. 11. The Apollonian metric is defined for x, y ∈ G by αG (x, y) = sup log |a, x, y, b| = sup log a,b∈∂G

a,b∈∂G

|a − y||x − b| . |a − x||y − b|

(5.30)

All the above Möbius invariant metrics are closely related to each other and coincide if the domain is G ∈ {Bn , Hn } . In some domains there are radical differences; for instance if G = Rn \{0} , then αG (−e1 , e1 ) = 0 and hence αG is not a metric, it is only a pseudometric. Of the above four Möbius invariant metrics, σG , δG , ρG , αG , the Apollonian metric αG is probably the most widely studied. Exercise 5.19 Assume that a ≥ 0 and define b by ch b = 1 + 12 a . Show that   √   √  log 1 + max a, a ≤ b ≤ log 1 + a + a   √  ≤ 2 log 1 + max a, a . Exercise 5.20 Let G = Rn \ {0} and sG as defined in (5.26). Explicitly, we see by (5.27) that sG (x, y) =

|x − y|2 , x, y ∈ G . 2|x||y|

Define ρG as in (5.29). Applying 5.19 show that jG (x, y) ≤ 2 ρG (x, y) ≤ 4 jG (x, y) for x, y ∈ G .

5.3 Properties of Generalized Hyperbolic Metrics The metrics kG , jG , σG , δG , ρG have several features in common and they can be regarded as generalizations of the hyperbolic metric to a general subdomain G of Rn . They are examples of what we call hyperbolic type metrics. If m is one of the

5.3 Properties of Generalized Hyperbolic Metrics

79

above five metrics, then it has the following properties: • • • •

G = ∪M>0 Bm (x, M) for all x ∈ G and Bm (x, M) is a compact subset of G . Hence the metric space (G, m) is proper. For each x ∈ G : ∩M>0 Bm (x, M) = {x}. The numerical value of m(x, y) depends strongly on the location of the points x, y with respect to the boundary For small radii M , the balls Bm (x, M) are similar to euclidean ball, see Chaps. 11, 12.

Hyperbolic type metrics will have a central role in our work below. These properties suggest, among other things, that balls of small radii could be compared with euclidean balls and balls of large radii with the domain G itself. Exercise 5.21 The logarithmic spiral [176, p. 23] in R2 has a parametric representation r(ω) = AeBω in polar coordinates where A and B are constants and A > 0 . It was shown by G. Martin and B. G. Osgood [351] that the geodesic segments of kG , G = Rn \ {0} , can be obtained as follows. Assume that x, y ∈ G and that the angle ϕ between the segments [0, x] and [0, y] satisfies 0 < ϕ < π . Then the triple 0, x, y determines a 2 -dimensional plane  and the geodesic segment of kG connecting x to y is a logarithmic spiral in  with equation ω |x|  log ; 0≤ω≤ϕ. r(ω) = |y| exp ϕ |y| Knowing this equation show (by integrating the element of length along this curve) that  |x| kG (x, y) = ϕ 2 + log2 , G = Rn \ {0} , (5.31) |y| holds for all x, y ∈ G . Making use of (5.31) study the set { z ∈ G : kG (e1 , z) = t } . Note the special case t = π . Remarks 5.22 (1) Many results of the euclidean trigonometry have their counterpart in the hyperbolic geometry of the unit disk. It is a natural question to ask whether some of the formulae from hyperbolic trigonometry, see [51, 450], would have their counterparts in the quasihyperbolic geometry in the form of inequalities. Practically nothing is known even in the case of simplest domains. A cosine inequality for the half plane was proved in [243]. S. Janson [261] has compiled a formulary which lists formulae for the euclidean, spherical and hyperbolic geometries. (2) In his PhD thesis R. Klén [287] has given an interpretation of the formula (5.31) as the Pythagorean theorem. Indeed, fix x, y ∈ G = R2 \ {0} with |x| ≤ |y|, z = x|y|/|x|. Now, for the quasihyperbolic triangle, whose sides are

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5 Generalized Hyperbolic Geometries

geodesic segments joining the vertices x, y, z , (5.31) yields the Pythagorean equality kG (x, y)2 = kG (x, z)2 + kG (z, y)2 . Exercise 5.23 Let G = Rn \{0} and f (x) = a 2 x/|x|2 for x ∈ G , where a > 0 . Show that   kG f (x), f (y) = kG (x, y) for x, y ∈ G . [Hint: Apply (5.31).] Show also that   jG f (x), f (y) = jG (x, y) for x, y ∈ G . [Hint: Apply (3.1).] Note that these assertions do not follow from 5.12. Exercise 5.24 (1) Let f : [0, ∞) → [0, ∞) be increasing with f (0) = 0 such that f (t)/t is decreasing on (0, ∞) . Show that f (s + t) ≤ f (s) + f (t) for s, t ≥ 0 . (2) Let (X, d) be a metric space and let f be as in part (1). Show that (X, f ◦ d) is a metric space, too. (3) Let (X, d) be a metric space and let d1 (x, y) = max{ d(x, y), d(x, y)α } , 0 < α ≤ 1 . Show that (X, d1 ) is a metric space, too. Note that the euclidean space R with the metric |x −y|α , 0 < α < 1 , has no rectifiable arcs, see 11.5. (4) Give an example of a metric space (Y, d) such that d β does not satisfy the triangle inequality for any β > 1 . (5) For a domain G  Rn and x, y ∈ G define [195] ∗ jG (x, y) = th

jG (x, y) . 2

(5.32)

∗ is a metric. Show also that Show that jG ∗ jG (x, y) =

|x − y| . |x − y| + 2 min{d(x), d(y)}

Exercise 5.25 Let G ⊂ Rn be a domain, x0 ∈ G, G1 = G \ {x0 }, t ∈ (0, 1/2]. Show that there is a constant c ≥ 1 such that for all x, y ∈ G \ B n (x0 , td(x0 )) kG1 (x, y) ≤ ckG (x, y). Remark 5.26 The property in 5.24 (1) is called subadditivity. Let 0 < a ≤ 1 ≤ b < ∞ and let f : [0, ∞) → [0, ∞) be a homeomorphism with f (0) = 0 and such that f (t)/t a is increasing whereas f (t)/t b is decreasing on (0, ∞) . Then it can be easily shown, see [29, 1.58(27), p. 340], that f satisfies the following

5.3 Properties of Generalized Hyperbolic Metrics

81

quasiadditivity property 21−b ≤

f (s) + f (t) ≤ 21−a f (s + t)

for all s, t > 0 . A simple example of a quasiadditive function is max{t a , t b } . Notes 5.27 The quasihyperbolic metric was introduced and studied by F. W. Gehring and his students and coauthors B. Palka [165], B. Osgood [164]. Soon after its first appearance in 1976, it became widely known and several interesting results can be found in [159, 164, 165, 349, 351]. This metric has found many applications to the theory of quasiconformal maps in euclidean, Banach and metric spaces [159, 531]. The metrics δG and ρG were first studied in 1988 [555, p. 116, pp. 38–39] and many important results were proven by P. Seittenranta in 1999 [483] and P. Hästö in 2005 [201]. J. Ferrand introduced the metric σG in an appendix to her 1988 paper [135] and her motivation was “to answer a question asked by M. Vuorinen.” The Apollonian metric of A.F. Beardon 1998 [53] has been studied by many authors, see 11.23. In a short note, the Apollonian metric had been discussed by D. Barbilian as early as in 1934 [48]; see also 11.23.

Chapter 6

Metrics and Geometry

In this chapter we shall consider some geometric issues related to the hyperbolic or quasihyperbolic metric. We begin with several comparison results for the quasihyperbolic metric. Here an important fact is that various metrics may be comparable in some but not in all domains. In particular, if the distance ratio metric and the quasihyperbolic metric are comparable in the sense that the former one has a lower bound in terms of the latter one, we are led to the notion of uniform domains which were introduced by O. Martio and J. Sarvas [368] in 1979. Many results of geometric function theory, originally stated for the unit disk or the unit ball, have been generalized to uniform domains: these domains have become ubiquitous. Another topic we shall discuss here is the problem of covering a set with a minimal or almost minimal number of balls. This is a classical topic of measure theory, where many types of covering theorems are used, see P. Mattila [377]. For our purposes hyperbolic and quasihyperbolic balls are most suitable. Indeed, for both geometries we have proved comparison theorems, which enable us to compare hyperbolic and quasihyperbolic balls to euclidean balls. Covering theorems can then be applied e.g. to estimate the growth of the hyperbolic volume of a ball as a function of the radius. More generally, the problem can be described as follows. Let X be a compact connected set in a domain D and let F be a covering of X by quasihyperbolic balls with radii equal  to a given constant. The problem is to extract a subcovering F1 of F with X ⊂ F1 and to give a quantitative upper bound for card F1 in terms of the parameters of the problem.

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_6

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6.1 Uniform Domains and Generalizations We know by 4.9 and 5.4 that if D = Bn , then an inequality similar to 5.7(2) holds for all x, y ∈ D. For a general domain D this is not true, i.e. the ratio

k (x, y) : x, y ∈ D , x = y A(D) = sup D jD (x, y) may be infinite. For instance, A(B2 \ [0, e1 )) = ∞. (For details, see 6.3.) Definition 6.1 A domain G in Rn , G = Rn , is called uniform, if there exists a number A = A(G) ≥ 1 such that kG (x, y) ≤ A jG (x, y) for all x, y ∈ G. The domain G is ϕ-uniform if there exists a homeomorphism ϕ : [0, ∞) → [0, ∞) , ϕ(0) = 0 , such that for all x, y ∈ G kG (x, y) ≤ ϕ(

|x − y| ). min{d(x , ∂G), d(y , ∂G)}

By 4.9 the unit ball Bn and the half-space Hn = Rn+ are uniform domains with the constant 2 and by a result of H. Lindén [335] the complement of the origin Rn \{0} is uniform with the constant π/ log 3 ; see Exercise 6.2. It follows from the definition that the class of uniform domains is invariant under similarity transformations in the same way as kG and jG are. It is not difficult to show that the image of a uniform domain under a bilipschitz mapping is again uniform. In their paper [368] Martio and Sarvas defined uniform domains in a different way, using curves connecting a pair of points under two constraints: (1) the curves avoid the boundary as much as possible, (2) the euclidean lengths of the curves are (nearly) minimal. The equivalence of their definition with the above one was proven by Gehring and Osgood [164] with a slight difference. In [164], uniform domains were characterized in terms of the inequality kG ≤ c1 jG + c2 , c1 , c2 > 0 . In [552, 2.49] the class of ϕ-uniform domains was introduced for the purpose of comparing various metrics and conformal invariants. It easily follows from [164], see [552, 2.50(2)] for details, that in the special case ϕ(t) = log(1 + t), ϕuniform domains are uniform in the sense of 6.1. Thus, in the Gehring–Osgood characterization of uniform domains, the constant c2 can be chosen to be 0 by increasing the other constant c1 . A very important class of planar Jordan domains, the quasidisks, provide many examples of uniform domains [159, 530, 533]. All convex domains are ϕ-uniform with ϕ(t) = t, t > 0 . The class of ϕ-uniform domains is wider than uniform domains: the strip domain {x ∈ R2 : 0 < x2 < 1} is not uniform but as a convex domain it is ϕ-uniform. Various properties of ϕ-uniform domains were studied in [209, 290].

6.1 Uniform Domains and Generalizations

85

Exercise 6.2 Show that G = Rn \ {0} is a uniform domain and that kG (x, y) ≤ A jG (x, y) for x, y ∈ G where A2 = 1 +

 π 2 ≈ 21.5 . log 2

[Hint: Let x, y ∈ G and let ϕ ∈ [0, π] be the angle between x and y. By a property |x−y| of the bisector of an angle in a triangle or by 3.5 (2) sin 12 ϕ ≤ |x|+|y| and hence |x − y| |x − y| ≤π |x| + |y| |x| + |y|  |x − y|  π π log 1 + ≤ j (x, y) . ≤ log 2 |x| + |y| log 2 G

ϕ ≤ 2 arcsin

By 4.6(2) and (5.31) we obtain the desired inequality.] The best constant in place of A is π/ log 3 ≈ 2.86 as shown by H. Lindén [335]. He also found estimates for the uniformity constants of some other domains. Exercise 6.3 Show that G = B2 \ [0, e1 ) is not a uniform domain. [Hint: For t ∈ 1 (0, 10 ) let xt = ( 14 , t) and yt = ( 14 , −t), Y = { (0, y) : y > 0 }. Show that kG (xt , yt ) ≥ kG (xt , Y ) −→ ∞ 1 ).] when t → 0 (cf. (4.7)), while jG (xt , yt ) = log 3 for all t ∈ (0, 10

Exercise 6.4 Suppose that there exists C ≥ 1 such that for all x, y ∈ G   kG (x, y) ≤ C sup kRn \{z} (x, y) : z ∈ ∂G . Show that G is uniform. [Hint: Apply 6.2 and (4.30).] Remarks 6.5 (1) J. Väisälä proved in [532, Thm 6.16] that the class of ϕ-uniform domains agrees with the class of uniform domains if the homeomorphism ϕ is slow, i.e. it satisfies the condition limt →∞ ϕ(t)/t = 0 . (2) Suppose that the domain G ⊂ Rn satisfies the following condition: There exists a constant C ≥ 1 such that every pair of points x, y ∈ G can be joined by a curve γ ⊂ G such that (a) (γ ) ≤ C |x − y| , (b) d(γ , ∂G) ≥ (1/C) min{d(x), d(y)} [549, 2.19(2)], [552, 2.50(1)]. Then we readily see that G is ϕ-uniform with ϕ(t) = C 2 t . (3) F. John [271] introduced a class of domains, nowadays usually known as John domains, which may have non-smooth, even locally nonrectifiable, boundaries. This class of domains was studied in [462], [465, p. 246], [466, p. 31], [368], and later by many people; see the surveys [408, 533]. Using a method based on counting of Whitney cubes, see below 6.2.1, D.A. Trotsenko [513] proved

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6 Metrics and Geometry

that the Hausdorff dimension of the boundary of a John domain is less than the dimension of the space. Growth bounds for the number of Whitney cubes of generation k = 1, . . . yield information about the metric size of ∂G [374]. Whitney decomposition is an important tool in real analysis and measure theory in Rn ; see [498, p. 167], [108, I.2, Thm 2.1], [171, p. 11].

6.1.1 Ptolemy’s Theorem Ptolemy’s theorem says that given a quadrilateral with vertices a, b, c, d ∈ R2 in this order, there holds |a − c||b − d| ≤ |a − b||c − d| + |b − c||a − d| with equality when all the points are located on the same circle or line [60, 10.9.2], [105, p. 42]. For a Jordan curve J and a, b, c, d ∈ J in the plane define p(a, b, c, d) =

a − b||c − d| + |b − c||a − d| = |a, c, b, d| + |a, c, d, b| , |a − c||b − d| p(C) = sup{p(a, b, c, d) : a, b, c, d ∈ J } ,

where the supremum is taken over all quadrilaterals with vertices on J . Then p(C) is called the Ptolemy constant of the Jordan curve J . It is clear that the Ptolemy constant is invariant under Möbius transformations. The Ptolemy constant is a handy tool in the theory of quasiconformal mappings: the well-known Ahlfors characterization of quasicircles [13, 159] makes use of it. In his unpublished Licentiate Thesis in 1996, P. Seittenranta proved, among other things, that for a triangle T with the least angle α p(T ) = 1/ sin(α/2) . E. Harmaala and R. Klén have carried out further investigations of Jordan curves and found estimates of their Ptolemy constants [196] and for the uniformity constants of Jordan domains. Remarks 6.6 (1) Since 1978, when uniform domains were introduced by O. Martio and J. Sarvas [368], these domains have found many interesting applications, e.g. in P. Jones’ works [273, 274] on extension operators of function spaces. An exposition of these results occurs in [159], with several equivalent definitions of plane uniform domains.

6.1 Uniform Domains and Generalizations

87

(2) The above variant of the definition of a uniform domain is based on a comparison property of the metrics kG and jG , see [164] and [552, 2.49]. Suppose now that m1 , m2 are metrics on a domain and that these metrics are “comparable” in some suitable sense, e.g. that for some homeomorphisms ϕ1 , ϕ2 : [0, ∞) → [0, ∞) we have for all x, y ∈ G ϕ1 (m1 (x, y)) ≤ m2 (x, y) ≤ ϕ2 (m1 (x, y)) . Such domains could be called (ϕ1 , ϕ2 )-uniform domains. As yet, very little is known about the geometric properties of these domains. For concrete choices of ϕ1 , ϕ2 and the metrics m1 , m2 these (ϕ1 , ϕ2 )-uniform domains lead to comparison problems between metrics studied for instance in [244] and [290]. A concrete research topic would be to look for conditions under which the union of two intersecting (ϕ1 , ϕ2 )-uniform domains is again in the same class. (3) The above definition of uniform domains applies only to proper subdomains of Rn . Using the above idea and the Ferrand metric σG and the Möbius metric δG we can define a class of Möbius uniform domains as follows. A domain G in n R with card(∂G) ≥ 2 is Möbius uniform if there exists a constant c ≥ 1 such that for all x, y ∈ G σG (x, y) ≤ c δG (x, y) . It is easy to see that if the domain G is a subset of Rn , then it is uniform iff it is Möbius uniform [483, 4.5]. (4) Uniform domains in a different meaning were studied by R. Näkki [405]. His class of domains is invariant under chordal isometries. Exercise 6.7 Show that for x , y ∈ G = Rn \ {0} we have |x − y| log 3 ≤ jG (x, y) . |x| + |y| Exercise 6.8 Let f : Rn → Rn be an L-bilipschitz mapping, that is |x − y|/L ≤ |f (x) − f (y)| ≤ L |x − y| for all x, y ∈ Rn , and let G ⊂ Rn be uniform. Show that f G is uniform. [Hint: Using the definitions show that   kG (x, y)/L2 ≤ kf G f (x), f (y) ≤ L2 kG (x, y) . Then deduce from (5.6) that   jG (x, y)/L2 ≤ jf G f (x), f (y) ≤ L2 jG (x, y) .]

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Exercise 6.9 (1) Let D = Rn \ {0}. Show that 1 + 2 q(x, y) ≤ exp kD (x, y) for x, y ∈ D. (2) Let z ∈ Rn and G = Rn \ {z}. Show that q(x, y) ≤

1 2

c (exp kG (x, y) − 1)

   for all x, y ∈ G, where c = 1 + 12 |z| |z| + 4 + |z|2 . [Hint: Let D = Rn \ {0} and h(x) = x − z. Then by part (1) and 3.29 kG (x, y) = kD (x − z, y − z) ≥ log(1 + 2 q(x − z, y − z)) ≥ log(1 + 2 q(x, y)/c) where c is as above.] Conclusion: If G is a proper subdomain of Rn , then (cf. 5.4) exp kG (x, y) ≥ 1 + 2 q(x, y)/A where A depends only on min{ |z| : z ∈ ∂G }. Exercise 6.10 Prove that sh x (1)  < th x < x < sh x < 12 sh 2x, x > 0. 2 2 sh x + ch x Use this to conclude that for 0 < a < b  √ a+b 2 a 2 + b2 b−a < ab < < < . (2) 1/a + 1/b log b − log a 2 2 Exercise 6.11 Prove the following statements for x, y > 0 . (1) For α ∈ [0, π] and x = y (x − y)2 x 2 + y 2 − 2xy cos α ≤ . α 2 + (log x − log y)2 (log x − log y)2 (2) For x = y we define |x − y|  g(x, y) = √ 2 1 + x 1 + y 2 | log x − log y|

6.1 Uniform Domains and Generalizations

89

and g(x, x) = lim g(x, y) = y→x

x . 1 + x2

Then g(x, y) ≤ 1/2. We now compare the metrics q and kG . We start with the case G = Rn \ {0} and thereafter study the case G = Rn \ {z}. Theorem 6.12 For G = Rn \ {0} and x, y ∈ G, x = y, we have 1 q(x, y) ≤ . kG (x, y) 2 Moreover, the constant 1/2 is the best possible. Proof Let us denote α = (x, 0, y) and assume |x| = |y|. By the definition of q(x, y) and k(x, y) and Exercise 6.11 

q(x, y) =  k(x, y)

|x|2 + |y|2 − 2|x||y| cos α   1 + |x|2 1 + |y|2 α 2 + log2 (|y|/|x|)

≤ 

||x| − |y|| 1 ≤ .  2 1 + |x|2 1 + |y|2 | log(|y|/|x|)|

If |x| = |y|, then q(x, y) 2|x| sin(α/2) |x| 1 = ≤ ≤ . 2 2 k(x, y) (1 + |x| )α 1 + |x| 2 Let us finally show that the constant 1/2 is the best possible. By choosing |x| = 1 = |y| we have q(x, y) sin(α/2) 1 = → kG (x, y) α 2  

as α → 0 and the assertion follows. Next the same result in the general punctured space will be studied. For the proof of Theorem 6.14, we need the following technical lemma. Lemma 6.13 For given a ≥ 0, (r + s + a)2 = max r,s≥0 (1 + r 2 )(1 + s 2 )



a+

2 √ 4 + a2 . 2

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Proof Let f (r, s) =

(r + s + a)2 , (1 + r 2 )(1 + s 2 )

r, s ∈ [0, +∞).

It is clear that f (r, +∞) = f (+∞, r) = 1/(1 + r 2 ) for each r ∈ [0, +∞). A simple calculation implies that max f (0, r) =

r∈[0,+∞)

max f (r, 0) = f (1/a, 0) = 1 + a 2 .

r∈[0,+∞)

By differentiation, √ ∂f ∂f −a + 4 + a 2 =0= ⇒r =s= ≡ r0 . ∂r ∂s 2 We have  f (r0 , r0 ) =

a+

2 √ 4 + a2 ≥ 1 + a 2. 2

Since f is differentiable in [0, +∞) × [0, +∞), f (r, s) ≤ f (r0 , r0 ) for all (r, s) ∈   [0, +∞) × [0, +∞). Theorem 6.14 Let z ∈ Rn . For G = Rn \ {z} and x, y ∈ G, x = y, we have  q(x, y) |z| + 1 + |z|2 ≤ . kG (x, y) 2  Moreover, the expression (|z| + 1 + |z|2 )/2 is the best possible. Proof By (5.2) and (3.6), q(x, y) =  k(x, y)  = ≤

1 + |x|2



|x − y|  1 + |y|2 θ 2 + log2 (|y − z|/|x − z|)

|x − z|2 + |y − z|2 − 2|x − z||y − z| cos θ θ2

+ log (|y − z|/|x − z|) 2

1 |x − z| − |y − z|   log |x − z| − log |y − z| 1 + |x|2 1 + |y|2

1 |x − z| + |y − z|   2 1 + |x|2 1 + |y|2  |x| + |y| + 2|z| |z| + 1 + |z|2 , ≤   ≤ 2 2 1 + |x|2 1 + |y|2 ≤



1  1 + |x|2 1 + |y|2

6.1 Uniform Domains and Generalizations

91

where the first inequality follows from Exercise 6.11 (1), the second inequality is the mean inequality (a − b)/(log a − log b) ≤ (a + b)/2 [29, 1.58(4)(b), p.78] also 6.10(2) and the last one follows from Lemma 6.13.  From the above chain of inequalities it is easy to see that the upper bound (|z|+ 1 + |z|2 )/2 can be obtained  2 when x, y → (− 1 + |z| + |z|)z/|z|.   The next result gives another comparison result, for the metrics q and jG in the complement of the origin. Theorem 6.15 For G = Rn \ {0} and x, y ∈ G, x = y, we have 1 q(x, y) ≤ jG (x, y) log 3 with equality for x and y such that x = −y and |x| = 1. In particular, the constant 1/ log 3 is the best possible. Proof See Exercises 6.7, 3.9. The sharpness follows choosing |x| = 1 and y = −x. Then we have q(x, y)/jG(x, y) = 1/ log 3 and the assertion follows.   Corollary 6.16 For all x, y ∈ G = Rn \ {0} π (1) 2q(x, y) ≤ kG (x, y) ≤ jG (x, y) log 3 (2) (log 3)q(x, y) ≤ jG (x, y) ≤ kG (x, y). Proof The assertion follows from Exercise 6.2 and Theorems 6.12 and 6.15.

 

Note that the constants in Corollary 6.16 are best possible. The second inequality of (1) holds with equality for y = −x. In (2) the first inequality holds for y = −x, |x| = 1, and the second inequality holds with equality for (x, 0, y) = 0. These results can be generalized to the domain Rn \ {z}, z ∈ Rn , see [287, 3.103.13]. Exercise 6.17 Let f : G → G = f (G) , G, G ⊂ Rn , be a homeomorphism such that for some C > 0 and all x, y ∈ G, kG (f (x), f (y)) ≤ CkG (x, y). Suppose that b ∈ ∂G and that bi ∈ G with bi → b, f (bi ) → β, i → ∞, and let E = ∪Bk (bi , 1). Here Bk (x, M) stands for the quasihyperbolic ball. Prove that f (x) → β when x → b, x ∈ E. Note: By topology, for each sequence (bi ) tending to a boundary point b of G such that the image sequence also has a limit γ , it follows that γ ∈ ∂G . Exercise 6.18 Let G = B2 \ {0}. (1) For 0 < r < 1/2 compute the quasihyperbolic area w.r.t. kG of the annulus {z : r < |z| < 1/2} . (2) For 1/2 < r < 1 compute the quasihyperbolic area w.r.t. kG of the annulus {z : 1/2 < |z| < r} .

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6.2 Whitney Squares and (a, b, s)-Admissible Families 6.2.1 Whitney Decomposition The Whitney decomposition of a domain in Rn is an important tool in real and harmonic analysis. Here we use it to illustrate the relative geometry of a planar domain. A bounded domain G ⊂ R2 can be represented as a countable union of nonoverlapping closed squares. With natural changes, using cubes in place of squares, a similar representation also holds for the case of domains G ⊂ Rn , n ≥ 2 [498, p. 167, Thm 1]. We consider here only the case of dimension n = 2 . The Whitney decomposition theorem says that there exist closed squares Qkj , k ∈ Z, 0 ≤ j ≤ Nk , having pairwise disjoint interiors and sides parallel to the coordinate axes, such that (i) every square Qkj , 0 ≤ j ≤ Nk , has sidelength 2−k , (ii) G = ∪k,j Qkj , (iii) d(Qkj ) ≤ d(Qkj , ∂G) < 4d(Qkj ) . The squares Qkj are called Whitney squares and the squares Qkj , 0 ≤ j ≤ Nk , form the generation k of squares. In the planar case, the construction of this decomposition is given in [149, p. 21]. Whitney squares can be viewed as approximations for quasihyperbolic balls since there are universal constants c1 < c2 such that each square contains a quasihyperbolic ball of radius c1 and is contained in a quasihyperbolic ball of radius c2 ; see (5.15). The authors are indebted to D. Marshall who helped us to generate the accompanying figure with his software (Fig. 6.1).

Fig. 6.1 Whitney squares

6.2 Whitney Squares and (a, b, s) -Admissible Families

93

Looking at the figure and keeping in mind the properties (i)–(iii) of the Whitney decomposition we see that the integer sequence Nk , k = 1, 2, . . . , contains a lot of information about the geometry of the domain G . Apparently, the union of the k k kth generation Whitney squares ∪N j =0 Qj , closely approximates the shape of the  boundary ∂G . If G is a bounded planar domain, then m(G) = Nk 2−pk < ∞ holds with p = 2 and we see that for all large enough k we have the trivial upper bound Nk < 2kp m(G) with p = 2 . If this inequality holds with p < 2 , and all large enough k , then we have nontrivial information about ∂G . Under some geometric hypotheses on G this growth bound for Nk yields a bound for the Hausdorff dimension of ∂G [374, 513]. This idea can also be reversed: If F = ∂G is a compact set of measure zero and t > 0 then (under some geometric conditions) we can find an upper bound for Nk in terms of the Hausdorff dimension of F . This idea also yields an estimate for the (n − 1)-dimensional Hausdorff measure of the parallel set {z : d(z, F ) = t} [276]. The Whitney decomposition can also be applied to produce upper bounds for the quasihyperbolic distance between two points x, y in a given planar domain G : choose a collection of Whitney squares of minimal cardinality p such that there is a curve γ contained in the union of the squares with x, y ∈ γ . Then the number p is comparable to an upper bound for kG (x, y) . We now formulate this idea in a precise way, but using, for a fixed number s ∈ (0, 1) , chains of balls B n (z, sd(z)), z ∈ G , in place of Whitney squares.

6.2.2

(a, b, s)-Admissible Families

Let G be a proper subdomain of Rn , a, b ∈ G, and s ∈ (0, 1). A family F = { B n (xi , ri ) : i = 1, . . . , p } of balls in G is said to be (a, b, s)–admissible [549, 550] if the following two conditions are satisfied: (1) a ∈ B n (x1 , sr1 ) , b ∈ B n (xp , srp ) , (2) B n (xj , srj ) ∩ B n (xj +1 , srj +1 ) = ∅ , j = 1, . . . , p − 1 .

(6.1)

We shall show that the smallest possible number of balls in an (a, b, s)admissible family is roughly proportional to kG (a, b) , with a constant of proportionality c(s) (Fig. 6.2). The case G = Hn will be studied first. To this end note that by (4.6)   1+t , t ∈ (0, 1) ρ B n (x, txn ) = log 1−t for x = (x1 , . . . , xn ) ∈ Hn .

(6.2)

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6 Metrics and Geometry

Fig. 6.2 Admissible family

Lemma 6.19 Let a, b ∈ Hn , s ∈ (0, 1) , and c = log 1+s 1−s . (1) There is an (a, b, s)-admissible family containing at most 1 + ρ(a, b)/c balls. (2) Every (a, b, s)-admissible family contains at least ρ(a, b)/c balls. Proof (1) Choose an integer p ≥ 1 such that (p − 1) log

1+s 1+s ≤ ρ(a, b) < p log . 1−s 1−s

(6.3)

Select points y0 = a , yj ∈ J [a, b] such that ρ(y0 , yj ) = j log 1+s 1−s , j = 1, . . . , p − 1 , and set yp = b . Let B n (xj , sxj n ) be chosen so that S n−1 (xj , sxj n ) is perpendicular to J [yj −1 , yj ] and yj −1 , yj ∈ S n−1 (xj , sxj n ) , j = 1, . . . , p . (In other words, B n (xj , sxj n ) = Bρ (zj , M) , where 2M = log 1+s 1−s and zj ∈ J [yj −1 , yj ] and ρ(zj , yj −1 ) = ρ(zj , yj ) = M , 1 ≤ j ≤ p − 1 .) Here xj n is the nth coordinate of xj . In view of (6.2) and (6.3) the family { B n (xj , xj n ) : j = 1, . . . , p } is the desired (a, b, s)-admissible family. (2) Suppose that { B n (zj , rj ) : j = 1, . . . , m} is (a, b, s)-admissible. By (6.2) we get ρ(a, b) ≤

m m !   !   1+s , ρ B n (zj , srj ) ≤ ρ B n (zj , szj n ) = m log 1−s j =1

j =1

from which the desired lower bound follows.

 

6.2 Whitney Squares and (a, b, s) -Admissible Families

95

Before formulating an analogue of Lemma 6.19 for Bn we make a few observations about hyperbolic balls. By (4.21) Bρ (x, M) = B n (y, r) with # (1 + |x|)t " 1 r = ∈ th 2 M, th M ; t = th 12 M , 2 1 − |y| 1 + |x|t for all x ∈ Bn (see Exercise 4.4) and hence     B n y, (th 12 M)(1 − |y|) ⊂ Bρ (x, M) ⊂ B n y, (th M)(1 − |y|) .

(6.4)

It follows from (6.4) that log

  1+s 1+s ≤ ρ B n (z, s(1 − |z|) ) ≤ 2 log . 1−s 1−s

(6.5)

Lemma 6.20 Let a, b ∈ Bn , s ∈ (0, 1) , and c = log 1+s 1−s . (1) There is an (a, b, s)-admissible family containing at most 1 + ρ(a, b)/c balls. (2) Every (a, b, s)-admissible family contains at least ρ(a, b)/(2c) balls. Proof As in the proof of Lemma 6.19 we cover J [a, b] by { B ρ (xj , M) : 1 j = 1, . . . , p } , p ≤ 1 + 2M ρ(a, b) where M is chosen so that B ρ (xj , M) ⊂ n B (yj , s(1 − |yj |) ) . By (6.4) we may choose M = 12 log 1+s 1−s . The proof of (1) follows now as in Lemma 6.19(1). The proof of (2) is similar to the proof of part (2) of Lemma 6.19 except that here we use the two-sided inequality (6.5) instead of the equality (6.2).   In the next lemma we prove a counterpart of Lemma 6.20 for an arbitrary domain. Lemma 6.21 Let G be a proper subdomain of Rn , a, b ∈ G and 0 < s < 1 . (1) There is an (a, b, s)-admissible family containing at most 1 + kG (a, b)/d1(s) balls, d1 (s) = 2 log(1 + s) . (2) Every (a, b, s)-admissible family contains at least kG (a, b)/d2(s) balls, 1 d2 (s) = 2 log 1−s . Proof (1) Fix a quasihyperbolic geodesic segment JG [a, b] . Choose points zj ∈ JG [a, b] , j = 1, . . . , p with kG (a, z1 ) = M , kG (zj , zj +1 ) = 2M , j = 1, . . . , p − 2 , zp = b , kG (zp−1 , zp ) < 2M , where M > 0 will be chosen soon and 2M(p − 1) ≤ kG (a, b) . We wish to choose M such that BkG (zj , M) ⊂ B n (zj , sd(zj ) ) , j = 1, . . . , p . In view of Exercise 5.11 it suffices to choose M such that log(1 + s) = M . It is clear that the family { B n (zj , d(zj )) : j = 1, . . . , p } is (a, b, s)-admissible and that p ≤ 1 + kG (a, b)/d1(s) , d1 = 2 log(1 + s) .

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(2) It follows from Exercise 5.11 that for all y ∈ G   kG B n (y, sd(y)) ≤ 2 log

1 . 1−s

The proof follows from this inequality exactly in the same way as in part (2) of   Lemma 6.19.

6.3 Harnack Functions We shall next give an immediate application of Lemma 6.21 to positive functions satisfying the Harnack inequality. Definition 6.22 ([550]) Let G be a proper subdomain of Rn and let u : G → R+ ∪ {0} be continuous. We say that u satisfies the Harnack inequality in G if there exist numbers s ∈ (0, 1) and Cs ≥ 1 such that max u(z) ≤ Cs min u(z) Bx

(6.6)

Bx

holds true whenever B n (x, r) ⊂ G and Bx = B n (x, sr) . A function satisfying (6.6) is called a Harnack function. The above definition does not require smoothness or any other regularity properties beyond continuity of u . It is well known that non-negative harmonic functions satisfy (6.6) [169, p. 16]. Lemma 6.23 Let u : G → R+ ∪ {0} be a Harnack function. Then u(x) ≤ Cs1+t u(y) , t =

kG (x, y) d1 (s)

for x, y ∈ G where d1 (s) = 2 log(1 + s) . If G = Hn or G = Bn , then we can replace t by ρ(x, y)/ log 1+s 1−s . Proof Fix x, y ∈ G and an (x, y, s)-admissible family { B n (xi , ri ) : i = 1, . . . , p } with p ≤ 1 + kG (x, y)/d1(s) (see 6.20). Let zj ∈ B j ∩ B j +1 , Bj = B n (xj , srj ) , j = 1, . . . , p − 1 . By (6.6) we get p−1

u(x) ≤ Cs u(z1 ) ≤ Cs2 u(z2 ) ≤ · · · ≤ Cs which is the desired inequality.

p

u(zp−1 ) ≤ Cs u(y) ,  

Let u be as in Lemma 6.23. It should be noticed that, by virtue of 6.23, either u vanishes identically or u is strictly positive in G .

6.3 Harnack Functions

97

In view of Lemma 6.23 one can use the term Harnack chain of balls joining a and b in place of (a, b, s) -admissible family [33, p. 261], [549]. Corollary 6.24 Let f : (G, kG ) → (R, | |) be uniformly continuous as a mapping between metric spaces. Then there is a number α such that |f (x) − f (y)| ≤ 1 + α kG (x, y) for all x, y ∈ G . Proof Because f is uniformly continuous, there exists a number t0 such that |f (x) − f (y)| ≤ 1 for x, y ∈ G , kG (x, y) ≤ t0 . If kG (x, y) > t0 we can exploit the method of the proof of 6.21 to show that |f (x) − f (y)| ≤ 1 + kG (x, y)/t0 . Hence we may choose α = 1/t0 . The details are left as an easy exercise for the   reader. The hyperbolic volume of a (Lebesgue) measurable set E in Bn is defined by  mh (E) = E

2n dm(x) (1 − |x|2 )n

(6.7)

(cf. Chap. 4). Let ωn−1 be the (n − 1)-dimensional area of S n−1 . Integration in polar coordinates yields s mh (B n (s)) = 2n ωn−1 0

t n−1 2n ωn−1  s n−1 dt < . 2 n (1 − t ) n−1 1−s

(6.8)

The last inequality holds because s 0

t n−1 dt ≤ s n−1 (1 − t 2 )n s

< s n−1

s 0

dt (1 + t)n (1 − t)n

$ (1 − t)1−n dt . = s n−1 n (1 − t) n−1 s

0

0

Since t n−1 ≥ 22−n t for t ∈ ( 12 , 1) we obtain s n

n

mh (B (s)) > 2 ωn−1 2

2−n 1/2

tdt 22(1−n)ωn−1 1 (1 − s)1−n − > 2 n (1 − t ) n−1 n−1 (6.9)

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6 Metrics and Geometry

for s ∈ ( 12 , 1) . Finally, for x ∈ Bn and M > 0 , by the invariance of mh under the action of GM(Bn ) and by (4.22) and (6.8) we get mh (Bρ (x, M)) = mh (Bρ (0, M)) = mh (B n (th 12 M)) ≤

(6.10)

2n ωn−1  th 12 M n−1 2n ωn−1 M(n−1) n−1 1 e < th ( 2 M) . 1 n − 1 1 − th 2 M n−1

For what follows we shall need a lemma about coverings by families of euclidean balls [320, p. 197, Lemma 3.2.]. We shall give such a lemma here with a slightly more general formulation. Covering theorems of this sort are very useful in analysis, in particular, in measure theory [377]. For a related result see [108, Theorem 1.1.]. Lemma 6.25 Let (X, d) be any one of the metric spaces (Rn , | |) , (Bn , ρBn ) , or (Hn , ρHn ) , and let BX (z, r) = { y ∈ X : d(y, z) < r } . Let A be a non-empty subset of X , F = { BX (z, r(z)) : z ∈ A} and suppose sup{ r(x) : x ∈ A } < ∞ . There exists a number  c(n) depending only on n and a countable subfamily F1 ⊂ F such that (1) A ⊂ F1 and (2) each x ∈ A belongs to at most c(n) elements of F1 . Let A ⊂ X , A = ∅ , where (X, d ) is a metric space, and for t > 0 set k

% BX (xj , t) , xj ∈ A . pX (A, t) = inf k : A ⊂

(6.11)

j =1

Because the space X is usually specified by the context, we write p(A, t) = pX (A, t) . Note that if A ⊂ X is non-empty and compact then pX (A, t) < ∞ and { BX (xj , t) : j = 1, . . . , pX (A, t) } , xj ∈ A , is a covering of A . Lemma 6.26 Let (X, d ) be any one of the metric spaces in 6.25, let mX = m for X = Rn , mX = mh if X = Bn or X = Hn , and let A ⊂ X , A = ∅ , be compact. There is a number d1 = 1/mX (BX (y, t)) depending only on n and t such that  % BX (z, t) d1 mX (A) ≤ p(A, t) ≤ c(n)d1mX z∈A

where c(n) is as in 6.25. The simple proof of this lemma is based on a standard volume-comparison argument and on Lemma 6.25, and left as an easy exercise for the reader.   Exercise 6.27 Show that mh |x|≤r Bρ (x, M) ≤ d2 (n, M)(1−r)1−n , where mh is the hyperbolic measure of (Bn , ρBn ) . [Hint: |x|≤r Bρ (x, M) = B n (R) where 1+R M 1+r −M , and one may apply (6.8).] 1−R = e 1−r (see (4.16)). Hence 1 −R ≥ (1 −r)e

6.3 Harnack Functions

99

Exercise 6.28 Apply 6.26 and 6.27 to show that for r near 1 d3 (1 − r)1−n ≤ p(B n (r), M) ≤ c(n) d1 d2 (1 − r)1−n where d3 depends only on n and M . [Hint: Apply also (6.9).] Exercise 6.29 For ϕ ∈ (0, 12 π) let C(ϕ) = { z ∈ Rn : z · en = |z| cos ϕ } and   ϕ At = C(ϕ) ∩ B n (1) \ B n ( 1t ) , t > 1 . Show that &

ϕ ρHn (At )

(1 + u)2 √ 1 2 ≤ log t + √ 4 cos2 ϕ t

' ,

2 2 where u2 = sin2 ϕ + ( tt −1 +1 ) cos ϕ . [Hint: Consider the smallest euclidean ball B ϕ containing At , and find an upper bound for ρ(B) .]

Remark 6.30 For n = 2 , the hyperbolic area of Bρ (0, r) is 4π sh2 ( 12 r) [51, p. 132, Theorem 7.22.]. Note that for r → 0 this is approximately πr 2 , the euclidean area of B 2 (r) . Exercise 6.31 In every ball B n (x, r) in Rn one can define a hyperbolic metric ρr by making use of a formula similar to (4.18). Generalizing (4.17), we have the n equality ρr (x, x + ae1 ) = log 1+a/r 1−a/r for 0 ≤ a < r . Assume that z ∈ B ,  be the hyperbolic metric M > 0 , and ρ is the hyperbolic metric of Bn , and let ρ of Bρ (z, M) . Let a ∈ Bρ (z, M) with ρ a, ∂Bρ (z, M) ≥ b > 0 . Find an upper bound for ρ (z, a) in terms of ρ(z, a) and b . [Hint: By the invariance of ρ  and ρ we may assume that z = 0 , whence Bρ (z, M) = B n (th 12 M) .] Exercise 6.32 Let G be a proper subdomain of Rn and F a connected subset of G with d(F, ∂G) > 0 . Applying the covering Lemma 6.25 show that  kG (F ) ≤ c n,

d(F )  0 , the number M( )1/(1−n) is called the extremal length of . We take the extremal length to be ∞ if M( ) = 0 . Lemma 7.1 The p -modulus Mp is an outer measure in the space of all curve families in Rn . That is, (1) Mp (∅) = 0 , (2) 1 ⊂ 2 implies Mp ( 1 ) ≤ Mp ( 2 ) , ∞ ∞ %  ! i ≤ Mp ( i ) . (3) Mp i=1

i=1

Let 1 and 2 be curve families in Rn . We say that 2 is minorized by 1 and write 2 > 1 if every γ ∈ 2 has a subcurve belonging to 1 . Lemma 7.2 1 < 2 implies Mp ( 1 ) ≥ Mp ( 2 ) . The curve families 1 , 2 , . . . are called separate if there exist disjoint Borel sets Ei in Rn such that if γ ∈ i is locally rectifiable then γ χi ds = 0 where χi is the characteristic function of Rn \ Ei . Lemma 7.3 If 1 , 2 , . . . are separate and if < i for all i , then Mp ( ) ≥

!

Mp ( i ) .

Lemma 7.4 Let G be a Borel set in Rn and = { γ : γ is a curve in G with (γ ) ≥ r } . If r > 0 then Mp ( ) ≤ m(G)r −p . Proof Because ρ = 1r χG ∈ F( ) the proof follows from (7.1).

 

Corollary 7.5 If is the family of non-constant curves in a Borel set G ⊂ Rn with m(G) = 0 , then Mp ( ) = 0 .  Proof If j = { γ ∈ : (γ ) ≥ j1 } , j = 1, 2, . . . , then = j and the proof follows from 7.1(3) and 7.4.  

7 The Modulus of a Curve Family

105

Curve families with zero p -modulus are sometimes called p –exceptional. We next give a general criterion for a curve family to be p -exceptional, which is a generalization of 7.5. Lemma 7.6 A curve family is p -exceptional if and only if there exists an admissible function ρ ∈ F( ) such that 

 Rn

ρ p dm < ∞ and

ρ ds = ∞ γ

for every locally rectifiable γ ∈ . Proof If ρ satisfies the above conditions, then k −1 ρ ∈ F( ) for every k = 1, 2, . . . and thus  Mp ( ) ≤ k −p ρ p dm −→ 0 Rn

as k → ∞ . Hence is p -exceptional. Conversely, let Mp ( ) = 0 and choose a  p sequence ρk ∈ F( ) such that Rn ρk dm < 4−k , k = 1, 2, . . . . Writing ρ(x) =

∞ !

2k ρk (x)p

1/p

k=1

we infer that

 Rn

ρ p dm < ∞ . On the other hand 

 ρ ds ≥ γ

for all k = 1, 2, . . . i.e.

 γ

2k/p ρk ds ≥ 2k/p γ

ρ ds = ∞ for each locally rectifiable curve γ in .  

Corollary 7.7 If is a curve family in Rn and r = { γ ∈ : (γ ) < ∞ } , then M( ) = M( r ) . Proof Set ρ(x) = 1 for |x| < 2 and ρ(x) = 1/(|x| log |x|) for |x| ≥ 2 . By direct computation  Rn

ρ n dm = 2n n +

ωn−1 0 on |γ | and it is clear that γ ρ ds = ∞ . If γ ∈ ∞ is unbounded

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7 The Modulus of a Curve Family

we choose x ∈ |γ | \ B n (2) . It follows that 

 ρ ds ≥ γ

∞

|x|

dr =∞ r log r  

as desired. n

For E, F, G ⊂ R we denote by (E, F ; G) the family of all closed nonconstant curves joining E and F in G . More precisely, a non-constant path n γ : [a, b] → R belongs to (E, F ; G) iff (1) one of the end points γ (a), γ (b) belongs to E and the other to F , and (2) γ (t) ∈ G for a < t < b . n

Remark 7.8 If G = Rn or R we often denote (E, F ; G) by (E, F ) . Curve families of this form are the most  important for what follows. The  following  subadditivity property is useful. If E = ∞ E and c (F ) = M (E, F ) = j p j =1 E  n cF (Ej ) , see 7.1(3).More precisely if G ⊂ R is a cF (E) , then cF (E) ≤  domain and F ⊂ G is fixed, then cFG (E) = Mp (E, F ; G) is an outer measure defined for E ⊂ G . In a sense which will be made precise later on, cE (F ) describes the mutual size and location of E and F . Assume now that D is an n open set in R and that F ⊂ D . It follows from 7.1(2) that       Mp (F, ∂D; D \ F ) ≤ Mp (F, ∂D; D) ≤ Mp (F, ∂D) . On the other hand, because (F, ∂D; D) < (F, ∂D) and (F, ∂D; D \ F ) < (F, ∂D; D) , 7.2 yields       Mp (F, ∂D) = Mp (F, ∂D; D) = Mp (F, ∂D; D \ F ) . (7.2) As the relatively complicated definition (7.1) of the p -modulus suggests, it is usually a very difficult task to find Mp ( ) when is given. In fact, the real number Mp ( ) is known for very few curve families. If has a simple structure, then one can sometimes compute Mp ( ) in two steps. First, applying Hölder’s  inequality and Fubini’s theorem one proves a lower bound for Rn ρ n dm when ρ is an admissible function. Second, one shows that this lower bound is attained by some particular admissible function ρ1 . Making use of this method one can compute the modulus of a cylinder and of a spherical ring (for details see Väisälä [524, pp. 20–23]).

7.1 Basic Properties of the Modulus 7.1.1 The Cylinder Let E ⊂ { x ∈ Rn : xn = 0 } be a Borel set, h > 0 , F = E + hen and denote G = { x ∈ Rn : (x1 , . . . , xn−1 , 0) ∈ E, 0 < xn < h } (Fig. 7.1). Then G is a

7.1 Basic Properties of the Modulus

107

Fig. 7.1 Left: A cylinder (7.1.1). Right: Remark 7.9

cylinder with bases E and F and, as shown in [524]   Mp (E, F ; G) = mn−1 (E) h1−p = m(G) h−p .

7.1.2 The Spherical Ring Let 0 < a < b , D = B n (b) \ B n (a) and let Y be a Borel set in S n−1 . Let γy = { ty : a ≤ t ≤ b } , y ∈ Y . Then [524]  b 1−n M( ) = mn−1 (Y ) log ; = { γy : y ∈ Y } , a    b 1−n M (S n−1 (b), S n−1 (a); D) = ωn−1 log . a

(7.3) (7.4)

By (7.2) the formula (7.4) holds also if D is replaced by Rn . Letting a → 0 we see by 7.2 that   M (S n−1 (b), {0}) = 0 .

(7.5)

It follows from (7.5), 7.1, and 7.2 (see the proof of 7.5 or [524, p. 23]) that the family of all non-constant curves γ passing through a prescribed point x0 ∈ Rn is n-exceptional. Remark 7.9 The inequality of Lemma 7.4 is sharp: if G is the cylinder in 7.1.1 with bases E and F then 7.4 holds as an equality. However, this is not usually the case. Applying (7.4) we shall now give an example in which Lemma 7.4  gives a very crude estimate. Let E = S n−1 , F = S n−1 (3) , Gt = B n (4) \  S n−1 (2) ∪ B n (2e1 , t) , and t = (E, F ; Gt ) , t ∈ (0, 12 ) . In this particular case, Lemma 7.4 yields for M( t ) an upper bound independent of t . Let t =

108

7 The Modulus of a Curve Family

(S n−1 (2e1 , t), S n−1 (2e1 , 1); Gt ) . Because t < t , we get by (7.4) M( t ) ≤ ωn−1 (log(1/t))1−n i.e. M( t ) → 0 as t → 0 . In conclusion, keeping E and F fixed and letting the  domain Gt vary so that E, F ⊂ Gt and m(Gt ) is constant, one can make M (E, F ; Gt ) arbitrarily small, while this fact is not reflected in the form of the upper bound 7.4. The most valuable property of the n-modulus is invariance under conformal mappings, which is the content of the next lemma. n We first extend the definition (7.1) to curve families in R when p = n . Let n be a curve family in R . If contains a constant curve, we set M( ) = ∞ . Denote ∞ = { γ ∈ : ∞ ∈ γ } . If does not contain a constant curve, we set M( ) = M( \ ∞ ) . It follows from (7.5), 7.1, and 7.2 that M( ∞ ) = 0 . Hence for curve families in Rn , this extended definition of n-modulus coincides with (7.1). It should be pointed out that we have not included p -modulus, p = n , in this extended definition (see 7.19). n

Lemma 7.10 Let D and D  be domains in R and let f : D → D  be a conformal mapping. Then M(f ) = M( ) for each curve family in D where f = {f ◦ γ : γ ∈ }. It is easy to see that 7.10 is false for the p -modulus, p = n , even if f is the stretching x → 2x . Consider e.g. the curve family  = (E, F ; G) where E = { z ∈ Bn : zn = 0 } , F = E + {hen } , h > 0 , and 2 G = { x ∈ Rn : x12 + · · · + xn−1 < 1 , 0 < xn < 1 } .

Then by 7.1.1 0 < Mp () = m(G)h−p = m(f G)(2h)−p = Mp (f ) = 2n−p Mp () for p = n . An immediate application of the conformal invariance 7.10 is the following counterpart of (7.4) in the hyperbolic and spherical geometries. Corollary 7.11 Let 0 < a < b and x ∈ Bn and   (a, b) =  ∂Bρ (x, a), ∂Bρ (x, b); Bρ (x, b) \ Bρ (x, a) . Then th(b/2) . th(a/2) n If z ∈ R , 0 < r < s < 1 , and (r, s) = ( ∂Bq (x, r), ∂Bq (x, s)) then ⎡ ⎛  ⎞⎤1−n   s 1 − r2 ⎠⎦ (2) M (r, s) = ωn−1 ⎣log ⎝ . r 1 − s2 (1) M((a, b)) = ωn−1 L(a, b)1−n ;

L(a, b) = log

7.1 Basic Properties of the Modulus

109

Proof Let Tx ∈ M(Bn ) be as defined in 3.2. Then Tx (x) = 0 and we obtain by (4.23) Tx Bρ (x, c) = Bρ (0, c) = B n (th 12 c) , c > 0 .

(7.6)

Now 7.10 together with (7.6) and (7.4) yields M((a, b)) = M(Tx (a, b)) = ωn−1 L(a, b)1−n . n

For the proof of (2), let tx ∈ M(R ) be a spherical isometry with tx (x) = 0 , see (3.23). Then by (3.24) or 3.15(1)    tx Bq (x, r) = Bq (0, r) = B n r/ 1 − r 2 .  

The proof of (2) follows from this equality, 7.10, and (7.4).

Next we shall discuss various symmetry properties of the modulus. If A ⊂ Rn+ we denote by A∗ the symmetric image { (x1 , . . . , xn−1 , −xn ) ∈ Rn : (x1 , . . . , xn ) ∈ A } of A in ∂Rn+ . The next three lemmas will be given without proofs. For the proofs of 7.12, 7.13, 7.14 see [154, 576], and [548], respectively. n

Lemma 7.12 Let E and F be disjoint compact sets in R+ and let E ∗ and F ∗ be the symmetric images of E and F in ∂Rn+ . If 1 and 2 are the families of n curves joining E to F in Rn+ and E ∪ E ∗ to F ∪ F ∗ in R , respectively, then Mp ( 2 ) = 2Mp ( 1 ) . Lemma 7.13 If ( j ) is an increasing sequence of curve families, i.e. j ⊂ j +1 , j = 1, 2, . . . , and p > 1 , then lim Mp ( j ) = Mp (

j →∞



j ) .

Applying this lemma one can prove the following symmetry property of the modulus. Lemma 7.14 Let p > 1 and let E and F be subsets of Rn+ . Then   Mp (E, F ; Rn+ ) ≥

1 2

  Mp (E, F ) .

Corollary 7.15 Let E and F be sets in R   M (E, F ) ≤ c(n, a) < ∞ .

n

with q( E, F ) ≥ a > 0 . Then n

Proof √ By the hypothesis there exists a ball Bq (z, r) in R \ (E ∪ F ) with r < 1/ 2 and with the spherical √ diameter q(Bq (z, r)) = a . By easy computation z be the antipodal (see 3.15(3)) q(Bq (z, r)) = 2r 1 − r 2 and hence r ≥ 12 a . Let 

110

7 The Modulus of a Curve Family

point defined in (3.7) and t √ a spherical isometry with t ( z) = 0 as defined in (3.23). By (3.9) tE , tF ⊂ Bq (0, 1 − r 2 ) . Because t is a spherical isometry we obtain d(tE, tF ) ≥ q(tE, tF ) = q(E, F ) ≥ a . Next observe that (see 3.15(1))   Bq (0, 1 − r 2 ) = B n ( r −2 − 1 ) = B . These last two relations together with 7.14, 7.10, and 7.4 yield         M (E, F ) = M t ((E, F )) = M (tE, tF ) ≤ 2 M (tE, tF ; B)  2 2 n/2 ≤ 2a −n n (r −2 − 1)n/2 ≤ 2a −n n −1 , a  

as desired. n

Lemma 7.16 Let 1 , 2 , . . . be separate curve families in R with j < for all j = 1, 2, . . . . If p > 1 , then Mp ( )1/(1−p) ≥

∞ !

Mp ( j )1/(1−p) .

j =1

Proof Let {Ej } be a family of disjoint Borel sets associated with the collection { j } , let E = Ej , and let χEj be the characteristic function of Ej . Fix ρj ∈ F( j ) and set σj = ρj χEj . Then it is easy  to see that σj ∈ F( j ) . Now choose a sequence (aj ) so that aj ∈ [0, 1] and aj = 1 and define a Borel function ρ by ρ=

∞ !

aj σj =

j =1

∞ !

aj ρj χEj .

j =1

We show that ρ ∈ F( ) . Fix a locally rectifiable γ ∈ and for each j a subcurve γj ∈ j . We obtain  ρ ds = γ

 ! γ

≥

! j

j



aj σj ds =

! j

 σj ds ≥

aj γj

! j

 aj

σj ds γ

aj = 1 .

7.1 Basic Properties of the Modulus

111

Hence ρ ∈ F( ) and we obtain  Mp ( )≤

≤

 ρ dm = p

Rn

p

ρ dm= E

∞  !

!

j =1 Ej

p p aj ρj

∞  ! ! j =1 Ej

 dm =

j

ai ρi χEi

dm =

p p aj ρj

dm ≤

∞ !

p aj



j =1

j

∞  !

p p

j =1 Ej

i

! Rn

p

aj ρj dm

p

Rn

ρj dm .

Taking the infimum yields Mp ( ) ≤

∞ !

p

aj Mp ( j ) .

(7.7)

j =1

We now apply this last inequality to prove the assertion. Clearly we may assume Mp ( ) > 0 , which implies by 7.2 that Mp ( j ) ≥ Mp ( ) > 0 . (If Mp ( ) = 0 , the left side of the inequality is ∞ and there is nothing to prove.) We may also assume that Mp ( j ) < ∞ for all j , because ∞ !

Mp ( j )

1/(1−p)

=

∞ ! ∗

j =1

where

∗

Mp ( j )1/(1−p)

j =1

refers to summation over terms with Mp ( j ) < ∞ . Denote tk =

k !

Mp ( j )1/(1−p)

−1

, aj = Mp ( j )1/(1−p) tk

j =1

for j = 1, . . . , k and k = 1, 2, . . . whence k !

aj = 1 .

j =1

The above inequality (7.7) (with aj = 0 for j ≥ k + 1 ) gives Mp ( ) ≤

p tk

k !

Mp ( j )

p/(1−p)

Mp ( j ) =

j =1

Letting k → ∞ yields the desired result.

k !

Mp ( j )1/(1−p)

1−p .

j =1

 

112

7 The Modulus of a Curve Family

As an example of application we consider the following simple particular case of 7.16. Let r1 = 1 < r2 < · · · < rj < · · · < a and = ( S n−1 , S n−1 (a) ) , i = ( S n−1 (ri ), S n−1 (ri+1 ) ) for i = 1, 2, . . . . It follows from (7.4) that in this particular case 7.16 yields (when p = n) log a ≥

∞ ! j =1

log

ri+1 = log a0 ri

where a0 = lim rj ≤ a . If we choose the sequence (rj ) so that a0 = a , then equality holds. Hence 7.16 is sharp. For the proof of the next result the reader is referred to [402, p. 82], [150, pp. 514– 515], [161, Thm 4.3.5, p. 128]. Lemma 7.17 Let s ∈ (0, 1) and 1 = ( [0, se1 ], S n−1 ; Bn ) , 2 = ( [0, se1 ], [ 1s e1 , ∞); Rn ) . Then Mp ( 1 ) = 2p−1 Mp ( 2 ) for p > 1 . The next result, Gehring’s upper bound for the modulus of a curve family joining the complementary components of a ring domain, will have interesting applications later on in this book. This result was conjectured by the third author and a proof was supplied by F. W. Gehring [552, 2.58]. Lemma 7.18 Let 1 = ( [0, e1 ], [t 2 e1 , ∞)) and 2 = ( [0, e], [t 2 e1 , ∞)) where e ∈ S n−1 and t > 1 . Then M(2 ) ≤ M(1 ) . Proof Denote 11 = ( [0, e1 ], S n−1 (t)) , 21 = ( [0, e], S n−1 (t)) , and 12 = 22 = ( S n−1 (t), [t 2 e1 , ∞)) . Obviously M(11 ) = M(21 ) , M(12 ) = M(22 ) . Let f be the inversion in S n−1 (t) . Because 12 = f 11 we obtain by 7.10 M(11 ) = M(f 11 ) = M(12 ) . Next, 7.16 yields M(2 )1/(1−n) ≥ M(21 )1/(1−n) + M(22 )1/(1−n) = 2 M(11)1/(1−n) while the fact that 1 is symmetric yields by 7.17 M(11 ) = 2n−1 M(1 ) .

7.1 Basic Properties of the Modulus

113

Fig. 7.2 Proof of Lemma 7.18

The desired inequality follows from the last two relations (Fig. 7.2).

 

The family of all non-constant curves passing through a fixed point is nexceptional as was pointed out in the paragraph following (7.5). One can show that such a family is not p -exceptional if p > n (see [179, Chapter 3], [381]). We shall require this result in the following form, which is sometimes called the spherical cap inequality. For this result we introduce first an extension of the definition (7.1) of the p -modulus. Suppose that S is a euclidean sphere in Rn with radius r and is a family of curves in S . We equip S with the restriction of the euclidean metric of Rn to S and with the (n − 1)-dimensional Hausdorff measure mn−1 with mn−1 (S) = ωn−1 r n−1 . Let A( ) be the set of all non-negative Borel-measurable functions ρ : S → R ∪ {∞} with  ρ ds ≥ 1 γ

for all locally rectifiable (with respect to the metric ds ) curves γ in and set  MSn ( )

=

inf

ρ∈A( ) S

ρ n dmn−1 .

For ϕ ∈ (0, π) let C(ϕ) = { z ∈ Rn : z · en ≥ |z| cos ϕ } . Lemma 7.19 Let S = S n−1 (r) , ϕ ∈ (0, π] , let K be the spherical cap S∩C(ϕ) , and let E and F be non-empty subsets of K . (1) Then   bn MSn (E, F ; K) ≥ r where bn is a positive number depending only on n . (2) If K = S , i.e. ϕ = π , then bn may be replaced by cn = 2n bn in the above inequality.

114

7 The Modulus of a Curve Family

The proof of 7.19 (see [524, 10.9]) is based on an application of Hölder’s inequality and Fubini’s theorem. A similar method yields also the following improved form of 7.19 [463, p. 57, Lemma 3.1.], [166, p. 20, Lemma 3.8.]. Lemma 7.20 Assume that E , F , and K are as in 7.19(1). If ϕ ∈ (0, 12 π) , then   dn MSn (E, F ; K) ≥ ϕr where dn depends only on n .

7.1.3 Constants and Dimension Throughout the book we will denote by cn the number in 7.19(2). The number bn = 2−n cn has the following expression ⎧ 1 1−2n ⎪ ⎪ , ωn−2 In1−n , b2 = ⎨bn = 2 2π  π/2 2−n ⎪ ⎪ ⎩In = sin n−1 t dt .

(7.8)

0

Because

2 πt

≤ sin t ≤ t for 0 ≤ t ≤ 12 π , it follows from (7.8) that (n − 1)

 π 1/(n−1) 2

≤ In ≤ (n − 1)

π 2

for n ≥ 2 . We shall rarely need the values of these constants; for numerical values the reader may look at [29, p. 458]. Some bounds for these constants are given in [29, pp.41–44], [444]. During the past two decades many authors have studied the function and applied their results to estimate various constants such as the volume n of the unit ball Bn [30]. The behavior of the constants when n → ∞ is interesting: one can show, for instance, that 2n cn → 0 , n → 0 , and ωn−1 = n n → 0 , when n → ∞ . By (7.1), any  admissible function ρ yields an upper bound for Mp ( ) , that is Mp ( ) ≤ Rn ρ p dm . The problem of finding lower bounds for Mp ( ) is much more difficult because then we need a lower bound for Rn ρ p dm for every admissible ρ . The next important lower bound for the modulus follows by integration from 7.19 and 7.20. Lemma 7.21 Let 0 < a < b and let E , F be sets in Rn with E ∩ S n−1 (t) = ∅ = F ∩ S n−1 (t)

7.2 Comparison Principle for the Modulus

115

for t ∈ (a, b) . Then   b M ( E, F ; B n (b) \ B n (a) ) ≥ cn log . a Equality holds if E = (ae1 , be1 ) , F = (−be1 , −ae1 ) . Corollary 7.22 If E and F are non-degenerate continua with 0 ∈ E ∩ F then   M (E, F ) = ∞ . Proof Apply 7.21 with a fixed b such that S n−1 (b) ∩ E = ∅ = S n−1 (b) ∩ F and   let a → 0 . We next give a typical application of Lemma 7.21. Unlike 7.21 this application fails to give a sharp bound, but it yields adequate bounds in many cases (see e.g. Chap. 8). A sharp version of 7.23, which requires some information about spherical symmetrization, will be given in Chap. 9 (see 9.20 and 9.21). Lemma 7.23 Let t > r > 0 and let E ⊂ B n (r) be a connected set containing at least two points. Then   2t + d(E) . M (S n−1 (t), E) ≥ cn log 2t − d(E) Proof Fix u, v ∈ E with |u − v| = d(E) = d and choose h ∈ GM(B n (t)) with h(u) = −se1 = −h(v) . By (4.25) d(E) = |u − v| ≤ 2 th 14 ρ(u, v) = 2 th 14 ρ(h(u), h(v)) = 2s , where ρ refers to the hyperbolic metric of B n (t) . Applying 7.21 to the annulus B n (te1 , t + s) \ B n (te1 , t − s) with E = hE and F = S n−1 (t) we obtain     t +s M (S n−1 (t), E) = M (S n−1 (t), hE) ≥ cn log t −s 2t + d(E) , ≥ cn log 2t − d(E) which proves the assertion.

 

7.2 Comparison Principle for the Modulus We shall frequently apply the following lemma when proving lower bounds for the moduli of curve families. This lemma will be called the comparison principle for the modulus. In the applications of this lemma, the sets F3 and F4 will often be chosen to be non-degenerate continua (that is continua containing at least two

116

7 The Modulus of a Curve Family

distinct points) while the sets F1 and F2 will usually be very “small” sets when compared to F3 and F4 . n

Lemma 7.24 Let G be a domain in R , let Fj ⊂ G , j = 1, 2, 3, 4 , and let ij = (Fi , Fj ; G) , 1 ≤ i, j ≤ 4 . Then   M( 12 ) ≥ 3−n min{ M( 13 ), M( 24 ), inf M (|γ13|, |γ24 |; G) } , where the infimum is taken over all rectifiable curves γ13 ∈ 13 and γ24 ∈ 24 . Proof By 7.1(1) we may assume that Fj = ∅ , j = 1, 2, 3, 4 . Fix ρ ∈ F ( 12 ) . If  ρ ds ≥ 13

(7.9)

γ13

for every rectifiable γ13 ∈ 13 or  ρ ds ≥ γ24

1 3

(7.10)

for every rectifiable γ24 ∈ 24 , then it follows from 7.7 and (7.1) that  Rn

ρ n dm ≥ 3−n min{ M( 13 ), M( 24 ) } .

(7.11)

If both (7.9) and (7.10) fail to hold we select rectifiable curves γ13 ∈ 13 and γ24 ∈ 24 . Because ρ ∈ F( 12 ) it follows that  ρ ds ≥ 1 γ13 ∪ α ∪γ24

for every locally rectifiable α ∈  = (|γ13|, |γ24 |; G) . Because both (7.9) and (7.10) fail to hold it follows from the last inequality that  ρ ds ≥ α

1 3

for each locally rectifiable α ∈  . Hence  Rn

  ρ n dm ≥ 3−n M() ≥ 3−n inf M (|γ13|, |γ24 |; G)

(7.12)

where the infimum is taken over all rectifiable curves γ13 ∈ 13 and γ24 ∈ 24 . In every case either (7.11) or (7.12) holds, and the desired inequality follows (Fig. 7.3).  

7.2 Comparison Principle for the Modulus

117

Fig. 7.3 Proof of Lemma 7.24

n

Corollary 7.25 Let Fj ⊂ R and ij = (Fi , Fj ) , 1 ≤ i, j ≤ 4 . Then M( 12 ) ≥ 3−n min{ M( 13 ), M( 24 ), δn (r) } where r = min{ q(F1 , F3 ), q(F2 , F4 ) } and   δn (r) = inf M (E, F ) . n

Here the infimum is taken over all continua E , F in R such that q(E) ≥ r , q(F ) ≥ r . It is clear √that δn (0) = 0√in 7.25. In fact, this follows from 7.11(2) if we choose r ∈ (0, 1/ 2 ) , set s = 1 − r 2 , and let r → 0 . We are going to show that δn (r) > 0 for r > 0 . To this end the following corollary will be needed. Corollary 7.26 If x ∈ Rn , 0 < a < b < ∞ , and F1 , F2 ⊂ B n (x, a) , F3 ⊂ Rn \ B n (x, b) , ij = (Fi , Fj ) , then

(1) M( 12 ) ≥ 3−n min M( 13 ), M( 23 ), cn log ab , (2) M( 12 ) ≥ d(n, b/a) t ; t = min{ M( 13 ), M( 23 ) } . Moreover, if t > 0 then there exists λ > 1 depending only on t, b/a , and n such that λ ) ≥ d(n, b/a)t/4 (3) M( 12 λ consists of the curves in where 12 12 contained in the spherical annulus n n A(λ) = B (x, bλ) \ B (x, a/λ) . Proof We apply the comparison principle 7.24 with G = Rn and F3 = F4 to get a lower bound for M( 12 ) . It follows from 7.21 that the infimum in the lower bound of 7.24 is at least cn log ab and thus (1) follows. For the proof of (2) we observe that

118

7 The Modulus of a Curve Family

by Lemma 7.2 and the spherical ring domain formula (7.4)    b 1−n max M( 13 ), M( 23 ) ≤ A = ωn−1 log . a By part (1) we get  

1 b cn log min M( 13 ), M( 23 ) M( 12 ) ≥ 3−n min M( 13 ), M( 23 ), A a   ≥ d(n, b/a) min M( 13 ), M( 23 )   where d(n, b/a) = 3−n min 1, A1 cn log(b/a) ∈ (0, 1) . √ 0j = Fj ∩ A( λ) , j = 1, 2 , and For part (3) we apply part (1) with the sets F observe that the family of all curves joining F1 and F2 in A(λ) has the desired lower bound for its modulus for a suitable √ 1−n λ which we will now choose. Fix λ > 1 such that ω (log λ) = d t/4 where d = d(n, b/a) < 1 is n−1 √ as in part (2). Then λ > b/a by the choice of the sets Fj , j = 1, 2, 3 and by the formula for the spherical ring domain, (7.4) and by Lemma 7.2. By part (1) 3 b

. M((F1 , F2 )) ≥ 3−n min M( 13 ), M( 23 ), cn log 4 a Again by the spherical ring domain estimate M((F1 , F2 ; A(λ))) ≥ M((F1 , F2 )) −

dt dt − . 4 4

Then part (2) yields the desired conclusion dt dt 3 λ M( 12 = . ) ≥ M((F1 , F2 ; A(λ))) ≥ dt − 4 2 4

 

Lemma 7.27 For n ≥ 2 there are positive numbers d and D with the following properties. (1) If E, F ⊂ Bn (s) are connected and d(E) ≥ s t , d(F ) ≥ s t , then M (E, F ) ≥ d t . n (2) If E, F ⊂ R are connected and q(E) ≥ t , q(F ) ≥ t , then M (E, F ) ≥ δn (t) ≥ D t . Proof (1) By 7.23 we obtain   4s + ts ≥ 12 cn (log 2)t M (S n−1 (2s), E) ≥ cn log 4s − ts

7.2 Comparison Principle for the Modulus

119

  and similarly M (S n−1 (2s), F ) ≥ 12 cn (log 2)t . Applying 7.26(1) with F1 = F , F2 = E , and F3 = S n−1 (2s) and the above estimates we get   M( 12 ) ≥ 3−n min 12 cn (log 2)t , cn log 2 ≥ d t where d = 12 · 3−n cn log 2 and the inequality t ≤ 2 was applied. (2) Observe first that both the first and last expressions in the asserted inequality remain invariant under spherical isometries (see 7.10). By performing a preliminary spherical isometry if necessary we may assume that −re1 ∈ E , re1 ∈ F , and r ∈ [0, 1] (cf. 3.15(1)). Let E1 (F1 ) be that component of E ∩ B n (2) (of F ∩ B n (2) , resp.) which contains −re1 ( re1 ). Then √ d(E1 ) ≥ q(E1 ) ≥ min{ t, q(S n−1 , S n−1 (2)) } ≥ t/ 10 , √ and√likewise d(F1 ) ≥ t/ 10 . The proof of (2) follows from (1) with D = d/ 10 .   By means of spherical symmetrization, which will be introduced in Chap. 9, one can give a different proof of 7.27(1) (see 9.26). Exercise 7.28 Let E and F be non-degenerate continua in Bn . Find a lower bound for M (E, F ; Bn ) in terms of n , ρ(E) , ρ(F ) , and ρ(E, F ) . [Hint: Fix a1 ∈ E , a2 ∈ F with ρ(a1 , a2 ) = ρ(E, F ) and let x ∈ J [a1 , a2 ] be such that ρ(a1 , x) = 12 ρ(E, F ) . Let Tx ∈ M(Bn ) be as defined in Lemma 3.19. By conformal invariance 7.10     M (E, F ; Bn ) = M (Tx E, Tx F ; Bn ) . Now one can find a lower bound for the euclidean diameters d(Tx E) , d(Tx F ) in terms of ρ(E) , ρ(F ) , and ρ(E, F ) , see (4.21)–(4.23). After this apply 7.26 with a = 1 , b = 2 , F1 = Tx E , F2 = Tx F , and F3 = S n−1 (2) . The desired result follows now from a symmetry property of the modulus, see 7.14.] Exercise 7.29 For E ⊂ Rn , x ∈ Rn , and 0 < r < t set   Mt (E, r, x) = M (S n−1 (x, t), E ∩ B n (x, r)) , M(E, r, x) = M2r (E, r, x) .

(7.13)

It follows from 7.2 that Mt (E, r, x) ≤ Ms (E, r, x) for 0 < r < s ≤ t . Also a converse inequality is true: Mt (E, r, x) ≤ Ms (E, r, x) ≤ a(n, r, s, t)Mt (E, r, x)

(7.14)

where a depends only on the parameters indicated. Prove (7.14) by applying 7.24 with F1 = E ∩ B n (x, r) , F2 = S n−1 (x, t) , F3 = S n−1 (x, s) , F4 = S n−1 (x, r) .

120

7 The Modulus of a Curve Family

Remark 7.30 The method described above fails to give the best possible constant a in (7.14). The sharp result, due to Martio and Sarvas [367], yields the inequality Ms (E, r, x) ≤ bn−1 Mt (E, r, x) , b =

log(t/r) log(s/r)

(7.15)

for 0 < r < s ≤ t . Here equality holds for E = B n (x, r) . The proof of (7.15) makes use of a radial quasiconformal mapping [524, p. 49] of B n (x, s) \ B n (x, r) onto B n (x, t) \ B n (x, r) .

7.2.1 The Modulus of a Ring n

n

A domain D in R is termed a ring, if R \ D has exactly two components. If the components are C0 and C1 we write D = R(C0 , C1 ) . The (conformal) modulus of a ring R(C0 , C1 ) is defined by mod R(C0 , C1 ) =

 M((C , C )) 1/(1−n) 0 1 . ωn−1

(7.16)

The capacity of R(C0 , C1 ) is M((C0 , C1 )) . A ring is a special case of a condenser, which we shall define in Chap. 9. In the two-dimensional case the modulus of a ring R has the following geometric interpretation: mod R = t if and only if R can be mapped conformally onto the annulus { z ∈ R2 : 1 < |z| < et } . Owing to this geometric interpretation the modulus of a ring is often convenient to use in the two-dimensional case. In the multidimensional case there is no such geometric interpretation for the modulus of a ring because of the rigidity of the class of conformal mappings in Rn , n ≥ 3 (cf. 3.44). On the other hand there is also a geometric way of looking at the capacity of a particular ring, the so-called Grötzsch ring, which is applicable to all dimensions n ≥ 2 . Indeed, as we shall see below, there is a close connection of two conformal invariants, the hyperbolic metric and the capacity of the Grötzsch condenser (see (7.17) and (9.12)). For this reason we shall prefer the capacity to the modulus of a ring.

7.3 Grötzsch and Teichmüller Rings n

The complementary components of the Grötzsch ring RG,n (s) in Rn are B and [se1 , ∞] , s > 1 , while those of the Teichmüller ring RT ,n (s) are [−e1, 0] and [se1 , ∞] , s > 0 . We shall need two special functions γn (s) , s > 1 , and τn (s) , s > 0 , to designate the moduli of the families of all those curves which connect the complementary components of the Grötzsch and Teichmüller rings in Rn ,

7.3 Grötzsch and Teichmüller Rings

121

Fig. 7.4 Left: Grötzsch ring, cap RG,n = M( s ) = γn (s) . Right: Teichmüller ring, cap RT ,n = M(s ) = τn (s)

respectively (Fig. 7.4).  γn (s) = M( s ) = γ (s) ,

(7.17)

τn (s) = M(s ) = τ (s) .

The subscript n is omitted if there is no danger of confusion. We shall refer to these functions as the Grötzsch capacity and the Teichmüller capacity. Lemma 7.31 For s > 1 , γn (s) = 2n−1 τn (s 2 − 1) . The functions γn and τn are decreasing. Furthermore, lims→1+ γn (s) = ∞ and lims→∞ γn (s) = 0 . Proof Let 1 = ( [0, 1s e1 ] , [se1 , ∞] ) , 2 = ( [0, 1s e1 ] , S n−1 ) , 3 = ( S n−1 , [se1 , ∞] ) . It follows from conformal invariance that M( 2 ) = M( 3 ) = γn (s) and from 7.17 that γn (s) = 2n−1 M( 1 ) = 2n−1 τ (s 2 − 1) . For each fixed n ≥ 2 the functions γn and τn are decreasing as follows easily from 7.1(3). The limit values of γn follow from 7.21 and (7.4).   For the sake of completeness we set γn (1) = τn (0) = ∞ and γn (∞) = τn (∞) = 0 . Exercise 7.32 Show that  (s − t)(r − u)    (1) M ( [re1, se1 ], [te1 , ue1 ] ) = τ , r < s < t < u, (r − s)(t − u)     st − 1 (2) M ( S n−1 , [se1 , te1 ] ) = γ , 1 < s < t < ∞. t −s (3) We can map the Grötzsch ring RG,n (s), s > 1 , by a Möbius transformation n

n

onto a ring in R with complementary components B and [ae1, ∞] ∪ √ [−ae1, ∞] where a = s + s 2 − 1 . (4) We can map the Teichmüller ring RT ,n (s), s > 0 , by a Möbius transformation n

onto a ring in R with complementary √ components [−e1 , e1 ] and [be1 , ∞] ∪ [−be1, ∞] where b = 1 + 2s(1 + 1 + 1/s ) .

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7.4 Hypergeometric Functions and Elliptic Integrals 7.4.1 Elliptic Integrals and γ2 (s) The plane Grötzsch ring can be mapped conformally onto {x ∈ R2 : t < |x| < 1} by an elliptic function. As shown in [29, 6.26], [215], [326, II.2] γ2 (s) =

2π , μ(1/s)

t = exp(−μ(1/s)) ,

(7.18)

for s > 1 where μ(r) =

√  1 π K( 1 − r 2 ) , K(r) = [(1 − x 2 )(1 − r 2 x 2 )]−1/2 dx 2 K(r) 0

for 0 < r < 1 . The function K(r) is called a complete elliptic integral of the first kind and its values can be found in tables [1, 91] and algorithms for numerical computation are given in all standard software packages. For a brief discussion of these computational topics, see [29, pp. 457–473]. √ The function K(r) satisfies the following Landen identities where K (t) = K( 1 − t 2 ) ⎧  √    2 r ⎪  1−r ⎪ ⎪ K = (1 + r) K(r) = K ⎨ 1+r 1+r   √   ⎪ 1−r 2 r 1 ⎪   ⎪ K (1 + r) K (r) = K = . ⎩ 1+r 2 1+r

(7.19)

The modulus μ(r) satisfies the following three functional identities ⎧   1 2 ⎪ μ(r)μ r = 4π , ⎪ ⎪ ⎪ ⎪  ⎨ 1−r 1 2 = 2π , μ(r)μ 1+r ⎪ ⎪ √ ⎪   r    ⎪ 2 ⎪ ⎩μ(r) = 2μ 2 r = 1 μ . 1+r 2 1 + r

(7.20)

√ where r  = 1 − r 2 . Many inequalities for μ(r) are given in [326] and [29, Chapter 5]. By [326, p. 62] the following inequalities hold for 0 < r < 1 √ 1 1 + 3r  (1 + r  )2 2(1 + r  ) 4 log < log < log < μ(r) < log < log . r r r r r (7.21) From (7.21) it follows that limr→0+ μ(r) = ∞ whence, by virtue of the functional identities (7.20), limr→1− μ(r) = 0 (Fig. 7.5). For the sake of completeness we set

7.4 Hypergeometric Functions and Elliptic Integrals Fig. 7.5 The functions μ(r) , 0 < r ≤ 1 , and μ(1/r) , r > 1

123

4

3

2

1

0

1

0.5

0

1.5

2

μ(0) = ∞ and μ(1) = 0 . By (7.18) and (7.20) we obtain γ2 (s) =

4 s − 1 μ , s>1. π s+1

(7.22)

Exercise 7.33 Verify the following identities π 2π = , √ √ √ μ(1/ t) μ( (#" 1 + t −√ t )2 ) #  " 1 +√ (2) τ2 (t) = 2 τ2 4 t + t (1 + t) 1 + t + t (1 + t)     2 1−r 2 (3) μ r μ = π2 . 1+r

(1) τ2 (t) =

7.4.2 The Gaussian Hypergeometric Function For a ∈ R and n = 1, 2, . . . , the shifted factorial function is (a, n) = a(a + 1) · · · (a + n − 1) , (a, 0) = 1 . Clearly (1, n) = n! . For a, b, c ∈ R, c = 0, −1, −2, . . . r ∈ (−1, 1) the Gaussian hypergeometric function is defined by [1, Ch 15] F (a, b; c; r) =

∞ ! (a, n)(b, n) n=0

(c, n)n!

rn .

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7 The Modulus of a Curve Family

The function K(r) can be expressed in terms of it as follows: K(r) =

π 1 1 F ( , ; 1; r 2) . 2 2 2

This connection with hypergeometric functions enables one to derive e.g. the differentiation formulae for K(r) from those of the hypergeometric functions. These and many other formulae can be found in [29, pp.474–475], [22] and below in Theorem B.1. It is a basic property of the hypergeometric function F (a, b; a + b; r), a, b > 0 that it has a logarithmic singularity at r = 1 , see [29, 1.48-1.52] and formula (B.18) below. It turns out that the hypergeometric functions F (a, b; a + b; r) satisfy Landen inequalities which in the case a = b = 12 yield the Landen identities (7.19) as shown by S.-L. Qiu, X.-Y. Ma and Y.-M. Chu [443]. Also in [29, pp. pp.84–86] some refined versions of (7.21) are given.

7.4.3 Landen Transformation For r ∈ (0, 1) let L(r, 0) = r and √ 2 L(r, p) , L(r, p + 1) = 1 + L(r, p)

 L(r, −p − 1) =

L(r, −p)  1 + 1 − L(r, −p)2

2 , (7.23)

for p = 0, 1, 2, . . . . Then L(r, p) and L(r, −p) are the ascending and descending Landen sequences, resp. Now the third identity in (7.20) can be written as follows μ(r) = 2p μ(L(r, p)) .

(7.24)

7.4.4 Arithmetic-Geometric Mean A highly interesting study of the complete elliptic integral K(r) and Gauss’ arithmetic–geometric mean is contained in [85]. The arithmetic-geometric mean of two positive numbers a, b > 0 is defined as AG(a, b) = lim an = lim bn , where a0 = a, b0 = b, an+1 = 12 (an + bn ), bn+1 = for all r ∈ (0, 1) K(r) =

√ an bn . By Gauss’ theorem,

π . √ 2AG(1, 1 − r 2 )

7.4 Hypergeometric Functions and Elliptic Integrals

125

This identity provides a very efficient method for the numerical approximation of K and μ [29, 58]. Exercise 7.34 In the study of distortion theory of quasiconformal mappings in Chap. 16 below the following special function will be useful ϕK,n (r) =

1 γn−1 (Kγn (1/r))

for 0 < r < 1 , K > 0 . (Note: Lemma 9.15 below shows that γn is strictly decreasing and hence that γn−1 exists.) Show that ϕAB,n (r) = ϕA,n (ϕB,n (r)) and −1 ϕA,n (r) = ϕ1/A,n (r) and that ϕK,2 (r) = ϕK (r) = μ−1

1

K μ(r)



.

Verify also that

√ 2 r , (1) ϕ2 (r) = 1 + r √ 2 (2) ϕK (r)2 + ϕ1/K 1 − r 2 = 1 . Exploiting (1) and (2) find ϕ1/2 (r) . Show also that 1 − r 

1 − ϕK (r) = , 1+r 1 + ϕK (r)   2√r  2 ϕK (r) (4) ϕK . = 1+r 1 + ϕK (r)

(3) ϕ1/K

Exercise 7.35 Verify the following identities for K, t > 0   1 (1) τ2−1 τ2 (t)/K = −1 , τ2 (Kτ2 (1/t)) 4 (2) τ2 (t) = . τ2 (1/t)

7.4.5 Modular Equations For r ∈ (0, 1) and a positive integer m, equations of the form μ(r) = mμ(s) are called modular equations or order m . Thus the solution is s = ϕm (r) . In the case m = 2 the modular equation yields a duplication formula for μ(r) and, more generally, in the case m = 2p , p = 1, 2, . . . , the solution s is given by the Landen transformation formula (7.24). It is surprising that also for many other integers m the solutions to modular equations are given by algebraic functions. A.M. Legendre’s modular equation corresponds to the case m = 3 [29, (5.37),

126

7 The Modulus of a Curve Family

(10.50)] and it can be written as follows (αβ)1/4 + ((1 − α)(1 − β))1/4 = 1, with α = r 2 , β = ϕ1/3 (r)2 . Symbolic computation software can be used to solve this equation for β . The above (α, β) parametrization of the modular equation was introduced by the Indian mathematical genius S. Ramanujan who found numerous algebraic identities satisfied by ϕK (r) for several integral values of K . See for example [29, 10.51], [104]. For a positive integer p > 0 , the algebraic number kp defined by  √ K( 1 − kp2 )/K(kp ) = p is called the p th singular value of K [85, pp. 139, 296], [29, p. 93, 5.39]. A symbolic computation method for finding k10 is given in [515, pp. 1127 (l), 1169].

7.4.6 Jacobi’s Infinite Products C.G.J. Jacobi’s theory of elliptic functions [260] contains numerous product expansions for functions expressed in terms of K(r) or related functions. For a recent treatment of the history of elliptic functions, see [476]. An example of such a formula, Jacobi’s inversion formula for μ, is for y > 0 [29, Thm 5.24(2)] μ−1 (y)2 = 1 −

8 ∞  1 1 − q 2n−1 n=1

1 + q 2n−1

, q = exp(−2y) .

Jacobi’s work also yields some other inversion formulae, for instance in terms of theta functions, see (B.27). Observe that from (7.20) it follows easily that μ−1 (y)2 + μ−1 (

π2 2 ) = 1. 4y

Newton’s method can also be used to numerically invert μ , see [29, pp. 92,438]. Jacobi’s work also yields the following refinement of the lower bound in (7.21) μ(r) > arth

√ √ 4  r > log((1 + r  )2 /r) > log((1 + 3r  )/r) ,

for r ∈ (0, 1) , r  =

√ 1 − r 2 , [29, (5.30), 5.31(3)].

(7.25)

7.4 Hypergeometric Functions and Elliptic Integrals

127

7.4.7 Ramanujan’s Approximation of μ(r) [29, 5.51], [64, p. 91, (2.4)] There are several methods to approximate μ numerically. For instance we could use the Landen transformation or the arithmetic-geometric mean iteration or the formula (7.21). Indeed, because   1 + 3r  2(1 + r  ) 1 + 3r  − log →0 log / log r r r when r → 0 , it is clear that for a given ε > 0 there exists r0 > 0 such that the relative approximation error of (7.21) on (0, r0 ] is smaller than ε . S. Ramanujan proved that  10 + 36 + a(r)2 1 , μ(r) ≈ h(r) ≡ log 2 a(r)

(7.26)

with a(r) = − log(1 − r 2 ) . Then h : (0, 1) → (0, ∞) is a decreasing homeomorphism with h(t) → ∞ as t → 0 . This approximation is remarkably good, the absolute error on the interval (0, 1/2) is of the order 10−8 . The above functional identities, e.g. (7.20) and 7.33(2), are restricted to the two-dimensional case. For the multidimensional case n ≥ 3 there is no explicit expression like (7.18) for γn (s) or τn (s) and no functional identities are known for γn (s) or τn (s) except the basic relationship 7.31. The well-known upper and lower estimates for γn (s) and τn (s) will be given in Chap. 9. Next we shall show that for all dimensions n ≥ 2 the Teichmüller capacity τn (s) satisfies certain functional inequalities. For a detailed study of these special functions see [29]. Lemma 7.36 The following functional inequalities hold: n−1 2 (1) τ (s) ≤ γ (1 + 2s) = 2 τ (4s + 4s) , s > 0 , (2) τ (s) ≤ 2τ 2s + 2s 1 + 1/s , s > 0 ,  s(1 + t)  , 0 0 . Proof The first and second inequalities follow from 7.36(1),(2), respectively.

 

Corollary 7.37 is a special case of Theorem 9.19 below. Remark 7.38 It follows from (7.21) that log(1/r) < μ(r) < log(4/r) for all r ∈ (0, 1) where both bounds have the correct asymptotic behavior as r → 0+ . For r → 1− the second inequality is very weak since μ(r) → 0 as r → 1− . An improved two-sided inequality for μ(r) can be obtained as follows. Exploiting (7.21) together with the functional identity (7.20) we obtain π2 4 log 2(1+r) r

< μ(r) =

π2 π2 < . 4μ(r  ) 4 log 1+3r r

This inequality together with (7.21) implies for r ∈ (0, 1) 1 + 3r  π2 max log , r 4 log 2(1+r) r

2

2 π2 2(1 + r  ) , < μ(r) < min log . r 4 log 1+3r r (7.27)

The asymptotic behavior in (7.27) is correct at both ends r = 0 and r = 1 . Exercise 7.39 Let A, B, C, D be distinct points on the unit circle S 1 in the stated order and 2α and  2β the lengths  of the arcs AB and CD , respectively. Find the least value of M (AB, CD) . [Hint: |A − C||B − D| = |A − B||C − D| + |B − C||A − D| by Ptolemy’s theorem [105, p. 42], [60, 10.9.2].]

7.4 Hypergeometric Functions and Elliptic Integrals

129

Remark 7.40 The function μ has several interesting properties which are given in [27] and [29]. For instance the inequalities  μ

  √  ab ab  < μ(a) + μ(b) ≤ μ ≤ 2μ ab     1+a b (1 + a )(1 + b )

√ hold for a, b ∈ (0, 1) where a  = 1 − a 2 . It follows from (7.20) that the second and third inequalities hold as equalities for a = b . For further results of this type see Á. Baricz [49]. Corollary 7.37 applied to τ2 yields by 7.33(1) the following two-sided inequality for the function μ :  √     √   √ μ 1/ 1 + t ≤ 2μ 1/ 1 + t ≤ 4μ 1/ 1 + t , t > 0 , which can also be deduced from the identities (7.20). Exercise 7.41 Show that for s > 0 and r ∈ (1, 1 + s) the inequality  1/(1−n) τn (s)1/(1−n) ≥ γn (r)1/(1−n) + γn (1 + s)/r holds with equality if r =

√ 1+s.

Exercise 7.42 Let r ∈ (0, 1), e ∈ S n−1 , E = [−re, re], E1 = [−re1 , re1 ] , and F = [−e1 , ∞] ∪ [e1 , ∞] . Show that M((E, F )) ≤ M((E1 , F )) . Exercise 7.43 Let x, y ∈ Bn , x = y and M ∈ (0, ρ(x,y) 2 ) . Show that M((Bρ (x, M), Bρ (y, M); Bn )) ≥ d1 (n, M)ρ(x, y)1−n , where d1 (n, M) > 0 . Exercise 7.44 Let f : Bn → Bn be a homeomorphism mapping each sphere centered at 0 onto another sphere centered at 0 (such a mapping is called a radial mapping) and with the property that for some K ≥ 1, M( )/K ≤ M(f ( )) ≤ KM( ) whenever is the family of all curves connecting the boundary components of a spherical annulus centered at 0. Show that for all x ∈ Bn |x|1/α ≤ |f (x)| ≤ |x|α , α = K 1/(1−n) . Exercise 7.45 The enclosed picture, Fig. 7.6, displays four ring domains, each of which consists of the region between two squares with parallel sides and the same center. For each ring, compute the following estimates for the modulus of the family of all curves joining the boundary components of the ring: (1) an upper bound using a separating annulus, (2) a lower bound using an annulus separated by the ring,

130

7 The Modulus of a Curve Family

Fig. 7.6 Ring domains in Exercise 7.45

(3) an upper bound using Lemma 7.16. (4) (optional) other upper and lower bounds. An explicit formula for this modulus is given in [70, p.239],[185, (7.1),p.294]. Exercise 7.46 Let G = Bn \ {0}, f : G → G = f (G), be a homeomorphism with the property that there exist curves αj : [0, 1) → G, j = 1, 2, such that αj (t) → 0, f (αj (t)) → βj ∈ ∂G , t → 1. Show that β1 = β2 if there exists C ≥ 1 with kG (f (x), f (y)) ≤ CkG (x, y) for all x, y ∈ G. Show that β1 = β2 also holds if there exists K ≥ 1 such that M( ) ≤ KM(f ) ≤ K 2 M( ) for all curve families in G. Notes 7.47 Most results in this chapter are standard and are well represented in the literature, e.g. in [524]. The origin of some less standard results is indicated above in connection with each result. Next we shall make some additional remarks on the results of this chapter. Lemmas 7.6 and 7.16 are from [145], 7.12, 7.13, and 7.14 from [154, 548, 576]. The comparison principle 7.24 has its roots in [364, 3.11], but under this name it was introduced and developed by R. Näkki [405] and the third author [551]. An account of the properties of the function μ(r) is given in [326]

7.4 Hypergeometric Functions and Elliptic Integrals

131

and [29]. The inequalities in 7.36 and 7.37 are from [556]. For further properties of τn (s) the reader is referred to Chap. 9 and to [27, 29]. Notes 7.48 The extremal length method of L. V. Ahlfors and A. Beurling [16] has its roots in the length-area method whose use is widespread throughout geometric function theory. Some historical comments about the origin of the lengtharea method are made by L. V. Ahlfors [10, p. 50, 81] and by J. Jenkins in [265, 266, 268]. According to Jenkins the first result of this type is due to H. Bohr in 1918 and slightly later results are due to W. Gross, G. Faber and R. Courant. G. V. Kuz’mina published a survey of Jenkins’ work [318], with many historical comments. In 1928 H. Grötzsch [179] published his well-known work on quasiconformal mappings and in a subsequent series of papers developed his strip method, giving applications to a variety of problems. In his dissertation in 1955 J. Hersch [215] established connections among harmonic measure, extremal length, and other conformal invariants. These developments are also reviewed in the clearly organized survey of A. Baernstein II [42]. The book of J.B. Garnett and D.E. Marshall [149] is comprehensive study of the harmonic measure and it also applies extremal length to problems of geometric function theory. A survey of some function-theoretic applications of the extremal length is given by B. Rodin in [474], which contains also a good bibliography of the subject. See also the bibliography in the book of G. V. Kuz’mina [317]. In his books [412, 413] M. Ohtsuka gives several function-theoretic applications of the extremal length. B. Fuglede [145] was the first to consider, in 1957, the p -modulus in the multidimensional case. He also considered the modulus of a surface family as well as the modulus of a system of measures (see also P. Mattila [376]). Later these notions were developed mainly in connection with the theory of quasiconformal mappings (see O. Lehto and K. I. Virtanen [326], F. W. Gehring [150, 151, 156] and J. Väisälä [519, 521, 524, 527]). See also the books of P. Caraman [94, pp. 46– 70], A. V. Sychev [509, pp. 26–35], O. Martio, V.I. Ryazanov, U. Srebro, and E. Yakubov [366, pp. 291–344], and F.W. Gehring, G.J. Martin and B.P. Palka [161, pp. 77–149]. The notion of p -capacity, which is closely connected with that of p modulus (see Chap. 9) has been studied by many authors in the setup of nonlinear potential theory (see the references given at the end of Chaps. 8 and 9). The paper of C. Loewner [339] is one of the first papers dealing with conformal capacity in space. For extensive bibliographies on quasiconformal maps and moduli of curve families, see P. Caraman [94, pp. 46–70], C. Andreian-Cazacu [31], and R. Kühnau [312].

Chapter 8

The Modulus as a Set Function

n

Let  E and  F be compact disjoint non-empty sets in R and M(EF ) = M (E, F ) . In the case when E and F are connected, we have seen in Chap. 7 that M(EF ) > 0 if d(E), d(F ) > 0 and in Chap. 9 we shall improve this result; we shall prove a two-sided bound for M(EF ) in terms of min{d(E), d(F )} . d(E, F ) In the case when E or F or both are disconnected the quantity M(EF ) is elusive and we shall give in this chapter upper and lower bounds for it. Some of the ingredients of our methods can be described roughly speaking as follows. A preliminary step is to identify the degenerate case when M(EF ) = 0. This could happen e.g. if one of the sets is a discrete set. We can detect this degenerate case by using the test set n

T = {z ∈ R : q(z, E ∪ F ) ≥ q(E, F )/3} and requiring that min{M(ET ), M(F T )} > 0 . This test, however, depends on both E and F which we prefer to avoid and we want to check the “size” of each set independently of the other set. The main step is to find “an analytic handle” or a characteristic of a set to measure the “size” of a compact set E and we introduce a set function c(·) for that purpose; it works both in the degenerate and the nondegenerate cases. Using this set function we prove two-sided bounds for M(EF ) .

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_8

133

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8 The Modulus as a Set Function

In view of the conformal invariance of the n-modulus 7.10, one would like to find estimates which reflect this invariance property in the following  way:The estimate should give the same lower/upper bound for M (E, F ) and M (hE, hF ) n whenever h ∈ GM(R ). In most estimates (see e.g. 7.27(1) or 8.1 below) this requirement is not completely met, the estimate remaining invariant only under the n action of a subgroup of GM(R ), e.g. under similarity transformations or under chordal isometries. Some aspects of this problem will be discussed in Chaps. 9 and 10. The main result of the present chapter is the following theorem. Theorem 8.1 For n ≥ 2 there exist positive numbers d1 , . . . , d4 and a set function n c(·) in R such that n

n

n

(1) c(E) = c(hE) whenever h : R → R is a spherical isometry and  E ⊂ R . n ≤ (2) c(∅) = 0, A ⊂ B ⊂ R implies c(A) ≤ c(B) and c ∞ j =1 Ej ∞ n d1 j =1 c(Ej ) if Ej ⊂ R . n

(3) If E ⊂ R is compact, then c(E) > 0 if and only if cap E > 0. Moreover n c(R ) ≤ d2 < ∞. n (4) c(E)  ≥ d3 q(E)  if E ⊂ R is connected and E = ∅. n (5) M (E, F ) ≥ d4 min{ c(E), c(F ) }, if E, F ⊂ R . Furthermore, for n ≥ 2 and t ∈ (0, 1) there exists a positive number d5 such that n (6) If E, F ⊂ R and q(E, F ) ≥ t,   M (E, F ) ≤ d5 min{c(E), c(F )} . It should be emphasized that the main interest in Theorem 8.1 lies in the inequalities (5) and (6). The definition of the condition cap E > 0 in 8.1(3) will be postponed until Chap. 9; it rules out e.g. countable sets. The geometric significance of this condition will become clear in Chap. 9 in connection with the definition 9.1 of the capacity of a condenser. Lemma 9.11 says that sets E with c(E) = 0 have zero Hausdorff dimension whereas Remark 9.13(2) shows that the converse is not true. We shall next give the reader some idea about the set function c(·). To this end define (see (7.13))  n  Mt (E, r, x) = M ( S n−1 (x, t), B n (x, r) ∩ E; R ) , M(E, r, x) = M2r (E, r, x)

(8.1)

n

whenever E ⊂ R , x ∈ Rn , and 0 < r < t. Moreover, let E −1 = { x/|x|2 : x ∈ E } and a(E) = max{ M(E, 1, 0), M(E −1 , 1, 0) }

(8.2)

8 The Modulus as a Set Function

135

n

for E ⊂ R . It follows from the results of this chapter that there are numbers γ1 and γ2 depending only on the dimension n such that γ1 a(E) ≤ c(E) ≤ γ2 a(E) .

(8.3)

In what follows we shall give a construction of the set function c(E). We remark that there may also be many other methods of constructing c(E): it is clear by (8.3) that any method which yields a set function differing from a(E) by at most a multiplicative constant is adequate also for constructing c(E). n For the next lemma we recall that the balls Bq (x, r) of the metric space (R , q) were defined in (3.8). Lemma 8.2 Let r > 1 and t ∈ (0, 1) be such that q( S n−1 , S n−1 (r) ) ≥ 2 t. There is a number b(t) depending only on n, r, and t such that the following holds. If  n E ⊂ B and Gt = x∈E Bq (x, t), then     M (E, ∂Gt ) ≤ b(t) M (E, S n−1 (r)) . Proof Let F1 = E , F2 = S n−1 (r) , and F3 = ∂Gt = F4 . Because q(F1 , F3 ) ≥ t and q(F2 , F4 ) ≥ t it follows from the comparison principle 7.25 and 7.27(2) that M( 12 ) ≥ 3−n min{ M( 13 ), M( 23 ), D t }

(8.4)

where ij = (Fi , Fj ). We shall first find a lower bound for M( the √ 23 ). From √ √ choice of t it follows that t < 1/ 2 and hence q(Bq (z, t)) = 2t 1 − t 2 ≥ t 2 (see √ 3.15). It follows that F3 contains a continuum of euclidean diameter at least t 2. Hence we deduce by 7.8 and 7.23 that √ cn t log 2 Dt 2r + t 2 √ ≥ ≥ . M( 23 ) ≥ cn log r r 2r − t 2

(8.5)

Here D is the number in 7.27(2) and (8.4). Since d(|γ |) ≥ q(|γ |) ≥ t and |γ | ⊂ B n (r) for γ ∈ (F1 , F3 ; Gt ) =  and M() = M( 13 ) (cf. (7.2)) we get by 7.4 M( 13 ) ≤ n

rn . tn

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8 The Modulus as a Set Function

By (8.4) and (8.5) Dt

M( 12 ) ≥ 3−n min M( 13 ) , r

Dt n+1 1 M( 13 ) , ≥ 3−n min M( 13 ) , M( 13 ) = n+1 n r b(t)   b(t) = 3n / min 1 , Dt n+1 /(n r n+1 ) .  

as desired.

8.1 The Construction of c(E) n

n

For E ⊂ R , 0 < r < t < 1 , and x ∈ R , denote (cf. (8.1)) ⎧ ⎨mt (E, r, x) = M(∂Bq (x, t), E ∩ Bq (x, r)) , ⎩m(E, x) = m (E, 1/√2, x) ; s = 1 √3 . s

(8.6)

2

We define (see (3.7) and (3.9))  c(E, x) = max{ m(E, x) , m(E,  x) } , n

c(E) = inf{ c(E, x) : x ∈ R } .

(8.7)

Remark 8.3 By 7.11(2)  ⎧ &  '1−n 2 ⎪ ⎪ ⎨mt (E, r, x) ≤ ωn−1 log t 1 − r , r 1 − t2 ⎪ ⎪ √ ⎩ n m(E, x) ≤ m(R , x) = ωn−1 (log 3 )1−n .

(8.8)

√ If F ⊂ Bq (x, r), where r ∈ (0, 1/ 2 ], by (8.8) we obtain  1 1−n m(F, x) ≤ ωn−1 log 3(1 − r 2 ) . r

(8.9)

Hence c(F, x) → 0 as r → 0 (Fig. 8.1). Note that equality holds in (8.9) if F = Bq (x, r). Exploiting 3.10(1) one can simplify the upper bound in (8.8). Lemma 8.4 There exists a number d1 ≥ 1 depending only on n such that c(E, x) ≤ d1 c(E, y)

8.1 The Construction of c(E)

137

Fig. 8.1 Function c(E) defined in (8.7)

n

n

for x, y ∈ R and E ⊂ R . In particular, c(E) ≤ c(E, x) ≤ d1 c(E) . √ Proof Let U = Bq (x, 1/ 2 ). By 7.11(2) or by (8.8) we obtain √ √     M (∂Bq (x, 12 3 ), ∂U ) = M (∂Bq (x, 12 ), ∂U ) = ωn−1 (log 3 )1−n = a . (8.10) n

Fix x, y ∈ R . In what follows we shall assume that c(E, x) = m(E, x) .

(8.11)

The other case c(E, x) = m(E,  x ) can be dealt with exactly in the same way; even the constants will be the same in the other case. Let √ √ E1∗ = E ∩ Bq (x, 1/ 2 ) ∩ Bq (y, 1/ 2 ) , √ E2∗ = (E \ E1∗ ) ∩ Bq (x, 1/ 2 ) . It follows from 7.8 and (8.11) that either     2 M (∂V , E1∗ ) ≥ c(E, x) or 2 M (∂V , E2∗ ) ≥ c(E, x) √ √ where V = Bq (x, 12 3 ). In the first case denote F1 = E1∗ , F2 = ∂Bq (y, 12 3 ), √ F3 = ∂V and F4 = ∂Bq (y, 1/ 2 ). In the second case let F1 = E2∗ , F2 =

138

8 The Modulus as a Set Function

√ √ ∂Bq ( y , 12 3 ), F3 = ∂V , and F4 = ∂Bq ( y , 1/ 2 ). In both cases (see 3.15(1) and (3.6)) √     min q(F1 , F3 ) , q(F2 , F4 ) ≥ q ∂V , ∂Bq (x, 1/ 2 ) √ √ 3−1 n−1 n−1 =δ. = q(S ( 3 ), S ) = √ 8 Let D > 0 be the constant in 7.27(2). We obtain by 7.25, 7.27(2), and (8.10)   c(E, y) ≥ M( 12 ) ≥ 3−n min M( 13 ), M( 24 ), Dδ   ≥ 3−n min 12 c(E, x) , a , Dδ . Because c(E, x) ≤ a by (8.8) and (8.10) we obtain from this inequality   c(E, y) ≥ 3−n min 12 c(E, x) , Dδ ≥ c(E, x)/d1 ; √   d1 = 3n / min 12 , Dδ(log 3 )n−1 /ωn−1 ,  

which yields the desired bound. n

Lemma 8.5 If E ⊂ R , then √  3 1−n c(E) ≤ ωn−1 log √ . 2 q(E) √ Proof Assume first that q(E) ≥ 1/ 2. In this case √  √ 1−n 3 1−n c(E) ≤ c(E, 0) ≤ ωn−1 (log 3 ) ≤ ωn−1 log √ 2 q(E) √ by (8.8). Assume next that q(E) ≤ 1/ 2. In this case E ⊂ Bq (z, q(E)), z ∈ E, and the proof follows from (8.9).   Corollary 8.6 There exists a number d3 > 0 depending only on n with the n following property. If E ⊂ R is connected, then c(E) ≥ d3 q(E) . Proof It follows from the definition (8.7) that c(E) = c(hE) whenever h is a spherical isometry. Hence both sides of the asserted inequality remain invariant under spherical isometries. By performing an auxiliary spherical isometry if necessary we n may assume that 0 ∈ E. Then E ∩ B has a connected component E1 with 0 ∈ E1

8.1 The Construction of c(E)

139

and hence by (3.6) √ √ d(E1 ) ≥ q(E1 ) ≥ min{ 1/ 2 , q(E)} ≥ q(E)/ 2 . By 7.23 we obtain (see (8.1), (8.7), and 3.15(1)) c(E, 0) ≥

M√3 (E1 , 1, 0)

√ √ √ 2 3 + q(E)/ 2 ≥ cn log √ √ ≥ cn q(E)/ 6 . 2 3 − q(E)/ 2

√ The proof with d3 = cn /(d1 6 ) follows now from 8.4.

 

Lemma 8.7 There exists a number d4 > 0 depending only on n with the following n property. If E, F ⊂ R , then   M (E, F ) ≥ d4 min{c(E), c(F ) } . n

x } with m(E, z) = c(E, x) and denote Proof Fix x ∈ R . Let z ∈ {x,  √ √ F1 = E ∩ Bq (z, 1/ 2 ) , F3 = ∂Bq (z, 12 3 ) . Let w ∈ {x,  x } be such that m(F, w) = c(F, x) and denote √ √ F2 = F ∩ Bq (w, 1/ 2 ) , F4 = ∂Bq (w, 12 3 ) . We see that (cf. 3.15) min{ q(F1 , F3 ), q(F2 , F4 ) } ≥ q(S

n−1

√ ( 3 ), S n−1 ) =

√ 3−1 =δ. √ 8

Set ij = (Fi , Fj ). Let D > 0 be as in 7.27(2). It follows from the comparison principle 7.25 and 7.27(2) (see also 7.8) that   M (E, F ) ≥ M( 12 ) ≥ 3−n min{ c(E, x), c(F, x), Dδ } ≥ d4 min{ c(E, x), c(F, x) } ≥ d4 min{ c(E), c(F ) } √ where d4 = 3−n min{ 1, Dδ(log 3 )n−1 /ωn−1 } and √ the second last inequality follows from the fact that c(E, x), c(F, x) ≤ ωn−1 (log 3 )1−n (cf. (8.8)).   n

Lemma 8.8 Let E, F ⊂ R be sets with q(E, F ) ≥ t > 0. Then   M (E, F ) ≤ d5 min{ c(E), c(F ) } where d5 depends only on n and t.

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8 The Modulus as a Set Function

√ √ Proof Let E1 = E ∩Bq (0, 1/ 2 ), E2 = E \E1 , F1 = F ∩Bq (0, 1/ 2 ), F2 = F \ F1 . Let 1 = (E1 , F1 ), 2 = (E1 , F2 ), 3 = (E2 , F1 ), and 4 = (E2 , F2 ). By 7.8   M (E, F ) ≤ 4 max{ M( j ) : j = 1, 2, 3, 4 } . Without loss of generality we may assume that the maximum on the right side of this inequality is equal to M( 2 ) because in the other cases the proof will be similar. Let E1t =



   Bq (x, 18 t) : x ∈ E1 , F2t = Bq (x, 18 t) : x ∈ F2 .

If γ ∈ 2 , then clearly |γ | ∩ ∂E1t = ∅ = |γ | ∩ ∂F2t and hence by 7.2 1 4

       M (E, F ) ≤ M( 2 ) ≤ min M (E1 , ∂E1t ) , M (F2 , ∂F2t ) .

(8.12)

  We shall now find an upper bound for M (E1 , ∂E1t ) . A simple calculation shows that √ √ 1 3−1 n−1 n−1 > . q(S ( 3 ), S ) = √ 4 8 n

Since E1 ⊂ B we get by 8.2 √     M (E1 , ∂E1t ) ≤ M (E1 , S n−1 ( 3 )) b(t/8) , 3 √   b(t/8) = 3n / min 1 , Dt n+1 (8 3 )n+1 n .   A similar estimate holds for M (F2 , ∂F2t ) as well. As a result we obtain in view of (8.12), (8.7), and 8.4   M (E, F ) ≤ 4 b(t/8) min{ c(E, 0) , c(F, 0) } ≤ 4 d1 b(t/8) min{ c(E) , c(F ) } and the assertion follows with d5 = 4 d1 b(t/8).

    Corollary 8.9 If E, F ⊂ R with q(E, F ) ≥ t > 0, then M (E, F ) ≤ d6 . √ n Proof By (6.12) and 8.4 c(E) ≤ c(R ) = ωn−1 (log 3 )1−n = d2 . The proof with   d6 = d2 d5 follows from 8.8. n

Recall that a different proof of 8.9 was given in 7.15.

8.2 Metric Concentration of Sets

141

8.1.1 Proof of Theorem 8.1 Part (1) is clear by the definition of c(·). Part (2) follows from (8.7), 8.4, and 7.8: c

∞ %

∞ ∞ ∞  %  ! ! Ej ≤ c Ej , 0 ≤ c(Ej , 0) ≤ d1 c(Ej ) .

j =1

j =1

j =1

j =1

The other assertions in (2) follow from 7.8. The proofs of (4), (5), and (6) were given in 8.6, 8.7, and 8.8, respectively. The proof of (3) follows from (5), (6), and the definition of a set with positive capacity, which will be given in Chap. 9 (see 9.9).   Exercise 8.10 Find a positive lower bound for c(B n (x, r)) . Exercise 8.11 Applying (7.14) and the results of this chapter show that (8.3) holds.  ∞ n−1 −k  S (2 ) and E(t) = { z ∈ Rn : q(z, E) < Exercise 8.12 Let E = {0}∪   k=1 1−n t }. Show that M (E, ∂E(t)) ≥ α t log 1t for small t where α depends only on n. [Hint: Apply (7.4).] Conclusion: The function b(t) in 8.2 must grow so fast that b(t)t n−1 / log 1t → 0 as t → 0. From the proof of 8.2 it follows that the rate of growth of b(t) is at most t −1−n , and the best rate of growth will be given in 8.22.

8.2 Metric Concentration of Sets 8.2.1 Thickness and Capacity √ The condition c(E) > 0 tells us that given a half space B q (z, 1/ 2) of the Möbius n space R , then either this half space or its complementary half space contains most of the set E in the sense of capacity. We now introduce a density condition which enables us describe in a quantitative way the local capacity concentration of a set. n

Definition 8.13 (Metric Thickness Condition [554]) We say that a set E ⊂ R satisfies a metric thickness condition if there exist positive numbers r0 ∈ (0, 14 ] and δ > 0 such that for all r ∈ (0, r0 ] m2r (tz E, r, 0) ≥ δ . n

and for every point z ∈ R . n

Definition 8.14 (Uniform Perfectness) Let E ⊂ R be a closed set containing at least two points and let α > 0 . We say that E is α-uniformly perfect if there is no n ring domain D ⊂ R \ E separating E such that mod(D) > α ; E is uniformly perfect if it is α-uniformly perfect for some α > 0 (cf. [428–430] for n = 2 and [517] for metric spaces).

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8 The Modulus as a Set Function

The next theorem shows that these two ways to measure the concentration of a set, the uniform perfectness and the metric thickness condition, are essentially the same. In the planar case there are numerous characterizations of uniform perfectness, see 8.20. Theorem 8.15 ([264, Thm 4.4, Cor. 4.2]) n

(1) If E ⊂ R is α-uniformly perfect, then it satisfies the metric thickness condition (8.13) with the parameters r0 = 14 q(E) , δ = δ(α, n) > 0 . n (2) If E ⊂ R satisfies (8.13) with r0 = 14 q(E) , then it is α-uniformly perfect with the parameter α = α(δ, r0 ) > 0 . n (3) If E ⊂ R is α-uniformly perfect, then its Hausdorff dimension satisfies H − dim(E) ≥ c(α, n) > 0 . Examples 8.16 n

(1) A set E ⊂ R is uniformly perfect if it consists of a finite number of components, each of which is connected compact set containing at least two points. Indeed, we find a lower bound for the capacity of a ring domain separating E using Theorem 8.1. This lower bound yields an upper bound for the modulus of a separating ring domain, required by the definition of uniform perfectness. (2) Let (rj ), (sj ) be two sequences of numbers such that 0 < sj +1 < sj for all j , n limj →∞ sj = 0 , and such that the balls B (sj en , rj ), j = 1 = 1, 2, . . . , are disjoint and let E = {0} ∪ (∪∞ j =1 B (sj en , rj )) . n

Then for some choice of the sequences (rj ), (sj ) the set E is uniformly perfect, for another choice E is not uniformly perfect. See Remark 8.19. (3) Some self-similar sets such as the Cantor middle-third set are uniformly perfect in the plane. (4) The metric thickness condition is sometimes expressed in different way, in terms of a density condition for capacity, see [555, pp. 178–181]. Some density conditions stem from potential theory, where an important question is to study whether a boundary point of a domain is a regular point for the Dirichlet problem; see [192, 354, 355, 367]. n

Exercise 8.17 Let rk ∈ (0, 2−k−3 ), let Bk = B (2−k en , rk ), k = 2, 3, . . . and let E = {0} ∪ (∪Bk ) . (1) Show that the numbers rk can be chosen so that M((E, ∂Hn )) = ∞. (2) Show that, for every ε > 0, the numbers rk can be chosen so that M((E, ∂Hn )) ≤

∞ ! k=1

M((Bk , ∂Hn )) < ε.

8.2 Metric Concentration of Sets

143

(3) Assume that the numbers rk have been chosen as in (2) with ε = 1. Show that M((Er , ∂Hn )) → 0 n

when r → 0 where Er = E ∩ B (r). n

Exercise 8.18 For α > 0 we denote by I (α) the class of compact subsets E of B with  dm < ∞. A= n d(x, E)α B (2)\E

Then, for example, {0} ∈ I (α) when α < n, and S n−1 ∈ I (α) when α < 1. Fix E ∈ I (α), denote Ek = {x ∈ Rn : 2−k−1 ≤ d(x, E) ≤ 2−k }, k = 1, 2, . . . , and for p > 0 let p be the family of all curves in (E, S n−1 (2); Rn ) with (γ ∩ Ek ) ≥ 2−kp . Show that M( p ) = 0 for p < α/n. An example of a domain not in class I (α) is given in [465, p. 248]. Remark 8.19 It can be proved that a set E ⊂ Hn satisfying the condition M((E, ∂Hn )) < ∞ in Exercise 8.17(2) is not uniformly perfect [555, Lemma 14.14]. See also [354, 367], [547, pp.34–40]. Remark 8.20 Uniformly perfect sets have an important role in geometric function theory, see Ch. Pommerenke [428, 429], T. Sugawa [501] and L. Keen–N. Lakic [283, Ch 15]. For instance, in planar domains with uniformly perfect boundaries, the hyperbolic and quasihyperbolic distances are comparable [283, Ch 15], [343, 501, 503]. Numerous characterizations for uniformly perfect sets are given in the monograph of J. Garnett–D. Marshall [149, pp. 343–345]. See also F. Avkhadiev– K.-J. Wirths [38]. Structure conditions opposite to uniform perfectness were recently studied by R. Stankewitz, T. Sugawa, and H. Sumi [497]. For hyperbolic distance in domains without uniformly perfect boundary, see A. Beardon–D. Minda [54] and T. Sugawa–M. Vuorinen–T. Zhang [505].

8.2.2 Conformal Invariants in the Plane Some of the widely used conformal invariants in geometric function theory in the plane are the modulus of a quadrilateral [326], the modulus of a ring domain [313, 326], the hyperbolic distance between two given points in a simply-connected domain [212, 283], and the harmonic measure ω(z0 , A, D) of a set A ⊂ ∂D in a simply-connected domain, evaluated at z0 ∈ D [149, 212, 410, 447, 516]. The interrelations between these powerful tools of classical function theory are studied in [10, 149, 230]. On the other hand, there are very few cases, when the analytic formulae of these expressions can be found in the literature.

144

8 The Modulus as a Set Function

8.2.3 Reduced Modulus Yet another conformal invariant is defined as follows. Given a simply connected domain D ⊂ C , a ∈ D , there exists by Riemann’s mapping theorem a unique conformal map w = f (z) of D onto a disk {w : |w| < R} such that f (a) = 0, f  (a) = 1 [542, p. 16]. The number R is called the conformal radius of D with respect to the point a and it is denoted by R(D, a) . The conformal radius is connected to the so-called reduced modulus m(D, a) of the domain D with respect 1 to a , m(D, a) = 2π log R(D, a) [542, Thm 2.2.1, pp. 32–33], and to the Robin 2

constant and to the logarithmic capacity of R \ D ; see V.N. Dubinin [121, pp. 26–27], G.M. Goluzin [172, p. 310], R. Nevanlinna [410, p. 128], N.S. Landkof [320, p.172]. The reduced modulus was studied by O. Teichmüller [511, p. 211], [510, p. 9], and it is also mentioned in [10, p. 78], [412, pp. 238–241], [314, pp. 3–6], [149, pp.162–166], [68], and in the extensive survey [318, p. 189]. See also the literature references in Remark 9.13(5). In the spatial case, the reduced modulus was studied by B.E. Levitskii [332] and D. Betsakos and S. Pouliasis [69]. In his lecture notes I. P. Mityuk [391] systematically studied these notions, in particular, the reduced modulus. Unfortunately these Russian lecture notes, kindly sent to us by B.E. Levitskii, are not widely available.

8.3 Tubular Neighborhoods We are now going to prove an improved form of Lemma 8.2 and shall show that the function b(t) in 8.2 can be chosen so that its rate of growth is at most t −n . It follows from Exercise 8.12 that the power −n cannot be replaced by 1 − n (see also 8.23). Here our main interest is to study the capacity of a “massive” set with respect its tubular t-neighborhood as a function of t ; for additional results see Remark 9.13(4). The following discussion is based on a Poincaré inequality type result of Yu. G. Reshetnyak [463, p. 60, Lemma 3.3.], and the proof of Lemma 8.22 below is also due to him. For the proof we need also some results from the early parts of Chap. 9. In particular, Lemma 9.6 will be useful. Lemma 8.21 ([463, p. 60, Lemma 3.3.]) Let u be a function of class C0∞ (Rn ) such that u(x) = 0 for |x| ≥ r > 0. Then the inequality 

 |u| dm ≤ (2r) n

Rn

holds.

n Rn

|∇u|n dm

8.3 Tubular Neighborhoods

145

Lemma 8.22 Let E be a compact set in B n (R) and let E(t) = E +B n (t) for t > 0. Then     M (∂E(t), E) ≤ a(t) M (∂E(1), E) for t > 0 where a(t) = a(1) for t ≥ 1 and a(t) ≤ a1 t −n for t ∈ (0, 1), and a1 depends only on n and R. Proof Fix  > 0. In view of (9.1)–9.6 there exists a function u ∈ C0∞ (E(1)) with u(x) ≥ 1 for x ∈ E and    |∇u|n dm ≤  + M (∂E(1), E) =  + cap(E(1), E) . Rn

There exists a constant b1 depending only on n and for each t ∈ (0, 1] a (realvalued) C0∞ (E(1))-function ϕt : E(1) → [0, 1] with the properties (see e.g. [498, p. 171]) (a) ϕt (x) = 1 for x ∈ E , (b) ϕt (x) = 0 for x ∈ E(1) \ E(t) , (c) |∇ϕt (x)| ≤ b1 /t . The function v(x) = u(x)ϕt (x) is admissible for the definition of cap(E(t), E) (see 9.2), and hence    M (∂E(t), E) = cap(E(t), E) ≤ |∇v|n dm . Rn

  Since |∇v(x)|n ≤ 2n |∇u(x)|n ϕt (x)n + |∇ϕt (x)|n |u(x)|n we get by the properties (a) and (c) of ϕt   cap(E(t), E) ≤ 2n |∇u|n dm + (2b1 )n t −n |u(x)|n dm . Rn

Rn

Moreover, by Lemma 8.21 we obtain for t ∈ (0, 1] 

 |u(x)| dm ≤ 2 (R + 1) n

Rn

n

n Rn

|∇u(x)|n dm .

Hence  cap(E(t), E) ≤ a(t)

Rn

|∇u|n dm ≤ a(t)(  + cap(E(1), E) ) ,

a(t) = 2n (1 + 2n b1n (R + 1)n t −n ) for t ∈ (0, 1]. For t ≥ 1 we define a(t) = a(1). Because  > 0 is arbitrary the proof follows from this last inequality in view of 9.6 and 7.2.  

146

8 The Modulus as a Set Function

We already know by Exercise 8.12 that the inequality a(t) ≤ a1 t −n of Lemma 8.22 provides the best possible integer power for the growth of a(t). Next, we shall show that this rate t −n of growth for a(t) is in fact attained. Example 8.23 We shall show that there exists a constant b1 > 0 and for arbitrarily small t ∈ (0, 1) a set E = Et in Rn such that   M(E, t) = M (E, ∂E(t)) ≥ b1 t −n

(8.13)

where E(t) = E + B n (t). Let Q = [0, 1]n−1 × {0} and let s ∈ (0, 1). It follows from 7.1.1 that M(Q, s) ≥ s 1−n .

(8.14)

Fix k ≥ 4 and let Qj = Q + 2−k j en , j = 0, . . . , 2k . Set Ek = t ∈ (2−k−2 , 2−k−1 ) we obtain by 7.3 and (8.14)

2k

j =0 Qj .

For

M(Ek , t) ≥ (2k + 1)t 1−n ≥ 14 t −n . In conclusion, we have proved (8.13) with b1 = 14 . n

Exercise 8.24 Let E, F ⊂ R(2, 1), R(a, b) = B n (a) \ B (b), a > b > 0, and δ = M((E, F, Rn )) > 0. Find a number c > 1, c = c(n, δ), such that M((E, F, R(2c, 1/c)) ≥ δ/2. Exercise 8.25 Let E ⊂ Bn and δ = M((S n−1 (2), E; Rn )) > 0. Find a number c > 1, c = c(n, δ), such that M((S n−1 (2), E; R(2c, 1/c))) ≥ δ/2. Exercise 8.26 Let t > r > s > 0, E ⊂ B n (s) and a = (E, S n−1 (a)). Show that M(r ) ≤ cM(t ), where c is only dependent on n, r, s and t. n

Exercise 8.27 Let s ∈ (0, √1 ] and z ∈ R . Use the proof of Theorem 8.1 to find an 2

upper bound for c(Bq (z, s)) in terms of n and s. Your bound should tend to 0 when s → 0. Exercise 8.28 Show that for given ε > 0 there are numbers r1 > s1 > r2 > s2 > ... such that M((E, F ; Rn )) < ε, when E = ∪S n−1 (rj ) and F = ∪S n−1 (sj ). Remark 8.29 By Example 8.23 we see that, in general, the growth rate t −n of the function a(t) in Lemma 8.22 is best possible. However, under special assumptions about the metric structure of the set E, this growth rate may be refined. See Remark 9.13(4) and [322] and [215].

8.3 Tubular Neighborhoods

147

Remark 8.30 Modulus estimates in the chordal metric appear in [326, I.6.5], [524, Chapter 12], [495], [364, Lemma 3.11.], and in [405]. The construction of the set function c(E) in Theorem 8.1 is based on [551]. H. Renggli [452] and W. P. Ziemer [577] have also constructed some set functions related to moduli of curve families. By a result of H. Wallin [563] a compact set E in the plane is of logarithmic capacity zero if and only if c(E) = 0 . We expect that in the case c(E) > 0 there are inequalities between the aforementioned conformal invariants and c(E) .

Chapter 9

The Capacity of a Condenser

In the present chapter we shall introduce, as a special case of curve families and their moduli, the notion of a condenser and its capacity, and we shall examine various properties of condensers. An important property of the capacity of a condenser is that it decreases under a special geometric transformation called symmetrization. Of the several kinds of symmetrization discussed in the literature (see e.g. [150, 426, 479], [463, p. 74], [43, 121]) we shall consider only spherical symmetrization. An immediate consequence of the above-mentioned monotonicity is the fact that condensers obtained as a result of spherical symmetrization are of extremal character—their capacities yield lower bounds for the capacities of a wide class of condensers in Rn . The extremal condensers of Grötzsch and Teichmüller are of particular importance, and the well-known estimates for the capacities of these condensers are given in this chapter. One of the main themes of this chapter is the relationship of the capacity of a condenser to its geometric structure. The hyperbolic and quasihyperbolic geometries are useful instruments in the study of this interrelation in Chaps. 9 and 10. In this context the hyperbolic and quasihyperbolic geometries are useful for proving estimates for the capacity only of ring domains with non-degenerate complementary components. Definition 9.1 For j = 1, . . . , n let Rjn = { x ∈ Rn : xj = 0 } and let Tj : Rn → Rjn be the orthogonal projection Tj x = x − xj ej . Let D ⊂ Rn be an open set and u : D → R a continuous function. The function u is called absolutely continuous on lines, abbreviated as ACL, if for every cube Q with Q ⊂ D, the set Aj ⊂ Tj D ⊂ Rjn of all points z ∈ Tj Q such that the function t → u(z + tej ), z + tej ∈ Q, is not absolutely continuous as a function of a single variable [233, p. 282], satisfies mn−1 (Aj ) = 0 for all j = 1, . . . , n. By well-known properties of absolutely continuous functions of a single variable the derivative exists almost everywhere and is Borel-measurable (see [233, p. 285], [524, pp. 87–89]). From this fact and from Fubini’s theorem it follows that an ACL © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_9

149

150

9 The Capacity of a Condenser

function u : D → R has partial derivatives with respect to every variable x1 , . . . , xn a.e. (with respect to n-dimensional Lebesgue measure) in D. We say that an ACL function u : D → R is ACLp , p ≥ 1, if ∂u(x)/∂xj ∈ Lp (K), j = 1, . . . , n, whenever K ⊂ D is compact. A vector-valued function is said to be ACL (ACLp ) if and only if each coordinate function is in this class. Definition 9.2 Let A ⊂ Rn be open and let C ⊂ A be compact. The pair E = (A, C) is called a condenser. Its p-capacity is defined by  p-cap E = inf u

Rn

|∇u|p dm ,

(9.1)

where the infimum is taken over the family of all non-negative ACLp functions u with compact support in A such that u(x) ≥ 1 for x ∈ C. Here  ∇u(x) =

∂u ∂u (x) , . . . , (x) ∂x1 ∂xn

 .

A function u with these properties is called an admissible function. It follows from (9.1) that p-cap E is invariant under translations and orthogonal maps. Without alteration of the real number p-cap E, one can take the infimum in (9.1) over several other classes of functions as can be shown by approximation. For instance one may take functions u ∈ C ∞ (A) with compact support in A and u(x) ≥ 1 for x ∈ C (see [363, 366]). The following monotone property of condensers is a consequence of the definition. If (A, C) and (A , C  ) are condensers with A ⊂ A and C ⊂ C  , then p-cap (A , C  ) ≥ p-cap (A, C) .

(9.2)

The p-capacity of (A, C) reflects the metric structure of the pair C, Rn \ A as we shall see later on. If p = n we denote n-cap(A, C) simply by cap(A, C) and call it the capacity or conformal capacity of the condenser (A, C). The next lemma, due to Gehring [151, p.367], [524, Theorem 37.1] deals with continuity of the capacity with respect to set. Several convergence theorems of this type are given in [219, Thm 2.2, p.28], [161, pp. 135–140], [366, pp. 291–344]. Lemma 9.3 Let C ⊂ Rn be a compact non-empty set and B(C, r) = ∪z∈C B n (z, r), for r > 0 . Then p-cap (B(C, r), B(C, s)) → p-cap (B(C, r), C) as s → 0 .

9 The Capacity of a Condenser

151

Proof Write A = B(C, r) . There is u ∈ C0∞ (A) such that u|C = 1 and 

 |∇u|p dm
1} is a neighborhood of C and contains B(C, s) for small s . For these s we have  p-cap (A, C) ≤ p-cap (A, B(C, s)) ≤ |∇(qu)|p dm 

 =q

|∇u|p dm) .

|∇u| dm < q (

p

p

p

Because q ∈ (0, 1) was arbitrary, the proof follows. Rm ,

 

where D ⊂ is open, is said to be absolutely An ACL function u : D → continuous on the rectifiable curve α : [a, b] → D iff f ◦ α 0 : [0, (α)] → Rm is absolutely continuous as a function of one variable. We shall make use of the following result of B. Fuglede [145], [524, 28.1, 28.2]. p

Rn

Lemma 9.4 Let D be an open set in Rn and let f : D → Rm be ACLp . Then the family of all locally rectifiable paths in D having a closed subpath on which f is not absolutely continuous, is p-exceptional. Lemma 9.5 Let G be a domain in Rn , let u : G → R be an ACLp function, −∞ < a < b < ∞, and let A, B ⊂ G be non-empty sets such that u(x) ≤ a for x ∈ A and u(x) ≥ b for x ∈ B. Then   Mp (A, B; G) ≤ (b − a)−p

 |∇u|p dm . G

Proof Define an ACLp function v : G → R by v(x) =

u(x) − a , x∈G. b−a

Then v(y) ≥ 1 for y ∈ B and v(y) ≤ 0 for y ∈ A. Let  = { γ ∈ (A, B; G) : γ is rectifiable } and u = { γ ∈  : v is not absolutely continuous on a closed subpath of γ } . Fix γ ∈  \ u with the normal representation γ 0 : [0, c] → G, c = (γ ), and with γ 0 (0) ∈ A, γ 0 (c) ∈ B. Then γ 0 has a Lipschitz constant 1 and (γ 0 ) (t) = 1

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9 The Capacity of a Condenser

a.e. in [0, c] (see [524, 2.4]). By [524, 1.3] we get 

c

1 ≤ v(γ 0 (c)) − v(γ 0 (0)) ≤ 0



c

≤ 0

(v ◦ γ 0 ) (t) dt

 0  0 |∇v| ds . (∇v ◦ γ )(t) (γ ) (t) dt =

(9.3)

γ

Since v is ACLp , |∇v| is a Borel function and thus |∇v| ∈ F ( \ u ) in view of (9.3). By 9.4 we obtain   Mp (A, B; G) ≤



|∇v|p dm = (b − a)−p



G

|∇u|p dm G

 

as desired. Let (A, C) be a condenser where A is bounded. It follows from 9.5 that    Mp (C, ∂A; A) ≤ |∇u|p dm A

for every u in ACLp with C ⊂ {z ∈ A : u(z) ≥ 1} and ∂A ⊂ {z ∈ Rn : u(z) ≤ 0}. Thus   Mp (C, ∂A; A) ≤ p-cap (A, C) . Also the converse inequality holds true according to the following result of W. P. Ziemer [576], but the proof is long and will be omitted. For a complete proof see [161, Thm 5.2.3, p. 164]. Theorem 9.6 If E = (A, C) is a bounded condenser in Rn , then   p-cap E = Mp (C, ∂A; A) . Remark 9.7 By 7.8 the curve family on the right side of Theorem 9.6 may be replaced by some other families as well. We shall need 9.6 mainly in the case p = n. We now show that 9.6 holds also for unbounded condensers if p = n. Let (A, C) be an unbounded condenser, let z ∈ C and r > 1 + d(C), and Ar = A ∩ B n (z, r 2 ). By the monotone property of the capacity, by 9.6, 7.8, and 7.4 we obtain   cap(A, C) ≤ cap(Ar , C) = M (C, ∂Ar ; Ar )     ≤ M (C, ∂A; A) + M (C, S n−1 (z, r 2 ))   ≤ M (C, ∂A; A) + ωn−1 (log r)1−n .

9 The Capacity of a Condenser

153

  Letting r → ∞ shows that cap(A, C) ≤ M (C, ∂A; A) . The converse inequality follows from 9.5 and (9.3). In conclusion, we have proved that the equality   cap(A, C) = M (C, ∂A; A)

(9.4)

holds whenever (A, C) is a condenser in Rn , whether A is bounded or not. n

n

We now extend the definition of a condenser to R . Assume that C ⊂ R is n n compact and that there exists an open set A ⊂ R with A = R and C ⊂ A. n Then we say that (A, C) is a condenser in R and define its (n-)capacity by (9.4). In view of 9.6 this extended definition is compatible with the definition (9.1) in case n A ⊂ Rn . (We shall not need the p-capacity, p = n, of a condenser in R .) n Now let (A, E) be a condenser in Rn or R . It follows from (9.4) and the conformal invariance of the modulus 7.10 that cap(A, E) is a conformal invariant. Likewise, by virtue of (9.4), many properties of cap(A, E) may be derived directly from the properties of the modulus. In particular, we shall often make use of Remark 7.8 specialized to condensers. If p = n, then p-cap(A, E) is not invariant under conformal mappings, while n-cap(A, E) has this invariance property. The corresponding property of the modulus immediately yields this conclusion. Lemma 9.8 Let (A, F ) be a condenser with A ⊂ Bn . Then there are positive numbers a1 depending only on n, and a2 depending only on n and d(F, ∂A) such that a1 c(F ) ≤ cap(A, F ) ≤ a2 c(F ) . Proof By the proof of 8.7 and 9.6   n cap(A, F ) ≥ d4 min c(R \ A, 0) , c(F, 0) . Since A ⊂ Bn , n

c(R \ A, 0) = ωn−1 (log

√ 1−n n 3) = c(R , 0) ≥ c(F, 0)

(cf. (8.7), (8.8)) and thus we obtain the desired lower bound cap(A, F ) ≥ d4 c(F ) . For the upper bound let t = q(F, Rn \ A). Then by 3.9 t ≥ F ⊂ A ⊂ Bn . By the proof of 8.4 and 8.8

1 2

d(F, Rn \ A) since

cap(A, F ) ≤ 4 b(t/8) min{ c(F, 0) , c(Rn \ A, 0) } ≤ 4 d1 b(d(F, ∂A)/16) c(F ) ,

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9 The Capacity of a Condenser

which yields the desired upper bound. (Note that a slightly better upper bound can   be derived from 8.22.) Definition 9.9 A compact set E in Rn is said to be of capacity zero, denoted cap E = 0, if there exists a bounded open set A with E ⊂ A and cap(A, E) = 0. n n A compact set E ⊂ R , E = R , is said to be of capacity zero if E can be mapped by a Möbius transformation onto a bounded set of capacity zero. Otherwise E is said to be of positive capacity, and we write cap E > 0. Remark 9.10 By conformal invariance the second part of the above definition is independent of the choice of Möbius transformation. We next show that the first part of the definition is independent of the choice of bounded open set A with A ⊃ E. Indeed, if Aj ⊃ E, j = 1, 2, are both bounded, say Aj ⊂ B n (R), j = 1, 2, then by 8.1(4)  n n d2 = c(R ) ≥ c(R \ Aj ) ≥ d3 / 2 + R 2 ; j = 1, 2 . This inequality together with 8.1(5),(6) yields for j = 1, 2    d4 min c(E), d3 / 2 + R 2 ≤ cap(Aj , E) ≤ d5 min { c(E), d2 } where d2 , d3 , d4 are positive numbers depending only on n and where d5 depends also on q (E, (∂A1 ) ∪ (∂A2 )). In other words, cap(Aj , E) = 0 if and only if c(E) = 0. Hence the condition cap E = 0 is independent of the choice of open bounded set A with E ⊂ A (see also [455, Lemma 2]). This argument also shows that in the above definition of cap E = 0, E ⊂ Rn compact, one can replace the bounded set A by a ball B n (r) with r ≥ d(0, E) + 2d(E), say. It should be observed that we have only defined the conditions cap E = 0 and cap E > 0 for a compact set E and that in the latter case the “capacity” of E will not be specified as a real number. In view of 9.7 and (7.5) countable compact sets are examples of sets of capacity zero. The following lemma shows that sets of capacity zero are always very thin [463, p. 72]. The definition of the Hausdorff dimension and the α-dimensional Hausdorff measure can be found in [133] and [377]. Lemma 9.11 Suppose that F is a compact set in Rn of capacity zero. Then for every α > 0, the α-dimensional Hausdorff measure α (F ) of F is zero. In particular, intF = ∅, and F is totally disconnected. Exercise 9.12 Let E ⊂ Bn be compact. Suppose that mn (Ek ) = ak , Ek = {x ∈ Rn \ E : 2−k−1 < d(x, E) < 2−k }, k = 1, 2, . . . . Use Lemma 7.16 to find an upper bound for M((E, S n−1 (2))). Apply your bound to give a sufficient condition for cap E = 0 in terms of the numbers (ak ).

9.1 Spherical Symmetrization

155

Remarks 9.13 (1) We do not know whether, in the case of Lemma 9.3, one could prove concrete estimates for the speed of convergence |p-cap (B(C, r), B(C, s)) − p-cap (B(C, r), C)| → 0 when s → 0 . (2) In the dimension n = 2 the logarithmic capacity is often used in complex analysis. H. Wallin [563] has proved that a compact set is of logarithmic capacity zero if and only if it is of capacity zero in the sense of the above definition (n = 2). He has also constructed a compact Cantor-type set E in Rn of positive capacity (in the sense of 9.9) with α (E) = 0 for all α > 0. See also V. G. Maz’ya–V. P. Khavin [382]. (3) Various sufficient or necessary conditions for capacity zero can be found in the literature [382, 564], [463, p. 71], [374, 459, 526]. (4) For a compact set C ⊂ Rn and t > 0 let E(t) = z∈E B n (z, t). The capacity cap(E(t), E) was discussed above in Exercise 8.12 and Lemma 9.3. J. Väisälä [526] has shown that if cap E > 0 then cap(E(t), E) → ∞ when t → 0. V. Heikkala [215, Thm 4.6] showed that this convergence may be arbitrarily slow and gave credit to Jan Maly for the idea behind this result. Heikkala also proved a lower bound for this convergence in the case when E is uniformly perfect. Later on, J. Lehrbäck [322] completed his results. (5) M. Tsuji [516] introduced the hyperbolic capacity of a planar set. It is closely connected with the condenser capacity, see [121, Thm 1.22, p. 23], [125, 126], [66, Thm 1], [284].

9.1 Spherical Symmetrization Since ancient times it is well-known that among all plane domains with area A and perimeter L , the isoperimetric inequality L2 ≥ 4πA holds with equality only for the case of a circle. This inequality expresses an inequality between two domain functionals, the area A and the perimeter L . Extremal problems in various areas of mathematics and science often lead to situations where the extremal case exhibits some symmetry; see the classical books of G.H. Hardy, J.E. Littlewood, and G. Pólya [190] and of W.K. Hayman [211]. Isoperimetric inequalities were systematically studied by G. Pólya and G. Szegö in their landmark work [426] where they, in particular, studied several variants of the classical isoperimetric problem under constraints. E.g. they studied this problem for triangles. One of the main themes of their work was to apply symmetrization methods to find lower bounds for condenser capacity. Extremal problems where symmetry has a role have many applications to analysis and geometric function theory, see A. Baernstein [42, 43], C. Bandle

156

9 The Capacity of a Condenser

[47], V.N. Dubinin [121], and I.P. Mitjuk [391]. Here we will study geometric transformations called symmetrizations under which the condenser capacity is decreased (Fig. 9.1). n If x0 ∈ Rn , E ⊂ R and if L is a ray from x0 to ∞, then the spherical symmetrization E ∗ of E in L is defined as follows: (1) x0 ∈ E ∗ iff x0 ∈ E, (2) ∞ ∈ E ∗ iff ∞ ∈ E, (3) for r ∈ (0, ∞), E ∗ ∩ S n−1 (x0 , r) = ∅ iff E ∩ S n−1 (x0 , r) = ∅, in which case E ∗ ∩ S n−1 (x0 , r) is a closed spherical cap centered on L with the same mn−1 measure as E ∩ S n−1 (x0 , r). Let (A, C) be a condenser and x0 ∈ Rn . Denote by C ∗ and B the spherical symmetrizations of C and Rn \ A in two opposite rays L1 and L2 emanating from x0 , and let A∗ = Rn \ B. Then it is easy to verify that (A∗ , C ∗ ) is a condenser [479]. An important property of spherical symmetrization is given in the following theorem [150, 479] (Fig. 9.2). Its complete proof is given in [161, pp. 167–180]. Theorem 9.14 If (A, C) is a condenser, then for p ≥ 1 p-cap (A, C) ≥ p-cap (A∗ , C ∗ ) . This inequality is sharp in the sense that there is equality if (A∗ , C ∗ ) = (A, C) (e.g. x0 = 0, C = [0, e1 ], A = B n (2) and L1 is the positive x1 -axis). Note that the minorant p-cap(A∗ , C ∗ ) in 9.14 depends on the choice of the center of symmetrization, the point x0 , in an essential  way. instance, if n ≥ 3,   For ∞ and if E ∗ is the Ej = { x ∈ S n−1 (2−j ) : x3 = 0 }, E = {0} E j =1 j spherical symmetrization of E in the positive x1 -axis (in which case x0 = 0), then E ∗ = { 0, 12 e1 , 14 e1 , . . . } and clearly cap(Bn , E ∗ ) = 0. It is left as an exercise for the

Fig. 9.1 Spherical symmetrization of a set E

C Rn /

A x0

Rn / A *

∞

Fig. 9.2 Spherical symmetrization in Theorem 9.14

C* x0

∞

9.2 Grötzsch and Teichmüller Condensers

157

reader to find a spherical symmetrization with center = 0 which provides a strictly positive minorant for cap(Bn , E). D. Betsakos [68, Lemma 1] proved a lower bound for the capacities of bounded condensers (A, C) where the compact set has positive n-dimensional Lebesgue measure m(C) > 0 , based on another type of symmetrization.

9.2 Grötzsch and Teichmüller Condensers Let us recall the Grötzsch and Teichmüller rings RG,n (s) and RT ,n (s) which were introduced in Chap. 7. They can also be understood as condensers in the following way:  n RG,n (s) = Rn \ {te1 : t ≥ s} , B , s ∈ (1, ∞) ,   RT ,n (s) = Rn \ {te1 : t ≥ s} , [−e1 , 0] , s ∈ (0, ∞) . We define functions  = n and  = n by modRG,n (s) = log (s) and modRT ,n (s) = log (s). In other words (cf. (7.17))  cap RG,n (s) = ωn−1 (log (s))1−n = γn (s) ,

(9.5)

cap RT ,n (s) = ωn−1 (log (s))1−n = τn (s) .

√ Lemma 9.15 The functions log√ (t) − log(t + t 2 − 1) and log (t) − log t are increasing and (t − 1) = ( t )2 for t > 1. Moreover, the functions γn and τn are strictly decreasing. Proof Fix 1 < s < t . The Grötzsch ring RG,n (t) can be mapped by a Möbius n

n

transformation onto a ring R in R with √ complementary components B and [ae1, ∞] ∪ [−ae1, ∞] where a = t + t 2 − 1 , see 7.32(3). For c ∈ (1, a) the sphere S n−1 (c) splits this ring into two rings of which one, say R1 , is bounded. Then modR = log (t) and by 7.16 we have modR = log (t) ≥ log c + mod R2 . Choose c > 1 such that mod R2 = log (s) . Hence a/c = s +

√ s 2 − 1 , and

√ a t + t2 − 1 c= √ = √ . s + s2 − 1 s + s2 − 1 These calculations yield the desired monotone property log (t) − log(t +



t 2 − 1) ≥ log (s) − log(s +



s 2 − 1) .

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9 The Capacity of a Condenser

The increasing property of log (t) − log t follows easily from this. It follows, in particular, that  and  are strictly increasing and hence by (9.5) γn and τn are strictly decreasing. The asserted identity is the functional identity 7.31 rewritten.   By 9.15 the function log (t) − log t is increasing and therefore has a limit as t → ∞. We define a number λn by   log λn = lim log (t) − log t .

(9.6)

t →∞

This number is sometimes called the Grötzsch (ring) constant. Only for n = 2 the exact value of the Grötzsch constant is known, λ2 = 4 [326, p. 61, (2.10.)]. Observe that the definition (9.6) resembles the definition of the reduced modulus 8.2.3, [121], [149, pp. 162–168]. Various estimates for λn , n ≥ 3, are given in [150, p. 518], [94, pp. 239–241], [21]. For instance it is known that λn ∈ [4, 2en−1 ), 

∞

λn ≤ 4 exp 1

a(n, s) ds s

 ; a(n, s) =

 s 2 + 1  n−2 n−1

s2 − 1

−1,

1/n

and that λn → e as n → ∞. For proof of these results the reader is referred to [29, Ch. 12] and [161]. Some of them are summarized in the next lemma. n−1 Lemma 9.16 For each n ≥ 2 there exists √ a number λn ∈ [4, 2e ), λ2 = 4, such that the function log (t) − log(t + t 2 − 1) is increasing from (1, ∞) onto (0, log(λn /2)) . In particular, √ √ (t) ≤ λn (t + t 2√ − 1)/2 ≤√λn t , t > 1 , (1) t ≤ t + √t 2 − 1 ≤ √ (2) t + 1 ≤ √ ( t + 1 + t)2 ≤ (t) ≤ λ2n (√ t + 1 + t)2 /4 ≤ λ2n (t + 1) , t > 0 , 2(1 + 1 − r 2 ) 1 + 1 − r2 ≤ μ(r) ≤ log , r ∈ (0, 1) . (3) log r r 1/n

Furthermore, λn

→ e as n → ∞ and, in particular, λn → ∞ as n → ∞.

Proof The bounds 4 ≤ λn ≤ 2en−1 are given in [21, 150]. As in the proof of Lemma 9.15 we see that RG,n (t) is Möbius equivalent to a ring whose boundary √ n components are separated by the annulus A = B n (t + t 2 − 1) \ B with √ 2 mod A = log(t √ + t − 1) . The upper bound in (1) follows from the identity (t −1) = ( t )2 9.15 and the definition (9.6) of λn above. Inequality (2) follows from (1) and 9.15, and the last assertion of (2) is proved in [21]. Inequality (3) follows because λ2 = 4 and log 2 (t) = μ(1/t) .   Recall that a lower bound for μ(r) better than 9.16(3) was given in (7.25). The above two lemmas yield, in fact, several bounds for μ(r) as follows. Because λ2 = 4 , it follows that the function μ(r) + log r is increasing from (0, 1) onto (0, log 4) . This fact, combined with the functional identities (7.20) of μ(r) , yields several upper and lower bounds for μ(r) such as those in (7.21). See also [29, Thm 5.13]. Because of the functional identity (7.17) the properties of τ can be derived from those of γ and conversely. A simple argument similar to 7.36(3) shows that the

9.2 Grötzsch and Teichmüller Condensers

159

strictly decreasing function τ is continuous on (0, ∞). In what follows we may use these simple properties without notice. The following fundamental difference between dimensions n = 2 and n ≥ 3 should be observed: for n ≥ 3 no explicit expression like (7.18) is known for γn (s). It is an interesting open problem to find such a formula also for the multidimensional case. The Grötzsch and Teichmüller condensers have some important extremal properties which are connected with the spherical symmetrization. In what follows we shall often require a lower bound for the capacity of a ring domain in terms of the Teichmüller capacity τn (s) which follows from the spherical symmetrization Theorem 9.14. For this reason various estimates for γn (s) and τn (s) will be very useful—in fact they will be necessary for our later work in the multidimensional case n ≥ 3 when no exact formulae for τn (s) or γn (s) are known. Before giving these estimates we shall discuss qualitatively the behavior of τn (s) and γn (s). First we note that by (7.4) and 7.21 the limit values of γn (s) and τn (s) are ⎧ ⎪ lim γn (s) = 0 , ⎨ lim γn (s) = ∞ , s→∞ s→1+ (9.7) ⎪ ⎩ lim τn (s) = ∞ , lim τn (s) = 0 . s→0+

s→∞

For convenience we set γn (∞) = 0 = τn (∞) and γn (1) = ∞ = τn (0). Lemma 9.16 yields the inequalities ⎧ ⎨ωn−1 (log(λn s))1−n ≤ γn (s) ≤ ωn−1 (log s)1−n , 1−n  ⎩ω 2 ≤ τn (s − 1) ≤ ωn−1 (log s)1−n n−1 log(λn s)

(9.8)

for s > 1. In passing we shall show how this upper bound for γn (s) can be slightly improved. First, fix s > 1 and choose h ∈ GM(Bn ) with h[0, 1s e1 ] = [−ae1, ae1 ], √ a > 0. Then a = s − s 2 − 1 by 4.11 and by conformal invariance 7.10, 7.2, and 7.4     γn (s) = M (S n−1 , [0, 1s e1 ]) = M (S n−1 , [−ae1, ae1 ]) (9.9)   1−n 1−n 2 < ωn−1 (log s) . ≤ ωn−1 log(s + s − 1 ) We note that (9.9) yields a slightly better upper bound for γn (s) than (9.8). Note also that by combining the first inequality in (9.8) with (9.9) and letting s → ∞ yields λn ≥ 2 for all n ≥ 2. An even better upper bound for γn (s) will be given in Lemma 9.17. Each of the bounds for γn (s) in (9.8) is asymptotically sharp as s tends to ∞, but not of the correct order as s tends to 1, as can be seen from 9.17 below. The following

160

9 The Capacity of a Condenser

theorem due to G. D. Anderson [20] yields inequalities that are asymptotically sharp as s tends to 1. Theorem 9.17 For s ∈ (1, ∞) and n ≥ 2  1−n  (1) γn (s) ≤ ωn−1 μ(1/s)1−n < ωn−1 log(s + 3 s 2 − 1 ) , s + 1 s − 1  s + 1 (2) 2n−1 cn log ≤ γn (s) ≤ 2n−1 cn μ < 2n−1 cn log 4 . s−1 s + 1√ s−1 Moreover, if s ∈ (0, ∞) and a = 1 + 2(1 + 1 + s )/s, then (3) cn log a ≤ τn (s) ≤ cn μ(1/a) < cn log(4a) √ √ and (1 + 1/ s )2 ≤ a ≤ (1 + 2/ s )2 hold true. Furthermore, when n = 2, the first inequality in (1), the second inequality in (2), and the second inequality in (3) hold as identities. Proof (1) The proof of the inequality γn (s) ≤ ωn−1 μ(1/s)1−n will be omitted (see [20]). The second inequality follows from (7.21). (2) & (3) It is left as an easy exercise for the reader to verify that (2) and (3) are equivalent, that is, one can be derived from the other in view of 7.31. Hence it suffices to prove (2). Here we shall only prove the lower bound in (2); the proof of the upper √ bound will be omitted (see [21]). Let h be an inversion in S n−1 (−en , 2 ) which maps Bn onto Hn and 0 to en . By (4.20) h preserves hyperbolic distances. Because   γn (s) = M ([0, 1s en ], S n−1 ) 1 and ρBn (0, 1s en ) = log s+1 s−1 we see by (4.6) and (4.15) that h( s en ) = s−1 s+1 en . By conformal invariance, 7.17, and 7.21 we obtain

    s − 1 en , en , ∂Hn γn (s) = M  s+1   s−1     s − 1 en , en , − en , −en = 2n−1 M  s+1 s+1 ≥ 2n−1 cn log

s+1 , s−1

which is the desired lower bound. The proof for the bounds for a is elementary.

9.2 Grötzsch and Teichmüller Condensers

161

The assertions concerning the case n = 2 follow from (7.18), (7.20), (7.22), 7.33(1), and 7.1.3.   It should be observed that 9.17(1) yields a slightly better explicit bound than (9.9). Remarks 9.18 (1) The last inequalities in 9.17(1)–(3) can be improved in view of (7.25). (2) The inequality 9.17(3) can also be written as follows cn log(1 + t) ≤ τ

 4(1 + t)  t2

≤ cn μ

 1  , t >0. 1+t

We shall next summarize the preceding inequalities for γn (s). Let u1 (s) = ωn−1 μ(1/s)1−n ,

u2 (s) = 2n−1 cn μ

s − 1

v1 (s) = ωn−1 (log λn s)1−n , v2 (s) = 2n−1 cn log

s+1

,

s+1 . s−1

By (9.8) and 9.17 (Fig. 9.3) max{ v1 (s), v2 (s) } ≤ γn (s) ≤ min{ u1 (s), u2 (s) } .

(9.10)

8 6 4 2 0

0

0.25

0.5

0.75

1

Fig. 9.3 The graph of γ3 (1/r), 0 < r < 1, lies in the shaded region. Lower bounds are f (r) = 4π 4c3 log 1+r 1−r (dotted line) and g(r) = log2 (9.9002/r) (solid line). Note that λ3 < 9.9002 and c3 =   4b3 ≈ 0.59907, where b3 is as in (7.8). Upper bounds are F (r) = 4c3 μ 1−r 1+r (dotted line) and G(r) =

4π μ(r)2

(solid line)

162

9 The Capacity of a Condenser

For n = 2 the right side of (9.10) is sharp. In fact, it follows from (7.18) (or (7.22)) that for n = 2 the right side holds as equality for all s > 1. One can rewrite (9.10) for τn (s) using the functional identity 7.31. Recall (Exercise 7.33) that for n = 2 τn (t) has an explicit formula in terms of the well-known special function μ(r) . Therefore τ2 (t) satisfies several identities that follow from the identities of μ(r). For the dimensions n ≥ 3 no such identities are known for τn (t) but nevertheless some functional inequalities hold as the next theorem shows. These inequalities refine those in 7.36 and 7.37. Theorem 9.19 For n ≥ 2, x > 0, τn (cx) 1 < √ for 0 < c < 1, τn (x) c τn (cx) 1 √ < < 1 for 1 < c < ∞. τn (x) c c τn (x ) ≤ c1−n for 0 < c < 1. c≤ τn (x) τn (x c ) ≤ c for 1 < c < ∞. c1−n ≤ τn (x) c τn (x ) τn (x c ) = c, lim = c1−n for each c > 0. lim x→∞ τn (x) x→0 τn (x)

(1) 1 < (2) (3) (4) (5)

 

Proof See [29, Theorem 11.25, 11.27].

9.3 Hyperbolic Metric and Capacity As in Chap. 4 we let J [x, y] denote the geodesic segment of the hyperbolic metric joining x to y, x, y ∈ Bn . It is clear by conformal invariance that cap(Bn , J [x, y]) = cap(Bn , Tx J [x, y]) where Tx is as defined in 3.2. We obtain by (4.23) and (9.9) cap(Bn , J [x, y]) = γn



1 th

1 2 ρ(x, y)



 1−n ≤ ωn−1 − log th 14 ρ(x, y) .

(9.11)

Next by (9.11), 9.17(2), and (7.21) we get 2n−1 cn ρ(x, y) ≤ cap(Bn , J [x, y]) ≤ 2n−1 cn μ(e−ρ(x,y)) < 2n−1 cn (ρ(x, y) + log 4) .

(9.12)

For large values of ρ(x, y) (9.12) is quite accurate. For small ρ(x, y) one obtains better inequalities than (9.12) by combining 9.17(1) and (9.11).

9.3 Hyperbolic Metric and Capacity

163

Lemma 9.20 Let x, y ∈ Bn and let E ⊂ Bn be a continuum with x, y ∈ E. Then cap(Bn , E) ≥ cap(Bn , J [x, y]) = γn



1 th 12 ρ(x, y)

 .

Proof Let Tx be as in 3.2 and let ∗ denote spherical symmetrization in the positive x1 -axis. Then the center of symmetrization is the origin and by 9.14 we obtain cap(Bn , E) = cap(Bn , Tx (E)) ≥ cap(Bn , (Tx (E))∗ ) . By (4.23) we see that [0, (th 12 ρ(x, y))e1 ] ⊂ (Tx (E))∗ and the proof follows from   (9.11). Exercise 9.21 Show that 7.23 follows from 9.20. [Hint: We may assume that t = 1 in 7.23. Apply (9.12) and 4.9(1).] The next result gives a very useful lower bound for the capacity of a ring domain. Lemma 9.22 Let R = R(E, F ) be a ring in Rn and let a, b ∈ E, c, ∞ ∈ F be distinct points. Then  |a − c|    . cap R = M (E, F ) ≥ τ |a − b| Here equality holds for E = [−e1 , 0], a = 0, b = −e1 , F = [se1 , ∞), c = se1 , d = ∞. Proof Observe first that the right side remains invariant under the similarity transformation f (x) = (x −a)/|a −b|. Then |f (c)| = |a −c|/|a −b| and f (a) = 0 (Fig. 9.4). The spherical symmetrizations of f E and f F in the negative and positive x1 -axis, |a−c|  . Thus by 9.14 respectively, contain the complementary components of RT ,n |a−b| cap R ≥ τ

Fig. 9.4 Spherical symmetrization in Lemma 9.22

 |a − c|  |a − b|

164

9 The Capacity of a Condenser

 

and the assertion follows. n

Lemma 9.23 Let R = R(E, F ) be a ring in R and let a, b ∈ E, c, d ∈ F be distinct points. Then cap R ≥ τ (| a, b, c, d |) . Here equality holds if b = s1 e1 , a = s2 e1 , c = s3 e1 , d = s4 e1 , and s1 < s2 < s3 < s4 . Proof By (3.12) we may assume that a = 0, b = e1 , d = ∞, and |c| = | a, b, c, d |. The proof follows now from 9.22. The assertion concerning the equality follows   from 7.32(1). Corollary 9.24 Let R = R(E, F ) be a ring and let a, b ∈ E, c, d ∈ F be distinct points in Rn . Then cap R ≥ τ

 t −s  s(1 − t)

where s=

|a − b| , |a − b| + |a − c| + |c − d|

t=

|a − b| + |a − c| . |a − b| + |a − c| + |c − d|

Here equality holds for E = [0, se1 ], a = 0, b = se1 , F = [te1 , e1 ], c = te1 , d = e1 , and 0 < s < t < 1. Proof Because the points are finite, (3.6) and 9.23 yield cap R ≥ τ

 |a − c| |b − d|  |a − b| |c − d|

.

The desired inequality follows from this and the fact that |b − d| ≤ |b − a| + |a − c| + |c − d|. The statement concerning equality follows from 7.32(1).   Corollary 9.25 If R = R(E, F ) is a ring, then   1 (1) cap R ≥ τ , q(E)q(F )   4q(E, F ) . (2) cap R ≥ τ q(E)q(F ) Proof (1) Choose a, b ∈ E and c, d ∈ F so that q(a, b) = q(E) and q(c, d) = q(F ). Then 1 q(a, c) q(b, d) ≤ q(a, b) q(c, d) q(E) q(F )

9.3 Hyperbolic Metric and Capacity

165

and (1) follows from 9.23 because τ is decreasing. (2) Choose a ∈ E, c ∈ F such that q(a, c) = q(E, F ) and choose b ∈ E, d ∈ F such that q(a, b) ≥

1 2

q(c, d) ≥

q(E) ,

1 2

q(F ) .

With this choice of a, b, c, d the proof follows from 9.23. Lemma 9.26 Let E and F be disjoint continua in

Rn

 

with d(E), d(F ) > 0. Then

  M (E, F ) ≥ τ (4m2 + 4m) ≥ cn log(1 + 1/m) where m = d(E, F )/ min{d(E), d(F )} and cn is as in 7.21. Proof Fix a ∈ E, c ∈ F with |a − c| = d(E, F ) and b ∈ E, d ∈ F with |a − b| = 12 d(E) and |c − d| = 12 d(F ), respectively. By 9.23 we obtain  |a−c| |b−d|   |a−c| (|a−b| + |a−c| + |c−d|)    M (E, F ) ≥ τ ≥τ = τ (u) . |a−b| |c−d| |a−b| |c−d| Here   2 d(E, F ) d(E) + 2d(E, F ) + d(F ) ≤ 2m + 4m2 + 2m , u= d(E) d(F ) and the first inequality follows. The second one follows from 9.17(3). Corollary 9.27 Let E and F be disjoint continua in Then

Rn

 

with 0 < d(E) ≤ d(F ).

 d(E, F )    . M (E, F ) ≥ 21−n τ d(E) Proof The proof follows from 9.26 and 7.36.

 

Exercise 9.28 n

(1) Show that if R = R(E, F ) is a ring in R , then  q(F )  . cap R ≥ q(E)cn log 1 + q(E, F ) [Hint: Apply 9.25(2), 9.17(3), and (5.6).] (2) Let R(E, F ) be a ring. Use 9.25(2) to establish a lower bound for cap R(E, F ) similar to 7.27(2). We are going to generalize the formula (9.12), which relates the hyperbolic distance ρ(x, y) and the capacity of the condenser (Bn , J [x, y]) in a simple fashion.

166

9 The Capacity of a Condenser

Now we shall discuss instead of this particular condenser a general ring R(E, F ) and the hyperbolic distance will be replaced by the function  jG (x, y) = log 1 +

 |x − y| min{d(x), d(y)}

which was introduced in (4.27). Lemma 9.29 If R = R(E, F ) is a ring with ∞ ∈ E ∪ F , then   cap R ≥ cn min jRn \E (F ) , jRn \F (E) . If ∞ ∈ F , then cap R ≥ cn jRn \F (E) . Proof The proof follows immediately from 9.26, 9.22, and the definition of jG (A)   (see (4.28) and 4.10). Applying this lemma with E = S n−1 , F = J [x, y], x, y ∈ Bn we obtain in view of 4.9(1) cap(Bn , J [x, y]) ≥ cn jBn (J [x, y]) = cn jBn (x, y) ≥ 12 cn ρ(x, y) . Hence 9.29 implies (9.12) with a slightly different constant. Thus we may regard 9.29 as a generalization of (9.12). Remark 9.30 Lemma 9.29 has a converse which is valid even for disconnected sets E and F . Indeed one can show that for a given integer n ≥ 2 there exists a homeomorphism h1 : [0, ∞) → [0, ∞) with the following properties. If E and F are compact disjoint sets in Rn , then     M (E, F ) ≤ h1 (T ) , T = min jRn \E (F ) , jRn \F (E) . We shall outline a proof for this estimate. Clearly we may assume that 0 < d(E) ≤ d(F ). Set t = d(E, F ). Then 7.4 yields  d(E) + t n     M (E, F ) ≤ M (E, ∂(E + B n (t)) ≤ n , t while for d(E) < t we obtain by (7.4)    M (E, F ) ≤ ωn−1 log

t 1−n . d(E)

These two inequalities together imply the desired bound.

9.4 Dimension Cancellation and Special Functions

167

9.4 Dimension Cancellation and Special Functions The functions γn and τn as well as their inverses and various combinations of these will occur often in our study of distortion theory in later chapters. Of particular importance is the function ϕK : [0, 1] → [0, 1], which will occur in the quasiregular version of the Schwarz lemma as well as in its many applications. This function is defined as follows. For 0 < r < 1 and K > 0 we define a special function ϕK (r) =

1 γn−1 (Kγn (1/r))

= ϕK,n (r)

(9.13)

and set ϕK (0) = 0, ϕK (1) = 1. It is easy to see that ϕK : [0, 1] → [0, 1] is a homeomorphism. Next we shall derive some explicit estimates for ϕK . Recall first that by (9.8) ωn−1 (log λn s)1−n ≤ γn (s) ≤ ωn−1 (log s)1−n

(9.14)

for s > 1 and hence by 7.1.3 γn (s) → 0 when n → ∞ . What is noteworthy in the results below is that we shall prove bounds for ϕK,n (r) that are dimension-free, they do not depend on the dimension n , yet at the same time they are asymptotically sharp when K → 1 . We refer to this phenomenon as the “dimension cancellation” property. It is left as Exercise 9.40 for the reader to derive from (9.14) the following inequality t α /λn ≤ γn−1 (Kγn (t)) ≤ λαn t α

(9.15)

for all t > 1 and K > 0, where α = K 1/(1−n) . From (9.15) it follows that α r α λ−α n ≤ ϕK (r) ≤ λn r

(9.16)

holds for all K > 0 and r ∈ (0, 1). Note that the upper bound in (9.16) is not dimension-free, it is unbounded when n → ∞ . It is easy to see that 0 < A ≤ B < ∞ implies ϕA (r) ≤ ϕB (r). In particular, ϕ1/K (r) ≤ r = ϕ1 (r) ≤ ϕK (r) for K ≥ 1. We next improve the upper bound in (9.16) for K ≥ 1. The resulting explicit bound is sharp for K = 1. Exercise 9.31 Show that ϕK,n (r) mod RG,n (1/r) and α = K 1/(1−n) .

=

Mn−1 (α Mn (r)) where Mn (r)

Theorem 9.32 For n ≥ 2, K ≥ 1, and 0 ≤ r ≤ 1 (1) ϕK (r) ≤ λ1−α r α , α = K 1/(1−n) , n 1−β (2) ϕ1/K (r) ≥ λn r β , β = K 1/(n−1) .

=

168

9 The Capacity of a Condenser

Proof (1) Let K ≥ 1, r ∈ (0, 1), and M(r) = mod RG,n (1/r). Setting r  = ϕK (r) we have M(r  ) = α M(r) and r  ≥ r. These facts follow from the definitions (9.5) and (9.13). From the proof of 9.15 it follows that M(r) + log r is a decreasing function on (0, 1), so that M(r) + log r ≥ M(r  ) + log r  . Let λn ≥ 4 be as in (9.6). Since log λn ≥ M(r) + log r by (9.6) we obtain 0 ≤ log

λn λn − M(r) ≤ log  − M(r  ) r r

and, further, because 0 < α ≤ 1 α log

λn λn − α M(r) ≤ log  − M(r  ) . r r

This inequality yields r  ≤ λ1−α rα n for r ∈ (0, 1). Because this holds for r = 0 and r = 1, too, we have completed the proof of (1). (2) The proof of (2) is similar.   It is clear that r α ≤ r 1/K , α = K 1/(1−n), for K ≥ 1 and 0 ≤ r ≤ 1. This fact together with 9.32 and the next lemma, shows that ϕK (r) ≤ c(K)r 1/K for K ≥ 1, where c(K) depends only on K and where c(K) → 1 as K → 1. Lemma 9.33 For n ≥ 2, K ≥ 1, and α = K 1/(1−n) = 1/β the following three inequalities hold: (1) λ1−α ≤ 21−α K ≤ 21−1/K K , n 1−β (2) λn ≥ 21−β K −β ≥ 21−K K −K , d(β−1)

(3) 1/ϕ1/K,n ( 12 ) ≤ 1 + λn

, d ≥ d0 = 2 +

log 4 . log λn

Proof (1) It follows from 9.16 that (1 − α) log λn ≤ (1 − α)(n − 1) + (1 − α) log 2 . From 1 − e−x ≤ x, x ≥ 0, one can deduce that (1 − α)(n − 1) = (1 − K 1/(1−n))(n − 1) ≤ log K .

9.4 Dimension Cancellation and Special Functions

169

Because 1 − α ≤ 1 − 1/K we conclude that (1 − α) log λn ≤ log K + (1 − α) log 2 ≤ log K + (1 − 1/K) log 2 and the desired upper bound follows. (2) By the proof of (1) we obtain  −β λ1−β = λ(α−1)β ≥ 21−α K = 21−β K −β ≥ 21−K K −K n n as desired. (3) By Theorem 9.32 (2) and the arithmetic-geometric inequality β−1

1/ϕ1/K,n ( 12 ) ≤ 2β λn

= 2 · 1 · (2λn )β−1 ≤ 12 + (2λn )2(β−1) .

Elementary calculation shows that for d ≥ d0 (2λn )2(β−1) ≤ λd(β−1) . n By Lemma 9.16 we see that d0 ≤ 3 .

 

Next we shall prove a “dimension cancellation” property of the function ϕK,n , K > 0, by finding dimension-free minorant and majorant functions. Lemma 9.34 For K > 0 and 0 < r < 1 there exist positive numbers a1 and a2 in (0, 1) such that a1 ≤ ϕK,n (r) ≤ a2 for all n ≥ 2. In particular, a1 and a2 are independent of n. Proof By 9.17(2) we have A log

s − 1 s+1 ≤ γ (s) ≤ A μ , A = 2n−1 cn , s−1 s+1

(9.17)

for s > 1. Because γ is strictly decreasing we obtain from this γ −1 (t) ≤

1 + μ−1 (t/A) 1 − μ−1 (t/A)

and γ −1 (t) ≥ cth

t . 2A

These two inequalities together with (9.17) yield for r ∈ (0, 1)   1 − r    1  1 + μ−1 (K log T ) ≤ γ −1 Kγ ≤ b1 = cth 12 Kμ = b2 1+r r 1 − μ−1 (K log T )

170

9 The Capacity of a Condenser

where T = (1 + r)/(1 − r). Both bounds are independent of n. In view of (9.13) we may choose a1 = 1/b2 and a2 = 1/b1.    −1  Exercise 9.35 For n = 2, K > 0, t > 0, let αK (t) = τ2 τ2 (t)/K . Show that   A2 t , A = ϕK,2 αK (t) = . (9.18) 1 − A2 1+t Let tK = 2(4K)−K . Then for K ≥ 1 and t ∈ (0, tK ] ϕK,2 (t) ≤ 41−1/K t 1/K ≤ (see 9.16, 9.32). Conclude that for K ≥ 1 and αK (t) ≤

1 2

√ t/(1 + t) ∈ (0, tK ]

 t 1/K 4 · 161−1/K . 3 1+t

Next, applying 7.34(2) and 9.32 show that for K ≥ 1 and t > 0 αK (t) ≤ 16K−1/K (1 + t)K−1/K t 1/K . boundary Exercise 9.36 Let G be a uniform domain in Rn with connected   (recall 6.1). Show that if E is a connected subset of G, then M (E, ∂G) ≥ c kG (E) where c is a constant. Exercise 9.37 (1) Applying the functional identities of μ, one can write the constant b2 in the proof of 9.34 also in other ways. Show that a1 =

 π2  1 . = μ−1 b2 2K log T

(2) Find an improved form of 9.33 (1), (2) by replacing the inequality 1 − e−x ≤ x in the proof of 9.33 by the better inequality 1 − e−x ≤ th x (cf. 4.2(1)). Remarks 9.38 (1) It is proven in [29, Thm 10.24, Cor 10.33] that the function αK (t) in (9.18) satisfies for t > 0, K ≥ 1 λ(K) min{t K , t 1/K } ≤ αK (t) ≤ λ(K) max{t K , t 1/K } , where eπ(K−1) ≤ λ(K) = αK (1) ≤ ea(K−1) ,

a=

√ 4 K(1/ 2) = 4.376879... . π

9.4 Dimension Cancellation and Special Functions

171

Moreover, αK (s) is a quasiadditive function, that is, for all s, t > 0, K ≥ 1 , 21−K ≤

αK (s) + αK (t) ≤ 21−(1/K) . αK (s + t)

(2) The Grötzsch capacity γn (s) has several interesting properties which are studied in [27] and [29]. It is shown there that γn (1/ th(a + b)) ≤ γn (1/ th a) + γn (1/ th b) holds for all a, b > 0. Several inequalities involving the function ϕK,n (r) can also be found in [24, 26, 27] and [29]. One of these is rα ≤

(1 +

 2r α ≤ ϕK,n (r) ; r  = 1 − r 2 , α = K 1/(1−n) ,  α + (1 − r )

r  )α

which holds for all K ≥ 1, r ∈ [0, 1], and n ≥ 2. Exercise 9.39 Let Mn (r), r ∈ (0, 1), n ≥ 2, be as in 9.31. Then M2 (r) = μ(r). Show that the relationship √  (1) Mn (r)Mn 1 − r 2 = c for all r ∈ (0, 1) is equivalent to √ #2 #2 " (2) ϕK,n (r) + ϕ1/K,n 1 − r 2 =1 "

for all r ∈ (0, 1) and all K > 0. Recall that both (1) and (2) hold for n = 2 by (7.20) and 7.34. Next, applying 9.17 show that (1) is false for n ≥ 3 and, therefore, also (2) is false for n ≥ 3. Exercise 9.40 Recall first that by (9.8) ωn−1 (log(λn s))1−n ≤ γn (s) ≤ ωn−1 (log s)1−n for s > 1. Derive from this the following inequality t α /λn ≤ γn−1 (Kγ n (t)) ≤ λαn t α for all t > 1 and K > 0, where α = K 1/(1−n) . Exercise 9.41 Let E ⊂ Rn be compact, cap E > 0 and E(t) = ∪x∈E B n (x, t). Show that cap (E(t), E) → ∞, when t → 0. [Hint: Use Lemma 9.3.] Notes 9.42 The method of symmetrization has found many applications in geometry (see [60, 9.13]) and in various branches of analysis, e.g. in the study of isoperimetric inequalities (see [71, 231, 426]), and in real analysis. O. Teichmüller [510] applied these ideas to geometric function theory and proved a special case of Lemma 9.14 above. Other function-theoretic applications are given in [121, 211].

172

9 The Capacity of a Condenser

In R3 the conformal capacity was studied by C. Loewner [339], who applied his result to quasiconformal mappings. Many results of this chapter are connected with the fundamental results of F. W. Gehring [150, 151]. A multidimensional version of Teichmüller’s work on symmetrization is contained in [150] and [479]. See also [426]. The literature dealing with p-capacity is vast: the reader is referred to [144, 171, 219, 345, 355, 380–382, 459] and [161, 366, 500, 564], as well as to the bibliographies of these works. For the twodimensional case, see [284].  One of the main goals of this chapter is to find estimates for M (E, F ) in terms of geometric quantities such as min{ d(E), d(F ) } . d(E, F ) For 9.22–9.25 see [150] and [153]. For 9.29 and 9.30 see [552] and [556]. The natural setup and motivation for 9.32 is the Schwarz lemma [232, 364, 488, 565], which we shall study in Chap. 16. For n = 2 Theorem 9.32 is due to O. Hübner [246] and the same method appears also in [326, p. 64] and, in the n-dimensional context, in [24]. For a different proof (n = 2) see P. P. Belinski˘ı [59, p. 15]. In [80, pp.55 Rmk 4.15] a correction is pointed out for the inequality [59, (19) p. 16]. Also 9.33 and 9.34 were proved in [24]. For 9.26 see [552, 556], and [163]. From the vast literature dealing with condensers in the plane we mention [45, 121, 284, 317], and [516, Ch. III].

Chapter 10

Conformal Invariants

Inthe precedingchapters we have studied some properties of the conformal invariant M (E, F ; G) . In this chapter we shall introduce two other conformal invariants, the modulus metric μG (x, y) and its “dual” quantity λG (x, y), where G is a domain n in R and x, y ∈ G. The modulus metric μG is functionally related to the hyperbolic metric ρG if G = Bn , while in the general case μG reflects the “capacitary geometry” of G in a delicate fashion. Unlike the quasihyperbolic metric kG or the distance ratio metric jG , the metric μG ignores isolated boundary points of the domain. Indeed, if G is a domain and E ⊂ G is a compact set of capacity zero, then G \ E is a domain and μG (x, y) = μG\E (x, y) for x, y ∈ G \ E . This equality is no longer true if capE > 0 . The dual quantity λG (x, y) is also functionally related to ρG if G = Bn . For a wide class of domains in Rn , the so-called QED-domains, we shall find two-sided estimates for λG (x, y) in terms of rG (x, y) =

|x − y| . min{ d(x, ∂G), d(y, ∂G) }

10.1 Two Conformal Metrics n

If G is a proper subdomain of R , then for x, y ∈ G with x = y we define   λG (x, y) = inf M (Cx , Cy ; G) Cx ,Cy

(10.1)

where Cz = γz [0, 1) and γz : [0, 1) → G is a curve such that z ∈ |γz | and γz (t) → ∂G when t → 1, z = x, y (Fig. 10.1). It follows from 7.10 that λG is invariant under conformal mappings of G. That is, λf G (f (x), f (y)) = λG (x, y), if f : G → f G is conformal and x, y ∈ G are distinct. © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_10

173

174

10 Conformal Invariants

Fig. 10.1 Conformal invariants λG (left) and μG (right)

n

Remark 10.1 If card(R \ G) = 1, then λG (x, y) ≡ ∞ by 7.22. Therefore λG is n n of interest only in case card(R \ G) ≥ 2. For card(R \ G) ≥ 2 and x, y ∈ G, x = y, there are continua Cx and Cy as in (10.1) with C x ∩ C y = ∅ and thus   n M (Cx , Cy ; G) < ∞ by 7.15. Thus, if card(R \ G) ≥ 2, we may assume that the infimum in (10.1) is taken over continua Cx and Cy with C x ∩ C y = ∅. n

For a proper subdomain G of R and for all x, y ∈ G define   μG (x, y) = inf M (Cxy , ∂G; G) Cxy

(10.2)

where the infimum is taken over all continua Cxy such that Cxy = γ [0, 1] and γ : [0, 1] → G is a curve with γ (0) = x and γ (1) = y (Fig. 10.1). It is clear that μG is also a conformal invariant in the same sense as λG . It is left as an easy exercise for the reader to verify that μG is a metric if cap ∂G > 0. [Hint: Apply 7.8 and 8.1.] If cap ∂G > 0, we call μG the modulus metric or conformal metric of G. As pointed out in Chap. 9, the Grötzsch and Teichmüller condensers have important properties connected with the spherical symmetrization. Because μBn and λG , G = Rn \{0}, can be related to these condensers as we shall see below, we may think of μG and λG as generalized Grötzsch and Teichmüller extremal problems, respectively. Remark 10.2 Let D be a subdomain of G. It follows from 7.8 and (7.2) that μG (a, b) ≤ μD (a, b) for all a, b ∈ D and λG (a, b) ≥ λD (a, b) for all distinct a, b ∈ D. In what follows we are interested only in the non-trivial case n card(R \ G) ≥ 2. Moreover, by performing an auxiliary Möbius transformation, n we may and shall assume that ∞ ∈ R \ G throughout this chapter. Hence G will have at least one finite boundary point.

10.2 Ferrand’s Modulus Metric J. Ferrand proved in [330] that λG (x, y)−1/n is a metric. We shall call it Ferrand’s metric or Ferrand’s modulus metric to avoid confusion with the other metric named after her (cf. Chap. 5). It was proved in [27] that for the case G = Bn the expression

10.2 Ferrand’s Modulus Metric

175

λBn (x, y)1/(1−n) is a metric and in [555, page 193] the question was raised whether this holds for all domains. This question was solved independently by A. Yu. Solynin [492] and by J. Jenkins [267] for n = 2. The general case n ≥ 3 was solved by J. Ferrand [136] who proved the following theorem, given here without proof. Recall that a metric raised to power α ∈ (0, 1) is also a metric by Exercise 5.24. Theorem 10.3 Let G  Rn be a domain. Then λG (x, y)1/(1−n) defines a metric on G. In a general domain G, the values of λG (x, y) and μG (x, y) cannot be expressed in terms of well-known simple functions. For G = Bn they can be given in terms of ρ(x, y) and the capacity of the Teichmüller condenser. Theorem 10.4 The following identities hold for all distinct x, y {Bn , Hn }:     1 1 (1) μD (x, y) = 2n−1 τ = γ , sh2 12 ρD (x, y) th 12 ρD (x, y)   (2) λD (x, y) = 12 τ sh2 21 ρD (x, y) .

∈ D ∈

Proof Because of the Möbius invariance it is enough to consider the case D = Bn . (1) The proof of part (1) follows directly from 9.10, (9.20), and 7.31. n (2) Because the  1assertion is GM(B )-invariant, we may assume that x = re1 = −y and r = th 4 ρ(x, y) (see (4.23)). By a symmetry property 7.12 of the modulus and by 7.32(1) we obtain λBn (x, y)

  ≤ M ( [−e1 , −re1 ] , [re1 , e1 ] ; Bn )   = 12 M ( [− 1r e1 , −re1 ] , [re1 , 1r e1 ] ; Rn ) =

1 2

 τ

4r 2 (1−r 2 )2



=

1 2

  τ sh2 21 ρ(x, y) .

Hence it will suffice to prove the inequality “≥”. Let Cx , Cy be as in (10.1) and 0 <  < 12 (1 − |x|). Choose compact connected subsets E, F of Cx , Cy with x ∈ E, y ∈ F and d(E, S n−1 ) = d(F, S n−1 ) = . Let E s = E ∪ hE, F s = F ∪ hF where h(x) = x/|x|2 . By 10.1 we may assume that C x ∩ C y = ∅ and hence at most one of the sets E and F can contain 0. We may assume 0 ∈ F , hence F s is compact. Let Sym(F s ) denote the set obtained from F s by spherical symmetrization in the positive x1 -axis and let Sym(E s ) be the set obtained from E s by spherical symmetrization in the negative x1 -axis. By 9.14, 9.6, and 7.8 cap( Rn \ E s , F s ) ≥ cap( Rn \ Sym(E s ) , Sym(F s ) )     ≥ M ( [− 1r e1 , −re1 ] , [re1 , 1r e1 ]) − 2 M (Y1 , Y2 )

176

10 Conformal Invariants

where Y1 = [− 1r e1 , −re1 ] and Y2 = [(1 − )e1 , (1 − )−1 e1 ]. This inequality together with 7.32(1) yields   cap( Rn \ E s , F s ) ≥ τ sh2 12 ρ(x, y) − δ() where δ() → 0 as  → 0. Letting  → 0 and applying 7.12 yields   M (Cx , Cy ; Bn ) ≥ 12 τ sh2 12 ρ(x, y) . Since Cx and Cy were arbitrary sets with   the stated properties, the desired inequality λBn (x, y) ≥ 12 τ sh2 12 ρ(x, y) follows.   Remark 10.5 From 9.17(3) we obtain the following inequality for x, y ∈ Bn (exercise)  21  1 1 2 τ sh 2 ρ(x, y) ≥ −cn log th 4 ρ(x, y)   1 1 = 2cn arth e− 2 ρ(x,y) ≥ 2cn e− 2 ρ(x,y) . Here the identities 2 ch2 A = 1 + ch 2A and sh 2A = 2 ch A sh A were applied (see also 4.2(3)). Recall that sh2 21 ρ(x, y) =

|x − y|2 (1 − |x|2 )(1 − |y|2)

by (4.17). Similarly, by 9.17(3) we obtain also 1 2τ



   sh2 21 ρ(x, y) ≤ 12 cn μ th2 ( 14 ρ(x, y)) < 12 cn log = cn log

2 th

1 4 ρ(x, y)

4 th2 14 ρ(x, y)

.

Lemma 10.6 Let G be a proper subdomain of Rn , x ∈ G, d(x) = d(x, ∂G), Bx = B n (x, d(x)), let y ∈ Bx with y = x, and let r = |x − y|/d(x). Then the following two inequalities hold: 1  r2  1 > cn log , (1) λG (x, y) ≥ λBx (x, y) = τ 2  1− r 2  r 1 1 1−n (2) μG (x, y) ≤ μBx (x, y) = γ r ≤ ωn−1 log r . Proof (1) By 10.2, 10.4(2), and 10.5 we obtain λG (x, y) ≥ λBx (x, y) = = cn log

1+

1  r2  τ ≥ −cn log th( 14 (2 arth r)) 2 1 − r2

√ 1 − r2 1 > cn log . r r

10.2 Ferrand’s Modulus Metric

177

(2) The desired inequalities follow from 10.2 and (9.8).

 

We now prove some estimates for the conformal metric μG . Lemma 10.7 Let G be a proper subdomain of Rn , s ∈ (0, 1), x, y ∈ G. If kG (x, y) ≤ 2 log(1 + s), then   1 . (1) μG (x, y) ≤ γ th(kG (x, y)/(1 − s)) Moreover, there exist positive numbers b1 and b2 depending only on n such that (2) μG (x, y) ≤ b1 kG (x, y) + b2 for all x, y ∈ G. Proof (1) Choose a quasihyperbolic geodesic segment JG [x, y] connecting x to y and let z ∈ JG [x, y] with kG (x, y) = 2kG (x, z) = 2kG (y, z). Then by (5.4)  jG (x, z) = log 1 +

 |x − z| ≤ kG (x, z) ≤ log(1 + s) min{d(x), d(z)}

and hence x ∈ B n (z, s d(z)). Let Bz = B n (z, d(z)). By 5.7(1), (5.6), and (5.4) we obtain   |x − z|  |x − z| ≤ log 1 + d(z) − |x − z| (1 − s)d(z)  |x − z|  1 1 1 log 1 + ≤ j (x, z) ≤ k (x, z) . ≤ 1−s d(z) 1−s G 1−s G

 kBz (x, z) ≤ log 1 +

Because of the symmetric choice of the point z, we get a similar upper bound also for kBz (z, y). Hence kBz (x, y) ≤ kBz (x, z) + kBz (z, y) ≤

 1  1 kG (x, z) + kG (z, y) = k (x, y) . 1−s 1−s G

Denote by ρBz the hyperbolic metric of Bz (see 6.31). Now by 5.4 and the above results we get ρBz (x, y) ≤ 2 kBz (x, y) ≤

2 k (x, y) 1−s G

178

10 Conformal Invariants

and hence by 10.2, 10.4(1), and 7.31 μG (x, y) ≤ μBz (x, y) = 2n−1 τ



1



sh2 12 ρ(x, y)     1 1 ≤ γ =γ th(kG (x, y)/(1 − s)) th 12 ρ(x, y)

where ρ = ρBz . (2) Choose points x1 , . . . , xp+1 ∈ JG [x, y] with x1 = x, xp+1 = y and kG (xj , xj +1 ) = 2 log(1+s) for j = 1, . . . , p−1 and kG (xp , xp+1 ) < 2 log(1+ s) and p ≤ 1 + kG(x, y)/(2 log(1 + s)) (cf. the proof of Lemma 6.21(1)). Then by part (1) μG (x, y) ≤

p !

μG (xj , xj +1 ) ≤ p b2

j =1

where b2 = γ (1/ th[(2 log(1 + s))/(1 − s)]) . The desired result with b1 = b2 /(2 log(1 + s)) follows.   It should be observed that Lemma 10.7(2) is a generalization of the upper bound in (9.12) to the case of an arbitrary domain. The lower bound in (9.12) will next be generalized to the case of domains with connected boundary. Lemma 10.8 Let G be a domain in Rn such that ∂G is connected. Then for all a, b ∈ G, a = b, (1) μG (a, b) ≥ τ (4m2 + 4m) ≥ cn jG (a, b) where cn is the constant in 7.19 and m = min{d(a), d(b)}/|a − b|. If, in addition, G is uniform, then (2) μG (a, b) ≥ B kG (a, b) for all a, b ∈ G. Proof Statement (1) follows from 9.26 and (10.2), while (2) follows from (1)   and 3.8. We shall next study Teichmüller’s function and apply it to produce upper bounds for the Ferrand invariant.

10.3 Teichmüller’s Function For x ∈ Rn \ {0, e1 }, n ≥ 2, define   p(x) = inf M (E, F ) E,F

(10.3)

10.3 Teichmüller’s Function

179 n

where the infimum is taken over all pairs of continua E and F in R with the properties 0, e1 ∈ E and x, ∞ ∈ F . We call this Teichmüller’s function, because he had posed in 1938 [510] the problem of studying this function for the twodimensional case. In this case, the function p(z) has been carefully analyzed by G.V. Kuz mina [317, Ch. 5]. Its values can be expressed in terms of elliptic integrals of complex argument [317, Ch. 5], [29, Thm 15.36, p. 313]. In [29, p. 463], [216] numerical methods were used to study this function. We now proceed to study the general case n ≥ 2 . Lemma 10.9 The inequality   p(x) ≥ max τ (|x|) , τ (|x − e1 |) holds for all x ∈ Rn \ {0, e1 }. Equality holds if x = se1 and s < 0 or s > 1.  

Proof The proof follows directly from 9.14. The main result of this chapter is the following theorem. Theorem 10.10 For x ∈ Rn \ [0, e1 ], p(x) ≤ M((E, F ; Rn )) ≤ τn

 |x| + |x − e | − 1  1 , 2

where E is a circular arc with 0, e1 ∈ E, and F is a ray with x ∈ F. Both inequalities reduce to equality if x = se1 and s ∈ (−∞, 0) ∪ (1, ∞). n

n

Proof Let h : R → R be the Möbius transformation taking x, 0, e1 , ∞ onto −e1 , −y, y, e1 , respectively, where |y| < 1. With E1 = [−|y|e1, |y|e1 ], E  = [−y, y], and F  = [−e1 , ∞] ∪ [e1 , ∞], by Theorem 9.23 we have M((E1 , F  )) = τn



 (1 − |y|)2 . 4|y|

Next, by invariance of cross ratios, we get |x, ∞, 0, e1 | = | − e1 , e1 , −y, y| and |x, ∞, e1 , 0| = | − e1 , e1 , y, −y|, which give |x| =

|y − e1 |2 4|y|

and |x − e1 | =

|y + e1 |2 . 4|y|

Hence |x| + |x − e1 | − 1 =

(1 − |y|)2 . 2|y|

180

10 Conformal Invariants

Now, with E = h−1 (E  ), F = h−1 (F  ), we have M((E, F )) = M((E  , F  )). From (10.3) and Exercise 7.42 we get 



p(x) ≤ M((E, F )) ≤ M((E1 , F )) = τn

1 (|x| + |x − e1 | − 1) 2



 

implying the assertion. Corollary 10.11 For x ∈

Rn

n

\ B the following inequalities hold:

τn (|x − e1 |) ≤ p(x) ≤ τn (|x − e1 |/2) ≤

√ 2τn (|x − e1 |).

Proof The first inequality follows from 10.9, the second one from Theorem 10.10,   and the last one from 9.19. Corollary 10.12 Let G = Rn \ {0} and x, y ∈ G with x = y. Then  λG (x, y) ≤ τn

|x − y| + | |x| − |y| | 2 min{|x|, |y|}



 ≤ τn

 |x − y| . 2 min{|x|, |y|}

Proof Since all three members are invariant under similarity transformations, we may assume that y = e1 and |x| ≥ 1. Since min{|x|, |y|} = 1, the result follows from Theorem 10.10 and (10.1) with Cy = E and Cx = F .   Corollary 10.13 For x, y ∈ G = Rn \ {0}, x = y,  τn

|x − y| m



 ≤ λG (x, y) ≤ τn

|x − y| 2m



√ ≤ 2τn



 |x − y| , m

where m = min{|x|, |y|}. Proof The first inequality follows from Theorem 9.22, the second one from Corollary 10.12, and the third one from Theorem 9.19 and [29, Theorem 11.25 (1)].   Exercise 10.14 (1) Show that (|x − y| + |x| − |y|)/(2|y|) is invariant under any Möbius transformation fixing 0 and ∞. (2) Given s > 1, show that there exists r(s) > 2 such that, in the notation of Lemma 10.9 and Theorem 10.10, τn ((|x| + |x − e1 | − 1)/2) 1/2, y > 0} . Then for z ∈ I1/2 ∩ (C \ {0, 1}) the following functional identity holds: p(z) = 2p(w4 )

w=

√ √ z + z − 1,

√ where the√branches of the square roots are chosen so that 0 < arg( z) < π/2 and 0 < arg( z − 1) < π for z ∈ I1/2 . This theorem can be used to give an algorithm for numerical computation of p(z) as shown by V. Heikkala and M. Vuorinen [216]. Remarks 10.16 (1) For n = 2 the upper bound in Theorem 10.10 can be refined. It was proved in [494] that for n = 2 we have for z ∈ C \ {0, 1} p(z) ≤ τ a(z) = (s +



 (a(z)b(z) − 1)2  4a(z)b(z)

s 2 − 1)1/2 , s = |z| + |z − 1|, b(z) =



|z| +



|z − 1| .

(2) By Theorem 10.10 p(1 + t) = τ2 (t) for t > 0 . Note that in the limiting case z = 1 + t, t > 0 Theorem 10.15 reduces to the identity 7.33(2). Remark 10.17 In answer to a question posed by M. Vuorinen in Leningrad in 1985, G. V. Kuz mina studied [318, pp. 216–218] the counterpart of the Teichmüller function in G = B2 \ {0} . This question was motivated by the work on the paper of M. Lehtinen and M. Vuorinen [323], where [323, Lemma 2.8] gives a formula for λG (r, s) , 0 < r < s < 1 . The paper [128] gives a formula for the Teichmüller function of G and therefore also for the Ferrand modulus metric λ−1 G . Note that there are a few unfortunate typos in [128], reported to us by G. V. Kuz mina. It should be carefully checked whether the general formula [128, Cor. 2.1] and the particular case [323, Lemma 2.8] are compatible. For x ∈ Rn \ {0} we denote by rx a similarity transformation with rx (0) = 0 and rx (x) = e1 . Then it is easy to see that |rx (y) − e1 | = |x − y|/|x|. It follows immediately from the definitions (10.1) and (10.3) that λRn \{0} (x, y) = min{ p(rx (y)) , p(ry (x)) } . Next we deduce the following two-sided inequality for λRn \{0} (x, y).

(10.4)

182

10 Conformal Invariants

Corollary 10.18 Let G be a proper subdomain of Rn , x and y distinct points in G and m(x, y) = min{d(x), d(y)}. Then λG (x, y) ≤ inf λRn \{z} (x, y) ≤ z∈∂G

√   2 τ |x − y|/m(x, y) .

Proof The first inequality follows from 10.2. For the second one fix z0 ∈ ∂G with m(x, y) = d({x, y}, {z0}). Applying 10.13 to Rn \ {z0 } yields the desired result.   We next show that 10.18 fails to be sharp for a Jordan domain G in Rn . For t ∈ (0, 15 ) consider the family Gt = B n (−e1 , 1) ∪ B n (e1 , 1) ∪ B n (t) of Jordan domains. Then by 10.18 λGt (−e1 , e1 ) ≤

√ 2 τ (2)

for all t ∈ (0, 15 ). But this is far from sharp because in fact   λGt (−e1 , e1 ) ≤ M ( [−2e1, −e1 ] , [e1 , 2e1 ] ; Gt )  1 1−n −→ 0 ≤ ωn−1 log t as t → 0. However, for a wide class of domains, which we shall now consider, we obtain a lower bound which is a constant multiple of the upper bound 10.18.

10.4 QED Domains n

A closed set E in R is called a c-quasiextremal distance set or c-QED exceptional n set or c-QED set, c ∈ (0, 1], if for each pair of disjoint continua F1 , F2 ⊂ R \ E     n M (F1 , F2 ; R \ E) ≥ c M (F1 , F2 ) . n

(10.5)

n

If G is a domain in R such that R \ G is a c-QED set, then we call G a c-QED domain. If c = 1 then the set E is called a null set for extremal distances or a NED-set. Examples 10.19 (1) The unit ball Bn is a 12 -QED set by [163] or by the above Lemma 7.14. (2) If E is a compact set of capacity zero, then E is a 1-QED set. For instance all discrete sets are 1-QED sets. The class of all 1-QED sets contains all closed sets in Rn of vanishing (n − 1)-dimensional Hausdorff measure (see [163, 521]).

10.4 QED Domains

183

(3) B2 \ [0, e1 ) is not a c-QED set for any c > 0. (4) For more information about QED sets, see [102], [159, pp.158–161], [366, Ch 3.3, 13.10]. Theorem 10.20 Let G be a c-QED domain in Rn . Then λG (x, y) ≥ c τ (s 2 + 2s) ≥ 21−n c τ (s) where s = |x − y|/ min{d(x), d(y)}. Proof Let Cx and Cy be connected sets as in (10.1) with x ∈ Cx and y ∈ Cy . Let 1 = (Cx , Cy ; G) and 2 = (Cx , Cy ). We may assume that d(x, ∂G) ≤ d(y, ∂G). Fix u ∈ C x and v ∈ C y with |x − u| = d(x, ∂G) and |y − v| = d(y, ∂G) ≥ d(x, ∂G). Because |u − v| ≤ |u − x| + |x − y| + |y − v| we obtain by 9.17 and 7.36(1) M( 1 ) ≥ c M( 2 ) ≥ c τ   ≥ c τ |x − y|

 |x − y| |u − v|  |x − u| |y − v|

 |x − y| 1  1 + + |y − v| |x − u| |y − v| |x − u|

≥ c τ (s 2 + 2s) ≥ c τ (4s 2 + 4s) ≥ 21−n c τ (s)  

as desired. n

n

Let a and d be distinct points in R and D = R \ {a, d} and let mD (x, y) = exp δD (x, y) − 1 where δD (x, y) is the Seittenranta metric (5.22). Recall that by Theorem 5.16 δD (x, y) has both upper and lower bounds in terms of the jD metric, if D ⊂ Rn . √ n Theorem 10.21 1 ≤ λD (b, c)/τ (mD (b, c)) ≤ 2 for distinct b, c ∈ D = R \ {a, d}.  

Proof The proof follows readily from (5.19) and 10.18. n

n

Corollary 10.22 Let D ⊂ R be a c-QED domain with card(R \ D) ≥ 2. Then for distinct x, y ∈ D 21−n c τ (m) ≤ c τ (m2 + 2m) ≤ λD (x, y) ≤

√

2 τ (m)

where m = mD (x, y). Proof The proof follows from 10.20, 10.18, and 10.21.

 

184

10 Conformal Invariants

10.5 Capacitary Geometry We now discuss the geometry generated by the balls of the metrics μG and 1/(1−n) . We refer to these geometries by the common name capacitary geometry. λG 1/(1−n) . By Corollary 10.22 Consider first the case of the balls of the metric λG and by the comparison properties of the jG and δG metrics we see that the explicit ball inclusion results (5.15), Exercises 5.10 and 5.11, which compare jG balls with euclidean balls, give also information about the topology of λG balls. We have already seen that for μG we always have an upper bound in terms of the quasihyperbolic metric and for domains G with connected boundary we also have a lower bound in terms of jG . Now, in the same way as above, the explicit ball inclusion results (5.15), Exercises 5.10 and 5.11, give also information about the topology of μG balls. The case of a general domain is more subtle. First of all we exclude the case when capE = 0 , because then μG is degenerate, μG ≡ 0 . Let E ⊂ {0} ∪ Hn be as in Example 8.17(2) with  = 1 . Then G = Rn \ E is a domain containing the lower half space L = Rn \ Hn . For given points x, y ∈ L we see that M(([x, y], ∂G)) ≤ M((∂Hn , E)) < 1 . This means that for every x ∈ L L ⊂ BμG (x, 1) . In particular, the closure of the ball BμG (x, 1) is not a compact subset of G . A similar behavior can happen if the domain G has boundary points z ∈ ∂G such that ∂G is too thin close to z . The balls BμG (x, t) have compact closures in G if ∂G is thick enough in the sense of capacity. Indeed, the following theorem holds. Theorem 10.23 Let G ⊂ Bn be a domain and let ∂G satisfy the metric thickness condition 8.13 with parameters r0 ∈ (0, 1/4] and δ > 0 . Then for all x, y ∈ G μG (x, y) ≥ c jG (x, y) where c > 0 is a constant depending only on n, r0 , δ . Proof The proof follows from [554, Thm 2.27, (2.26))].

 

Recall that uniform perfectness 8.14 is equivalent to the metric thickness condition 8.13 by Theorem 8.15. From the basic properties of the j -metric it easily follows that under the assumptions of Theorem 10.23, for each x ∈ G and M > 0 the closures of the balls BμG (x, M) are compact.

10.5 Capacitary Geometry

185

Exercise 10.24 Let b, c ∈ Bn . Show that  ρ(b, c) ≤ log 1 +

 2 |b − c| , (min{1 − |b|, 1 − |c|})2

 ρ(b, c) ≥ log 1 +

 2 |b − c| . min{1 − |b|, 1 − |c|} (1 + min{1 − |b|, 1 − |c|})

Let d ∈ S n−1 . Show that  ρ(b, c) ≥ log 1 +

2 |b − c|  . |b − d| |c + d|

Remark 10.25 For n = 2 an explicit expression for the function p(x) can be deduced from [317, Theorem 5.2., p. 192]. This explicit expression is a real number determined by certain elliptic integrals with a complex argument. Because of this fairly complicated definition it is difficult to see how the exact value of p(x) changes with x or, say, with the angles α between [0, x] and [0, e1 ] and β between [0, e1 ] and √ [e1 , x], respectively. In [556, 4.3] it was conjectured that for n = 2 √ the constant √ 2 in 10.13 can be replaced by a smaller one, c = 1.1712 · · · = μ(1/ 3 )/μ(1/ 2 ), which would be sharp as shown in [556, 4.3]. Some bounds for λB2 \{0} (x, y) were found in [323] and [215]. See also [216] and [128]. Exercise 10.26 Mori’s ring RM,2 (α, β) in R2 has two complementary components C1 = { te1 : t ≥ 0 } and C0 = { (cos ϕ, sin ϕ) ∈ R2 : π − α ≤ ϕ ≤ π + β }, 0 < α ≤ β < π. Find an expression for cap RM,2 (α, β) in terms of Grötzsch capacity γ2 by mapping R2 \ C1 conformally onto H2 . (For n = 2 the function p(( 12 , y)) from (10.3) can be expressed in terms of the capacity of Mori’s ring, see [317, Theorem 5.2., p. 192].) Next we shall find an upper bound for the function αK,n (t) defined as   αK,n (t) = τn−1 τn (t)/K ,

t > 0, K > 0 .

(10.6)

It is easy to show using the basic functional identity 7.31 that αK,n (t) =

1 − B2 ; B2

 √  B = ϕ1/K,n 1/ 1 + t .

(10.7)

For n = 2 we can go one step further using the identity 7.34(2) and obtaining (9.18) as a result. Further from (9.18) one can easily deduce that αK,2 (t) has a majorant of the form A t 1/K , where A is a constant, as we have pointed out in (9.18). Although the multidimensional analogue of 7.34(2) is false (recall 9.39), we nevertheless can find a similar majorant for αK,n (t) valid for all dimensions n ≥ 2.

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10 Conformal Invariants

Theorem 10.27 For n ≥ 2, K ≥ 1, and t ∈ (0, 22−3K ) the following inequality holds   τn−1 τn (t)/K ≤ 43−1/K t 1/K .    √  Proof Let x = τn−1 τn (t)/K and b = log 1 + 2(1 + 1 + t )/t . By 9.17(3) we obtain √   cn b ≤ τn (t) = Kτn (x) ≤ cn Kμ 1 + 2(1 − 1 + x )/x and further x ≤

4 μ−1 (b/K) . (1 − μ−1 (b/K))2

The inequality log(1/r) < μ(r) < log(4/r) (cf. (7.21)) shows that e−u < μ−1 (u) < 4e−u for u > 0. Therefore μ−1 (b/K) < 12 for t ∈ (0, 22−3K ) and also  x ≤ 4

t

3

t + 2(1 +

1/K √ 1+t)

≤ 43−1/K t 1/K

holds for t ∈ (0, 22−3K ) as desired.

 

Exercise 10.28 Let f : Bn → f (Bn ) ⊂ Rn be a homeomorphism with the property that there exists a number K ≥ 1 such that for all x, y ∈ Bn μf (Bn ) (f (x), f (y)) ≤ KμBn (x, y), and let (bn ) be a sequence of points in Bn such that bk → b ∈ ∂Bn and f (bk ) → β. (It is known that ∂f Bn is connected.) Let ρ(ak , bk ) < M, for all k. Show that limk→∞ f (ak ) = β exists. Does the same conclusion hold for noninjective mappings? Exercise 10.29 Let f : Bn → Bn be a homeomorphism with f (0) = 0 and assume that there is K ≥ 1 such that for all distinct x, y ∈ Bn λBn (x, y)/K ≤ λf Bn (f (x), f (y)) ≤ KλBn (x, y). Prove that there are a, b, c, d > 0 such that a|x|b ≤ |f (x)| ≤ c|x|d for all x ∈ Bn . Notes 10.30 NED sets in the complex plane were introduced by L. V. Ahlfors and A. Beurling [16]. J. Väisälä [521] studied NED sets in n-space and finally F. W. Gehring and O. Martio [163] introduced QED sets. See also V. V. Aseev and A. V. Sychev [34] as well as J. Väisälä [529], T. Cheng and S. Yang [102]. The conformal metric μG was introduced and studied by I. S. Gál [147] and also by T. Kuusalo [315]. The invariant λG was introduced by J. Ferrand [135, 136, 330] and studied in [137]. The metric μG was recently applied in the study of the Ferrand problem 15.4.1. E.G. Prilepkina and A.S. Afanas eva-Grigoreva [441] have studied

10.5 Capacitary Geometry

187

the metric μG in the case when G is a spherical ring domain. For n = 2 the shape of the extremal ring for the Teichmüller function p(x) has been thoroughly examined (see G. V. Kuz’mina [317, p. 192, Theorem 5.2.]). This chapter relies mainly on the papers [552, 556], and [216].

Part IV

Intrinsic Geometry

According to Liouville’s theorem conformal maps f : D → D  between domains n D, D  ⊂ Rn , n ≥ 3, are of the form f = g|D where g ∈ GM(R ). Therefore conformal invariance in dimensions n ≥ 3 means, in fact, Möbius invariance. For our application, in addition to Möbius invariant metrics studied in Parts II and III, we need metrics which are quasi-invariant under bilipschitz or quasiconformal maps. Moreover, the metric should be simple enough so that the values of corresponding distances can be computed. All this leads us to introduce new metrics and to study, for a given domain D ⊂ Rn , topics such as: • Comparison of two metrics of D. • Comparison of a new metric to the hyperbolic metric when the domain D is the unit ball or a half space. • Are the metric balls of small radii similar to euclidean balls, are they e.g. convex, simply connected or do they perhaps have smooth boundary? We will study these issues for what we call hyperbolic type metrics of subdomains of the euclidean space—some examples discussed above are the quasihyperbolic, distance ratio, Apollonian, and Seittenranta metrics. These metrics take into account not only the distance between the points but also the position of the points relative to the boundary of the domain. Each of these metrics depends strongly, in its own way, on the geometric structure of the boundary of the domain. Thus the intrinsic geometry induced by the metric reflects the structure of the boundary and we compare these metrics for domains under various hypotheses on the boundary of the domain. Recall that some properties of several hyperbolic type metrics were listed at the end of Chap. 5. What comes to balls of small radii, we will prove results which give a concrete upper bound for the convexity radius, a real number such that if the radius is below this bound, then the ball is convex. Finally, we compare metric balls B(x, t) centered at x to euclidean balls B n (x, r), B n (x, R) with the same center in the sense that B n (x, r) ⊂ B(x, t) ⊂ B n (x, R) and, moreover, R/r → 1 when t → 0 . Finding concrete upper bounds for this quotient requires sometimes tedious calculations

190

IV

Intrinsic Geometry

which are relegated to exercises. It is possible that for some metric m of a planar polygonal domain D and for given point x ∈ D all metric balls Bm (x, t) centered at x have non-smooth boundaries. The convergence R/r → 1 as t → 0 means that metric balls will become more and more round when the radius decreases to 0 , and we will give examples of this behavior. We expect that many of the results of this part of the book will also hold for 1/(1−n) the metrics μG , λG of Chap. 10, but the necessary analytic tools are not currently available. Indeed, extending the results of this part to these two metrics offers challenges for future research. Moreover, A. Papadopoulos [419, pp. 42–48] lists twelve metrics commonly used in geometric function theory and analogous questions could be formulated in this setup, too. In Part V shall see that comparison inequalities between various metrics are very useful tools in the study of how mappings deform distances.

Chapter 11

Hyperbolic Type Metrics

In Chaps. 4 and 5 we studied the quasihyperbolic metric kD (5.2) and the distance ratio metric jD (4.27). These two metrics are generalizations of the hyperbolic metric (Chap. 4) and in this chapter we introduce other such generalizations. Note first that a simply connected domain D  R2 can be mapped by Riemann’s mapping theorem conformally onto the unit disk B2 . If this conformal map is f : D → B2 , then we can define the hyperbolic metric on D by [283] ρD (z, w) = ρB2 (f (z), f (w)) , z, w ∈ D . There is no counterpart of Riemann’s mapping theorem in Rn , n ≥ 3 , and hence we cannot use this idea to define a metric for simply connected domains in higher dimensions. What we can do is to introduce substitutes for the hyperbolic metric— we call these hyperbolic type metrics. We do not define this term precisely, the term is motivated by the fact that a hyperbolic type metric shares a few, but not necessarily all, properties of the hyperbolic metric. Before we proceed with the study of this topic let us formulate some properties of metrics which are useful in our work below. For this purpose we state these properties for some family of metrics M = {mD } defined for proper subdomains D n n 2 of R . Usually we assume that ∞ ∈ R \ D and that card(R \ D) ≥ 2 . (1) Monotonicity with respect to domain. Let D1 ⊂ D2 be proper subdomains of n R and x, y ∈ D1 . Then mD2 (x, y) ≤ mD1 (x, y) . (2) The distance mD (x, y) depends not merely on the euclidean distance |x − y| but also on the position of x and y with respect to the boundary of D . (3) Sensitivity to boundary variation: if A ⊂ D is a closed non-empty subset of the domain D such that also D \ A is a domain and x, y ∈ D \ A, are “close” to A, © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_11

191

192

11 Hyperbolic Type Metrics

and x = y, then mD (x, y) < mD\A (x, y) . (4) D = ∪M>0 Bm (x, M) for all x ∈ D and Bm (x, M) is a compact subset of D . For each x ∈ D: ∩M>0 Bm (x, M) = {x}. (5) For small radii M , the balls Bm (x, M) are similar to euclidean ball, see Chaps. 11, 12. (6) If the domains Dk ⊂ D, k = 1, 2, . . . , converge in the Hausdorff distance to D in the sense that H (∂D, ∂Dk ) → 0 when k → ∞ , then for x, y ∈ D mDk (x, y) → mD (x, y) . when k → ∞ . Here the Hausdorff distance H (∂D, ∂Dk ) is taken with respect to the chordal metric. (7) For D = Bn and x, y ∈ Bn , mBn (x, y) can be estimated in terms of ρBn (x, y) . (8) For mD1 , mD2 ∈ M and for a homeomorphism f : D1 → D2 in some class F of homeomorphisms mD1 (x, y) and mD2 (f (x), f (y)) are comparable explicitly. The class F might be e.g. Möbius transformations, conformal maps, bilipschitz maps, quasiconformal maps. Some of the basic questions of interest are • How are these metrics of class M related to other metrics such as the distance ratio metric? • How are these metrics transformed under well-known classes of mappings such as Möbius transformations, conformal maps, bilipschitz maps? The above list of eight properties of metrics collects observations about the properties of some of our metrics—all of our metrics may not posses these features. This list also suggests a heuristic principle: For a given unbounded domain G and given points x, y ∈ G , the value of mG (x, y) is mostly determined by the boundary points in B n (x, R) ∩ (∂G) for large R > |x − y| + d(x) . As we have seen earlier, metrics determined by merely one or two boundary points such as jG or δG may be very useful and in this chapter we will study such metrics. For some metrics every boundary point has influence on the value of mG (x, y) . For example the values jBn \{0} (x, y) and jBn (x, y) are quite different for x, y ∈ B n (1/10) \ {0} . But quantitative analysis of this influence is not easy. Often we study these questions in some special or simple domains and in this way obtain more accurate information than in the general case. Remarks 11.1 (1) Thin parts of the boundary are “invisible” for the modulus metric μG : If E ⊂ G is a compact set of capacity zero and x, y ∈ G \ E , then μG (x, y) = μG\E (x, y) . Therefore the modulus metric does not have property (3). Recall that by 10.5 also the closures of the balls BμG (x, r) need not be compact subsets of G .

11.1 Metrics Determined by One Boundary Point 2

193

2

(2) If G ⊂ R is a domain with card(R \ G) ≥ 3 , then the value of the hyperbolic distance hG (x, y) can be efficiently estimated in terms of sup hD (x, y) 2

where the supremum is taken over all domains D of the form D = R \{a, b, c} where a, b, c ∈ ∂G . See [67, 504], [283, Ch 14].

11.1 Metrics Determined by One Boundary Point Throughout this chapter we use the symbol G and D for a proper subdomain of Rn . Consider metrics on a domain G  Rn defined by an expression of the form |x − y| |x − y| = f (x, y, z) inf z∈G f (x, y, z) z∈G

m(x, y) = sup

where f : G × G × ∂G → (0, ∞) is a continuous function. By continuity we see that there is an extremal point z0 ∈ ∂G for which equality holds. The triangular ratio metric for x, y ∈ G is defined by |x − y| ∈ [0, 1]. |z − x| + |z − y| z∈∂G

sG (x, y) = sup

(11.1)

P. Hästö [188, Theorem 6.1] proved that sG satisfies the triangle inequality and developed a theory for metrics more general than sG . Very recently, the geometry of the balls of sG for some special domains was studied in [100, 237] and [195]. Note that for x, y ∈ G the set {z ∈ Rn : |x −z|+|z −y| = c} defines an ellipsoid with foci x and y, if c > |x − y|. The supremum in the definition (11.1) is attained at some point z ∈ ∂G. If the segment [x, y] ⊂ G, there exists a largest ellipsoid contained in G with focii x and y and z on its boundary and sG (x, y) < 1. If there exists z ∈ ∂G such that z ∈ [x, y], then by the triangle inequality |z − x| + |z − y| = |x − y| and sG (x, y) = 1 (Fig. 11.1). Let G be a proper subdomain of Rn . The Cassinian metric for x, y ∈ G is defined by cG (x, y) = sup z∈∂G

|x − y| . |z − x||z − y|

(11.2)

This metric was introduced by Z. Ibragimov [251]. The set {w ∈ R2 : |x − w||w − y| = t} defines in the planar case n = 2 a Cassinian oval. As in the case of the triangular ratio metric the supremum is attained by an extremal situation, in this case by the largest possible Cassinian oval contained in G with foci x and y. A Cassinian

194

11 Hyperbolic Type Metrics

Fig. 11.1 From left to right: visualization of the extremal point z ∈ ∂G for metrics sG , cG and vG

oval always encircles its foci and with small values of the parameter t it consists of two separate curves each curve enclosing one of the foci. If t > |x − y|2 /4 the Cassinian oval consists of a single curve enclosing its both foci [176, p. 79]. Let G be a proper subdomain of Rn such that ∂G is not a proper subset of a line. The visual angle metric for x, y ∈ G is defined by vG (x, y) = sup{α : α = (x, z, y), z ∈ ∂G} .

(11.3)

Equivalently, it can be also defined as α ∩∂G = ∅}, vG (x, y) = sup{α : Exy

α Exy = {z ∈ Rn : (x, z, y) ≥ α}.

(11.4)

This metric was introduced and studied very recently in [291]. The supremum in the definition is attained for z ∈ ∂G such that the angle (x, z, y) is largest. This means that for all x, y ∈ G, we have vG (x, y) ∈ [0, π] and vG (x, y) = π whenever there exists z ∈ ∂G such that z ∈ [x, y]. Lemma 11.2 The function vG : G × G → [0, π] is a similarity invariant pseudometric for every domain G  Rn and it is a metric unless ∂G is a proper subset of a line. Proof Let x, y ∈ G be arbitrary. Clearly (x, z, y) = 0 if and only if x = y or z is located on ray(x, x − y) or ray(y, y − x). Here for u = 0 ray(w, u) = {w + tu : t ≥ 0 }. Thus vG (x, y) = 0 implies x = y, unless ∂G is a proper subset of a line. We now prove the triangle inequality for vG . Let x, y, z ∈ G. Let w be an α ∩ ∂G, where E α is the envelope such that α is the supremum arbitrary point in Exy xy angle in the definition (11.4). Without loss of generality, we may assume that x , y , z , w are in R3 . Let r = min{|w − x|, |w − y|, |w − z|}/2, and the points x  , y  and z denote the intersections of S 2 (w, r) with [w, x], [w, y] and [w, z], respectively. Clearly (x, w, y) = (x  , w, y  ), (z, w, y) = (z , w, y  ) , (z, w, x) = (z , w, x  ).

11.1 Metrics Determined by One Boundary Point

195

Also, by considering the intrinsic metric of the sphere S 2 (w, r) (see [51, Corollary 18.6.10]), it is clear that (x  , w, y  ) ≤ (x  , w, z ) + (z , w, y  ).

(11.5)

Let β = (x, w, z) and γ = (z, w, y). By the definition of vG , it is clear that β ≤ vG (x, z) and γ ≤ vG (x, z). Then by the inequality (11.5), it now follows that vG (x, y) = α = (x, w, y) = (x  , w, y  ) ≤ (x  , w, z ) + (z , w, y  ) = (x, w, z) + (z, w, y) ≤ vG (x, z) + vG (z, y). This proves the triangle inequality. Similarity invariance is clear by definition   (11.4). For x, y ∈ Rn \ {z}, z ∈ Rn , we have vRn \{z} (x, y) = (x, z, y) ∈ [0, π] and it is easy to see that vRn \{z} is only a pseudometric. In Rn \ {z} the quasihyperbolic metric k and the distance ratio metric j are connected to v via (5.31) as follows  vRn \{z} (x, y) =

2 2 kR n \{z} (x, y) − log

|x| ≤ kRn \{z} (x, y) |y|

see 6.2 and Lindén [335] vRn \{z} (x, y) ≤

π jRn \{z} (x, y), log 3

where the equalities hold if x = z − y. In the case G = Hn an explicit formula for the visual angle metric can be given. We consider here the case n = 2, but generalization for any n > 2 is straightforward. Let x = (x1 , x2 ), y = (y1 , y2 ) ∈ H2 and x = y. Then the circle through x , y and tangent to ∂H2 has the center √ √ (x2 + y2 )|x − y|2 + 2 x2 y2 (x1 − y1 )|x − y| x1 y2 − x2 y1 + x2 y2 |x − y| +i z= y2 − x2 2(y2 − x2 )2 or √ √ x1 y2 − x2 y1 − x2 y2 |x − y| (x2 + y2 )|x − y|2 − 2 x2 y2 (x1 − y1 )|x − y| z= +i y2 − x2 2(y2 − x2 )2 

196

11 Hyperbolic Type Metrics

Fig. 11.2 Distance vH2 (x, y) = ω in two different cases

if x2 = y2 and w=

4x 2 + (x1 − y1 )2 x1 + y1 +i 2 2 8x2

if x2 = y2 (see Fig. 11.2). Therefore, √ ⎧ 2 x2 y2 |x − y| + (x1 − y1 )(x2 + y2 ) ⎪ ⎪ , x1 ≤ y1 , x2 < y2 , arccos √ ⎨ (x2 + y2 )|x − y| + 2 x2 y2 (x1 − y1 ) vH2 (x, y) = 4x 2 − (x1 − y1 )2 ⎪ ⎪ , x1 =  y1 , x2 = y2 . ⎩ arccos 22 4x2 + (x1 − y1 )2 (11.6) In particular, if x2 = y2 and y1 = −x1 > 0 , then vH2 (x, y) = 2 arctan

y1 . y2

(11.7)

If x1 = y1 , then vH2 (x, y) = arccos

√ 2 x2 y2 . x2 + y2

Next we introduce a point-pair function that has turned out to be a useful tool even though it is in general not a metric. Let G ⊂ Rn be a proper subdomain. For x, y ∈ G we define [100] pG (x, y) = 

|x − y| |x − y|2 + 4d(x)d(y)

.

11.1 Metrics Determined by One Boundary Point

197

Exercise 11.3 (1) Use (4.8) to show that for x, y ∈ Hn we have th

|x − y| ρHn (x, y) =  2 |x − y|2 + 4xn yn

and hence conclude that pHn (x, y) = th ρ(x,y) 2 . (2) Show that pHn is a metric whereas pBn is not a metric. Exercise 11.4 (1) Let G, G ⊂ Rn be domains and f : G → G be a homeomorphism. For x, y ∈ G define mf (x, y) = |f (x) − f (y)| . Show that mf is a metric. (2) Let m : R+ × R+ → R+ , R+ = (0, ∞), be defined by m(x, y) = | log(x/y)|. Show that m is a metric. (3) Suppose that u : (R+ , m) → (R+ , m) is uniformly continuous. Show that u(x) ≤ Ax B for some constants A and B for all x ≥ 1. Exercise 11.5 Let us define m(x, y) = |x − y|1/2 for x, y ∈ R. Show that m is a metric and [0, 1] is not rectifiable with respect to m. [Hint: The length of [0, 1] is sup

 n !

4 m(xi , xi+1 ) : n ∈ N, x1 = 0, xn+1 = 1, xk < xk+1 .]

i=1

Lemma 11.6 (1) If x, y ∈ G ⊂ Rn and G is convex, then sG (x, y) ≤ pG (x, y). Here equality holds for all x, y ∈ G if G = Hn . (2) For x, y ∈ G ⊂ Rn , pG (x, y) ≤

√

2sG (x, y).

Proof (1) Suppose that z ∈ ∂G is an extremal boundary point for the s-metric for which the equality holds in (11.1). We draw a line L through z, tangent to ∂G. Let y be the reflection of y in the line L. By geometry, |x − z| + |z − y| = |x − y| =



|x − y|2 + 4d1 (x)d1 (y),

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11 Hyperbolic Type Metrics

d1 (x) = d(x, L), d1 (y) = d(y, L) . Because G is convex it is clear that L is outside G, but d(x), d(y) are the shortest distances from x, y to ∂G, so obviously d(x) ≤ d1 (x), d(y) ≤ d1 (y), thus sG (x, y) =

|x − y| |x − y| =  |x − z| + |z − y| |x − y|2 + 4d1 (x)d1 (y)

≤ 

|x − y| |x − y|2 + 4d(x)d(y)

= pG (x, y).

The equality statement follows from Exercise 11.3. (2) Fix x, y ∈ G, z ∈ ∂G, such that d(x) = |x − z|. By symmetry we may assume d(x) ≤ d(y) and then sG (x, y) ≥

|x − y| . |x − y| + 2d(x)

(11.8)

Now by [29, 1.58 (13)] and (11.8) pG (x, y) ≤ 

√

|x − y| |x

− y|2

+ 4d(x)2

≤

√ 2|x − y| ≤ 2sG (x, y). |x − y| + 2d(x)

Combining 11.6(1) and 11.3 we see that for all x, y ∈ th

 

Hn

ρHn (x, y) = pHn (x, y) = sHn (x, y) . 2

(11.9)

∗ from 5.24(5). By (11.9) and 4.9(2) we see that Recall the notation jG

sHn (x, y) ≥ jH∗ n (x, y) .

(11.10)

11.2 Metrics Determined by Two Boundary Points Next we recall two metrics briefly mentioned in Chap. 5. Both metrics are defined in a domain with at least two boundary points. n n Let G be a subdomain of R with card(R \ G) ≥ 2 . The Apollonian metric is defined for x, y ∈ G by (Fig. 11.3) αG (x, y) = sup log |a, x, y, b| = sup log a,b∈∂G

a,b∈∂G

|a − y||x − b| . |a − x||y − b|

(11.11)

Lemma 11.7 ([51, Theorem 1.1]) The function αG is a metric if and only if ∂G is n not a proper subset of a sphere in R .

11.2 Metrics Determined by Two Boundary Points

199

r Fig. 11.3 Left: Apollonian metric αG . Right: Apollonian circles Sx,y

The Apollonian metric was first discussed by D. Barbilian in a short note [48], but forgotten for many years. In 1998 Beardon [53], unaware of Barbilian’s work, rediscovered this metric and systematically studied it. For x, y ∈ Rn and r > 0 we define the Apollonian ball and sphere, respectively, to be r r Bx,y = {z ∈ Rn : r|x − z| < |y − z|}, Sx,y = {z ∈ Rn : r|x − z| = |y − z|}.

n

The Seittenranta metric is defined for x, y ∈ G ⊂ R with card ∂G ≥ 2 by   |a − b||x − y| δG (x, y) = sup log(1 + |a, x, b, y|) = sup log 1 + |a − x||y − b| a,b∈∂G a,b∈∂G (11.12) and it always satisfies the triangle inequality Theorem 5.16, [483, Theorem 3.3]. From the definition it is also easy to verify that the Apollonian metric is monotone with respect to the domain, i.e. for all x, y ∈ G ⊂ G we have αG (x, y) ≤ αG (x, y).

(11.13)

The following proposition shows that Seittenranta’s metric is also monotone with respect to the domain. n

Proposition 11.8 ([483, Remark 3.2 (2)]) Let G  R and G ⊂ G be domains. Seittenranta’s metric is monotone with respect to the domain, i.e. for all x, y ∈ G we have δG (x, y) ≤ δG (x, y).

(11.14)

200

11 Hyperbolic Type Metrics

Lemma 11.9 ([53, 483]) (1) Let D be the upper half-space or the unit ball in Rn , then ρD = αD = δD . n n n (2) If G ⊂ R is a domain with card(R \ G) ≥ 2 and h ∈ GM(R ) , then for all x, y ∈ G αhG (h(x), h(y)) = αG (x, y) , δhG (h(x), h(y)) = δG (x, y) . Proof The proof for ρD = αD is given in [53] and the proof for ρD = δD follows from Theorem 5.16. The proof of (2) is self-evident.   We formulate next some inequalities involving these metrics. n

Proposition 11.10 ([483, Theorem 3.11]) Let G  R be an open set. Then for all x, y ∈ G we have   αG (x, y) ≤ δG (x, y) ≤ log eαG (x,y) + 2 ≤ αG (x, y) + log 3. We introduce next a result that can be used to estimate the size of the metric balls Bα and Bδ . n

Theorem 11.11 Let G  R , be a domain with card ∂G ≥ 2, x ∈ G and r > 0. Then for m ∈ {α, δ} 5 BmRn \{a,b} (x, r). BmG (x, r) = a,b∈∂G a=b

Proof We show that for all x, y ∈ G mG (x, y) =

sup

a,b∈∂G, a=b

mRn \{a,b} (x, y).

(11.15)

n

Because G ⊂ R \ {a, b} for all a, b ∈ ∂G by (11.13) and (11.14) we have mG (x, y) ≥ mRn \{a,b} (x, y) and thus mG (x, y) ≥

sup

a,b∈∂G, a=b

mRn \{a,b} (x, y).

On the other hand, for some a, b ∈ ∂G with a = b we have mG (x, y) = mRn \{a,b} (x, y) and (11.15) holds.

 

11.2 Metrics Determined by Two Boundary Points

201

In the next lemma we consider the metric (cf. Exercise 5.24(5)) ∗ jG (x, y) = th

jG (x, y) . 2

Lemma 11.12 Let G be a proper subdomain of Rn . If x, y ∈ G, then ∗ jG (x, y) =

|x − y| |x − y| + 2 min{d(x), d(y)}

and ∗ jG (x, y) ≤ sG (x, y) ≤

ejG (x,y) − 1 . 2

The first inequality is sharp for G = Rn \ {0} . Proof By symmetry we may assume that d(x) ≤ d(y) . For x, y ∈ G, let z ∈ ∂G be a point satisfying d(x) = |x − z| . For the equality claim we see that |x − y| |x − y|/d(x) ejG (x,y) − 1 = = j (x,y) |x − y| + 2d(x) |x − y|/d(x) + 2 eG +1 =

ejG (x,y)/2 − e−jG (x,y)/2 ∗ = jG (x, y) . ejG (x,y)/2 + e−jG (x,y)/2

For the first inequality we observe that by the triangle inequality sG (x, y) ≥

|x − y| |x − y| ∗ ≥ = jG (x, y) . |x − z| + |z − y| |x − y| + 2d(x)

The sharpness of the first inequality when G = Rn \{0}, follows if we choose x = 1, ∗ y = t > 1 . Then sG (x, y) = tt −1 +1 = jG (x, y) . For the second inequality, note that sG (x, y) ≤ ≤

|x − y| |x − y| ≤ √ d(x) + d(y) 2 d(x)d(y) ejG (x,y) − 1 |x − y| = . 2d(x) 2

 

Lemma 11.13 Let G be a proper subdomain of Rn . Then for all x, y ∈ G we have ∗ sG (x, y) ≤ 2jG (x, y) .

This inequality is sharp when the domain is G = Rn \ {0} . Proof Exercise.

 

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11 Hyperbolic Type Metrics

Lemma 11.14 If G is a proper subdomain of Rn , then for all x, y ∈ G, ∗ (x, y) ≤ pG (x, y) ≤ √ jG

w w2 + 1

≤

√ ∗ 2jG (x, y) ,

with w = (ejG (x,y) − 1)/2 . Both upper and lower bounds for pG (x, y) are sharp when the domain is G = Rn \ {0} . Proof Without loss of generality we may suppose that d(x) ≤ d(y) . Then by Lemma 11.12 the first inequality is equivalent to |x − y| |x − y| ≤ . |x − y| + 2d(x) |x − y|2 + 4d(x)d(y) This, in turn, follows easily from the inequality d(y) ≤ |x − y| + d(x) . For the second inequality observe that with w = (ejG (x,y) − 1)/2 pG (x, y) =

w |x − y|  =  2 2 2d(x) (|x − y|/(2d(x))) + d(y)/d(x) w + d(y)/d(x) √ ∗ 1+w ∗ w ≤√ jG (x, y) ≤ 2jG ≤ √ (x, y) . w2 + 1 w2 + 1

To see the sharpness of the first inequality in G = Rn \ {0} choose y = 1/x with x > 1. Then x2 − 1 1 1 ∗ = pG (x, ) . jG (x, ) = 2 x x −1+2 x For the sharpness of the last inequality again in G = Rn \ {0}, we choose y = −x . Then 1 pG (x, −x) = √ , 2

∗ jG (x, −x) =

Proposition 11.15 ([195, 2.4]) If G is a bounded domain G ∗ jG (x, y) ≥

|x − y| . d(G)

1 . 2 of Rn ,

  then for all x, y ∈

11.2 Metrics Determined by Two Boundary Points

203

Lemma 11.16 Let G be a proper subdomain of Rn , then for all x, y ∈ G, pG (x, y) ≤ sG (x, y) ≤ 2pG (x, y) , pG (x, y) (2) sG (x, y) ≤ , pG (x, y) = 1 . 1 − pG (x, y)

(1)

√1 2

Proof By symmetry we may suppose that d(x) ≤ d(y) . (1) The lower bound follows from Lemma 11.6 (2). For the upper bound observe that by Lemma 11.13 sG (x, y) ≤

2|x − y| 2|x − y| ≤  = 2pG (x, y) , |x − y| + 2d(x) |x − y|2 + 4d(x)d(y)

where the second inequality follows from the inequality d(y) ≤ d(x) + |x − y| . (2) The first inequality in Lemma 11.14 can be written as w ≤ pG (x, y) , 1+w

w = (ejG (x,y) − 1)/2 .

This inequality implies (2) because sG (x, y) ≤ w by Lemma 11.12. Lemma 11.17 For all x, y ∈

Bn

 

we have

pBn (x, y) ≤ th

ρBn (x, y) . 2

(11.16)

Proof The inequality (1 − |x|2)(1 − |y|2 ) = (1 − |x|)(1 − |y|)(1 + |x|)(1 + |y|) ≤ 4(1 − |x|)(1 − |y|), yields th

|x − y| ρBn (x, y) =  2 2 |x − y| + (1 − |x|2)(1 − |y|2) ≥ 

|x − y| |x − y|2 + 4d(x)d(y)

= pBn (x, y).

 

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11 Hyperbolic Type Metrics

Lemma 11.18 (1) For x, y ∈ Bn we have th

ρBn (x, y) ρBn (x, y) ρBn (x, y) ≤ sBn (x, y) ≤ pBn (x, y) ≤ th ≤ 2 th . 4 2 4

(2) For x, y ∈ Hn we have th

ρHn (x, y) ρHn (x, y) ρHn (x, y) ≤ sHn (x, y) ≡ pHn (x, y) ≡ th ≤ 2 th . 4 2 4

Proof (1) For the first inequality, by Lemmas 11.12 and 4.9, we have sBn (x, y) ≥ jB∗n (x, y) = th

ρBn (x, y) jBn (x, y) ≥ th . 2 4

The second and the third inequalities follow from Lemmas 11.6 and 11.17. For the last inequality, by 4.2 (2), th

ρBn (x, y) = 4

th(ρBn (x, y)/2) 1 ρBn (x, y) .  ≥ th 2 2 1 + 1 − th2 (ρBn (x, y)/2)

(2) For x, y ∈ Hn the proof is similar to part (1) and it follows from 11.12, 11.14, 11.6 and (11.9).   Lemma 11.19 (1) Let G be a proper subdomain of Rn . If x, y ∈ G, then th

jG (x, y) ≤ pG (x, y) ≤ th jG (x, y) . 2

(2) If G ⊂ Rn is a convex domain, x, y ∈ G and m = min{d(x), d(y)} , then th(jG (x, y)/2) ≤ sG (x, y) ≤ 

|x − y| |x − y|2 + 4m2

≤ thjG (x, y) .

Proof (1) For the second inequality, by symmetry we may assume that d(x) ≤ d(y) . Writing |x − y| = b, pG (x, y) = 

b b2

+ 4 d(x) d(y)

≤

b b2

+ 4 d(x)2

,

11.2 Metrics Determined by Two Boundary Points

205

we have  1+ e2jG (x,y) − 1 th jG (x, y) = 2j (x,y) =  e G +1 1+

b d(x) b d(x)

2 2

−1 +1

=

b 2 + 2b d(x) . b2 + 2b d(x) + 2d(x)2

Denote t = d(x) . Then the inequality pG (x, y) ≤ th jG (x, y) , is equivalent to √

b b2 + 4 t 2

≤

b2 + 2b t + 2t 2 , b2 + 2b t

and the last inequality is equivalent to 4b2 t 3 (2b + 3t) ≥ 0, which is true, since t = d(x) > 0 . The first inequality follows from Lemma 11.14. (2) Fix x, y ∈ G . Because G is convex, by 11.6 (1) and the proof of (1) we have sG (x, y) ≤ pG (x, y) ≤ 

|x − y| |x − y|2 + 4m2

≤ thjG (x, y) .

The first inequality in the claim follows from Lemma 11.14.

11.2.1 Special Values of sBn (1) For u ∈ ∂Bn we have |x − y| |x − y| |x − y| |x − y| ≤ ≤ = |x − u| + |y − u| |2u − (x + y)| |2|u| − |x + y|| 2 − |x + y| and hence sBn (x, y) ≤

|x − y| , 2 − |x + y|

with equality if 0, x, y are collinear. In particular, sBn (x, −x) = |x| . (2) We also have with z = x/|x|, x = 0 , sBn (x, 0) ≥ where equality holds by (1).

|x| |x| = |x − z| + |z| 2 − |x|

 

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11 Hyperbolic Type Metrics

2 (3) For a complex number a ∈ C = R let a¯ be its complex conjugate. Then if a ∈ B2 and Re (a) > 0, and a − 12 > 1/2 , we have sBn (a, a) ¯ = |a| [193].

Theorem 11.20 Let G  Rn be a domain. Then for all z ∈ G and all r ∈ (0, 1], the s-ball Bs (z, r) is euclidean starlike with respect to z . Proof Without loss of generality we may assume z = 0 . Fix 0 < r ≤ 1. Let y ∈ Bs (0, r). Since s(y, 0) < 1, [0, y] ⊂ G. Let x ∈ [0, y] and z ∈ ∂G. Since |x| ≤ |z| + |x − z|, |x| + |x − y| = |x|, and |y − z| ≤ |y − x| + |x − z|, we have |x| + |x − y| |y| |x| ≤ = |z| + |x − z| |z| + |x − z| + |x − y| |z| + |x − z| + |x − y| ≤

|y| ≤ s(y, 0) . |z| + |y − z|

Taking a supremum over all z ∈ ∂G we thus obtain s(x, 0) ≤ s(y, 0) < r , since x is an arbitrary point in [0, y] ⊂ Bs (0, r) (Fig. 11.4).

 

11.2.2 Ptolemy-Alhazen Problem and s-Metric The Greek mathematician Ptolemy (ca. 100–170) formulated a problem concerning reflection of light at a spherical mirror surface: Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer. Alhazen, also known as Ibn al-Haytham (ca. 965-1040), was a mathematician, astronomer, and physicist, who also studied this problem. Fig. 11.4 The s-metric disks Bs (z, t) are euclidean starlike with respect to the center z

11.2 Metrics Determined by Two Boundary Points

207

Let D = B2 be the unit disk {z ∈ C : |z| < 1}, and suppose that the circumference ∂D = {z ∈ C : |z| = 1} is a reflecting curve. In the two-dimensional case the problem reads: Given two points z1 , z2 ∈ D , find u ∈ ∂D such that (z1 , 0, u) = (0, u, z2 ) .

(11.17)

Here (z, u, w) denotes the radian measure in [0, π/2) of the angle with sides [u, z] and [u, w] . This equality condition for the angles says that the angles of incidence and reflection are equal, a light ray from z1 to u is reflected at u and goes through the point z2 . The point u is also the point of contact between the unit circle and an ellipse with foci z1 , z2 , contained in D . It is clear that this point u is the extremal point for the triangular ratio metric sD (z1 , z2 ) . Finding u requires solving a quartic equation. A computer program for finding the numerical value of sD (x, y) was given in [146]. Remark 11.21 In a general domain our metrics are not, as a rule, comparable. For instance if the domain G has isolated boundary points the metric vG may not have a ∗ . This can be seen by choosing G = D\{0} and observing lower bound in terms of jG ∗ −k −k−1 jG (2 e1 , 2 e1 ) ≥ c > 0 for all k ≥ 3 whereas vG (2−k e1 , 2−k−1 e1 ) → 0 when k → ∞. Under suitable conditions on the boundary one can find lower bounds for ∗ , see [195] and Chap. 12. P. Hästö [202] has also shown that α vG in terms of jG G can be estimated from below in terms of jG if the boundary is thick in the sense of [541]. Remarks 11.22 (1) It is natural to expect that spheres of small radii in the metric spaces considered here are similar to euclidean spheres. For instance, one could ask whether the quasihyperbolic spheres have smooth boundaries. This question was solved in the positive by Klén, Rasila, and Talponen [292]. (2) In every metric space, the diameter of a ball with radius r is at most 2r and equal to 2r for instance in the euclidean or hyperbolic spaces Bn and Hn . It is not known whether this holds for the quasihyperbolic metric kG of a domain in Rn . Beardon and Minda [55] have given an example where the diameter of the hyperbolic disk is less than twice the radius when G is the extended complex plane C minus three points. For the definition of the hyperbolic metric in this case, see [283]. (3) In the beginning of this chapter, eight properties of metrics were listed. We have not studied these properties systematically. For instance, one could hope to find a quantitative estimate for the speed of the convergence property. Recall the question posed in Remark 9.13(1). (4) For conformal invariants in plane some convergence questions were investigated by S. Rohde and M. Zinsmeister [475] and N. Lakic and G. Markowsky [319].

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11 Hyperbolic Type Metrics

Notes 11.23 Since the 1995 paper of A. F. Beardon [53] many authors studied the Apollonian metric e.g. F.W. Gehring and K. Hag [159], P. Hästö [199, 200], Z. Ibragimov [249, 250], and P. Hästö, Z. Ibragimov, D. Minda, S. Ponnusamy, S. Sahoo [208]. For the triangular ratio metric see P. Hästö [198, 201] and also [100, 195]. The visual angle metric was introduced in [291], and further studied in [194] and [567]. Several inequalities between the hyperbolic, Apollonian, and quasihyperbolic metric in plane domains can be found in [159]. The results of this chapter are taken mainly from [100, 192, 194, 195].

Chapter 12

Comparison of Metrics

We start the comparison of metrics with those ones we have considered in the earlier chapters, namely the chordal metric q, (3.5), the hyperbolic metric ρ, (4.8), (4.16), the distance ratio metric j , (4.27) and the quasihyperbolic metric k, (5.2). Preliminary comparison results for the quasihyperbolic and the distance ratio metrics were introduced in (5.4) and Lemma 5.7. We begin this chapter by considering the punctured space. The Punctured Space The domain Rn \ {z} when z ∈ Rn is called a punctured space. If for all x, y ∈ Rn \ {z} we have mRn \{z} (x, y) = mRn \{z } (x − z + z , y − z + z ) , then the selection of the punctured point z has no effect. This means that the metric mRn \{z} is translation invariant. In such cases we may as well choose z = 0 and consider the complement of the origin Rn \ {0}. In many cases our metrics and functions are not merely translation invariant but invariant under similarity transformations such as the functions kG , jG , pG , vG . As shown by H. Lindén, kG (x, y) ≤ πjG (x, y)/ log 3 for G = Rn \ {0} and x, y ∈ G [335, Theorem 1.6], see also Exercise 6.2. Recalling (5.4) we have jG (x, y) ≤ kG (x, y) ≤

π jG (x, y) log 3

(12.1)

for x, y ∈ G = Rn \ {0}.

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_12

209

210

12 Comparison of Metrics

12.1 The Unit Ball Let us consider the visual angle metric in Bn . For the purpose of illustration, assume that n = 2 and let x, y ∈ B2 and x = y. We define ellipses Ex = {z ∈ B2 : |x − z| + |z| = 1},

Ey = {z ∈ B2 : |y − z| + |z| = 1} ,

and denote Ex ∩ Ey = {z1 , z2 } (see Fig. 12.1). We choose z to be that one of the points z1 and z2 , which has the larger norm. Then vB2 (x, y) =

1 (x, z, y). 2

(12.2)

In particular, for x = 0 , y = 0, we have vB2 (0, x) = arcsin |x| ∈ (0, π/2)

(12.3)

and for |x| = |y| = 0, θ = 12 (x, 0, y) ∈ (0, π/2], we have vB2 (x, y) = 2 arctan

|x| sin θ . 1 − |x| cos θ

(12.4)

Our next aim is to compare vBn and ρBn . We prove three special cases Corollary 12.5, Corollary 12.6 and Lemma 12.7. Combining these three results we finally obtain a general result in Bn , Theorem 12.8. Before comparing vBn and ρBn we need a few auxiliary results. We next record a lemma from calculus. It has found recently numerous applications, see the bibliography of [30] and also [132]. Lemma 12.1 ((l’Hôpital Monotone Rule)[29, Theorem 1.25],[28, 2.2], [190, p. 106]) Let −∞ < a < b < ∞, and let f, g : [a, b] → R be continuous on [a, b], differentiable on (a, b). Let g  (x) = 0 on (a, b). Then, if f  (x)/g  (x) is increasing

Fig. 12.1 Distance vB2 (x, y) in general case

12.1 The Unit Ball

211

(decreasing) on (a, b), so are [f (x) − f (a)]/[g(x) − g(a)] and [f (x) − f (b)]/[g(x) − g(b)]. If f  (x)/g  (x) is strictly monotone, then the monotoneity in the conclusion is also strict. Lemma 12.2 r (1) The function f1 (r) ≡ arcsin arth r is strictly decreasing from (0, 1) onto (0, 1).

arcsin r (2) The function f2 (r) ≡ log(1/(1−r)) is strictly decreasing from (0, 1) onto (0, 1). cr 2cr √ (3) The function f3 (r) ≡ arctan − arsh 1−c 2 is strictly decreasing 1−c

from (0, 1) onto (arctan c − log

1+c 1−c

1−r 2

, 0) for c ∈ (0, 1).

arctan r is strictly decreasing from (0, ∞) onto (4) The function f4 (r) ≡ arch (1+2r 2 )

(0, 1/2).  

Proof Exercise. Lemma 12.3 Let α ∈ (0, π). Then the function f (θ ) = (1 + cos(α + θ ))(1 + cos(α − θ )) is strictly decreasing from (0, π − α) onto (0, (1 + cos α)2 ).

 

Proof Exercise.

The geometric property expressed by the next lemma will be used to estimate the metrics vBn and ρBn in a special case. For x ∈ Rn , y ∈ Rn \ {0} we denote the line through x in the direction y by L(x, y) = {x + ty : t ∈ R} .

(12.5)

Note that in Chap. 4, e.g. in Exercises 4.3, 4.22–4.23 this symbol was used in a slightly different meaning. Lemma 12.4 Let a ∈ Bn be a point and P be a hyperplane through 0 and a. Let n C = S(a, r) be a circle centered at a with radius r in B ∩ P and tangent to S n−1 at the point z. Let two distinct points x  , y  ∈ C be such that |x  | = |y  | and (x  , z, y  ) = α ∈ (0, π). Then for arbitrary two distinct points x , y ∈ C with (x, z, y) = α, there holds (1 − |x|2 )(1 − |y|2 ) ≤ (1 − |x  |2 )(1 − |y  |2 ). n

Proof Without loss of generality, we may assume that B ∩ P = B2 . Choose two distinct points x , y ∈ C and (x, z, y) = α. By symmetry, we may also assume

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12 Comparison of Metrics

Fig. 12.2 The 3 cases in the proof of Lemma 12.4

that |x| ≤ |y|, and the triples (x, z, y) and (x  , z, y  ) are labeled in the positive order on C, respectively (see Fig. 12.2). It is easy to see that r = 1 − |a| and [x  , y  ] ⊥ L(0, z), namely, x  , y  are symmetric with respect to L(0, z). For the inequality, we divide the proof into three cases. π 2

Case 1. 0 < α < 



Let θ = (0, a, x+y 2 ) ∈ [0, π − α). It is clear that

(0, a, x +y 2 ) = 0. Then x=a−

a a (1 − |a|)ei(α−θ) and y = a − (1 − |a|)e−i(α+θ). |a| |a|

Hence by Lemma 12.3, we have (1 − |x|2 )(1 − |y|2 ) = 4|a|2(1 − |a|)2f (θ ) ≤ 4|a|2(1 − |a|)2 f (0), where f (θ ) = (1 + cos(α + θ ))(1 + cos(α − θ )). Namely, (1 − |x|2 )(1 − |y|2 ) ≤ (1 − |x  |2 )(1 − |y  |2 ). Case 2. α = π2 Let θ = (z, a, y) ∈ (0, π/2]. It is clear that (z, a, x  ) = (z, a, y  ) = π2 . Then x =a+

a a (1 − |a|)e−i(π−θ) and y = a + (1 − |a|)eiθ . |a| |a|

Hence we have (1 − |x|2 )(1 − |y|2 ) = 4|a|2(1 − |a|)2 sin2 θ ≤ 4|a|2(1 − |a|)2 sin2 Therefore, we have (1 − |x|2 )(1 − |y|2 ) ≤ (1 − |x  |2 )(1 − |y  |2 ).

π . 2

12.1 The Unit Ball

Case 3.

π 2

(z, a,

213

< α < π Let θ = (z, a, x+y 2 ) ∈ [0, π − α). It is clear that

x  +y  2

x =a+

) = 0. Then a a (1 − |a|)e−i(π−α+θ) and y = a + (1 − |a|)ei(π−α−θ). |a| |a|

By an argument similar to Case 1, we have (1 − |x|2 )(1 − |y|2 ) = 4|a|2(1 − |a|)2f (θ ) ≤ 4|a|2(1 − |a|)2 f (0), where f (θ ) = (1 + cos(α + θ ))(1 + cos(α − θ )). Namely, (1 − |x|2 )(1 − |y|2 ) ≤ (1 − |x  |2 )(1 − |y  |2 ).  

By Cases 1–3, the proof is complete. Corollary 12.5 Let x , y , x  , y  be as in Lemma 12.4. Then ρBn (x, y) ≥ ρBn (x  , y  ).

We next compare the metrics vBn (x, y) and ρBn (x, y) when one of the points {x, y} is the origin. Lemma 12.6 Let x ∈ Bn . Then vBn (0, x) ≤

1 ρBn (0, x), 2

and vBn (0, x) ≤ jBn (0, x). The constant is sharp.

1 2

in the first inequality is the best possible and the second inequality

Proof For all x ∈ Bn \ {0} and x = 0, by (4.14) we have ρBn (0, x) = 2arth |x| and jBn (0, x) = log

1 . 1 − |x|

By (12.3), Lemma 12.2(1)–(2), we obtain the inequalities, which are sharp if |x| →   0+ . The third and the last special case in comparing the metrics vBn and ρBn is when the points x and y are at the same distance from the origin.

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12 Comparison of Metrics

Lemma 12.7 Let x , y ∈ Bn and |x| = |y|. Then vBn (x, y) ≤ ρBn (x, y), and the inequality is sharp. Proof Let |x| = |y| ∈ (0, 1) and θ = 12 (x, 0, y) ∈ (0, π/2]. Then by (4.16) ρBn (x, y) = 2arsh

2|x| sin θ . 1 − |x|2

By (12.4) and making substitution of r = sin θ and c = |x| in Lemma 12.2(3), we have vBn (x, y) ≤ ρBn (x, y). For the sharpness, let |x| = |y| = 1 − 1/t and sin θ = e−t (t > 0). Then by l’Hôpital’s Rule vBn (x, y) lim = lim t →+∞ ρBn (x, y) t →+∞

arctan

√

1−1/t

et [1−(1−1/t ) 1−e−2t ] 2(1−1/t ) arsh et [1−(1−1/t )2 ]

1 − (1 − 1/t)2 √ = 1. t →+∞ 2[1 − (1 − 1/t) 1 − e −2t ]

= lim

 

Applying Lemma 12.6, we obtain the result. Theorem 12.8 For all x, y ∈

Bn vBn (x, y) ≤ ρBn (x, y).

Proof For x , y ∈ Bn , x = y and |x| = |y|, by Lemma 12.4, Lemma 12.7, Corollary 12.5 there exist x  , y  ∈ Bn such that |x  | = |y  |, |x − y| = |x  − y  | and vBn (x, y) = vBn (x  , y  ) ≤ ρBn (x  , y  ) ≤ ρBn (x, y). By Lemma 12.7, the inequality holds for all x , y ∈ Bn and it is sharp.

 

If x, y and 0 are collinear, then we actually get an equality for the metrics vBn and ρBn . Lemma 12.9 For all x, y ∈ Bn tan vBn (x, y) = sh

ρBn (x, y) 2

12.1 The Unit Ball

215

if and only if x and y are collinear with 0. Proof It suffices to consider the 2-dimensional case. ω ∩ S 1 , where E ω is the envelope such that For x , y ∈ B2 and x = y. Let z ∈ Exy xy ω is the supremum angle in the definition (11.4), i.e., vBn (x, y) = (x, z, y) = ω. Then there exists exactly one circle S 1 (a, 1 − |a|) which passes through x , y , z and is tangent to S 1 . For the convenience of proof, we suppose that the three points x , z , y are labelled in the positive order on S 1 (a, 1 − |a|). By symmetry, we may assume that |x| ≤ |y|. By geometric observation, vB2 (x, y) ∈ (0, π/2) if 0 , x , y are collinear or one of the two points x , y is 0 (cf. Fig. 12.2) or tan vB2 (x, y) = sh ρ(x,y) 2 . It is clear that |x − y| = 2(1 − |a|) sin ω. Let θ = (0, a, x+y 2 ), then by the proof of Case 1 in Lemma 12.4, we get (1 − |x|2 )(1 − |y|2 ) = 4|a|2(1 − |a|)2f (θ ), where f (θ ) is as in Lemma 12.3 by taking α = ω. Hence, we have tan vB2 (x, y) = sh

 ρB2 (x, y) ⇔ |a| f (θ ) = cos ω. 2

Let s = |a| and | x+y 2 | = t in the sequel. If one of the two points x , y is 0, then 0 ∈ S 1 (a, 1/2) and θ = ω. Hence tan vB2 (0, y) = sh

 ρB2 (0, y) ⇔ s f (ω) = cos ω, 2

and the last equality clearly holds. If 0 , x , y are collinear, we consider the following two cases. Case 1. 0 ∈ B 2 (a, 1 − s) By the law of cosines cos(ω + θ ) =

s 2 + (1 − s)2 − ((1 − s) sin ω + t)2 2s(1 − s)

cos(ω − θ ) =

s 2 + (1 − s)2 − ((1 − s) sin ω − t)2 . 2s(1 − s)

and

Since ((1 − s) sin ω + t)((1 − s) sin ω − t) = 1 − 2s,

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12 Comparison of Metrics

we get t 2 = (1 − s)2 sin2 ω − (1 − 2s). Thus we have 4s 2 (1 − s)2 f (θ ) = (1 − (1 − s)2 sin2 ω − t 2 )2 − 4t 2 (1 − s)2 sin2 ω = 4(1 − s)2 cos2 ω. Therefore,  s f (θ ) = cos ω. Case 2. 0 ∈ B2 \ B 2 (a, 1 − s) By the law of cosines cos(ω + θ ) =

s 2 + (1 − s)2 − (t + (1 − s) sin ω)2 2s(1 − s)

cos(ω − θ ) =

s 2 + (1 − s)2 − (t − (1 − s) sin ω)2 . 2s(1 − s)

and

Since (t + (1 − s) sin ω)(t − (1 − s) sin ω) = 2s − 1, we have, by an argument similar to the proof of Case 1,  s f (θ ) = cos ω. By Cases 1–2, we conclude that 0 , x , y are collinear implies tan vB2 (x, y) = sh ρ(x,y) 2 . Next, suppose that 0 , x , y are noncollinear and tan vB2 (x, y) = sh ρ(x,y) 2 . Then there exist two points x  , y  ∈ S 1 (a, 1 − s) such that 0 , x  , y  are collinear and |x  − y  | = |x − y|. Then by the above proof and the monotonicity of f , we have tan vB2 (x, y) = tan vB2 (x  , y  ) = sh

ρ 2 (x, y) ρB2 (x  , y  ) = sh B , 2 2

which is a contradiction. Therefore, if neither of the two points x , y is 0 and tan vB2 (x, y) = sh ρ(x,y) 2 , then 0 , x , y are collinear. This completes the proof.  

12.1 The Unit Ball

217

Lemma 12.9 suggests that we should compare the visual angle metric with the quantity   ρBn (x, y) , ρB∗ n (x, y) = arctan sh 2

x, y ∈ Bn

∗ is a metric and Möbius instead of the hyperbolic metric. Next we show that ρD invariant in the unit ball and the upper half space.

Lemma 12.10 Let D ∈ {Bn , Hn }. Then for x, y ∈ D   ρD (x, y) ∗ ρD (x, y) = arctan sh 2 is a Möbius invariant metric. Proof The function f : x → arctan(sh(x/2)) is increasing on [0, ∞), f (x)/x is ∗ (x, y) is a decreasing on (0, ∞), and f (0) = 0. Therefore, by Exercise 5.24(2), ρD metric, and the Möbius invariance immediately follows by the Möbius invariance of ρD (x, y).   The following result shows a relation between ρB∗ n and vBn . Theorem 12.11 For all x, y ∈ Bn ρB∗ n (x, y) ≤ vBn (x, y) ≤ 2ρB∗ n (x, y). The first inequality holds as an equality if x and y are collinear with 0. The constant 2 in the second inequality is the best possible. Proof It suffices to consider the 2-dimensional case. Fix x , y ∈ B2 with x = y. In the same way as in the proof of Lemma 12.9, we have the point z ∈ S 1 and the angle ω such that vBn (x, y) = (x, z, y) = ω. We also have the circle S 1 (a, 1 − |a|) which is through x , y , z and tangent to S 1 . By Lemma 12.4, there exist x  , y  ∈ S 1 (a, 1 − |a|) such that (x  , z, y  ) = (x, z, y) and |x  | = |y  |. For the convenience of proof, we suppose that the three points x , z , y are labeled in the positive order on S 1 (a, 1 − |a|), and so are the the points x  , z , y  . Without loss of generality, we may still assume that |x| ≤ |y| (cf. Fig. 12.2). By the proof of Lemma 12.4, we have (1 − |x  |2 )(1 − |y  |2 ) = 4|a|2(1 − |a|)2(1 + cos ω)2 and |x  − y  | = |x − y| = 2(1 − |a|) sin ω.

218

12 Comparison of Metrics

Therefore, by Lemma 12.4 tan

ω |x  − y  | = |a|  2 (1 − |x  |2 )(1 − |y  |2 ) ≤ 

|x − y| (1 − |x|2 )(1 − |y|2 )

= sh

ρB2 (x, y) . 2

(12.6)

Thus we have proved the right-hand side of the inequality. For the sharpness, let x = (1 − t) + i t and y = (1 − t) − i t (0 < t < 1). Then x , y ∈ S 1 (1 − t, t) and |x − y| = 2t. Therefore, we have v 2 (x, y) π = lim lim B∗ + + t →0 ρ 2 (x, y) t →0 2 B

 arctan

1 1−t

−1

= 2.

To prove the left-hand side of the inequality, we only need to consider is always nonnegative. The equality clearly vBn (x, y) ∈ (0, π/2) because sh ρ(x,y) 2 holds if one of the two points x , y is 0 by Lemma 12.9. We consider the case x = 0 in the sequel. Let x  , y  be two points such that the three points x  , z , y  are labeled in the positive order on S 1 (a, 1 − |a|), and |x  − y  | = |x − y|, and 0 , x  , y  are collinear. Then by the definition of the visual angle metric, Lemma 12.9 and Lemma 12.3, we get vB2 (x, y) = vB2 (x  , y  ) = ρB∗ 2 (x  , y  ) ≥ ρB∗ 2 (x, y). Therefore, the left-hand side of the inequality is proved, and the equality is clear by Lemma 12.9. This completes the proof.   Corollary 12.12 For all x, y ∈ Bn |x − y|(2 − |x − y|) , vBn (x, y) ≤ 2 arctan  2 (1 − |x|2 )(1 − |y|2) √ and the equality holds if |x| = |y| = ( 2 sin(θ + (0, π/2).

π −1 4 ))

and θ =

1 2 (x, 0, y)

∈

Proof We still consider the 2-dimensional case. Let a be as in the proof of Lemma 12.11. Since |a| + |a − x| = |a| + |a − y| = 1, we get |a| =

2 − (|a − x| + |a − y|) 2 − |x − y| ≤ . 2 2

Then by (12.6), we have proved the inequality.

12.2 The Upper Half Space

219

For the equality, let |x| = |y| > 0 and θ = 12 (x, 0, y) > 0. Then |x − y|(2 − |x − y|) 2|x| sin θ (1 − |x| sin θ )  . = 2 2 1 − |x|2 2 (1 − |x| )(1 − |y| ) By (12.4) and (12.7) the equality holds if |x| = |y| =

1 sin θ+cos θ .

(12.7)  

12.2 The Upper Half Space We move on to the next domain, which is the upper half space. We compare first the visual angle metric and the hyperbolic metric. To obtain our first result we need two lemmas. Lemma 12.13 Let a ∈ Hn . Let C = S(a, r) be a circle centered at a with radius n r in H and tangent to ∂Hn at point z. Let two distinct points x  , y  ∈ C such that  |x − z| = |y  − z| and (x  , z, y  ) = α ∈ (0, π). Then for arbitrary two distinct points x , y ∈ C with (x, z, y) = α, there holds d(x , ∂Hn )d(y , ∂Hn ) ≤ d(x  , ∂Hn )d(y  , ∂Hn ).

(12.8)

2

Proof Without loss of generality, we may assume that C ⊂ H and z = 0. Choose two distinct points x , y ∈ C and (x, z, y) = α. By symmetry, we may also assume that |x| ≤ |y|, and the triples (x, z, y) and (x  , z, y  ) are labeled in the positive order on C, respectively (see Fig. 12.3). It is clear that r = |a|, [x  , y  ] ⊥ L(0, a), namely, x  , y  are symmetric with respect to L(0, a). Furthermore, inequality (12.8) reduces to Im x Im y ≤ Im x  Im y  . In the same way as in the proof of Lemma 12.4, we also divide the proof into three cases.

Fig. 12.3 The 3 cases in the proof of Lemma 12.13

220

12 Comparison of Metrics

Case 1. 0 < α < (2a, a,

x  +y  2

π 2

Let θ = (2a, a, x+y 2 ) ∈ [0, π − α). It is clear that

) = 0. Then x = a(1 + ei(α+θ) ) and y = a(1 + e−i(α−θ) ).

Moreover, Im x Im y = |a|2f (θ ), where f (θ ) = (1 + cos(α + θ ))(1 + cos(α − θ )), by Lemma 12.3, we have Im x Im y ≤ Im x  Im y  . Case 2. α = π2 Let θ = (0, a, x) ∈ (0, π/2]. It is clear that (0, a, x  ) = (0, a, y  ) = π2 . Then x = a(1 − e−iθ ) and y = a(1 − ei(π−θ) ). Moreover, Im x Im y = |a|2 sin2 θ ≤ |a|2 sin2

π , 2

and hence Im x Im y ≤ Im x  Im y  . Case 3.

π 2

(0, a,

< α < π Let θ = (0, a, x+y 2 ) ∈ [0, π − α). It is clear that

x  +y  2

) = 0. Then x = a(1 − e−i(π−α−θ) ) and y = a(1 − ei(π−α+θ) ).

Moreover, Im x Im y = |a|2f (θ ), where f (θ ) = (1 + cos(α + θ ))(1 + cos(α − θ )), by Lemma 12.3 we have Im x Im y ≤ Im x  Im y  . By Cases 1–3, the proof is complete. Corollary 12.14 Let x , y , x  , y  be as in Lemma 12.13. Then ρHn (x, y) ≥ ρHn (x  , y  ).

 

12.2 The Upper Half Space

221

Lemma 12.15 Let x , y ∈ Hn . Then tan vHn (x, y) = sh

ρHn (x, y) 2

if and only if L(x, y − x) is perpendicular to the boundary ∂Hn . The next result compares the metrics vHn and ρHn . To be more precise, we use ∗ instead of ρ n . ρH n H Proof It suffices to consider the 2-dimensional case. ω ∩∂H2 , where E ω is the envelope such that For x , y ∈ H2 and x = y. Let z ∈ Exy xy ω is the supremum angle in Definition 11.4, i.e., vH2 (x, y) = (x, z, y) = ω. Then there exists exactly one circle S 1 (a, r) which passes through x , y , z and is tangent to ∂H2 . For the convenience of proof, we suppose that the three points x , z , y are labeled in the positive order on S 1 (a, r). Without loss of generality, we may assume that z = 0 and |x| ≤ |y|. By geometric observation, vH2 (x, y) ∈ (0, π/2) if L(x, y− x) is perpendicular to ∂H2 (cf. Fig. 12.3) or tan vH2 (x, y) = sh ρ(x,y) 2 . It is clear that |x − y| = 2|a| sin ω. Let θ = (2a, a, x+y 2 ). By the proof of Case 1 in Lemma 12.13, we have Im x Im y = |a|2 f (θ ), where f (θ ) is as in Lemma 12.3 by taking α = ω. Since ρ 2 (x, y) sh H = 2



1 |x − y| (chρH2 (x, y) − 1) = √ , 2 2 Im x Im y

by Lemma 12.3, we have tan vH2 (x, y) = sh

 ρH2 (x, y) ⇔ f (θ ) = cos ω ⇔ θ = π/2 ⇔ Re x = Re y. 2

This completes the proof.

  ∗ (x, y) = arctan sh ρ(x,y) for x , y ∈ Hn . Then Theorem 12.16 Let ρH n 2

 

∗ ∗ ρH n (x, y) ≤ vHn (x, y) ≤ 2ρHn (x, y).

The equality holds in the left-hand side if and only if L(x, y − x) is perpendicular to the boundary ∂Hn and in the right-hand side if and only if L(x, y − x) is parallel to the boundary ∂Hn . Proof It suffices to consider the 2-dimensional case. For x , y ∈ H2 and x = y. In the same way as in the proof of Lemma 12.15, we have the point z ∈ ∂H2 and the angle ω such that vH2 (x, y) = (x, z, y) = ω.

222

12 Comparison of Metrics

We also have the circle S 1 (a, r) which is through x , y , z and tangent to ∂H2 . By Lemma 12.13, there exist x  , y  ∈ S 1 (a, r) such that (x  , z, y  ) = (x, z, y) and Im x  = Im y  . For the convenience of proof, we suppose that the three points x , z , y are labeled in the positive order on S 1 (a, r), and so are x  , z , y  . Without loss of generality, we still assume that z = 0 and |x| ≤ |y| (cf. Fig. 12.3). By the proof of Lemma 12.13, we have |x  − y  | = |x − y| = 2|a| sin ω and Im x  Im y  = |a|2(1 + cos ω)2 . Therefore, by Lemma 12.13 tan

|x  − y  | ρ 2 (x, y) |x − y| ω =  , = sh H ≤ √   2 2 2 Im x Im y 2 Im x Im y

which implies the right-hand side of the inequality with equality if and only if Im x = Im y by the proof of Lemma 12.13. To prove the left-hand side of the inequality, we only need to consider vH2 (x, y) ∈ (0, π/2) since sh ρ(x,y) is always nonnegative. Let x  , y  be two 2   points such that x , z , y are labeled in the positive order on S 1 (a, 1 − |a|) and |x  − y  | = |x − y|, and L(x  , y  − x  ) is perpendicular to ∂H2 . Then by the definition of the visual angle metric, Lemma 12.15 and Lemma 12.3, we get ∗   ∗ vH2 (x, y) = vH2 (x  , y  ) = ρH 2 (x , y ) ≥ ρH2 (x, y).

Thus we prove the left-hand side of the inequality, and the equality holds if and only if Re x = Re y. This completes the proof.   Next we show that the hyperbolic metric gives an upper bound for the visual angle metric in the upper half space. We consider first a special case and then generalize the result. Lemma 12.17 Let x , y ∈ Hn and d(x , ∂Hn ) = d(y , ∂Hn ). Then vHn (x, y) ≤ ρHn (x, y), and the inequality is sharp.

12.3 General Domains

223

Proof It suffices to prove the 2-dimensional case. Let x , y be two distinct points in H2 . Since both metrics are invariant under translations, we may assume that y1 = Re y = −Re x > 0 and y2 = Im y = Im x > 0. Then 



y1 ρH2 (x, y) = arch 1 + 2 y2 By (11.7) and making substitution of r =

y1 y2

2  .

in Lemma 12.2(4), we have

vH2 (x, y) ≤ ρH2 (x, y) and the inequality is sharp if r → 0.

 

Lemma 12.18 For all x, y ∈ Hn vHn (x, y) ≤ ρHn (x, y). and the inequality is sharp. Proof For x , y ∈ Hn , x = y and d(x , ∂Hn ) = d(y , ∂Hn ), by Lemma 12.13, Lemma 12.17 and Corollary 12.14, there exist x  , y  ∈ Hn such that d(x  , ∂Hn ) = d(y  , ∂Hn ), |x − y| = |x  − y  | and vHn (x, y) = vHn (x  , y  ) ≤ ρHn (x  , y  ) ≤ ρHn (x, y). By Lemma 12.17, the inequality holds for all x , y ∈ Hn and it is sharp.

 

Next we consider the relation between the triangular ratio metric and the distance ratio metric.

12.3 General Domains We compare hyperbolic type metrics in general domains. We begin with the triangular ratio metric and the distance ratio metric. Recall first that by Lemma 11.13 we have ∗ sG (x, y) ≤ 2jG (x, y) = 2 th

jG (x, y) ≤ jG (x, y) 2

for all domains G  Rn and all x, y ∈ Rn . We next refine this result.

224

12 Comparison of Metrics

Theorem 12.19 Let G be a proper subdomain of Rn . Then for all x, y ∈ G we have sG (x, y) ≤ and the constant

1 log 3

1 jG (x, y) log 3

≈ 0.91 is the best possible.

Proof Let us fix the points x and y. By rescaling the domain we may assume that |x − y| = 1. We can also assume that d(x) ≤ d(y), because otherwise we can swap the points. We denote t = d(x) > 0. Now   1 jG (x, y) = log 1 + t and we divide the proof into two cases: t ≤ 12 and t > 12 . We assume first that t ≤ 12 . Now jG (x, y) ≥ log 3 and since sG (x, y) ≤ 1 we have sG (x, y) ≤ 1 ≤

jG (x, y) . log 3

We assume then that t > 12 . We want to maximize sG (x, y) in terms of t. Now sG (x, y) ≤ sB n (x,t )∪B n(y,t )(x, y) =

1 |x − y| = |x − z| + |y − z| 2t

for some z ∈ ∂(B n (x, t) ∪ B n (y, t)), and we want to find a lower bound for the function   1 jG (x, y) 1 ≥ 2t log 1 + f (t) = , t> . sG (x, z) t 2 By 4.2(6) g(t) = (log(1 + t))/t is decreasing for t, so it is increasing for 1t , thus f (t) is increasing. We conclude f (t) > f ( 12 ) = log 3 and the claimed inequality is proved. The constant log1 3 can be easily verified to be the best possible by investigating the domain G = Rn \ {0}. For a point x ∈ G selecting y = −x gives sG (x, y) = 1 and jG (x, y) = log 3.   Next we compare the distance ratio metric with the point-pair function p.

12.3 General Domains

225

Theorem 12.20 If x, y ∈ G ⊂ Rn , then pG (x, y) ≤

√ ∗ 1 2 jG (x, y) ≤ √ jG (x, y). 2

Proof The first inequality follows from Lemma 11.14 and the second one is a basic property of th.   Our next aim is to compare hyperbolic type metrics in different domains. We compare metrics first in two domains, the unit ball Bn and the punctured space Rn \ {e1 }. Lemma 12.21 For 0 < b ≤ a the function f (x) =

log(1 + ax) , x ∈ (0, ∞), log(1 + bx)

is decreasing onto (1, a/b). Proof Since f  (x) =

a 1+ax

log(1 + bx) −

b 1+bx

log(1 + ax)

log (1 + bx) 2

the inequality f  (x) ≤ 0 is equivalent to 1 + ax 1 + bx log(1 + bx) ≤ log(1 + ax). b a

(12.9)

Now we show that the function g(c) =

1 + cx log(1 + cx) c

is increasing on (0, ∞), which implies (12.9) and the assertion. This is clear because 0 < b ≤ a and g  (c) =

cx − log(1 + cx) >0 c2

as log(1 + y) < y for y > 0. Taking limit when x → 0 or x → ∞, resp. we see   that the limits are the indicated values. Theorem 12.22 Let t ∈ (0, 1) and m ∈ {j, p, s}. There exists a constant cm = cm (t) > 1 such that for all x, y ∈ Bn with |x|, |y| < t we have mBn (x, y) ≤ cm mRn \{e1 } (x, y).

226

12 Comparison of Metrics

Moreover, cm (t) → 1 as t → 0 and cm (t) → ∞ as t → 1, for all m ∈ {j, p, s}. 1 Proof We denote m1 = mBn , m2 = mRn \{e1 } and find upper bound for m m2 , which gives us cm . Let us start with m = j . We denote z = |x − y| ∈ [0, 2t) and obtain by Lemma 12.21     z z log 1 + 1−t log 1 + min{1−|x|,1−|y|} j1  ≤   = z j2 log 1 + log 1 + z

min{|x−e1 |,|y−e1 |}

 log 1 +  ≤ lim z→0 log 1 +

z 1−t z 1+t

1+t



 = lim

z→0

1+t 1+t +z = = cj , 1−t +z 1−t

where the second equality follows from l’Hôpital’s rule. Obviously cj → 1 as t → 0 and cj → ∞ as t → 1. Let us now consider m = p. Now p12 p22

=

4t 2 + 4(1 + t)2 |x − y|2 + 4|x − e1 ||y − e1 | 2t 2 + 2t + 1 ≤ = |x − y|2 + 4(1 − |x|)(1 − |y|) 0 + 4(1 − t)2 t 2 − 2t + 1

and we can choose  cp =

2t 2 + 2t + 1 . t 2 − 2t + 1

Clearly cp → 1 as t → 0 and cp → ∞ as t → 1. Next we set m = s and obtain by geometry s1 2(1 + t) 1+t |x − e1 | + |y − e1 | ≤ = = cs . = s2 infz∈∂Bn |x − z| + |z − y| 2(1 − t) 1−t Again it is clear that cs → 1 as t → 0 and cs → ∞ as t → 1.

 

Next we want to estimate how does a hyperbolic metric change, when we remove a point from the original domain. Theorem 12.23 Let G ⊂ Rn , x ∈ G, t ∈ (0, 1) and m ∈ {j, p, s}. Then there exists a constant cm = cm (t) such that for all y, z ∈ G \ B n (x, tdG (x)) we have mG\{x} (y, z) ≤ cm mG (y, z). Moreover, the constant is best possible as t → 1. This means that cj , cp , cs → 2 as t → 1.

12.3 General Domains

227

Proof We denote G = G \ {x} and will find an upper bound for mGG(y,z) . We consider first the case m = j . If dG (y) = dG (y) and dG (z) = dG (z), then there is nothing to prove as jG (y, z) = jG (y, z) and we can choose cj = 1. We consider next two cases: dG (y) = dG (y), dG (z) = dG (z) and dG (y) = dG (y), dG (z) = dG (z). Let us assume dG (y) = dG (y) and dG (z) = dG (z) (or by symmetry we could as well assume dG (y) = dG (y) and dG (z) = dG (z)). Now m (y,z)

   log 1 + min{d |y−z| log 1 + (y),d (z)} jG (y, z)   G G  =   = |y−z| jG (y, z) log 1 + min{dG (y),dG(z)} log 1 +

|y−z| min{|y−x|,dG (z)} |y−z| min{dG (y),dG(z)}

 .

Let us assume that dG (z) ≤ dG (y). If dG (z) ≤ |y − x| then jG (y, z) = jG (y, z) and there is nothing to prove. If dG (z) ≥ |y − x| then     |y−z| |y−z| log 1 + log 1 + |y−x| t dG (x) jG (y, z)  ≤    = |y−z| jG (y, z) log 1 + dG (z) log 1 + d|y−z| G (z)     log 1 + t|y−z| log 1 + t|y−z| dG (x) dG (x)  ≤ .   ≤ |y−z| log 1 + d|y−z| log 1 + (y) |y−x|+d (x) G G If |x − y| ≤ dG (x) then we have by Lemma 12.21     |y−z| |y−z| log 1 + log 1 + jG (y, z) t dG (x) t dG (x)  ≤   ≤ lim |y−z| |y−z| |y−z|/d (x)→0 jG (y, z) G log 1 + 2dG (x) log 1 + 2d G (x) ≤

lim

|y−z|/dG (x)→0

2+ t+

|y−z| dG (x) |y−z| dG (x)

=

2 . t

If |x − y| ≥ dG (x) again by Lemma 12.21     |y−z| |y−z| log 1 + log 1 + jG (y, z) |y−x| |y−x|   ≤   ≤ lim |y−z| |y−z|/|y−x|→0 log 1 + |y−z| jG (y, z) log 1 + 2|y−x| 2|y−x| ≤

lim

|y−z|/|y−x|→0

2+ 1+

|y−z| |y−x| |y−z| |y−x|

= 2.

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12 Comparison of Metrics

Let us then assume dG (y) ≤ dG (z). Now dG (y) = dG (y) implies |y − x| < dG (y) and thus     |y−z| |y−z| log 1 + log 1 + jG (y, z) |y−x| |y−x|   ≤ . = (12.10) |y−z| |y−z| jG (y, z) log 1 + log 1 + |y−x|+dG (x)

dG (y)

If |x − y| ≤ dG (x) we have by (12.10) and Lemma 12.21     log 1 + t|y−z| log 1 + t|y−z| dG (x) dG (x) jG (y, z) ≤    ≤ lim |y−z| |y−z|/dG (x)→0 log 1 + |y−z| jG (y, z) log 1 + 2d 2dG (x) G (x) ≤

2+

lim

|y−z|/dG (x)→0

t+

|y−z| dG (x) |y−z| dG (x)

=

2 . t

If dG (x) ≤ |x − y| we have by (12.10) and Lemma 12.21     |y−z| |y−z| log 1 + log 1 + jG (y, z) |y−x| |y−x|  ≤    ≤ lim |y−z| |y−z| |y−z|/|y−x|→0 jG (y, z) log 1 + 2|y−x| log 1 + 2|y−x| ≤

lim

2+

|y−z|/|y−x|→0

1+

|y−z| |y−x| |y−z| |y−x|

= 2.

Let us then assume dG (y) = dG (y) and dG (z) = dG (z). Now we may assume by symmetry that |y − x| ≤ |z − x| and thus     |y−z| |y−z| log 1 + log 1 + |y−x| |y−x| jG (y, z)  ≤    = |y−z| |y−z| jG (y, z) log 1 + min{dG (y),dG(z)} log 1 + |y−x|+d (x) G and this is exactly the same as (12.10) so we know that jG (y, z) 2 ≤ . jG (y, z) t Putting all this together gives us cj = 2t . Let now m = p. If dG (y) = dG (y) and dG (z) = dG (z), then there is nothing to prove as pG (y, z) = pG (y, z) and we can choose cp = 1. We consider next two cases: dG (y) = dG (y), dG (z) = dG (z) and dG (y) = dG (y), dG (z) = dG (z).

12.3 General Domains

229

Let us assume dG (y) = dG (y) and dG (z) = dG (z) (or by symmetry we could as well assume dG (y) = dG (y) and dG (z) = dG (z)). Now 2 (y, z) pG  2 (y, z) pG

=

|y − z|2 + 4dG (y)dG (z) |y − z|2 + 4dG (y)dG (z) = |y − z|2 + 4dG (y)dG (z) |y − z|2 + 4|y − x|dG (z)

≤

|y − z|2 + 4(|x − y| + dG (x))dG (z) |y − z|2 + 4|y − x|dG (z)

= 1+

4dG (x)dG (z) 4dG (x)dG (z) ≤1+ |y − z|2 + 4|y − x|dG (z) 0 + 4tdG (x)dG (z)

1 = 1+ . t Let us then assume dG (y) = dG (y) and dG (z) = dG (z). Now 2 (y, z) pG  2 (y, z) pG

=

|y − z|2 + 4dG (y)dG (z) |y − z|2 + 4dG (y)dG (z) = |y − z|2 + 4dG (y)dG (z) |y − z|2 + 4|y − x||z − x|

≤

|y − z|2 + 4(|x − y| + dG (x))(|x − z| + dG (x)) |y − z|2 + 4|y − x||z − x|

= 1+

4(|x − y|dG (x) + |x − z|dG (x) + dG (x)2 ) |y − z|2 + 4|y − x||z − x|

≤ 1+

4(|x − y|dG (x) + |x − z|dG (x) + dG (x)2 ) 4|y − x||z − x|

= 1+

|x − z|dG (x) dG (x)2 |x − y|dG (x) + + |y − x||z − x| |y − x||z − x| |y − x||z − x|

≤ 1+

|x − z|dG (x) dG (x)2 |x − y|dG (x) + + |y − x|tdG (x) tdG (x)|z − x| tdG (x)tdG (x)

= 1+

1 2 2t + 1 + 2 =1+ . t t t2

Combining the cases we obtain cp = t +1 t . Let us finally consider the case m = s. Now infu∈∂G |y − u| + |u − z| sG (y, z) = sG (y, z) infu∈∂G |y − u| + |u − z| and if the infimum in the denominator is obtained at a point u ∈ ∂G, then there is nothing to prove as sG (y, z) = sG (y, z) and we can choose cs = 1. If this is not the

230

12 Comparison of Metrics

case, then infu∈∂G |y − u| + |u − z| infu∈∂G |y − u| + |u − z| sG (y, z) = = sG (y, z) infu∈∂G |y − u| + |u − z| |y − x| + |x − z| ≤

2dG (x) |x − y| + dG (x) + |x − z| + dG (x) =1+ |y − x| + |x − z| |y − x| + |x − z|

≤ 1+

2dG (x) 1 =1+ 2tdG (x) t

and we can choose cs = 1 + 1t . We see easily that cj , cp , cs → 2 as t → 1. We show next that the constants cj , cp and cs are best possible. In all three cases we consider G = Rn \ {0}. We start with the case m = j . Let a > 0. For points x = e1 , y = (1 + t)e1 and z = (1 + t + a)e1 we have   log 1 + at jG (y, z)   = a jG (y, z) log 1 + 1+t and log(1 + a) jG (y, z) → jG (y, z) log(1 + a2 ) as t → 1. The asymptotic behavior is clear since log(1 + a) →2 log(1 + a2 ) as a → 0. We √ next consider the case m = p. √ Let a ∈ (0, t]. For points x = e1 , y = (1 + t 2 − a 2 )e1 + ae2 and z = (1 + t 2 − a 2 )e1 − ae2 we have |y − z| = 2a and 2 (y, z) pG  2 (y, z) pG

=

|y − z|2 + 4dG (y)dG (z) = |y − z|2 + 4dG (y)dG (z)

4a 2

  2  √ 2 2 2 +4 a + 1+ t −a 4a 2 + 4t 2

Now 2 (y, z) pG  2 (y, z) pG

4a 2 →

 2   √ 2 2 +4 a + 1+ 1−a 4a 2 + 4

√ 4a 2 + 8 + 8 1 − a 2 = 4a 2 + 4

.

12.3 General Domains

231

as t → 1 and √ 4a 2 + 8 + 8 1 − a 2 →4 4a 2 + 4 as a → 0. We √ finally consider the case m = √s. Let a ∈ (0, t]. For points x = e1 , y = (1 + t 2 − a 2 )e1 + ae2 and z = (1 + t 2 − a 2 )e1 − ae2 we have |y − z| = 2a and  sG (y, z) = sG (y, z)

2a/(2t) 2 =  √ 2 2 2 2a/(2 a + 1 + t − a ) 

 2 √ a2 + 1 + t 2 − a2 t



2   a2 + 1 + 1 − a2

→ as t → 1 and 

2     a2 + 1 + 1 − a2 = 2 + 2 1 − a2 → 2  

as a → 0.

Our next goal is to prove Theorem 12.26. We formulate a few auxiliary results. Lemma 12.24 Let D  Rn be a domain. Then for x, y ∈ D sin

vD (x, y) ≤ sD (x, y), 2

vD (x, y) ≤ πsD (x, y).

(12.11) (12.12)

Proof Fix θ ∈ (0, π) such that θ sD (x, y) = sin . 2

(12.13)

Then the ellipsoid θ E = {z ∈ Rn : |z − x| + |z − y| < |x − y|/ sin } 2 is contained in D. Hence by domain monotonicity vD (x, y) ≤ vE (x, y) = θ,

(12.14)

232

12 Comparison of Metrics

and by (12.13) and (12.14) we see that sin vD (x,y) ≤ sD (x, y). Moreover, 2 θ vD (x, y) ≤ ≤π sD (x, y) sin(θ/2) where the last inequality follows from Jordan’s inequality.

 

Lemma 12.25 If D is a convex subdomain of Rn , and x, y ∈ D, then sD (x, y) ≤ vD (x, y). Proof If vD (x, y) ≥ π2 , then sD (x, y) ≤ 1 ≤ vD (x, y). Assume that vD (x, y) < π2 . Let z ∈ ∂D be such that vD (x, y) = (x, z, y). Denote by r the radius of the circle passing through x, y, z. Since D is convex, the convex hull of the set {z ∈ Rn : (x, z, y) > vD (x, y)} is contained in D and hence also the ellipsoid E = {z ∈ Rn : |z − x| + |z − y| < 2r} is contained in the domain D, see Fig. 12.4. We easily conclude that sin vD (x, y) = |x−y| 2r . Moreover, by the domain monotonicity property of the s-metric, sD (x, y) ≤ sE (x, y) =

|x − y| = sin vD (x, y) ≤ vD (x, y). 2r

 

Theorem 12.26 Let D  Rn be a convex domain. Then for all x, y ∈ D we have sD (x, y) ≤ vD (x, y) ≤ πsD (x, y). Proof The proof follows from Lemmas 12.24 and 12.25.

 

Remark 12.27 The visual angle metric and the triangular ratio metric both highly depend on the boundary of the domain. If we replace the convex domain D in Theorem 12.26 with G = B2 \ {0}, then the visual angle metric and the triangular ratio metric are not comparable in G. To this end, we consider two sequences of points xk = (1/k, 0), yk = (1/k 2 , 0) (k = 2, 3, · · · ). Then sG (xk , yk ) =

k−1 . k+1

By (12.9), we get vG (xk , yk ) = vB2 (xk , yk )     1 k . < arctan = arctan √ k+1 (k + 1) 1 + k 2

12.3 General Domains

233

z

r x

y

E

Fig. 12.4 Proof of Lemma 12.25, vD (x, y) = θ = (x, z, y)

Therefore, k−1 sG (xk , yk )  → ∞ , as k → ∞ .  ≥ 1 vG (xk , yk ) (k + 1) arctan k+1 In general it is not true that vG has a lower bound in terms of sG . For instance, this fails for G = Bn \ {0} , by the above remark and [194, Remark 2.18]. The nonlinearity condition in the next theorem is similar to the thickness condition in [541], and it ensures a lower bound for vG in terms of sG . For the case n = 2 an example of a domain satisfying the nonlinearity condition is the snowflake domain. Definition 12.28 Suppose that G ⊂ R2 is a domain. We say that ∂G , satisfies the nonlinearity condition, if there exists δ ∈ (0, 1) , such that for every z ∈ ∂G and for every r ∈ (0, d(G))  and for every line L with L ∩ B(z, r) = ∅ , there exists w ∈ B(z, r) ∩ ∂G \ y∈L B(y, δr) .

234

12 Comparison of Metrics

Theorem 12.29 Let G ⊂ R2 be a domain such that ∂G satisfies the nonlinearity condition. If x, y ∈ G and sG (x, y) < 1 then 

δ sG (x, y) vG (x, y) > arctan 6

 .

Proof Fix x, y ∈ G . We may assume that d(x) ≤ d(y) . Choose z0 ∈ ∂G such that |x − z0 | = d(x) . Let r = d(x) + |x − y| . Then B(z0 , r) ⊂ B(m, t) , m = (x + y)/2 for t = 2r . By the nonlinearity condition as we see in Fig. 12.5, vG (x, y) ≥ y−x y−x (x, w, y) = α , w = m + teiθ ( |y−x| ) , θ = arcsin δrt . Writing w1 = m + t |y−x| , β = (w, y, w1 ) and γ = (w, x, w1 ) we see that δr tan β = √ 4r 2 − δ 2 r 2 − |x − y|/2

and

δr tan γ = √ 4r 2 − δ 2 r 2 + |x − y|/2

and hence tan α = tan (β − γ ) =

Fig. 12.5 Proof of Theorem 12.29

δr|x − y| . 4r 2 − |x − y|2 /4

12.3 General Domains

235

Therefore vG (x, y) ≥ α = arctan

Then sG (x, y) ≤

δr|x − y| 4r 2 − |x − y|2/4

= arctan

δ(d(x) + |x − y|)|x − y| 4(d(x) + |x − y|)2 − |x − y|2 /4

= arctan

δ(1 + |x − y|/d(x))|x − y|/d(x) . 4(1 + |x − y|/d(x))2 − (|x − y|/2d(x))2

|x−y| 2d(x)

. A simple calculation shows that the function f (t) =   (1+t )t 1 1  (t) = 2 is increasing for t > 0 , since f + > 0. 2 2 2 2 4(1+t ) −(t /2) (4+3t ) (4+5t ) On the other hand, g(t) = f (t)/t is decreasing for t > 0 . Hence, for 0 < t ≤ 2 , 3 1 1 g(t) ≥ g(2) = 35 > 12 and f (t) ≥ 12 t. Therefore arctan

δ(1 + |x − y|/d(x))|x − y|/d(x) = arctan(f (|x − y|/d(x))δ) 4(1 + |x − y|/d(x))2 − (|x − y|/2d(x))2   δ ≥ arctan(f (2sG (x, y))δ) ≥ arctan sG (x, y) 6  

and the proof is complete. Lemma 12.30 Let G ⊂ B n (x, d(x)) . Then

Rn

be a proper subdomain of

Rn ,

x ∈ G and y ∈

|x − y| |x − y| = . |x − w| d(x) w∈∂G

sin(vG (x, y)) ≤ sup

Proof Fix x ∈ G and y ∈ B n (x, d(x)) . For each w ∈ ∂G we have by elementary geometry (x − w, y − w) ≤ θ ;

sin θ =

|x − y| . |x − w|

Taking supremum over all w ∈ ∂G we obtain |x − y| |x − y| = . |x − w| d(x) w∈∂G

sin(vG (x, y)) ≤ sup

Theorem 12.31 Let G be a proper subdomain of sG (x, y) ≤

R2 .

For x, y ∈ G ,

|x − y|/d(x)  . 1 + cos(vG (x, y)) + (|x − y|/d(x))2 − sin2 (vG (x, y))

 

236

12 Comparison of Metrics

Proof We may assume that d(x) ≤ d(y) . We first consider the case of ∂G ∩ [x, y] = ∅ . It is clear in this case that sG (x, y) = 1 and vG (x, y) = π , and the desired inequality holds as an equality. Next, we assume that ∂G ∩ [x, y] = ∅ . Let E be the interior of the envelope which defines the visual angle metric between x and y . Then D = B 2 (x, d(x)) ∪ B 2 (y, d(x))) ∪ E is a subdomain of G . Let w0 ∈ ∂D ∩ S 1 (x, d(x)) ∩ ∂E . By use of the law of cosine in the triangle xyw0 we get  |x −w0 |+|w0 −y| = (1+cos(vG (x, y)))d(x)+ |x − y|2 − d(x)2 sin2 (vG (x, y)). A simple geometric observation gives sD (x, y) = =

|x − y| |x − w0 | + |w0 − y| |x − y|/d(x)  . 1 + cos(vG (x, y)) + (|x − y|/d(x))2 − sin2 (vG (x, y))

Then the domain monotonicity of the s-metric yields the desired inequality sG (x, y) ≤ sD (x, y) .   Remarks 12.32 (1) If |x − y|/d(x) > 1 , then the square root in Theorem 12.31 is clearly welldefined. In the case |x − y|/d(x) ≤ 1 it follows from Lemma 12.30 that the square root is well-defined, too. (2) The inequalities in Theorem 12.31 are sharp in the following sense: If vG (x, y) = 0 , then sG (x, y) ≤ |x − y|/(|x − y| + 2d(x)) which together with Lemma 11.12 actually gives sG (x, y) = |x − y|/(|x − y| + 2d(x)) . If sG (x, y) = 1 , then the inequality actually gives vG (x, y) = π . Definition 12.33 Let δ ∈ (0, 1/2) . We say that a domain G ⊂ Rn satisfies condition H (δ) if for every z ∈ ∂G and all r ∈ (0, d(G)/2) there exists w ∈ B n (z, r) ∩ (Rn \ G) such that B n (w, δr) ⊂ B n (z, r) ∩ (Rn \ G) . Note that the condition H (δ) excludes domains whose boundaries have zero angle cusps directed into the domain. For instance the domain B2 \ [0, 1] does not satisfy the condition H (δ) . A similar condition has been studied also in [374] and [276] and sometimes this condition is referred to as the porosity condition. For instance, domains with smooth boundaries are in the class H (δ) (Fig. 12.6).

12.3 General Domains

237

Fig. 12.6 Condition H (δ)

Theorem 12.34 Let G ⊂ R2 be a domain satisfying the condition H (δ) . Then for all x, y ∈ G we have sin vG (x, y) ≥

δ ∗ j (x, y) . 2 G

Proof Fix x, y ∈ G . By symmetry we may suppose that d(x) ≤ d(y) . Denote r = d(x) and choose a point z ∈ ∂G such that r = |x − z| . By the condition H (δ) there exists w ∈ R2 \ G such that B 2 (w, δr) ⊂ B 2 (z, r) ∩ (R2 \ G) . Denote 2 G1 = R2 \ B (w, δr) . By the monotonicity of vG with respect to the domain we have vG (x, y) ≥ vG1 (x, y) . Geometrically, vG1 (x, y) can be found by considering the circle through x, y  . In order to find externally tangent to B 2 (w, δr) . Suppose this circle is B 2 ( c, R)  . By elementary geometry a lower bound for vG1 we need an upper bound for R  ≤ R where B 2 (c, R) corresponds to the case when y = y1 = x + x−w |x − y| . R |x−w| Then |x − y1 | = |x − y| . Using the power of the point w with respect to the circle ∂B 2 (c, R) we have δr(δr + 2R) = |x − w||y1 − w| = |x − w|(|x − w| + |x − y|) and hence 2R =

|x − w| (|x − w| + |x − y|) − δr. δr

We next utilize the law of Sine to obtain R=

|x − y1 | |x − y| = . 2 sin vG1 (x, y1 ) 2 sin vG1 (x, y1 )

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12 Comparison of Metrics

Observing that |x − w| ≤ |x − z| + |z − w| ≤ d(x) + (1 − δ)d(x) we have sin vG1 (x, y) ≥ ≥

where t =

|x−y| d(x)

|x − y| 2−δ δ (|x

− w| + |x − y|) − δr |x − y|

2−δ δ ((2 − δ)d(x) + |x

− y|) − δd(x)

=

δ t · 2 − δ t + 4−4δ 2−δ

≥

t δ ejG (x,y) − 1 δ δ ∗ · = · j (x,y) (x, y) ≥ jG G 2−δ t +2 2−δ e 2 +1

.

 

Exercise 12.35 Show that Theorem 12.22 does not hold for the visual angle metric. Exercise 12.36 Show that Theorem 12.23 does not hold for the visual angle metric. Notes 12.37 This chapter is based on [100, 194, 195, 237, 287, 291] and [297].

Chapter 13

Local Convexity of Balls

In this chapter we study some geometric properties of metric balls for small radii. It is natural to expect that balls of small radii are like euclidean balls whereas the geometric structure of ∂G strongly influences on the shape of balls for large radii. It turns out, as suggested in [559], that balls of small radii have a number of properties not valid for large radii. In fact we give a quantitative bound T0 for the radius such that balls of radii less than T0 are convex, for instance. We assume that G is a proper subdomain of Rn and m is a metric on G. Question 13.1 Does there exist a constant T0 > 0 such that Bm (x, T ) = {z ∈ G : m(x, z) < T }, is convex (in euclidean geometry) for all T ∈ (0, T0 )? Is ∂Bm (x, T ) smooth for T < T0 ? A similar question can be also posed for other geometric conditions such as starlikeness and close-to-convexity, which we define next. Definition 13.2 Let G ⊂ Rn be a domain and x ∈ G. We say that G is starlike with respect to x if for each y ∈ G the line segment [x, y] from x to y is contained in G (this notion was introduced by R. Nevanlinna, see W.K. Hayman [211, p.14]). The domain G is strictly starlike with respect to x for x ∈ G if G is bounded and each ray from xmeets ∂G at exactly one point. Definition 13.3 A domain G ⊂ Rn is close-to-convex if Rn \G can be covered with non-intersecting half-lines. By half-lines we mean sets {x ∈ Rn : x = ty + z, t > 0} and {x ∈ Rn : x = ty + z, t ≥ 0} for z ∈ Rn and y ∈ Rn \ {0}.

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_13

239

240

13 Local Convexity of Balls

13.1 Distance Ratio Metric We begin our study of local convexity properties with the distance ratio metric. We first consider the complement of the origin and then generalize the result to all domains. Theorem 13.4 Let x ∈ Rn \ {0}. Then the j -metric ball Bj (x, r) is (1) convex if and only if r ∈ (0, log 2]. (2) strictly convex if and only if r ∈ (0, log 2). Proof (1) By similarity we can assume x = e1 and by symmetry it is sufficient to consider only the case n = 2. We will consider ∂Bj (1, r) for fixed r and show that it consists of two circular arcs. By definition we have for z ∈ ∂Bj (1, r) j (1, z) = r =

log(1 + |z − 1|), |z| ≥ 1, log (1 + |z − 1|/|z|) , |z| < 1,

which is equivalent to er − 1 =

|z − 1|, |z| ≥ 1, |1 − 1/z| , |z| < 1.

We denote D2 = B 2 (0, 1) and D1 = R2 \ D2 . Now U1 = ∂Bj (1, r) ∩ D1 is an arc of a circle with center 1 and radius er − 1. Similarly U2 = ∂Bj (1, r) ∩ D2 is a circle that goes through points 1/(er ) and 1/(2 − er ) and  has center  on the real axis. This means that the center of the circle is c = 1/ er (2 − er ) and the radius of the circle is |er − 1|/|er (2 − er )|. If r ≤ log 2, then c > 1 and Bj (1, r) is convex. If r > log 2, then c < 0 and Bj (1, r). (2) We have c ∈ (1, ∞), where c is as above. Therefore Bj (x, r) is strictly convex. In the case r = log 2 we have c = ∞ and Bj (x, r) is not strictly convex.   We consider Question 13.1 for starlikeness instead of convexity for the distance ratio metric. Theorem 13.5 For x ∈ Rn \ {0} the j -metric √ ball Bj (x, r) is strictly starlike with respect to x if and only if r ∈ (0, log(1 + 2)). Proof Because for each r > 0 the j -balls of radii r are similar we may assume without loss of generality that x = e1 . By symmetry it is sufficient to consider the case n = 2 and the part of ∂Bj (1, r) that is above the real axis. If r ≥ log 3, then Bj (1, r) = B 2 (1, t) \ B 2 (c, s), where c, t and s are given in the proof of Theorem 13.4 and B 2 (c, s) ⊂ B 2 (1, t). Therefore Bj (1, r) can be starlike with respect to 1 only for r < log 3.

13.1 Distance Ratio Metric

241

Let us assume r < log 3. By the proof of Theorem 13.4 Bj (1, r) = B 2 (1, t) \ Let us denote the point of intersection of S 1 (1, t) and S 1 (c, s) above the real axis by z. Now z is also the point of intersection of the unit circle and the boundary ∂Bj (1, r). Let us denote by l the line that goes through points 1 and z. Now Bj (1, r) is strictly starlike with respect to 1 if and only if l ∩ B 2 (1, t) ∩ B 2 (c, s) = ∅. If z is a tangent of S 1 (c, s), then the circles S 1 (1, t) and S 1 (c, s) are perpendicular and r has the largest value such that Bj (1, r) is starlike with respect to 1. By the proof of Theorem 13.4 we have c = −1/er (er − 2), r = |1 − z| = er − 1, |1 − c| = (er − 1)2/er (er − 2) and s = |z − c| = (er − 1)/er (er − 2). Let us assume that z is a tangent of S 1 (c, s). Now by the Pythagorean theorem B 2 (c, s).

(er − 1)2 (er − 1)4 = (er − 1)2 + 2r r , 2 − 2) e (e − 2)2

e2r (er

which is equivalent to e2r − 2er − 1 = 0 and therefore √ 2).

r = log(1 +

 

We generalize Theorems 13.4 and 13.5 from Rn \ {0} to all proper subdomains G ⊂ Rn . The generalization is based on the fact that BjG (x, r) can be represented as intersection of BjRn \{z} (x, r), where z goes through all the points in ∂G. Theorem 13.6 For G ⊂ Rn and x ∈ G the j -metric ball Bj (x, r) √ is convex if r ∈ (0, log 2) and strictly starlike with respect to x if r ∈ (0, log(1 + 2)). Proof Let x ∈ G be arbitrary. We claim that A = BjG (x, r) =

5

BjRn \{z} (x, r) = B.

(13.1)

z∈∂G

Let y ∈ B. We can choose z ∈ ∂G with jRn \{z } (x, y) = min jRn \{z} (x, y). z∈∂G

Because z ∈ ∂G we have jG (x, y) ≤ jRn \{z } (x, y) and therefore y ∈ A. On the other hand, let y ∈ A. By definition there is a point z ∈ ∂G with min{|x −  z |, |y − z |} = minz∈∂G {|x − z|, |y − z|}. Now jRn \{z } (x, y) ≤ jG (x, y) and y ∈ B. By Theorem 13.4 each BjRn \{z} (x, r) is convex for 0 < r ≤ log 2 and (13.1) BjG (x, r) is an intersection of convex domains and therefore it is convex. If r ∈ (0, log 2], then Bj√ G (x, r) is convex and therefore also starlike with respect to x. If r ∈ (log 2, log(1 + 2)], then  Bj (x, r) = B \

% z∈∂G

 Az ,

242

13 Local Convexity of Balls

  where B = B n x, (er − 1)d(x) and Az = B n (cz z, rz ) for cz = |z|/(er (2 − er )) and rz = |z||1 − e−r |/|er − 2|. Let us assume that Bj (x, r) is not strictly starlike with respect to x. Now there exist a, b ∈ B such that b ∈ (x, a), a ∈ Bj (x, r) and b∈ / Bj (x, r). Now b ∈ B n (cz z, rz ) for some z ∈ ∂G. By the proof of Theorem 13.5 a ∈ B n (cz z, rz ), which is a contradiction.   Exercise 13.7 Show that the idea of the proof of Theorem 13.6 cannot be used for the quasihyperbolic metric by giving an example of a domain G ⊂ Rn , a point x ∈ G and a radius r > 0 such that 5 BkRn \{z} (x, r) ⊂ BkG (x, r). z∈∂G

Next we consider convex domains and show that in convex domains the distance ratio metric balls are also convex. Theorem 13.8 Let r > 0, G ⊂ Rn be a convex domain and x ∈ G. Then j -metric balls Bj (x, r) are convex. Proof By Theorem 13.6 we need to consider only the case r > log 2. Let us divide G into two parts D1 = {z ∈ G : d(z) ≥ d(x)} and D2 = G \ D1 . We will first show that convexity of G implies the convexity of D1 . Let us assume that D1 is not convex. There exist a, b ∈ D1 such / D1 and d(a) = d(x) =  that c = (a + b)/2 ∈ d(b). Now B n a, d(x) and B n b, d(x) does not contain any points of ∂G, but B n (c, s) for some s < d(x) contains at least one point of ∂G. Therefore G is not convex, which is a contradiction. Let us consider Bj (x, r) ∩ D1 . By definition of the j -metric we have for y ∈ ∂Bj (x, r) ∩ D1   |x − y| = d(x) er − 1   and therefore ∂Bj (x, r) ∩ D1 is a subset of S n−1 (x, s), where s = d(x) er − 1 . By convexity of D1 the domain Bj (x, r) ∩ D1 is convex. Let us then show that each chord with end points in Bj (x, r) ∩ D2 is contained in Bj (x, r). By definition for y ∈ ∂Bj (x, r) ∩ D2 we have d(y) =

|x − y| . er − 1

(13.2)

/ Bj (x, r). If z ∈ D1 , Let us assume y1 , y2 ∈ Bj (x, r) ∩ D2 and z = (y1 + y2 )/2 ∈ then z ∈ Bj (x, r) because Bj (x, r) ⊂ B n (x, s). Therefore we may assume z ∈ D2 \ Bj (x, r). By (13.2) we have d(yi ) > |x − yi |/(er − 1) for i ∈ {1, 2} and d(z) < |x − z|/(er − 1). Since r > log 2 we have c = 1/(er − 1) < 1. Now the boundary ∂G is outside B n (y1 , c|x − y1 |) ∪ B n (y2 , c|x − y2 |) and has a point in B n (z, c|x − z|), see Fig. 13.1.

13.1 Distance Ratio Metric

243

Fig. 13.1 Line l, euclidean balls B1 = B n (y1 , c|x − y1 |), B2 = B n (y2 , c|x − y2 |) and B3 = B n (z, c|x − z|) and points y1 , y2 and z

We will show that for c < 1 the domain G is not convex. Let us denote by l a line that is a tangent to balls B n (y1 , c|x − y1 |) and B n (y2 , c|x − y2 |). Because d(yi , l) = c|x − yi | for i ∈ {0, 1} we have d(z, l) =

c|x − y1 | + c|x − y2 | . 2

(13.3)

By the triangle inequality x − y1 x − y2 |x − y1 | |x − y2 | ≤ |x − z| = + + 2 2 2 2 and by (13.3) d(z, l) =

c (|x − y1 | + |x − y2 |) ≥ c|x − z|. 2

Now the domain G is not convex, which is a contradiction, and each chord with end points in Bj (x, r) ∩ D2 is contained in Bj (x, r). Since each chord with end points in Bj (x, r) ∩ D2 is contained in Bj (x, r), Bj (x, r) ∩ D2 ⊂ B n (x, s), D1 is convex and ∂Bj (x, r) ∩ D1 ⊂ S n−1 (x, s) the j -ball Bj (x, r) is convex.   We show that in starlike domains the distance ratio metric balls are also starlike with respect to the center of starlikeness. Theorem 13.9 Let r > 0 and G ⊂ Rn be a starlike domain with respect to x ∈ G. Then the j -metric balls Bj (x, r) are starlike with respect to x. √ Proof By Theorem 13.6 we need to consider r > log( 2 + 1) which is equivalent √ to er − 1 > 2. Let us divide G into two parts D1 = {z ∈ G : d(z) ≥ d(x)} and D2 = G \ D1 . Similarly as in the proof of Theorem 13.8 the boundary ∂Bj (x, r)∩D1 is a subset of a sphere S n−1 (x, s) and Bj (x, r) ∩ D2 ⊂ B n (x, s). Therefore it is sufficient to show that for each y ∈ Bj (x, r) ∩ D2 the line segment [x, y] is in Bj (x, r).

244

13 Local Convexity of Balls

We will show that all chords [x, y] for y ∈ Bj (x, r) ∩ D2 are contained in  Bj (x, r).  Let us assume, on the contrary, that there exist points y1 , y2 ∈ ∂Bj (x, r) ∩ D2 with y1 ∈ (x, y2 ) and z = (y1 + y2 )/2 ∈ / B j (x, r). Let us first assume z ∈ D1 . Now jG (x, z) > jG (x, y2 ) is equivalent to |x − z|/d(x) > |x − y2 |/d(y2). By the selection of y1 and y2 we have |x − z| < |x − y2 | and d(x) > d(y2 ) implying |x − z|/d(x) < |x − y2 |/d(y2), which is a contradiction. Let us then assume z ∈ D2 . Now |x − y1 | |x − y2 | |x − z| = = er − 1 < d(y1 ) d(y2 ) d(z)     and therefore the boundary ∂G does not intersect B n y1 , d(y1 ) or B n y2 , d(y2 )  and contains a point in B n z, d(z) , see Fig. 13.2. This means that G is not starlike with respect to x, which is a contradiction.   Finally we consider close-to-convex domains. We begin with the complement of the origin and generalize the result to all close-to-convex domains. √ Theorem 13.10 Let G = Rn \ {0} and x ∈ G. For r ∈ (0, log(1 + 3)] the √j metric ball Bj (x, r) is close-to-convex and in the case n = 2 for r > log(1 + 3) it is not close-to-convex. r) is Proof By symmetry of G we may assume x = e1 . By Theorem 13.6, Bj (x,√ starlike with respect to x and therefore also close-to-convex for r ≤ log(1 + 2). √ Therefore we assume r > log(1 + 2). Now by the definition of the j -metric (see proof of Theorem 13.4) n

Bj (x, r) = B n (e1 , er − 1) \ B (c, s), where c=

Fig. 13.2 Points  y1 and y2 and circles B n y1 , d(y1 ) ,   B n z, d(z) and   B n y2 , d(y2 )

e1 r e (2 − er )

and s =

er − 1 . er (er − 2)

13.1 Distance Ratio Metric

245

By geometry Bj (x, r) is close-to-convex if and only if r ≤ r0 , where r0 is such that 

e r0 − 1 er0 (er0 − 2)

2

 + 1−

which is equivalent to r0 = log(1 + Rn

1 r 0 e (2 − er0 )

2 = (er0 − 1)2 ,

√ 3) and the assertion follows.

  √ and ri > 0 be such that |xi | ≥ r/ 2, ri < |xi |

Lemma  13.11 Let r > 0, xi ∈ and ri2 + |xi |2 ≥ r whenever |xi | < r. Then

B = B n (0, r) \

∞ %

Bi

i=i

is close-to-convex and connected, where Bi = B n (xi , ri ).   √ be a domain and x ∈ G. For r ∈ (0, log(1 + 3)] the Theorem 13.12 Let G ⊂ j -metric ball Bj (x, r) is close-to-convex and connected. Proof Exercise.

Rn

Proof By 13.6 the j -metric√ball Bj (x, r) is starlike with respect to x√and thus closeto-convex for r ≤ log(1 + 2). Therefore we assume r > log(1 + 2). By (13.1) Bj (x, r) =

5

BjRn \{z} (x, r).

(13.4)

z∈∂G

We will prove that Bj (x, r) has the same form as B in Lemma 13.11. By (13.1)  n Bj (x, r) = B n (x, r0 ) \ ∞ i=1 B (xi , ri ), where r0 = d(x)(er − 1),   1 and |xi | = c 1 + r r e (e − 2) ri = c

er − 1 er (er − 2)

(13.5) (13.6) (13.7)

for some c ≥ d(x). √ Firstly, we have |xi | ≥ r0 / 2 because by (13.5) and (13.6) √ √ r0 er − 1 1 1 ≤ = er − 1 + ≤ 3 − √ ≤ 2, r 1 |xi | 1−e 1 + er (er −2) 3 where the second inequality follows from the fact that the function er −1+1/(1−er ) √ is increasing on (0, log(1 + 3)) .

246

13 Local Convexity of Balls

Secondly, we have ri < |xi | since by (13.6) and (13.7) ri 1 er − 1 1 1 |xi | = r r = r r + r < r r +1= . c e (e − 2) e (e − 2) e e (e − 2) c  Finally, we have

ri2 + |xi |2 ≥ r0 because by (13.5), (13.6) and (13.7)

ri2 + |xi |2 =

(er − 1)2 (e2r − 2er + 2) ≥ 3 ≥ (er − 1)2 = r02 , e2r (er − 2)2

where the first inequality follows from the fact that the function ((er − 1)2 (e2r − 2er + 2))/(e2r (er − 2)2 ) is decreasing on interval (log(1 + Lemma 13.11.

√

2), ∞). Now the assertion follows from  

Exercise 13.13 Let G = B n (0, 1) ∪ B n (e1 , 1/4) ∪ B n (2e1 , 1). Show that the j metric ball Bj (0, log 3) is connected but the j -metric sphere {z ∈ G : jG (0, z) = log 3} is disconnected. Exercise 13.14 Give an example of a simply connected domain G ⊂ Rn , a point x ∈ G and radius r > 0 such that the j -metric ball Bj (x, r) is not connected. Exercise 13.15 Show that for x ∈ Rn \ {0} and r > log 3 the j -metric ball Bj (x, r) is not close-to-convex. Exercise 13.16 Show that for x ∈ R3 \ {0} and r ≤ log 3 the j -metric ball Bj (x, r) is close-to-convex. √ Exercise 13.17 Show that Bj ( 3i, log(29/10)) is disconnected in C \ {−1, 1}.

13.2 Quasihyperbolic Balls We consider local convexity properties of the quasihyperbolic metric and extend our convexity results for j-balls to the case of quasihyperbolic balls in the complement of the origin. By (5.31) we have a parametric representation in the case n = 2 √  √2 2  2 2 x = (|x| cos ϕ, |x| sin ϕ) = e± r −ϕ cos ϕ, e± r −ϕ sin ϕ ,

(13.8)

for x ∈ ∂Bk (1, r) and −r ≤ ϕ ≤ r. By using this presentation we will prove the following result.

13.2 Quasihyperbolic Balls

247

Theorem 13.18 For r > 1 and z ∈ G = Rn \ {0} the quasihyperbolic disk Bk (z, r) is not convex. Proof We consider first the case n = 2. We may assume z = 1 and let x ∈ ∂Bk (z, r) be arbitrary. Assume r > 1. By (13.8) we have √  √2 2  2 2 x = e± r −ϕ cos ϕ, e± r −ϕ sin ϕ , where −r ≤ ϕ ≤ r. If r > π/2, then the claim is clear by symmetry because Re x = e−r > 0 for ϕ = 0 and Re x < 0 for ϕ = ±r. We will show that the function √ 2 2 f (ϕ) = e− r −ϕ cos ϕ is concave in the neighborhood of ϕ = 0 and the function √ 2 2 g(ϕ) = e− r −ϕ sin ϕ   is increasing in 0, min{r, π2 } . This will imply non-convexity of Bk (z, r). First,   √ ϕ sin ϕ  − r 2 −ϕ 2 g (ϕ) = e cos ϕ +  r 2 − ϕ2 and this is clearly non-negative for 0 < φ < min{r, π2 }. Therefore g(ϕ) is increasing. Second, by a straightforward computation we obtain 

f (ϕ) = e

√

−

r 2 −ϕ 2



ϕ cos ϕ  − sin ϕ r 2 − ϕ2



and √   2 2  e− r −ϕ r 2 − r 2 − ϕ 2 (r 2 − 2ϕ 2 ) cos ϕ + 2ϕ(ϕ 2 − r 2 ) sin ϕ  . f (ϕ) = ( r 2 − ϕ 2 )3 

Now f  (0) = 0 and f  (0) = e−r (1/r − 1) < 0 and therefore f (ϕ) is concave in the neighborhood of ϕ = 0. Consider next the general case. Fix y ∈ Rn \ {0} such that y = t z for all t ∈ R. Now Bk (z, r) ∩ span(0, y, z) is not convex by the case n = 2 and therefore the quasihyperbolic ball Bk (z, r) cannot be convex.   Next we introduce a geometric lemma, which is used to prove the next theorem.

248

13 Local Convexity of Balls

Lemma 13.19 Let the domain G ⊂ Rn be symmetric about a line l (i.e. is invariant under any rotation about l) and G∩L be strictly convex for any plane L with l ⊂ L. Then G is strictly convex.  

Proof Exercise.

Theorem 13.20 For 0 < r ≤ 1 and z ∈ Rn \ {0} the quasihyperbolic disk Bk (z, r) is strictly convex. Proof Let us first consider the case n = 2. Let z = 1 and x ∈ ∂Bk (z, r). By symmetry it is sufficient to consider the upper half C of ∂Bk (z, r), which is given by x = x(s) = (es cos ϕ, es sin ϕ), where r ∈ (0, π), s ∈ [−r, r] and ϕ = ϕ(s) = and therefore for s ∈ (−r, r) x  (s) =

√

(13.9)

r 2 − s 2 . Now ϕ  (s) = −s/ϕ(s)

 es  a(s), b(s) , ϕ(s)

where a(s)  = ϕ(s)cos ϕ(s) + s sin ϕ(s) and b(s) = ϕ(s) sin ϕ(s) − s cos ϕ(s). Now t (s) = a(s), b(s) is a tangent vector of C for s ∈ [−r, r]. Equality t (s) = 0 is equivalent to s 2 = −ϕ(s)2 , which never holds. Since t (s) = 0 for all s ∈ [−r, r] the angle α(s) = arg t (s) is a continuous function on (−r, r). We need to show that α(s) is strictly decreasing on [−r, r]. Since tan α(s) = b(s)/a(s) and cot α(s) = a(s)/b(s) it suffices to show that d(s) = b  (s)a(s) − a  (s)b(s) < 0 on (−r, r). By a straightforward computation d(s) = −

r 2 (1 + s) ϕ(s)

(13.10)

and α(s) is strictly decreasing on [−r, r]. Since t (s) → (0, r) as s → −r and t (s) → (0, −r) as s → r, α(s) decreases from π/2 to −π/2 on C and the assertion follows. Let us next consider the general case n > 2. By (5.31) the quasihyperbolic ball Bk (x, r) is symmetric about the line that contains x and 0. By Lemma 13.19 and the case n = 2, Bk (x, r) is strictly convex for 0 < r ≤ 1.   In the planar case n = 2, Theorem 13.20 was generalized in [540, Main theorem 7.7 (1)]. Theorem 13.21 Let G ⊂ R2 be a proper subdomain. For 0 < r ≤ 1 and z ∈ G the quasihyperbolic disk Bk (z, r) is strictly convex. Next we consider starlikeness and close-to-convexity of the quasihyperbolic balls in the complement of the origin.

13.2 Quasihyperbolic Balls

249

Let us define a constant κ as the solution of the equation    cos p2 − 1 + p2 − 1 sin p2 − 1 = e−1

(13.11)

for p ∈ [1, π]. Remark 13.22 The number κ occurs also in the paper of P.T. Mocanu in 1960 [393]. Later V. Anisiu and P.T. Mocanu showed [4, page 99] that if f : D → C is an analytic function in the unit disk, f (0) = 0 and  f (z) f  (z) ≤ κ, then f is starlike with respect to 0. Theorem 13.23 The quasihyperbolic disk BkRn \{0} (x, r) is strictly starlike with respect to x if and only if 0 < r ≤ κ. Proof Because of symmetry we will consider ∂Bk (x, r) only above the real axis and because of invariance under similarity it is sufficient to consider only the case x = 1. By Theorem 13.20 we need to consider r ∈ (1, π). Let us denote by l(s) a tangent line of the upper half of ∂Bk (1, r) where s is as in the proof of Theorem 13.20. The angle between l(s) and the real axis is described by the function α(s) defined in the proof of Theorem 13.20. The function α  (s) has the same sign as d(s) in (13.10) and therefore α  (s) is positive on (−r, −1) and negative on (−1, r). We need to find r such that l(s), s ∈ [−r, r], goes through point 1 exactly once. In other words, we need to find r such that l(−1) goes through 1. Define functions a(s) and b(s) as in the proof of Theorem 13.20. Since a(−1) = 0 for r ∈ (1, π), the tangent line l(−1) goes through 1 if and only if b(−1) x2 = , a(−1) x1 − 1

(13.12)

where x1 = es cos ϕ(s) and x2 = es sin ϕ(s). The equation (13.12) is equivalent to √ √ √ e cos r 2 − 1 + e r 2 − 1 sin r 2 − 1 − 1 √ √ √ √ = 0, (e − cos r 2 − 1)( r 2 − 1 cos r 2 − 1 − sin r 2 − 1) which holds if and only if r = κ. We next show that r = κ is the only solution of (13.11) on (1, π). We define a function h(x) = cos x + x sin x − e−1 and show that it has only one root on √ 2 (0, π − 1). Since h (x) = x cos x, h(0) = 1 − e−1 > 0 and  h( π 2 − 1) < h(11π/12) < 0 √ the function h has only one root on (0, π 2 − 1) and the assertion follows.

250

13 Local Convexity of Balls

Assume n > 2 and choose x ∈ Rn \ {0} and r ∈ (0, κ]. Suppose, on the contrary, that there exist y ∈ ∂Bk (x, r) and z ∈ (x, y) such that z ∈ ∂Bk (x, r). Now z ∈ ∂Bk (x, r) ∩ span(0, x, y) and therefore Bk (x, r) is not strictly starlike with respect to x. This is a contradiction by the case n = 2.   Exercise 13.24 Show that the quasihyperbolic ball Bk (x, r) in R2 \{0} is not simply connected for r > π. Equation (13.11) gave the radius of starlikeness for quasihyperbolic balls in the complement of the origin. The following proposition gives the radius of close-toconvexity. √ √ √ Proposition 13.25 The function f (z) = cos z2 − 1 + z2 − 1 sin z2 − 1 has exactly one zero on (2, π).  

Proof Exercise.

Denote the unique zero of the function f in Proposition 13.25 by λ. We introduce a geometric lemma, which we use in the proof of the next theorem. Lemma 13.26 If the domain G ⊂ Rn is invariant under rotation about a line l and G ∩ L is close-to-convex for every plane L with l ⊂ L, then G is close-to-convex.  

Proof Exercise.

\ {0}, r > 0 and x ∈ G. The quasihyperbolic ball Theorem 13.27 Let G = Bk (x, r) is close-to-convex if and only if r ≤ λ. Rn

Proof Let us first consider the case n = 2. We may assume that r ∈ (2, π], because for r ∈ (0, 2] the quasihyperbolic disk is starlike with respect to x by Theorem 13.23 and thus close-to-convex. For r > π the quasihyperbolic disk is not simply connected by Exercise 13.24 and thus not close-to-convex. By symmetry of G we may assume x = e1 and it is sufficient to consider the upper half U of the boundary ∂Bk (x, r). By (5.31) U = {y = y(s) ∈ R2 : y(s) = (es cos ϕ(s), es sin ϕ(s))}, where ϕ(s) = s ∈ (−r, r)

√ r 2 − s 2 and s ∈ [−r, r]. Since ϕ  (s) = −s/ϕ(s) we obtain for y  (s) =

es (a(s), b(s)) , ϕ(s)

where a(s) = ϕ(s) cos ϕ(s) + s sin ϕ(s) and b(s) = ϕ(s) sin ϕ(s) − s cos ϕ(s). To prove the close-to-convexity we need to find the largest possible r such that the tangent vector (a(s), b(s)) of U is parallel to the real axis exactly once for s ∈ (−r, 0). Since b (s) =

−(1 + s)a(s) ϕ(s)

13.2 Quasihyperbolic Balls

251

and a(s) < 0 for all s ∈ (−r, 0) we have b (s) = 0 if and only if s = −1. Therefore we want to find the greatest r such that b(−1) = 0, but this condition is equivalent to f (z) = 0 for the function in Proposition 13.25 and the assertion follows.   The case n > 2 follows from Lemma 13.26 and the case n = 2. In the case of the distance ratio metric we were able to prove some results for a general proper subdomain of Rn using as a standpoint a result for the particular domain Rn \ {0} . For instance, the proof of Theorem 13.6 was based on the formula (13.1). Now one could ask whether Theorem 13.27 could be applied in the same way. However, by Exercise 13.7 this idea does not work for the quasihyperbolic metric. In [539] and [373] there are various results concerning the quasihyperbolic metric in convex domains. One of these results [373, Theorem 2.13] shows that the quasihyperbolic balls are convex in convex domains. Theorem 13.28 ([373, Theorem 2.13]) If G ⊂ Rn is a convex domain, then the quasihyperbolic ball Bk (x, r) is convex for every r > 0. Next we consider starlike domains and show that in this case also the quasihyperbolic balls are starlike. Theorem 13.29 If G ⊂ Rn is a starlike domain with respect to x ∈ G, then the quasihyperbolic ball Bk (x, r) is starlike with respect to x for every r > 0. Proof We need to show that the function f (y) = kG (x, y) is increasing along each ray from x to ∂G. To simplify notation we may assume x = 0. Let y ∈ G \ {x} be arbitrary and denote a geodesic segment from x to y by γ . Fix y  ∈ (x, y) and denote |y  | γ = cγ . |y|

γ =

Since G is starlike with respect to x the path γ  from x to y  is in G. Therefore 



kG (x, y ) ≤

γ

|dz| = d(z)

 γ

c|dz| . d(cz)

Since G is starlike with respect to x we have d(cz) ≥ cd(z) which is equivalent to c 1 ≤ . d(cz) d(z) Now kG (x, y  ) ≤

 γ

c|dz| ≤ d(cz)

 γ

|dz| = kG (x, y) d(z)

and f is increasing along each ray from x to ∂G.

 

252

13 Local Convexity of Balls

13.3 Apollonian Metric We turn our attention to the Apollonian metric. We recall the definition of the Apollonian ball and sphere: For x, y ∈ Rn and r > 0 we define r = {z ∈ Rn : r|x − z| < |y − z|}, Bx,y

r Sx,y = {z ∈ Rn : r|x − z| = |y − z|}.

For x, y ∈ Rn and c > 0, c = 1, we have [248, Lemma 2.2.3]  c Sx,y

=S

n−1

 y − c2 x c|x − y| , . 1 − c2 |1 − c2 |

(13.13)

In the case c = 1 the Apollonian ball is a half-space. Example 13.30 By 11.9 we have αHn = ρHn and since both metrics are Möbius invariant the metrics agree in all domains which can be obtained from Hn by a Möbius transformation. Especially, we have αBn = ρBn . By Theorem 5.16 we have δBn = ρBn and by Möbius invariance it is clear that αG = δG for G = f (Bn ), where f is a Möbius transformation. Especially αG = δG = ρG for G ∈ {Bn , Hn }. By (4.11) Bα (x, r) = Bδ (x, r) = B n (x + xn en (cosh r − 1), xn sinh r) for all x ∈ Hn , r > 0 and by (4.20)  Bα (x, r) = Bδ (x, r) = B n

x(1 − th2 2r ) (1 − |x|2 ) th 2r , 1 − |x|2 th2 2r 1 − |x|2 th2 2r



for all x ∈ Bn , r > 0. By Example 13.30, Theorems 13.4 and 13.5 we obtain the following result. Proposition 13.31 Let G ∈ {Bn , Hn } and x ∈ G. Then Bα (x, r) is strictly convex for all r > 0. We consider the Apollonian metric in the domain G = Rn \ {−e1 , e1 }. Note that αG is pseudometric in this domain. Especially, for x ∈ G and a = |x + e1 |/|x − e1 | we have αG (x, y) = 0 for 1/a

y ∈ Sea1 ,−e1 , if |x − e1 | ≤ |x + e1 |, and y ∈ S−e1 ,e1 , if |x − e1 | ≥ |x + e1 |. (13.14) Theorem 13.32 Let G = Rn \ {−e1 , e1 }, x ∈ G and r > 0. We denote d Bc = Bec1 ,−e1 , Bd = B−e 1 ,e1

13.3 Apollonian Metric

253

for c = er |x + e1 |/|x − e1 | and d = er |x − e1 |/|x + e1 |. Then ⎧ if c < 1 and d ≥ 1, ⎨ Bc \ Bd , Bα (x, r) = Rn \ (Bc ∪ Bd ), if c > 1 and d > 1, ⎩ Bd \ Bc , if c ≥ 1 and d < 1. Moreover, the complement of Bα (x, r) is always disconnected. Proof By definition αG (x, y) = r is equivalent to c|y − e1 | = |y + e1 |

or d|y + e1 | = |y − e1 |.

(13.15)

Equalities (13.15) determine Apollonian spheres with respect to points e1 and −e1 d and by (13.13) the Apollonian spheres are Sec1 ,−e1 and S−e . We denote Sc = 1 ,e1 c d a Se1 ,−e1 and Sd = S−e1 ,e1 . By (13.14) we see that Sa = Se1 ,−e1 , where a = |x + e1 |/|x − e1 |, is contained in Bα (x, r). Note that all the spheres Sa , Sc and Sd = 1/d d S−e = Se1 ,−e1 are Apollonian spheres and 1/d < a < c. Since Sa ⊂ Bα (x, r) 1 ,e1 and ∂Bα (x, r) = Sc ∩ Sd , the complement of Bα (x, r) is disconnected. We denote the convex hull of Sa by Ba . Let us assume that c < 1 and d ≥ 1. Now also a > 1. Because 1 + c2 2c 1 + a2 2a + < + 2 2 2 2 c −1 c −1 a −1 a −1 is equivalent to (aer + 1)/(aer − 1) < (a + 1)/(a − 1) and 1 + c2 2c 1 + a2 2a − > − 2 2 2 2 c −1 c −1 a −1 a −1 is equivalent to (aer − 1)/(aer + 1) < (a − 1)/(a + 1), it is clear that Sc ⊂ Sa . A similar argument shows that Ba ⊂ Bd . Let us then assume that c > 1 and d > 1. It is easy to verify that Bd ⊂ (−∞, 0)× Rn−1 and Bc ⊂ (0, ∞) × Rn−1 . Thus we have Bc ∩ Bd = ∅. Since Sa , Sc and Sd are Apollonian spheres with 1/d < a < c it is clear that Sa ∩ (Bc ∪ Bd ) = ∅. The case c ≥ 1 and d < 1 is proved similarly to the case c < 1 and d ≥ 1.   Exercise 13.33 Let y, z ∈ Rn with y = z, G = Rn \ {y, z}, x ∈ G and r > 0. Find a formula for Bα (x, r), similar to the formula in Theorem 13.32. Our next goal is to consider Apollonian metric balls in starlike domains. We begin with the following lemma for Apollonian balls. Lemma 13.34 Let x, y ∈ Rn with x = y and r ∈ (0, 1). Then % t ∈(0,1]

r r Bx,z = A ∪ Bx,y ,

254

13 Local Convexity of Balls

where z = x + t (y − x) and 2 |x − y| A = a ∈ Rn : (a, x, y) < arcsin r, |a − x| < √ . 1 − r2 Proof We show that %

  r Bx,x+s(y−x) = a ∈ Rn : (a, x, y) < arcsin r .

s>0 r For b ∈ Rn and c > 0 such that B n (b, c) = Bx,x+s(y−x) we show that the ratio of c and |x − b| is a constant. By (13.13)

c r|x − (x + s(y − x))| sr|x − y| = = |x − b| |1 − r 2 ||x − b| |1 − r 2 ||x − b| =

rs|x − y| = r. − s(y − x) + r 2 x|

|x(1 − r 2 ) − x

r Thus the union of the Apollonian balls Bx,x+s(y−x) is an angular domain with (a, x, y) < arcsin r. By the Pythagorean theorem and (13.13)

 |x − b| ≤

√  2   |x − y| r|x − y| 2 |x − y| 1 − r 2 y − r 2x = √ + = x− 2 2 2 1−r |1 − r | |1 − r | 1 − r2  

and the assertion follows.

Now we are ready to prove the counterpart of Theorems 13.9 and 13.29 for the Apollonian metric. Theorem 13.35 Let G ⊂ Rn be a starlike domain with respect to x ∈ G such that the complement of G is not contained in any (n − 1)-dimensional sphere. Then Bα (x, r) is strictly starlike with respect to x for all r > 0. Proof Let us assume that Bα (x, r) is not starlike with respect to x. Then there exist y, z ∈ G such that z is contained in the line segment (x, y), αG (x, y) = r and αG (x, z) = r  > r. By the definition |a − y| |x − b| + log sup = r. |a − x| a∈∂G a∈∂G |y − b|

αG (x, y) = log sup By geometry

|a − z| |a − y| < sup a∈∂G |a − x| a∈∂G |a − x| sup

13.4 Seittenranta Metric

255

and since r < r  we have |x − b| |x − b| > sup . |z − b| b∈∂G b∈∂G |y − b| sup



r and S r contain at least one point in ∂G and because r < r  we have Both Sx,y x,z r r  and thus S r ⊂ Sx,z Sx,z x,z ⊂ G. By Lemma 13.34, G is not starlike, which is a   contradiction.

13.4 Seittenranta Metric Next we consider local convexity properties of the Seittenranta metric. We begin with the punctured space and note that then δG = jG . Lemma 13.36 ([483, Remark 3.2 (3)]) For all a ∈ Rn , we have δG = jG , where G = Rn \ {a}. Example 13.37 Let G = Rn \ {a}, where a ∈ Rn . Then it is easily seen, as pointed out 13.36, that δG = jG , where jG is the distance ratio metric defined by  jG (x, y) = log 1+

|x−y| min{d(x, ∂G), d(y, ∂G)}



 = log 1+

|x−y| min{|x−a|, |y−a|}



for all x, y ∈ G. By 13.4 and (13.13) we have for all x ∈ Rn \ {a}  Bδ (x, r) =

1/(er −1)

B n (x, (er − 1)|x − a|) ∩ Bx,a B n (x, (er

− 1)|x

, for 0 < r ≤ log 2,

1/(er −1) − a|) \ B x,a ,

for r > log 2.

Example 13.37 together with Theorem 13.4 gives us the following proposition. Proposition 13.38 Let a ∈ Rn , G = Rn \ {a}, x ∈ G. Then Bδ (x, r) is (strictly) convex and for r ∈ (0, √ log 2] (r ∈ (0, log 2)) √ and (strictly) starlike with respect to x for r ∈ (0, log(1 + 2)] (r ∈ (0, log(1 + 2))). The following proposition is an immediate consequence of Example 13.30. Proposition 13.39 Let G ∈ {Bn , Hn } and x ∈ G. Then Bδ (x, r) is strictly convex for all r > 0. We consider next the Seittenranta metric in the twice punctured space. As in the case of the Apollonian metric we use Apollonian balls.

256

13 Local Convexity of Balls

c Theorem 13.40 Let G = Rn \ {−e1 , e1 }, x ∈ G and r > 0. Then for Bc = B−e 1 ,x d and Bd = Be1 ,x we have

⎧ Bc ∩ Bd , if c ⎪ ⎪ ⎨ if c Bc \ Bd , Bδ (x, r) = ⎪ if c B \ Bc , ⎪ ⎩ dn R \ (Bc ∪ Bd ), if c

≤ 1 and d ≤ 1 and d > 1 and d > 1 and d

≤ 1, > 1, ≤ 1, > 1,

where c = |x − e1 |(er − 1)/2 and d = |x + e1 |(er − 1)/2. Proof Let us denote β = |x − e1 |/|x + e1 |. Since |x − e1 ||y + e1 | < |x + e1 ||y − e1 | β is equivalent to y ∈ B−e1 ,e1 we have by definition   ⎧ 2|x − y| β ⎪ ⎪ , if y ∈ B−e1 ,e1 , ⎨ log 1 + |x − e1 ||y + e1 |   δ(x, y) = 2|x − y| ⎪ 1/β ⎪ ⎩ log 1 + , if y ∈ Be1 ,−e1 . |x + e1 ||y − e1 | β

c For y ∈ B−e1 ,e1 the equality δG (x, y) = r is equivalent to y ∈ S−e . Similarly, 1 ,x 1/β

for y ∈ Be1 ,−e1 the equality δG (x, y) = r is equivalent to y ∈ Sed1 ,x . Therefore it is c ∪ Sed1 ,x . clear that ∂Bδ (x, r) ⊂ S−e 1 ,x By (13.13) we can see that c ≤ 1 is equivalent to x ∈ Bc and c > 1 is equivalent to x ∈ / Bc . Similarly we observe that d ≤ 1 is equivalent to x ∈ Bd and d > 1 is equivalent to x ∈ / Bd . Since always x ∈ Bδ (x, r), the above observations imply the   assertion. By Theorem 13.40 we can find the radius of convexity for Seittenranta metric balls in the twice punctured space. Theorem 13.41 Let G = Rn \ {−e1 , e1 }, x ∈ G and r0 = log(1 + 2/ max{|x − e1 |, |x + e1 |}). Then Bδ (x, r) is convex if and only if r ∈ (0, r0 ]. Proof By Lemma 13.40 the metric ball Bδ (x, r) is convex if and only c ≤ 1 and d ≤ 1, which is equivalent to   2 2 2 , log 1 + |x − e1 | |x + e1 |   2 = log 1 + max{|x − e1 |, |x + e1 |}

 r ≤ min log 1 +

 

and the assertion follows. Exercise 13.42 Formulate Theorem 13.41 in G = with y = z.

Rn

\ {y, z}, where y, z ∈ Rn

13.4 Seittenranta Metric

257

Lemma 13.43 Let x ∈ Bn \ {0} and r > 0. Then the set  A = y ∈ Bn \ {0} : log 1 +

|x − y| |y|(1 − |x|)



2 max{rB , rC },

13.4 Seittenranta Metric

259

which implies the sharpness of r0 . Inequality rA > rB is equivalent to |x| > |y|, which is true by the selection of y. Because rC = log(1+2|x−y|/((1+|y|)(1−|x|))) it is easy to see that rA > rC is equivalent to |y| < 1, which is true as y ∈ G.   Notes 13.45 This chapter is based on [285, 286, 288] and [289]. Local convexity in Banach spaces is studied in [449].

Chapter 14

Inclusion Results for Balls

In this chapter we consider inclusion of balls defined by several metrics. Proposition 14.1 Let G ⊂ Rn be a domain and r ∈ (0, log 2). Then Bj (x, m) ⊂ Bk (x, r) ⊂ Bj (x, r) ⊂ Bk (x, M) where m = log(2 − e−r )

and M = log

1 . 2 − er

Moreover, the second inclusion is sharp and M/m → 1 as r → 0. Proof Clearly m > 0 and since r ∈ (0, log 2) we also have M > 0. By Exercise 5.10 and [483, Theorem 3.8] Bj (x, r) ⊂ B n (x, (er − 1)d(x))

(14.1)

and by (5.15) B n (x, (1 − e−r )d(x)) ⊂ Bk (x, r) ⊂ B n (x, (er − 1)d(x)).

(14.2)

By (5.4), (14.1) and (14.2) we have Bj (x, m) ⊂ B n (x, (em − 1)d(x)) = B n (x, (1 − e−r )d(x)) ⊂ Bk (x, r) ⊂ Bj (x, r) ⊂ B n (x, (er − 1)d(x)) = B n (x, (1 − e−M )d(x)) ⊂ Bk (x, M).

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_14

261

262

14 Inclusion Results for Balls

Fig. 14.1 Left: Inclusions Bj (e1 , m) ⊂ Bk (e1 , r) ⊂ Bj (e1 , r) from Proposition 14.1 and Theorem 14.7. Middle: Inclusions Bj (e1 , m) ⊂ Bq (e1 , r) ⊂ Bj (e1 , M) from Theorem 14.12. Right: Inclusions Bk (e1 , m) ⊂ Bq (e1 , r) ⊂ Bk (e1 , M) from Theorem 14.14

Fig. 14.2 Left: Inclusions Bj (3e2 /2, m) ⊂ Bk (3e2 /2, r) ⊂ Bj (3e2 /2, r) from Proposition 14.1 and Theorem 14.18. Middle: Inclusions Bj (3e2 /2, m) ⊂ Bq (3e2 /2, r) ⊂ Bj (3e2 /2, M) from Theorem 14.24. Right: Inclusions Bk (3e2 /2, m) ⊂ Bq (3e2 /2, r) ⊂ Bk (3e2 /2, M) from Theorem 14.27

Fix x ∈ G and let z ∈ ∂G be any point with d(x) = |x − z|. Choose y ∈ ∂Bk (x, r) ∩ [x, z]. Now d(y) = d(x) − |x − y| and j (x, y) = k(x, y) implying y ∈ ∂Bj (x, r). Thus, the inclusion Bk (x, r) ⊂ Bj (x, r) is sharp. By l’Hôpital’s rule, the assertion follows.   Proposition 14.1 is illustrated in Figs. 14.1 and 14.2. Exercise 14.2 Show that in Proposition 14.1 for r ∈ (0, log 2) 1+

r r ≤ ≤ 1 + 2r. 2 m

Lemma  14.3 (Lemma 3.13, [29, Lemma 7.16]) For x (0, 1/ 1 + |x|2 ) we have Bq (x, r) = B n (y, s)  and for x ∈ Rn and r ∈ (1/ 1 + |x|2 , 1) Bq (x, r) = Rn \ B n (y, s),

∈

Rn and r

∈

14 Inclusion Results for Balls

263

where x y= 2 1 − r (1 + |x|2 )

√ r(1 + |x|2) 1 − r 2 and s = . 1 − r 2 (1 + |x|2 )

The chordal ball Bq (x, r) is either  a half-space or the complement of a closed |x|2 . On the other hand, the chordal ball euclidean ball whenever r ≥ 1/ 1 + Bq (x, r) contains 0 whenever r > |x|/ 1 + |x|2 . Since we do not want either of the cases to occur it is natural to assume that r < rq (x) for   2 rq (x) = min{1/ 1 + |x| , |x|/ 1 + |x|2 } .

(14.3)

√ Note that the selection of r implies that r < 1/ 2 and √ 1 − r2 . √ < |x| < r 1 − r2 r

Theorem 14.4 Let G = Bn , x ∈ G and r > 0. Then Bj (x, m1 (r)) ⊂ Bρ (x, r), Bj (x, m2 (r)) ⊂ Bk (x, r), Bj (x, m3 (r)) ⊂ Bq (x, r),

 r < (1 − |x|)/ 2(1 + |x|2 ),

where  r m1 (r) = log 1 + 2 sh , 2  r , m2 (r) = log 1 + 2 sh 4   r m3 (r) = log 1 + √ . 1 − r2 Exercise 14.5 Show that r < m1 (r) < r, 2

r r < m2 (r) < , 4 2

4r < m3 (r), 5

where m1 (r), m2 (r) and m3 (r) are as in Theorem 14.4.

(14.4)

264

14 Inclusion Results for Balls

Theorem 14.6 For a proper subdomain G of Rn , r > 0 and y ∈ ∂Bk (x, r), let Jk [x, y] be a geodesic segment of the quasihyperbolic metric joining x and y. For z ∈ Jk [x, y] we have   |z − y| B n z, ⊂ Bk (x, r), 1+u where u = |z − y|/d(z). Proof Recall that the geodesic segment Jk [x, y] exists by Lemma 5.1. By the choice of z we have r = kG (x, y) = kG (x, z) + kG (z, y)   and by the triangle inequality for w ∈ Bk z, kG (z, y) we have kG (x, w) ≤ kG (x, z) + kG (z, w) < r. Now   Bk z, kG (z, y) ⊂ Bk (x, r). By (5.15)       B n z, 1 − e−kG (z,y) d(z) ⊂ Bk z, kG (z, y) and therefore     B n z, 1 − e−kG (z,y) d(z) ⊂ Bk (x, r).  By (5.4) kG (z, y) ≥ log 1 +

|z−y| d(z)

 and therefore

   1 − e−kG (z,y) d(z) ≥ 1 − for u =

|z−y| d(z) .

 d(z) |z − y| d(z) = d(z) + |z − y| 1+u

Now       |z − y| B n z, ⊂ B n z, 1 − e−kG (z,y) d(z) 1+u

and the claim is clear.

 

14.1 The Punctured Space

265

14.1 The Punctured Space We consider inclusions of hyperbolic type balls in the punctured space Theorem 14.7 Let G = Rn \ {0}, x ∈ G and r ∈ (0, π/2). Then Bj (x, m) ⊂ Bk (x, r), where  r . m = log 1 + 2 sin 2 Moreover, the inclusion is sharp and r/m → 1 as r → 0. Proof Let y ∈ Bj (x, m) and assume first that |x| ≤ |y|. Now |x − y| < |x|(em − 1) = 2|x| sin(r/2) = k(x, z) for z ∈ ∂Bk (x, r) and |z| = |x|. For all u ∈ ∂Bk (x, r) with |u| > |x| we have |x − u| > |x − z| by [286, Lemma 4.9] and therefore y ∈ Bk (x, r). If |x| > |y| we have y ∈ Bk (x, r) by the previous case and the fact that the quasihyperbolic and j -metric balls are invariant under inversion about origin. We show that m is sharp. We choose y ∈ ∂Bk (x, r) with |y| = |x|. Now     2|x| sin(r/2) |x − y| = log 1 + =m j (x, y) = log 1 + |x| |x| and m is sharp. Finally, we show that r/m → 1 as r → 0. By l’Hôpital’s rule we have lim

r→0

r 1 + 2 sin(r/2) = lim =1 m r→0 cos(r/2)  

and the assertion follows. Theorem 14.7 is illustrated in Fig. 14.1. Corollary 14.8 Let G = Rn \ {0}, x ∈ G and r ∈ (0, log 3). Then Bj (x, r) ⊂ Bk (x, M), where M = 2 arcsin

er − 1 . 2

Moreover, the inclusion is sharp and M/r → 1 as r → 0.

266

14 Inclusion Results for Balls

Exercise 14.9 Let m be as in Theorem 14.7. Show that for r ∈ (0, π/2) 1+

r r ≤ ≤ 1 + r. 3 m

Exercise 14.10 Let M be as in Corollary 14.8. Show that for r ∈ (0, 1/2) 1+

r M ≤ ≤ 1 + r. 2 r

Exercise 14.11 Show that the radius m1 = log(1 + 2 sin(r/2)) of Theorem 14.7 is better than the radius m2 = log(2 − e−r ) of Proposition 14.1. Theorem 14.12 Let G = Rn \ {0}, x ∈ G and r ∈ (0, rq (x)), where rq is as in (14.3). Then Bj (x, m) ⊂ Bq (x, r) ⊂ Bj (x, M), where    1 m = log 1 + r |x| + |x| and ⎧ ⎪ ⎪ ⎨ log

|x|(1 − r 2 (1 + |x|2)) , for |x| ≤ 1 √ |x| − r √1 − r 2 (1 + |x|2 ) M= ⎪ |x| + r 1 − r 2 (1 + |x|2 ) ⎪ ⎩ log , for |x| > 1. |x|(1 − r 2 (1 + |x|2)) Moreover, the inclusions are sharp and M/m → 1 as r → 0. Proof Let us first show that Bj (x, m) ⊂ Bq (x, r). We assume y ∈ Bj (x, m), which is equivalent to   1 |x − y| < r |x| + , min{|x|, |y|} |x| and thus

q(x, y)2
1, and s is depending only on x and r. By (14.5) and [285, p. 285] it is clear that for minimal M such that Bq (x, r) ⊂ Bj (x, M) we have ∂Bq (x, r) ∩ ∂Bj (x, M) ⊂ {z ∈ G : z = xt, t > 0}. In other words it is sufficient to show that and |y| − s ≥ |x|e−M .

|y| + s ≤ |x|eM

(14.6)

Before proving (14.6) we observe that since |y|2 − s 2 =

|x|2 − r 2 − r 2 |x|2 1 − r 2 − r 2 |x|2

we have that |x| ≤ 1 is equivalent to |x|/(|y| − s) ≥ (|y| + s)/|x| and |x| > 1 is equivalent to |x|/(|y| − s) < (|y| + s)/|x|. Now we have for |x| ≤ 1 |x| |x|(1 − r 2 (1 + |x|2 )) |y| + s ≤ = = eM √ |x| |y| − s |x| − r 1 − r 2 (1 + |x|2) and for |x| > 1 √ |y| + s |x| + r 1 − r 2 (1 + |x|2) |x| < = = eM |y| − s |x| |x|(1 − r 2 (1 + |x|2 )) and the inequalities of (14.6) hold true. We show that m is sharp. We choose y ∈ G with |y| = |x| and q(x, y) = r. Now |x − y| = r(1 + |x|2 ) and     r(1 + |x|2 ) |x − y| = log 1 + = m. j (x, y) = log 1 + |x| |x| We show that M is sharp. Let us first assume |x| ≤ 1. We choose y ∈ G with q(x, y) = r and |y| ≤ |z| for all z ∈ G with q(x, z) = r. Now q(x, y) = 

|x| − |y|  =r 1 + |x|2 1 + |y|2

268

14 Inclusion Results for Balls

implying √ |x| − r 1 − r 2 (1 + |x|2) . |y| = 1 − r 2 (1 + |x|2 )

(14.7)

Now j (x, y) = log(|x|/|y|) = M and M is sharp. Let us then assume |x| > 1. We choose y ∈ G with q(x, y) = r and |y| ≥ |z| for all z ∈ G with q(x, z) = r by (14.7) we have j (x, y) = log(|y|/|x|) = M. Finally, we show that M/m → 1 as r → 0. Let us first consider the case |x| ≤ 1. By l’Hôpital’s rule M r + |x| + r|x|2 = lim √ = 1. r→0 m r→0 1 − r 2 (1 − 2r 2 )|x| + r(1 − r 2 )(1 − |x|2 ) lim

In the case |x| > 1 we obtain by l’Hôpital’s rule M r + |x| + r|x|2 = lim √ = 1. r→0 m r→0 1 − r 2 (1 − 2r 2 )|x| − r(1 − r 2 )(1 − |x|2 ) lim

 

Theorem 14.12 is illustrated in Fig. 14.1. Corollary 14.13 Let G = Rn \ {0}, x ∈ G and r ∈ (0, log(1 + |x| + 1/|x|)). Then Bq (x, m) ⊂ Bj (x, r) ⊂ Bq (x, M), where

m=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(er − 1)|x| (1 + |x|2 )(e2r + |x|2 ) (er − 1)|x|

, if |x| ≤ 1,

(1 + |x|2 )(1 + |x|2e2r )

, if |x| ≥ 1,

and M=

|x|(er − 1) . 1 + |x|2

Moreover, the inclusions are sharp and M/m → 1 as r → 0. The assumption r < log(1 + |x| + 1/|x|) is equivalent to M < 1. Theorem 14.14 Let G = Rn \ {0}, x ∈ G and r ∈ (0, R) for R= 

2|x|  . 1 + |x|2 1 + 9|x|2

14.1 The Punctured Space

269

Then Bk (x, m) ⊂ Bq (x, r) ⊂ Bk (x, M), where    1 m = log 1 + r |x| + |x| and M = 2 arcsin f (r, x) with ⎧ r(1 + |x|2) ⎪ ⎪ , for |x| ≤ 1 ⎨ √ 2 f (r, x) = 2 1 − r |x| −2 2r ⎪ ⎪ √ r + r|x| ⎩ , for |x| > 1. 2 1 − r 2 |x| − 2r|x|2 Moreover, we have M/m → 1 as r → 0. Proof By (5.4) and Theorem 14.12 we have Bk (x, m) ⊂ Bj (x, m) ⊂ Bq (x, r) for r ∈ (0, rq (x)), where rq is as in (14.3). By Theorem 14.12 we have Bq (x, r) ⊂ Bj (x, t)

(14.8)

for r ∈ (0, rq (x)). By Theorem 14.7 Bj (x, t) ⊂ Bk (x, s)

(14.9)

for t ∈ (0, log 3) and s = 2 arcsin((et − 1)/2). Combining (14.8) and (14.9) gives M = s and r ∈ (0, R0 ) ⊂ (0, rq (x)), where 

|x|(eπ/2 − 1) 2|x|   , R0 = min  1 + |x|2 1 + eπ |x|2 1 + |x|2 1 + 9|x|2 We also have f (r, x) ∈ [0, 1] for r ∈ (0, R0 ) .

4 .

270

14 Inclusion Results for Balls

We show next that R = R0 , which is equivalent to 

eπ/2 − 1

≥

1 + eπ |x|2

2 1 + 9|x|2

(14.10)

.

Inequality (14.10) is equivalent to |x|2(5eπ/2 − 3) ≥ −(eπ/2 + 1), which is clearly true and thus R = R0 . Finally, by a straightforward computation using l’Hôpital’s rule we obtain M/m → 1 as r → 0.   Theorem 14.14 is illustrated in Fig. 14.1. Corollary 14.15 Let G = Rn \ {0}, x ∈ G and r ∈ (0, log(1 + |x| + 1/|x|)). Then Bq (x, m) ⊂ Bk (x, r) ⊂ Bq (x, M), where  m = min 

2|x| sin(r/2) (1 + |x|2 )(|x|2 + (1 + 2 sin(r/2))2 ) 4 2|x| sin(r/2)

,

 (1 + |x|2)(1 + |x|2 (1 + 2 sin(r/2))2 )) and M=

|x|(er − 1) . 1 + |x|2

Moreover, we have M/r → 1 as r → 0. Proof The assertion follows from Theorem 14.14. The assumption r < log(1 + |x| + 1/|x|) is equivalent to M < 1.   Theorem 14.16 Let G = Rn \ {0}, x ∈ G and r ∈ (0, rq (x)). Then Bj (x, m) ⊂ Bq (x, r)

and Bk (x, m) ⊂ Bq (x, r),

where   2r 2 . m = log 1 + √ 1 − r2 Corollary 14.17 Let G = Rn \ {0}, x ∈ G and r ∈ (0, log(1 + Bj (x, r) ⊂ Bq (x, M)

√

2)). Then

and Bk (x, r) ⊂ Bq (x, M),

14.2 The Upper Half-Space

271

where  M=

√ (er − 1)(er − 1 + 17 + er (er − 2)) √ . 2 2

14.2 The Upper Half-Space Next we consider inclusions in the upper half space. Theorem 14.18 Let G = Hn , x ∈ G and r > 0. Then Bj (x, m) ⊂ Bk (x, r), where   √ √ m = log 1 + 2 ch r − 1 . Moreover, the inclusion is sharp and r/m → 1 as r → 0. Proof Let y ∈ Bj (x, m) and denote y = yn en + y  . By (4.11) we have Bk (x, r) = B n (z, |x| sh r) for z = |x|en ch r. Let us first assume xn = d(x) ≤ d(y) = yn . Now y ∈ Bj (x, m) implies that |y  |2 < 2(ch r − 1)xn2 − (xn − yn )2 . Now |z − y|2 = (xn ch r − yn )2 + |y  |2 < (xn ch r − yn )2 + 2(ch r − 1)xn2 − (xn − yn )2 = xn2 (ch2 r + 2 ch r − 3) − 2xn yn (ch r − 1) ≤ xn2 (ch2 r + 2 ch r − 3) − 2xn2 (ch r − 1) = xn2 (ch2 r − 1) ≤ xn2 sh2 r and therefore y ∈ Bk (x, r). Let us then assume yn = d(y) ≤ d(x) = xn . Now y ∈ Bj (x, m) implies that |y  |2 < 2(ch r − 1)yn2 − (xn − yn )2 . Now |z − y|2 = (xn ch r − yn )2 + |y  |2 < (xn ch r − yn )2 + 2(ch r − 1)yn2 − (xn − yn )2 = xn2 (ch2 r − 1) + 2(ch r − 1)yn (yn − xn ) ≤ xn2 (ch2 r − 1) = xn2 sh2 r and therefore y ∈ Bk (x, r).

272

14 Inclusion Results for Balls

We show that m is sharp for y ∈ Hn such that j (x, y) = m and d(x) = d(y). Now |y  |2 = 2(ch r − 1)xn2 and |z − y|2 = xn2 (ch r − 1)2 + |y  |2 = xn2 (ch r − 1)2 + 2(ch r − 1)xn2 = xn2 sh2 r. Finally, by l’Hôpital’s rule √ √ r 2 ch r − 2 − 2 ch r − 1 = lim lim r→0 m r→0 sh r sh r 2 sh r + √ √ 2 ch r − 1 = lim r→0 ch r √ sh r ch r + 1 = lim =1 √ = lim √ √ r→0 2 ch r − 1 r→0 2  

and the assertion follows. Theorem 14.18 is illustrated in Fig. 14.2. Corollary 14.19 Let G = Hn , x ∈ G and r > 0. Then Bj (x, r) ⊂ Bk (x, M), where   (er − 1)2 M = arch 1 + . 2 Moreover, the inclusion is sharp and M/r → 1 as r → 0. Exercise 14.20 Let m be as in Theorem 14.18. Show that for r ∈ (0, 1) r ≤ 1 + r. m Exercise 14.21 Let M be as in Corollary 14.19. Show that for r ∈ (0, 1) 1+

r M ≤ . 2 r

√ √ Exercise 14.22 Show that the radius m1 = log(1 + 2 ch r − 1) of Theorem 14.18 is larger than the radius m2 = log(2 − e−r ) of Proposition 14.1. √ √ Exercise 14.23 Show that the radius m1 = log(1 + 2 ch r − 1) of Theorem 14.18 is larger than the radius m3 = log(1 + 2 sin(r/2)) of Theorem 14.7.

14.2 The Upper Half-Space

273

Theorem 14.24 Let G = Hn , x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r ∈ (0, rq (x)). Then Bj (x, m) ⊂ Bq (x, r) ⊂ Bj (x, M), where 



r(1 + |x|2 ) m = min log 1 +  |x| 1 − r 2 − r 2 |x|2



 4 r(1 + |x|2) , log 1 + √ 1 − r 2 |x| − r

and   2  r(1 + |x|2 ) r(1 + |x|2 ) , log 1 + . M = max log 1 + √ √ 1 − r 2 |x| − r |x|( 1 − r 2 − r|x|) Moreover, the inclusions are sharp and M/m → 1 as r → 0. Proof Lemma 14.3 gives concrete s > 0 and y ∈ Rn such that Bq (x, r) = B n (y, s). We start by proving the first inclusion. Let z ∈ Bj (x, m). We assume first that |x| = d(x) ≤ d(z) and show that in this case  m = log(1 + (r(1 + |x|2))/(|x| 1 − r 2 − r 2 |x|2 ) . By definition r(1 + |x|2 ) |x − z| < (em − 1)|x| =  1 − r 2 − r 2 |x|2 and therefore z ∈ B n (x, (em − 1)|x|) and q(x, z) < q(x, z1 ) for z1 = |x|en + |x|(em − 1)e1 . Now |x − z1 |  =r q(x, z) < q(x, z1 ) =  1 + |x|2 1 + |z1 |2

(14.11)

and the assertion follows. This also shows that m is sharp for d(x) ≤ d(z). Let us then assume d(z) ≤ d(x). Now for z ∈ ∂Bj (x, m) we have |x − z| = (em − 1)d(z) and thus for each plane p, which contains {z ∈ Rn : z = ten , t ∈ R}, the intersection ∂Bj (x, m) ∩ p is a plane curve with largest curvature at the point that is closest to ∂G. Therefore q(x, z) ≤ min{q(x, z1), q(x, z2 )}

274

14 Inclusion Results for Balls

√ √ for z1 = |x|en + |x|(em − 1)e1 and z2 = (|x| 1 − r 2 − r)/( 1 + r 2 + r|x|)en . By (14.11) we have q(x, z) < q(x, z1) = r. Because √ √ |x| 1 − r 2 − r |x| − r(1 + |x|2 ) 1 − r 2 |z2 | = √ = |y| − s ≤ 1 − r 2 (1 + |x|2 ) 1 + r 2 + r|x|

(14.12)

we have q(x, z) < q(x, z2 ) = r and the assertion follows. The inequality in (14.12) is equivalent to     (1 + |x|2)(r 2 (2|x| 1 − r 2 − r) + r(1 − 1 − r 4 )) + |x|( 1 + r 2 − 1 − r 2 ) > 0 √ √ 2 4 and holds true, √ √ because of the assumptions 2|x| 1 − r > r, 1 > 1 − r and 1 + r 2 > 1 − r 2 . Sharpness of m follows from the selection of z1 and z2 . Let us then prove the second inclusion. We assume that z ∈ ∂Bq (x, r) and |x| = d(x) ≤ d(z). By definition |x − z| = (eM − 1)|x| and therefore     r(1 + |x|2) |y| + s − |x| = log 1 + j (x, z) ≤ j (x, z3 ) = log 1 + √ |x| |x|( 1 − r 2 − r|x|) for z3 = (|y| + s)en . Let us assume z ∈ ∂Bq (x, r) and d(z) ≤ d(x) = |x|. By definition |x − z| = (eM − 1)|z| and therefore     |y| − s − |x| r(1 + |x|2) j (x, z) ≤ j (x, z2 ) = log 1 + = log 1 + √ |x| 1 − r 2 |x| − r for z2 = (|y| − s)en . Sharpness of M is clear by selection of z3 and z2 . We shall finally show that M/m → 1 as r → 0. We denote 

r(1 + |x|2) m1 = log 1 +  |x| 1 − r 2 − r 2 |x|2

 ,

  r(1 + |x|2 ) m2 = log 1 + √ 1 − r 2 |x| − r

and   r(1 + |x|2 ) , M1 = log 1 + √ 1 − r 2 |x| − r

 M2 = log 1 +

 r(1 + |x|2) . √ |x|( 1 − r 2 − r|x|)

14.2 The Upper Half-Space

275

Clearly M1 /m2 = 1 and by l’Hôpital’s rule  M1 (r 2 (1 + |x|2 ) − 1)(r + r|x|2 + |x| 1 − r 2 (1 + |x|2 )) = lim = 1, lim √ r→0 m1 r→0 |x| 1 − r 2 (2r 2 − 1) + r(r 2 − 1)(|x|2 − 1)  M2 −(r 2 (1 + |x|2) − 1)(r + r|x|2 + |x| 1 − r 2 (1 + |x|2 )) lim = lim √ = 1, r→0 m1 r→0 −|x| 1 − r 2 (2r 2 − 1) + r(r 2 − 1)(|x|2 − 1) √ M2 |x| − 2r 2 |x| + r 1 − r 2 (|x|2 − 1) = lim √ =1 lim r→0 m2 r→0 |x| − 2r 2 |x| − r 1 − r 2 (|x|2 − 1)  

and the assertion follows. Theorem 14.24 is illustrated in Fig. 14.2.

Corollary 14.25 Let G = Hn , x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r > 0. Then Bq (x, m) ⊂ Bj (x, r) ⊂ Bq (x, M), where ⎧ ⎪ ⎪ ⎨

|x|(er − 1)  , if |x| ≤ 1, 1 + |x|2 e2r + |x|2 m= r |x|(e − 1) ⎪ ⎪  , if |x| ≥ 1 ⎩ 1 + |x|2 1 + e2r |x|2 and ⎧ ⎪ ⎪ ⎨

√ |x|(er − 1)  , if |x| ≤ th(r/2), 2 2r 2 1 + |x| e + |x| M= √ |x|(er − 1) ⎪ ⎪  , if |x| ≥ th(r/2). ⎩ 2 r r 2 1 + |x| 1 + (2 + e (e − 2))|x| Moreover, the inclusions are sharp and M/m → 1 as r → 0. Exercise 14.26 Show that 

1

|x|

,√ f (r) = min  1 − r 2 |x| − r 1 − r 2 − r 2 |x|2

4 −1

is positive for r ∈ (0, rq (x)). Note that Theorem 14.24 improves Theorem 14.12 by Exercise 14.26.

276

14 Inclusion Results for Balls

Theorem 14.27 Let G = Hn , x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r ∈ (0, rq (x)). Then Bk (x, m) ⊂ Bq (x, r) ⊂ Bk (x, M), where 4 √ |x| + r 1 − r 2 (1 + |x|2) |x|(r 2 (1 + |x|2) − 1) m = min log , log √ |x|(1 − r 2 (1 + |x|2 )) r 1 − r 2 (1 + |x|2) − |x| 

and 

4 √ |x| + r 1 − r 2 (1 + |x|2 ) |x|(r 2 (1 + |x|2 ) − 1) M = max log . , log √ |x|(1 − r 2 (1 + |x|2 )) r 1 − r 2 (1 + |x|2) − |x| Moreover, the inclusions are sharp and M/m → 1 as r → 0. Proof Let us denote a = log

√ |x| + r 1 − r 2 (1 + |x|2 ) |x|(1 − r 2 (1 + |x|2))

|x|(r 2 (1 + |x|2 ) − 1) and A = log √ . r 1 − r 2 (1 + |x|2 ) − |x|

It is easy to verify that a = A is equivalent to |x| = 1, a < A is equivalent to |x| < 1 and a > A is equivalent to |x| > 1. Since k agrees with the usual hyperbolic metric in G we know that quasihyperbolic balls are euclidean balls. Therefore, both Bk (x, r) and Bq (x, r) are euclidean balls of form B n (sxn , t) for s, t > 0. For chordal metric the value of s is s1 = |x|/(1 − r 2 (1 + |x|2 )) and quasihyperbolic metric s2 = |x| ch a and s3 = |x| ch A. We will first show that for |x| < 1 we have s1 < s2 < s3 and for |x| > 1 we have s3 < s2 < s1 . In the case |x| = 1 we have s1 = s2 = s3 and the assertion follows. Let us first assume that |x| < 1. Since ch t is strictly increasing on (0, ∞) and a < A we have s2 < s3 . Inequality s1 < s2 is equivalent to 2f (x, r) < 1 + f (x, r)2 , for f (x, r) =

√ |x| + r 1 − r 2 (1 + |x|2 ) . |x|(1 − r 2 (1 + |x|2 ))

By selection of r we have f (x, r) > 0 and (14.13) implies that s1 < s2 .

(14.13)

14.2 The Upper Half-Space

277

Let us now prove the inclusion Bk (x, m) ⊂ Bq (x, r). Since s1 < s2 we have for the largest possible m that ∂Bk (x, m) ∩ ∂Bq (x, r) = {y}, where y ∈ ∂Bq (x, r) is such that d(y) > d(w) for all w ∈ ∂Bq (x, r), w = y. Now q(x, y) = k(x, y) and thus Bk (x, m) ⊂ Bq (x, r) and m is sharp. Let us then prove the inclusion Bq (x, r) ⊂ Bk (x, M). Since s1 < s3 we have for the largest possible m that ∂Bk (x, m) ∩ ∂Bq (x, r) = {z}, where z ∈ ∂Bq (x, r) is such that d(z) < d(w) for all w ∈ ∂Bq (x, r), w = z. Now q(x, z) = k(x, z) and thus Bq (x, r) ⊂ Bk (x, M) and M is sharp. The case |x| > 1 is similar to the case |x| < 1. We will finally show that M/m → 1 as r → 0. By l’Hôpital’s rule √ M |x| − 2r 2 |x| − r 1 − r 2 (|x|2 − 1) lim = lim = 1. √ r→0 m r→0 |x| − 2r 2 |x| + r 1 − r 2 (|x|2 − 1)

 

Theorem 14.27 is illustrated in Fig. 14.2. Corollary 14.28 Let G = Hn , x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r > 0. Then Bq (x, m) ⊂ Bk (x, r) ⊂ Bq (x, M), where  m = min 

|x|(er − 1) (1 + |x|2 )(e2r + |x|2 )

,

|x|(er − 1)

4

(1 + |x|2 )(1 + e2r |x|2 )

for  M = max 

|x|(er − 1) (1 + |x|2 )(e2r + |x|2 )

,

|x|(er − 1) (1 + |x|2 )(1 + e2r |x|2)

4 .

Moreover, the inclusions are sharp and M/m → 1 as r → 0. Theorem 14.29 Let G = Hn , x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r ∈ (0, rq (x)). Then Bj (x, m1 ) ⊂ Bq (x, r)

and Bk (x, m2 ) ⊂ Bq (x, r),

278

14 Inclusion Results for Balls

where  m1 = log 1 +

2r 1 − r2



  2r 2 and m2 = log 1 + √ . 1 − r2

Proof Radius m2 follows from Theorem 14.16. By Theorem 14.24 we √ need to show that the inequality m1 ≤ min{f (t), g(t)} √ holds for t ∈ (r/ 1 − r 2 , 1 − r 2 /r) and functions   r(1 + t 2 ) f (t) = log 1 + √ , t 1 − r 2 − r 2t 2

  r(1 + t 2 ) g(t) = log 1 + √ . 1 − r 2t − r

By elementary computation functions f  (t) and g  (t) are monotone and f  (tf ) = 0 √ √ is equivalent to tf = 1 − r 2 / 1 + r 2 and g  (tg ) = 0 is equivalent to tg = (1 + √ r)/ 1 − r 2 . Now     2r 2r ≥ log 1 + g(tg ) = log 1 + = f (tf ) 1−r 1 − r2  

and the assertion follows. Corollary 14.30 Let G = Then

Hn ,

x ∈ G with x1 = x2 = · · · = xn−1 = 0 and r > 0.

Bj (x, r) ⊂ Bq (x, M1 ) and for r ∈ (0, log(1 +

√ 2)) Bk (x, r) ⊂ Bq (x, M2 ),

where √ M1 =

2 + er (er − 2) − 1 er − 1

and  M2 =

√ (er − 1)(er − 1 + 17 + er (er − 2)) . √ 2 2

Notes 14.31 This chapter is based on [285, 286, 295] and [296]. M.R. Mohapatra and S.K. Sahoo [395] have proved various ball inclusion theorems and comparison theorems for several metrics, including e.g. the Cassinian and triangular ratio metrics. See also the paper of X. Zhang [573].

Part V

Quasiregular Mappings

Quasiregular mappings were introduced and studied by Yu. G. Reshetnyak [453– 465] and O. Martio, S. Rickman, J. Väisälä [363–365]. The fundamental topological and analytic properties of these mappings were proven by Reshetnyak, see [465, 466]. The study of quasiconformal and quasiregular mappings in this chapter will be based on the transformation formulae for the moduli of curve families under these mappings. We refer the reader to Rickman’s book [473, pp. 31–58] which covers both the topological properties of discrete and open maps as well as a thorough discussion of these transformation rules. In most cases it will be enough to make use of these transformation formulae specialized to the conformal invariants μG and λG . These special cases of the general transformation formulae are convenient to use because they together with the results of Chap. 10 provide immediate insight into some relevant geometric quantities. In the case of the conformal (pseudo)metric μG the transformation formula reads: a quasiregular mapping f : G → f G ⊂ Rn is a Lipschitz mapping between the (pseudo)metric spaces (G, μG ) and (f G, μf G ). From this result and a similar result for Ferrand’s modulus metric we derive several distortion and growth theorems for quasiregular mappings. To this end we shall make use of some results from Part III that will enable us to find simple estimates for the functions μG (x, y) and λG (x, y). Except for a few special cases such as G = Bn formulae for μG (x, y) and λG (x, y) are unknown, but one can give upper and lower bounds for them in terms of rG (x, y) =

|x − y| , d(x) = d(x, ∂G) , min{d(x), d(y)}

for a wide class of domains G (see 6.1 and 10.4). When G = Bn the transformation formulae for μG and λG yield two variants of the Schwarz lemma (see 16.2 and 16.22, respectively). A central theme of this part is a circle of ideas around the Schwarz lemma and its various generalizations, including a study of uniform continuity properties of quasiregular mappings.

Chapter 15

Basic Properties of Quasiregular Mappings

In this chapter we introduce quasiregular and quasiconformal mappings and their basic properties. These results are usually given as theorems—for proofs we refer the reader to the books of Reshetnyak [465] and Rickman [473]. We start by reviewing some properties of discrete open mappings.

15.1 Topological Properties of Discrete Open Mappings We shall survey some topological properties of discrete open mappings. A thorough discussion of this topic, including the definition of the degree of a mapping, requires machinery from algebraic topology (see [445]). In this subchapter no proofs will be given. n

Definition 15.1 The set Tn consists of all triples (y, f, D), where f : G → R n is a continuous mapping, G ⊂ R is a domain, D is a domain with D ⊂ G and n y ∈ R \ f ∂D. Lemma 15.2 There exists a unique function μ : Tn → Z, the topological degree, such that (1) (2) (3) (4)

n

y → μ(y, f, D) is a constant in each component of R \ f ∂D. |μ(y, f, D)| = 1 if y ∈ f D and f |D is one-to-one. μ(y, id, D) = 1 if y ∈ D and id is the identity mapping. Let (y, f, D) ∈ Tn and D1 , . . . , Dkbe disjoint domains such that (y, f, Di ) ∈ Tn and f −1 (y) ∩ D ⊂ ki=1 Di ⊂ D. Then μ(y, f, D) =

k !

μ(y, f, Di ) .

i=1

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_15

281

282

15 Basic Properties of Quasiregular Mappings

(5) Let (y, f, D), (y, g, D) ∈ Tn be such that there exists a homotopy ht : D → n R , t ∈ [0, 1], with h0 = f |D, h1 = g|D, and (y, ht , D) ∈ Tn for all t ∈ [0, 1]. Then μ(y, f, D) = μ(y, g, D). Lemma 15.3 (1) If (y, f, D) ∈ Tn and y ∈ f D, then μ(y, f, D) = 0. (2) If f is a constant c, then μ(y, f, D) = 0 for all y = c. (3) If f : D → Rn is differentiable at x0 ∈ D and Jf (x0 ) = det f  (x0 ) = 0, then there exists a neighborhood U of x0 such that (y, f, U ) ∈ Tn and μ(y, f, U ) = sign Jf (x0 ) for y ∈ f U . It follows from 15.3(3) that if f is a reflection in the plane xn = 0, then μ(y, f, Bn ) = −1 for y ∈ Bn . We next extend the Definition 3.7 of a sensepreserving C 1 -homeomorphism. n

Definition 15.4 A mapping f : G → R is called sense-preserving (orientationpreserving) if μ(y, f, D) > 0 whenever D is a domain with D ⊂ G and y ∈ f D \ f ∂D . If μ(y, f, D) < 0 for all such y and D , then f is called sensereversing (orientation-reversing). Reflection in a plane and inversion in a sphere are sense-reversing mappings [445, pp. 137–145]. n

n

Lemma 15.5 Let f : G → R and g : f G → R be mappings and set h = g ◦ f . If f and g are both sense-preserving or both sense-reversing, then h is sense-preserving. If one of the maps f and g is sense-reversing and the second one is sense-preserving, then h is sense-reversing. Remark 15.6 The approach to the degree theory in [445] is based on algebraic topology. An alternative approach can be based on Sard’s theorem and on approximation of continuous functions by C ∞ -functions, for which the degree μ(y, f, D) can be defined as the sum of the signs of the Jacobians, evaluated at the points of D ∩ f −1 (y) . See [110, 223, 463]. Lemma 15.7 Let (y, f, D) and (y, g, D) ∈ Tn be such that f |∂D = g|∂D and ∞ ∈ f D ∪ gD . Then μ(y, f, D) = μ(y, g, D) . For 15.5 see [522] and for 15.7 see [445, pp. 129–130]. The assumption ∞ ∈ f D ∪ gD in 15.7 cannot be dropped, as the example D = Bn , f = id , and g n an inversion in S n−1 , shows. The branch set Bf of a mapping f : G → R is defined to be the set of all points x ∈ G such that f is not a local homeomorphism at x . It is easily seen that Bf is a closed subset of G . We call f open if f A is n open in R whenever A ⊂ G is open, light if f −1 (y) is totally disconnected for all y ∈ f G , and discrete if f −1 (y) is isolated for all y ∈ f G . The next lemma is a fundamental property of discrete open mappings (see A. V. Chernavski˘ı [98, 99] and J. Väisälä [523]).

15.1 Topological Properties of Discrete Open Mappings

283

n

Lemma 15.8 Let f : G → R be discrete open. Then dim Bf = dim f Bf = dim f −1 f Bf ≤ n − 2 , where dim refers to the topological dimension. Remark 15.9 It is clear that G \ Bf is open, and from 15.8 and a well-known nonseparation property of sets of dimension ≤ n − 2 (see [247, p. 98]) it follows that G \ Bf is a domain. Stoilow’s theorem (see [326]) implies that Bf consists of isolated points for n = 2 . In the multidimensional case n ≥ 3 Bf never contains isolated points, as one can show by applying some properties of covering mappings (monodromy theorem). n

Let G ⊂ R be a domain. We denote by J (G) the collection of all subdomains D of G with D ⊂ G . n

Definition 15.10 Let f : G → R be discrete. Fix x ∈ G and a neighborhood U ∈ J (G) of x such that {x} = U ∩ f −1 (f (x)) . The number μ(f (x), f, U ) is denoted by i(x, f ) and called the local (topological) index of f at x . Now let f : G → Rn be discrete open. It follows from 15.8 that G \ Bf is connected. Hence i(x, f ) has a constant value, either +1 or −1 , in G \ Bf . In the first case f is sense-preserving, and in the second case sense-reversing. In both cases we have by 15.2(4) if D ∈ J (G) , y ∈ f D \ f ∂D , and D ∩ f −1 (y) = {x1 , . . . , xk } μ(y, f, D) =

k !

i(xj , f ) .

(15.1)

j =1

A domain D ∈ J (G) is said to be a normal domain of f : G → Rn if f ∂D = ∂f D . A normal neighborhood of x is a normal domain D such that D ∩ f −1 (f (x)) = {x} . It follows from 15.2(1) that μ(y, f, D) is a constant if D is a normal domain of f and y ∈ f D . This constant is denoted μ(f, D) . Let D be a normal domain of f , y ∈ f D , and f −1 (y) = {x1 , . . . , xk } . It follows from (15.10) that μ(f, D) =

k !

i(xj , f ) .

j =1

Exercise 15.11 If f : G → Rn is open and D ∈ J (G) , then ∂f D ⊂ f ∂D is always true (Fig. 15.1). In classical function theory (see [90, p. 84], [570]) the local topological index is usually called the winding number of a point. We shall next list several topological results about discrete open mappings without proofs. The proofs of Lemmas 15.12–15.14 are given in [363].

284

15 Basic Properties of Quasiregular Mappings

Fig. 15.1 Open mapping f in Exercise 15.11

f open

G

fG D

fD

Lemma 15.12 Suppose that f : G → Rn is open, that U ⊂ Rn is a domain, and that D is a component of f −1 U such that D ∈ J (G) . Then D is a normal domain, f D = U , and U ∈ J (f G) . If f : G → Rn , x ∈ G , and r > 0 , then the x -component of f −1 B n (f (x), r) is denoted by U (x, f, r) . Lemma 15.13 Suppose that f : G → Rn is a discrete and open mapping. Then limr→0 d(U (x, f, r)) = 0 for every x ∈ G . If U (x, f, r) ∈ J (G) , then U (x, f, r) is a normal domain and f U (x, f, r) = B n (f (x), r) ∈ J (f G) . Furthermore, for every point x ∈ G there is a positive number σx such that the following conditions are satisfied for 0 < r ≤ σx : (1) (2) (3) (4) (5) (6)

U (x, f, r) is a normal neighborhood of x . U (x, f, r) = U (x, f, σx ) ∩ f −1 B n (f (x), r) . ∂U (x, f, r) = U (x, f, σx ) ∩ f −1 S n−1 (f (x), r) if r < σx . n R \ U (x, f, r) is connected. n R \ U (x, f, r) is connected. If 0 < r < s ≤ σx , then U (x, f, r) ⊂ U (x, f, s) , and U (x, f, s) \ U (x, f, r) is a ring, i.e. its complement has exactly two components. If f : G → Rn , A ⊂ Rn and y ∈ Rn , denote N(y, f, A) = card(A ∩ f −1 (y)) , N(f, A) = sup{ N(y, f, A) : y ∈ Rn } , N(f ) = N(f, G) .

Here N(y, f, A) is called the multiplicity of y in A and N(f, A) the maximal multiplicity of f in A . Lemma 15.14 Suppose that f : G → Rn is sense-preserving, discrete, and open. (1) If D ∈ J (G) , then N(y, f, D) ≤ μ(y, f, D) for all y ∈ Rn \ f ∂D and N(y, f, D) = μ(y, f, D) for y ∈ Rn \ f A , A = ∂D ∪ (D ∩ Bf ) . (2) If D is a normal domain, N(f, D) = μ(f, D) . (3) If A ⊂ G is compact, N(f, A) < ∞ .

15.2 Path Lifting

285

(4) Every point x ∈ G has a neighborhood V such that if U is a neighborhood of x and if U ⊂ V , then N(f, U ) = i(x, f ) . (5) x ∈ Bf iff i(x, f ) ≥ 2 . It follows from 15.14(4) that the local index i(x, f ) of a sense-preserving discrete open mapping f can be defined in terms of the maximal multiplicity of f as follows i(x, f ) = lim N(f, B n (x, r)) . r→0

(15.2)

A trivial example is the function g : B2 → B2 , g(z) = z2 with i(0, g) = 2 . Remark 15.15 Let f : G → Rn be continuous, Aj ⊂ Rn , j = 1, 2, . . . . Then one can show that !  N(y, f, Aj ) ≤ N(y, f, Aj ) , !  N(f, Aj ) ≤ N(f, Aj ) . If A is a Borel set in G , then N(y, f, A) is measurable (cf. [445, pp. 216–219]).

15.1.1 An Open Problem Let f : G → Rn be discrete open, x0 ∈ G , t ∈ (0, d(x0 , ∂G)) , and assume that f S n−1 (x0 , t) = ∂f B n (x0 , t) , that is, B n (x0 , t) is a normal domain. Assume, further, that Bf ∩S n−1 (x0 , t) = ∅ and n ≥ 3 . Is it true that f |B n (x0 , t) is one-toone? For n = 2 we have the obvious counterexample g : B2 → B2 , g(z) = z2 . This problem is given in [50, p. 503, 7.66]. See also problem (4) at the end of this book.

15.2 Path Lifting n

n

Let f : G → R and let β : [a, b) → R be a path and let x0 ∈ G be such that f (x0 ) = β(a) . A path α : [a, c) → G is said to be a maximal lifting of β starting at x0 if: (1) α(a) = x0 . (2) f ◦ α = β|[a, c) . (3) If c < c ≤ b , then there does not exist a path α  : [a, c ) → G such that α = α  |[a, c) and f ◦ α  = β|[a, c) .

286

15 Basic Properties of Quasiregular Mappings n

n

If β : [a, b) → R is a path and if C ⊂ R , we write β(t) → C as t → b if the spherical distance q(β(t), C) → 0 as t → b . n

Lemma 15.16 Suppose that f : G → R is light and open, that x0 ∈ G , and n that β : [a, b) → R is a path such that β(a) = f (x0 ) and such that either limt →b β(t) exists or β(t) → ∂f G as t → b . Then β has a maximal lifting α : [a, c) → G starting at x0 . If α(t) → x1 ∈ G as t → c , then c = b and f (x1 ) = limt →b β(t) . Otherwise α(t) → ∂G as t → c . If f is discrete and if the local index i(α(t), f ) is constant for t ∈ [a, c) , then α is the only maximal lifting of β starting at x0 . This lemma is proved in [365, 3.12]. It follows from the lemma, in particular, that a locally homeomorphic mapping has a unique maximal lifting starting at a point. Remark 15.17 In the sequel Lemma 15.16 will be applied in the following situation. n Let f : G → Rn be non-constant quasiregular, x0 ∈ G , and let β : [0, 1] → R be a path with β(0) = f (x0 ) and β(1) ∈ ∂f G . Then 15.16 shows that β has a maximal lifting α : [0, c) → G starting at x0 with α(t) → ∂G as t → c . A mapping f : G → Rn is called proper if f −1 K is a compact subset of G whenever K is a compact subset of f G , and closed if f C is a (relatively) closed subset of f G whenever C is a (relatively) closed subset of G . Lemma 15.18 Let f : G → Rn be discrete open. Then the following conditions are equivalent: (1) f is proper. (2) f is closed. (3) N(f, G) = p < ∞ and for all y ∈ f G p=

k !

i(xj , f ) , {x1 , . . . , xk } = f −1 (y) .

j =1

For a proof of 15.18 see [370, 523, 547], and the references in these. As the simple example z → z2 shows, a maximal lifting of a path starting at a branch point need not be unique. The next lemma is a quantitative statement of this fact. For a proof see [467]. Lemma 15.19 Let f : G → Rn be discrete, open, and closed. Denote p = N(f, G) < ∞ and let β : [a, b) → f G be a path. Then there exist paths αj : [a, b) → G , 1 ≤ j ≤ p , for which (1) f ◦ αj = β , (2)  card { j : αj (t) = x } = |i(x, f )| for x ∈ f −1 |β| and t ∈ [a, b) , p −1 |β| . (3) j =1 |αj | = f Exercise 15.20 Let G, G ⊂ Rn , n ≥ 2 , and let f : G → G be a homeomorphism with the following property: There exists c1 ∈ (0, ∞) such that

15.2 Path Lifting

287

for every subdomain D ⊂ G and for all x, y ∈ D , jf D (f (x), f (y)) ≤ c1 jD (x, y).

()

Show that for each z ∈ G , 2 |f (x) − f (z)| : |x − z| = |y − z| = r ≤ c2 , H (f, z) = lim sup |f (y) − f (z)| r→0 where c2 ∈ (1, ∞) . Show that this inequality holds (possibly with a different constant c2 ) also if in (  ) jD and jf D are replaced by kD and kf D . Exercise 15.21 Let G, G ⊂ Rn be domains, and let f : G → G = f G be continuous. Then f is said to be open if it maps all open subsets onto open subsets of G , closed if it maps all closed subsets onto closed subsets of G , and proper if for every compact K ⊂ G also f −1 K is compact. Note the condition f G = G above, i.e., f is a surjective map. (1) Show that the map f : H → B2 \ {0}, H = {z ∈ C : Re z < 0}, f (z) = exp (z), is open but neither proper nor closed. (2) Prove: Let G, G ⊂ Rn be domains, and let f : G → G = f G be continuous, open, and closed. If y ∈ G , then f −1 (y) is compact. (3) Prove: Let G, G ⊂ Rn be domains, and let f : G → G = f G be continuous, discrete, open, and closed. If y ∈ G \ f (Bf ) and U is an open neighborhood of f −1 (y) in G , then there is an open neighborhood V of y in G such that f −1 V ⊂ U . [Hint: Lemmas 15.13 and 15.18.] Exercise 15.22 Let G, G ⊂ Rn be domains, and let f : G → G = f G be continuous, discrete, open, and closed. Then f is proper, i.e., for every compact E ⊂ G , also f −1 E is compact. Exercise 15.23 Let G, G ⊂ Rn be domains, and let f : G → G = f G be continuous. The cluster set of f at a point b ∈ ∂G is the set C(f, b) = {b ∈ Rn : ∃(bk ) ∈ G, bk → b, f (bk ) → b  } . It is clear that C(f, b) ⊂ G , and that for injective maps C(f, b) ⊂ ∂G . The cluster set C(f, b) is a singleton iff f has a limit at b . The cluster set is connected if there are arbitrarily small numbers t > 0 such that B(b, t) ∩ G is connected. We say that f is boundary preserving if C(f, b) ⊂ ∂G for all b ∈ ∂G . (1) Find for each b ∈ S 1 the cluster set C(f, b) of the analytic function f : B2 → B2 , with f (z) = exp g(z) when g(z) = −(1 + z)/(1 − z), z ∈ B2 . (2) Let G, G ⊂ Rn be domains, and let f : G → G = f G be open and continuous. Show that f is boundary preserving iff f is proper.

288

15 Basic Properties of Quasiregular Mappings

Exercise 15.24 Let f : B2 → B2 \ {0} ≡ G be the analytic function, f (z) = exp (g(z)) with g(z) = −(1 + z)/(1 − z), z ∈ B2 . For t ∈ (0, 1) estimate sup{kG (f (0), f (z)) : |z| = t}. [Hint: Consider kG (|f (0)|, |f (z)|). ] Remark 15.25 It follows easily from the definitions that an open continuous mapping f : G → Rn obeys the maximum principle, i.e. if D ∈ J (G) then max |f (x)| = max |f (x)| . ∂D

D

For further results concerning with discrete and open mappings see [103] and [512].

15.3 Analytic Properties of Quasiregular Mappings We study next some fundamental properties of quasiregular mappings. According to deep results of Yu. G. Reshetnyak [455, 463], a non-constant quasiregular mapping is discrete, open, and differentiable a.e., and it satisfies Lusin’s condition (N) [233, p. 288]. By definition, condition (N) holds if and only if sets of measure zero are mapped onto sets of measure zero. The proofs are beyond the scope of this book. Applying these results one can prove the transformation formulae, the so-called KO -and KI -inequalities, for the moduli of curve families under quasiregular mappings. For the proofs of these important results we refer the reader to S. Rickman’s book [473, Ch. II]. Of these the KO -inequality is due to O. Martio, S. Rickman, and J. Väisälä [363], while the KI -inequality was proved by E. A. Poletski˘ı [425] and in an improved form by J. Väisälä [525]. A simplified proof of Poletski˘ı’s result was given by M. Pesonen [424]. Let G ⊂ Rn be a domain. A mapping f : G → Rn is said to be quasiregular if f is ACLn and if there exists a constant K ≥ 1 such that |f  (x)|n ≤ KJf (x) , |f  (x)| = max |f  (x)h| , |h|=1

(15.3)

a.e. in G . Here f  (x) denotes the formal derivative of f at x (cf. Notation and terminology). The smallest K ≥ 1 for which this inequality is true is called the outer dilatation of f and denoted by KO (f ) . If f is quasiregular, then the smallest K ≥ 1 for which the inequality Jf (x) ≤ K l(f  (x))n , l(f  (x)) = min |f  (x)h| , |h|=1

(15.4)

holds a.e. in G is called the inner dilatation of f and denoted by KI (f ) . The maximal dilatation of f is the number K(f ) = max{ KI (f ), KO (f ) } . If

15.4 Curve Families and Quasiconformal Mappings

289

K(f ) ≤ K , f is said to be K – quasiregular. If f is not quasiregular, we set KO (f ) = KI (f ) = K(f ) = ∞ . It follows from linear algebra (see [524, p. 44] and [463, p. 22]) that KO (f ) ≤ KI (f )n−1 , KI (f ) ≤ KO (f )n−1

(15.5)

hold. Moreover, these inequalities are best possible. Lemma 15.27 Let f : G → Rn be a non-constant quasiregular mapping. Then (1) f is sense-preserving, discrete, and open , (2) f is differentiable a.e. , (3) f satisfies condition (N), i.e. if A ⊂ G and m(A) = 0 , then also m(f A) = 0 . For proofs of these results, see [455, 456, 463]. Next we extend the definition of a quasiregular mapping. n

n

Definition 15.28 Let G ⊂ R be a domain. A mapping f : G → R is called quasimeromorphic if either f G = {∞} or the set E = f −1 (∞) is discrete and f1 = f | G\(E∪{∞}) is quasiregular. We set K(f ) = K(f1 ) , KO (f ) = KO (f1 ) , and KI (f ) = KI (f1 ) . Definition 15.29 If f is a homeomorphism satisfying (10.2) and (15.4) with |Jf (x)| in place of Jf (x) , then f is called quasiconformal. Remark 15.30 For n = 2 and K = 1 the class of K -quasiregular maps coincides with the class of analytic functions. By 15.29 a quasiconformal mapping may be sense-reversing, while a quasiregular mapping in the sense of (15.3) is always sense-preserving. In his book [463] Reshetnyak replaces Jf (x) by |Jf (x)| in the definition of a quasiregular mapping and hence quasiregular maps in the sense of [463] may be sense-reversing. This is largely a question of technical convention, since by topology (see Lemma 15.8) each discrete open mapping is either sensepreserving or sense-reversing.

15.4 Curve Families and Quasiconformal Mappings We now give an alternative definition of a quasiconformal mapping. Let G, G n be domains in R and let f : G → G be a homeomorphism. Then f is K – quasiconformal if 3 M( ) K ≤ M(f ) ≤ K M( )

(15.6)

290

15 Basic Properties of Quasiregular Mappings

for every curve family in G . Moreover, the dilatations of f are defined as KI (f ) = sup

M(f ) , M( )

KO (f ) = sup

M( ) , M(f )

where the suprema are taken over all curve families in G such that M( ) and M(f ) are not simultaneously 0 or ∞ . Thus 3 M( ) KO (f ) ≤ M(f ) ≤ KI (f ) M( )

(15.7)

for every curve family in G . The equivalence of the two definitions 15.29 and (15.6) of a quasiconformal mapping is proved in [524] and also in [94, pp. 81–110]. The next example shows that (15.7) does not generalize directly to the case of quasiregular mappings. Examples 15.31 (1) Let fk (z) = zk , k ∈ N \ {0} , z ∈ C = R2 , and = (S 1 , S 1 (1/e)) . By (7.4) M( ) = 2π ,

3 M(fk ) ≤ 2π log(ek ) = 2π/k .

Moreover, K(fk ) = 1 because fk is analytic. If k ≥ 2 , we see that the left inequality of (15.7) fails to hold for (non-univalent) analytic functions. (2) Let Aj = { (x, y) ∈ R2 : x = j } , j = 0, 1 , f (z) = exp z , z ∈ R2 , and = (A0 , A1 ) . Then f ⊂ (S 1 (e), S 1 ) and M(f ) ≤ 2π/ log e = 2π by (7.4), whereas M( ) = ∞ by 7.1.1 or by 7.22 and 7.10. Since K(f ) = 1 also in this example, we see that the left inequality of (15.7) fails to hold for analytic functions. A fortiori, it fails to hold for quasiregular mappings. By inserting a multiplicity factor in the left side of (15.7) one obtains the KO inequality for quasiregular mappings [363]. Theorem 15.31 Suppose that f : G → Rn is a quasiregular mapping and that A is a Borel set in G such that N(f, A) < ∞ . If is a family of paths in A , M( ) ≤ N(f, A) KO (f ) M(f ) . Proof Set L(x, f ) = lim sup h→0

|f (x + h) − f (x)| |h|

for x ∈ G . Thus L(x, f ) = |f  (x)| whenever f is differentiable at x . It is easy to see that x → L(x, f ) is a Borel function.

15.4 Curve Families and Quasiconformal Mappings

291

Suppose that σ ∈ F (f ) . Define ρ : Rn → R ∪ {∞} by setting  ρ(x) =

σ (f (x)) L(x, f )

if x ∈ A ;

0

otherwise .

Let 0 be the family of all rectifiable paths γ ∈ such that f is absolutely continuous on γ . By Lemma 9.4 M( 0 ) = M( ) . From the formula concerning change of variables in integrals it follows that 

 ρ ds ≥

σ ds ≥ 1

f ◦γ

γ

for all γ ∈ 0 . Thus ρ ∈ F ( 0 ) . A more detailed proof is given in [363]. Hence we obtain   n M( ) = M( 0 ) ≤ ρ dm = σ (f (x))n L(x, f )n dm(x) Rn

≤ KO (f )

A



σ (f (x))n J (x, f ) dm(x) . A

Since f is ACLn , J (x, f ) is integrable over every domain D ∈ J (G) . Thus the transformation formula in [445, Theorem 3, p. 364] yields 

 σ (f (x)) J (x, f ) dm(x) = n

A∩D

Rn

σ (y)n N(y, f, A ∩ D) dm(y) 

≤ N(f, A)

Rn

σ n dm .

The theorem cited above is formulated in [445] for finite-valued functions, but we may apply it to min(k, σ n ) and then let k → ∞ . Since D ∈ J (G) is arbitrary, we obtain  M( ) ≤ N(f, A) KO (f ) σ n dm . Rn

Since this holds for every σ ∈ F (f ) , the theorem follows.

 

The right side of (15.7) holds for quasiregular mappings, too, as the following theorem shows. We shall mainly need the special case m = 1 of this result. The proof is omitted [525]. Theorem 15.32 Suppose that f : G → Rn is a non-constant quasiregular mapping, that is a path family in G , that  is a path family in Rn and that m is a positive integer such that the following condition is satisfied: There is a set E0 ⊂ G of measure zero such that for every path β : I → Rn in  there

292

15 Basic Properties of Quasiregular Mappings

are paths α1 , . . . , αm in with f ◦ αi ⊂ β for all i and such that for every x ∈ G \ E0 and t ∈ I , αi (t) = x for at most one i . Then M(  ) ≤

KI (f ) M( ) . m

In this result it is not required that f =  . As a matter of fact, in many applications f <  . If D is a normal domain of f , if  is a family of paths in f D , and if is the family of all paths α in D such that f ◦ α ∈  , then the condition in 15.32 is satisfied with m = N(f, D) , E0 = Bf by 15.18 and 15.34(1) below. Due to the relationship (9.4) between the conformal capacity and the modulus of a curve family, one can formulate the KO - and KI -inequalities for condensers as well. If f : G → Rn is discrete open and (A, C) is a condenser in G such that A is a normal domain of f , then (A, C) is called a normal condenser. Also the next result is from [525]. Theorem 15.33 Suppose that f : G → Rn is a non-constant quasiregular mapping. Then (1) cap(f A, f C) ≤

KI (f ) cap(A, C) M(f, C)

for all condensers (A, C) in G where !

M(f, C) = inf

y∈f C

i(x, f )

x∈C∩f −1 (y)

and (2) cap(A, C) ≤ KO (f ) N(f, A) cap(f A, f C) for all normal condensers (A, C) in G . In the next theorem we list some basic properties of quasiregular mappings. Theorem 15.34 Let f : G → Rn be a non-constant quasiregular mapping. Then (1) m(Bf ) = m(f Bf ) = 0 . (2) Jf (x) > 0 a.e. in G . (3) If g : G → Rn is a quasiregular mapping with f G ⊂ G , then KO (f ◦ g) ≤ KO (f )KO (g)

and KI (f ◦ g) ≤ KI (f )KI (g) .

Remark 15.35 Part (3) of 15.34 follows immediately from 15.33. Part (1) of 15.34 can be much improved, see [362, 461, 480].

15.4 Curve Families and Quasiconformal Mappings

293

In most of our later applications of the KO - and KI -inequalities, one may appeal to the following particular cases, which are the transformation formulae for μG and λG . Theorem 15.36 If f : G → Rn is a non-constant quasiregular mapping, then (1) μf G (f (a), f (b)) ≤ KI (f ) μG (a, b) ; a, b ∈ G . In particular, f : (G, μG ) → (f G, μf G ) is Lipschitz continuous. If N(f, G) < ∞ , then (2) λG (a, b) ≤ KO (f ) N(f, G) λf G (f (a), f (b)) for all a, b ∈ G with f (a) = f (b) . Proof (1) Fix a, b ∈ G and a curve α : [0, 1] → G such that α(0) = a , α(1) = b , and denote  = ( (f ◦ α)[0, 1], ∂f G ) (Fig. 15.2). Let be the family of all maximal liftings of the elements of  starting at |α| . That is, γ ∈ iff there exists β in  such that γ is a maximal lifting of β starting at a point of |α| . Then f <  ; by the definition (10.1) of the conformal invariant μG and by 7.2 and 15.32, μf G (f (a), f (b)) ≤ M(  ) ≤ M(f ) ≤ KI (f ) M( ) . Because |β|∩∂f G = ∅ for all β ∈  , it follows from 15.16 that |γ |∩∂G = ∅ for all γ ∈ . Then by 7.1(2)   M( ) ≤ M (|α|, ∂G; G) . The proof now follows from this and the preceding inequality since α is an arbitrary curve in G with α(0) = a , α(1) = b (see (10.2)). (2) Let βj : [0, 1) → f G be paths such that βj (t) → ∂f G , j = 1, 2 , as t → 1 , f (a) = β1 (0) , f (b) = β2 (0) and |β1 |∩|β2 |∩f G = ∅ . Let γj : [0, cj ) → G be a maximal lifting of βj , j = 1, 2 , with γ1 (0) = a , γ2 (0) = b . Since βj (t) → ∂f G as t → 1 it follows from 15.16 that γj (t) → ∂G as t → cj ,

Fig. 15.2 Proof of Theorem 15.36

294

15 Basic Properties of Quasiregular Mappings

j = 1, 2 . Let = (|γ1 |, |γ2 |; G) . By (10.1) and 15.31 λG (a, b) ≤ M( ) ≤ KO (f ) N(f, G) M(f ) . Because βj : [0, 1) → f G , j = 1, 2 , were arbitrary curves satisfying the conditions mentioned above and because f ⊂ (|β1 |, |β2 |; G) , the proof follows from the last inequality, (10.1), and 7.1(2).   2 Exercise 15.37 Let f : Bn → Z , Z = {x ∈ Rn : jn−1 =1 xj < 1} be K quasiregular, f (0) = 0 . Show that

|f (x)| ≤ AK(log

1 + |x| + B), 1 − |x|

where A, B depend only on n . [Hint: 7.20 and μ -metric.] Corollary 15.38 If f : G → G = f G is a quasiconformal mapping, then 3 (1) μG (a, b) KO (f ) ≤ μf G (f (a), f (b)) ≤ KI (f ) μG (a, b) , 3 (2) λG (a, b) KO (f ) ≤ λf G (f (a), f (b)) ≤ KI (f ) λG (a, b) hold for all distinct a, b ∈ G . Proof The right inequalities were proved in 15.36. Because KO (f −1 ) = KI (f ) , KI (f −1 ) = KO (f ) , the left inequalities also follow from 15.36.   Exercise 15.39 Let G ⊂ Rn be a domain x, y, z ∈ G with |x − y| = d(x)/2 and |x − z| > d(x). Find a lower bound for λG (x, z) in terms of λG (x, y) and kG (z, y). [Hint: You may reduce the former case ( λG (x, z) ) to the latter case ( λG (x, y) ) by use of an auxiliary quasiconformal mapping as follows. It is well-known [165] that for a domain D ⊂ Rn and x, y ∈ D there is a K quasiconformal mapping f : D → D with f (z) = z for all z ∈ ∂D with f (x) = y, K ≤ exp (c1 kD (x, y)) where c1 > 0 is a constant.]

15.4.1 Ferrand’s Problem J. Ferrand studied in [329] the question whether a homeomorphism satisfying the inequality in Corollary 15.38 (2) is quasiconformal. This question was then solved in the negative in the paper Ferrand–Martin–Vuorinen [137]. In [137] it was conjectured that isometries of μG , i.e. homeomorphisms for which 15.38 (1) hold as equality are conformal. Klén–Vuorinen–Zhang proved [298] that for all n ≥ 2 these isometries are quasiconformal for domains with connected boundary. The case of a general domain in the dimension n = 2 was established by D. Betsakos and S. Pouliasis [69]. For dimensions n ≥ 2 , the general case was proved by X. Zhang [574] and independently by S. Pouliasis and A. Solynin [433]. Similar questions

15.4 Curve Families and Quasiconformal Mappings

295

about the characterization of isometries can be formulated for other metrics as well. P. Hästö has extensively studied these questions, [204, 205], see also [206–208, 210]. According to 15.36, each quasiregular mapping f : G → f G is a Lipschitz mapping of the (pseudo)metric space (G, μG ) onto (f G, μf G ) . We shall employ   the inequalities of Chap. 10 for M (E, F ) , which enable us to give a geometric meaning to this general result in many interesting cases and to replace the metric space (G, μG ) by other metric spaces. Depending on the context, one may wish to replace (G, μG ) by some less abstract space such as (Bn , ρ) , (G, kG ) , (G, jG ) or even (Rn , | |) . One can derive numerous distortion results for quasiconformal and quasiregular mappings directly from 15.36, 15.38, and the estimates of Chap. 10. Examples of such results will be given in Chap. 16. We shall next give an application of 15.36 which yields a bound for the linear dilatation H (x, f ) defined by ⎧ L(x, f, r) ⎪ ⎪ H (x, f ) = lim sup , ⎪ ⎪ ⎨ r→0 l(x, f, r) ⎪ L(x, f, r) = max{ |f (x) − f (z)| : |x − z| = r } , ⎪ ⎪ ⎪ ⎩ l(x, f, r) = min{ |f (z) − f (x)| : |x − z| = r } ,

(15.8)

where 0 < r < d(x, ∂G) , whenever f : G → Rn is discrete and x ∈ G . Note that l(x, f, r) > 0 for all small enough r > 0 if f is discrete and hence H (x, f ) is well-defined for discrete maps. Theorem 15.40 If f : G → Rn is a non-constant quasiregular mapping and x ∈ G , then   H (x, f ) ≤ c n, KO (f ) i(x, f ) < ∞ . Proof We may assume that x = 0 = f (x) . Let σ0 be as in 11.10, U = U (0, f, σ0 ) , and choose t > 0 such that B n (3t) ⊂ U . For each r ∈ (0, t] choose xr , yr ∈ S n−1 (r) with |f (xr )| = L(0, f, r) , |f (yr )| = l(0, f, r) . Let Ar be the yr -component of f −1 [0, f (yr )] and Br the xr -component of f −1 [f (xr ), ∞) . Then 0 ∈ Ar and Br ∩∂U = ∅ by 15.16. Denote r = (Ar , Br ; U ) . By 7.8, 7.2, and 7.4 we obtain    3t 1−n M( r ) + ωn−1 log ≥ M (Ar , U ∩ Br ; Rn ) . r

(15.9)

Next, 9.14 and 7.32(1) yield  3t + r      M (Ar , U ∩ Br ; Rn ) ≥ M ( [0, re1], [−re1 , −3te1 ]) = τ . 3t − r (15.10)

296

15 Basic Properties of Quasiregular Mappings

If |f (xr )| > |f (yr )| then by 7.18 we obtain M(f r ) ≤ τ

  |f (x )| r −1 . |f (yr )|

(15.11)

This inequality holds trivially if |f (xr )| = |f (yr )| . By 15.31, 15.10, and 15.14 M( r ) ≤ KO (f ) i(x, f ) M(f r ) . We combine the latter inequality with (15.9) and (15.10) and let r → 0 . As a result we obtain   τ (1) ≤ KO (f ) i(0, f ) τ H (0, f ) − 1 , H (0, f ) ≤ 1 + τ −1



 τ (1) , KO (f ) i(0, f )  

as desired.

Corollary 15.41 If f : Rn → Rn is a quasiconformal mapping with f (0) = 0 , then   |f (x)| ≤ c n, KO (f ) |f (y)| for |x| = |y| . Proof The proof is similar to that of 15.40; in fact, it is slightly simpler.

 

Remark 15.42 Making use of the functional identity in 7.31 one can write the constant in 15.40 also as follows √ 2    −1  γ( 2) c n, KO (f ) i(x, f ) = γ . KO (f ) i(x, f ) This equality together with 9.34 and 15.40 shows that the linear dilatation H (x, f ) of a K -quasiregular mapping f has an upper bound depending only on K i(x, f ) . In particular, this upper bound is independent of n . Exercise 15.43 Let f : Rn → Rn be a K -quasiconformal mapping with f (0) = 0 and let m = min{ |f (x)| : |x| = 1 } , M = max{ |f (x)| : |x| = 1 } . Without appealing to 15.40 or 15.41 show that for some constant d(n, K) M/m ≤ d(n, K) .   [Hint: Let  =  S n−1 (m), S n−1 (M) . Because S n−1 ∩ f −1 S n−1 (m) = ∅ = S n−1 ∩ f −1 S n−1 (M) , 9.22 yields a lower bound for M( ) , = f −1  .]

15.4 Curve Families and Quasiconformal Mappings

297

Remark 15.44 A. Mori [398] proved that the linear dilatation of a K -quasiconformal mapping of a plane domain has an upper bound eπK . His result was extended to the multidimensional case by F. W. Gehring [151] who found the upper bound  K ω   n−1 1/(n−1) d(n, K) = exp 2 , √ γ( 2)

d(2, K) = eπK ,

for the linear dilatation of a K -quasiconformal mapping of a domain G in Rn . The bound in 15.40 and 15.42 yields a better bound c(n, K) with   √ 3 2 c(n, K) = γ −1 γ ( 2 ) K ≤ d(n, K)/10 . For these facts see [553] and [24]. With a different (larger) constant 15.40 was proved by Yu. G. Reshetnyak [461] and O. Martio, S. Rickman, and J. Väisälä [363]. n

m

Exercise 15.45 Let G ⊂ R be a domain and let f : G → R be a continuous function. Fix b ∈ ∂G . We say that f has a sequential limit α at b, if there exists a sequence (bk ) in G with bk → b and f (bk ) → α when k → ∞ . Let C(f, b) be the set of all sequential limits of f at b. Then obviously C(f, b) ⊂ f (G). (1) Show that f has a limit f (b) at b iff C(f, b) = {f (b)} . n (2) Show that C(f, b) = ∩∞ k=1 f (G ∩ B (b, 1/k)) , if b = ∞ . 2 2 (3) Let G = R+ = {(x, y) ∈ R : y > 0} and f : G → (0, ∞) be the function arg z ∈ (0, π) . What is C(f, 0)? Exercise 15.46 A mapping f : R → R is said to be Hölder continuous, if there exist constants C, β > 0 such that |f (x) − f (y)| ≤ C|x − y|β for all x, y ∈ R . Let f : R → R be Hölder continuous with exponent β > 1 . Show that f is a constant, equal to f (0) . Remark 15.47 The sharp upper bound λ(K) for the linear dilatation of a K quasiconformal mapping of R2 onto R2 was found by O. Lehto, K. I. Virtanen, and J. Väisälä [327]. For further results of this type see [214]. It can be shown that √ 2 (1/ 2 ) ϕK,2 λ(K) = c(2, K) − 1 = 2 √ ϕ1/K,2(1/ 2 ) (see [327] and 7.34(2)) and that λ(K) ∼ it is known (cf. [27, 29]) that

1 πK 16 e

for large values of K . Moreover,

eπ(K−1) ≤ λ(K) ≤ eπ(K−1/K) for K ≥ 1 . The question of finding a sharp bound for the linear dilatation in dimensions n ≥ 3 remained open for many years until 1990 when such a bound

298

15 Basic Properties of Quasiregular Mappings

for a quasiconformal mapping defined on the whole space Rn was given in [557]. See also [29, Chapter 14]. This result was then generalized by Seittenranta [482] to the case of a qc mapping defined on a subdomain of Rn . Notes 15.48 Complete proofs of the basic properties of quasiregular mappings can be found in the books of Reshetnyak [463] and Rickman [473]. Theorem 15.36 and Corollary 15.38 are from [552], Theorem 15.40 from [553].

15.4.2 Open Problem Let QCK (Rn ) = {f : Rn → Rn : f is K − qc and f (0) = 0, f (e1 ) = e1 } and Hn∗ (K) = sup{|f (x)| : |x| = 1, f ∈ QCK (Rn )} . It is well-known, e.g. that for K ≥ 1 [29, 14.8], [557, p. 627] √ Hn∗ (K) ≤ exp(4K(K + 1 K − 1)) . Then for a fixed n ≥ 2 , Hn∗ (K) → 1 when K → 1 . Is it true that, for a fixed K > 1 , limn→∞ Hn∗ (K) = 1 (cf. [557, (4.3)])? It is well-known [26, Lemma 4.28] that Hn∗ (K) ≥ λ(K 1/(n−1)) where λ is the function in Remark 15.47.

Chapter 16

Distortion Theory

In the present chapter we shall put into effective use the transformation formulae 15.36(1) and (2) for the conformal invariants λG and μG . Most results of this chapter are of the following general type: we combine the transformation formulae in 15.36 with some particular estimates for λG and μG proved in Chap. 10 and as a result obtain distortion theorems i.e. information about how distances between points are deformed under mappings. Besides the fundamental distortion theorems, the quasiregular variant of the Schwarz lemma, and the Hölder continuity, we prove several additional special distortion theorems.

16.1 The Schwarz Lemma and Quasiregular Maps We consider the distortion in terms of the hyperbolic metric under quasiconformal and related mappings. n

Theorem 16.1 Let E ⊂ R be a compact set of positive capacity and let f : Bn → n R \ E be a K -quasimeromorphic mapping. Then 1−n   aK bK  μBn (x, y) ≤ q f (x), f (y) ≤ − log th 14 ρ(x, y) c(E) c(E) for distinct x, y ∈ Bn where a and b depend only on n . Proof It follows from 8.1 and 10.2 that      μf Bn f (x), f (y) ≥ d4 min c(E) , q f (x), f (y)   ≥ d4 q f (x), f (y) min{ d3 , c(E) } .

© Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_16

299

300

16 Distortion Theory n

Because E is of positive capacity we deduce from 8.1 that 1 ≥ c(E)/c(R ) ≥ c(E)/d2 > 0, and therefore     μf Bn f (x), f (y) ≥ d4 c(E) q f (x), f (y) min{ d3 /d2 , 1 } . The proof follows now from 15.36(1), 10.4(1), and (9.11).

 

It follows from 16.1 and the monotone property 8.1(2) of   the set function c(E) that for fixed K and μBn (x, y), the distance q f (x), f (y) decreases if the set E becomes larger. In other words, the larger the set omitted by the mapping f , the n less f can oscillate as a mapping between metric spaces f : (Bn , μBn ) → (R , q). Later on we shall encounter a similar phenomenon with other metric spaces in place n of (Bn , μBn ) and (R , q). The next result is a counterpart of the Schwarz lemma for quasiregular mappings. We consider here the function ϕK = ϕK,n introduced in (9.13) and refer to this result briefly as the Schwarz lemma. Theorem 16.2 Let f : G1 → G2 , G1 , G2 ∈ {Bn , Hn } , be a non-constant K quasiregular mapping with f G1 ⊂ G2 and let α = KI (f )1/(1−n) . Then    α   1 th ρ (x, y) , (1) th 12 ρG2 f (x), f (y) ≤ ϕK th 12 ρG1 (x, y) ≤ λ1−α G n 1 2     (2) ρG2 f (x), f (y) ≤ KI (f ) ρG1 (x, y) + log 4 , hold for all x, y ∈ G1 , where λn is the constant in (9.6). Proof Because of the Möbius invariance it is enough to consider the case when G1 = G2 = Bn . Fix x, y ∈ Bn . Because f Bn ⊂ Bn it follows from 10.2, 10.4, and (9.20) that     μf Bn f (x), f (y) ≥ μBn f (x), f (y) = γ (1/ th b)   where b = 12 ρ f (x), f (y) . Similarly, by 15.36(1) and 10.4,   μf Bn f (x), f (y) ≤ KI (f ) μBn (x, y) = KI (f ) γ (1/ th a) where a = 12 ρ(x, y). These inequalities together with 9.32(1) imply (1). For the proof of (2) we note that by (9.12) and 15.36(1)     A ρ f (x), f (y) ≤ γ (1/ th b) ≤ KI (f ) A ρ(x, y) + log 4 where A = 2n−1 cn . Hence we have proved also (2).

 

16.1 The Schwarz Lemma and Quasiregular Maps

301

Corollary 16.3 Let f : Bn → Bn be a K -quasiregular mapping with f (0) = 0 and let α = KI (f )1/(1−n) . Then α 1−1/K K|x|1/K , (1) |f (x)| ≤ ϕK,n (|x|) ≤ λ1−α n |x| ≤ 2  KI (f ) (2) |f (x)| ≤ a−1 a = 4 1+|x| , a+1 , 1−|x|

for all x ∈ Bn . ≤ 21−1/K K by 9.34. Proof Apply (4.15) and 16.2 with y = 0 and recall that λ1−α n   P. Järvi [263] proved several improved variants of the Schwarz lemma for quasiregular mappings. In these results the multiplicity has a role: the bound is improved when the multiplicity grows and other parameters remain fixed. His paper also provides a solution to the problem 7 in [555, p. 193]. Exercise 16.4 Let D be a c-QED domain in Rn and f : Rn → Rn Kquasiconformal. Show that f D is also c -QED. Does the claim hold if f is only defined in D? Recall from (10.5) that a domain D is c-QED if there exists a constant c > 0 such that for all compact connected sets E, F ⊂ D, M((E, F ; D)) ≥ cM((E, F ; Rn )). Exercise 16.5 Find a counterpart of the Schwarz lemma for (1) K-quasimeromorphic mappings f : Bq (z, r) → Bq (w, s), f (z) = w . (2) K-quasiregular mappings f : Hn → Hn , f (en ) = en . Exercise 16.6 Let f : Bn → Bn be K-quasiregular and u(x) = 1 − |f (x)|. Show that the Harnack inequality holds for u. The following invariance property of 16.2 should be noted. The inequality of 16.2 yields the same upper bounds for     ρG2 f (x), f (y) and ρG2 (g2 ◦ f ◦ g1 )(x), (g2 ◦ f ◦ g1 )(x) whenever gj ∈ M(Gj ) , j = 1, 2 . This statement readily follows because hyperbolic distances are invariant under Möbius transformations and because the dilatations of a quasiregular mapping remain unchanged under pre- and postcomposition with Möbius transformations. It should also be observed that the explicit estimate 16.3(1) is sharp if K = 1. In 16.2 we assumed, when G1 = G2 = Bn , that f Bn ⊂ Bn and proved that f : (Bn , ρ) → (Bn , ρ) is uniformly continuous with a quantitative bound for its modulus of continuity. If, in addition, Bn \f Bn = ∅, one would expect a better result than 16.2. For instance, one could hope to replace the target space (Bn , ρ) in 16.2 by (f Bn , kf Bn ). In the particular case of Möbius transformations this indeed is possible by 5.15 and later on we shall prove that this is possible also for quasiconformal mappings. Now we are going to show that for quasiregular mappings and even for bounded analytic functions such an expectation is futile.

302

16 Distortion Theory

Example 16.7 Let g : B2 → B2 \ {0} = gB2 be the exponential function g(z) = 2 2 2 exp( z+1 z−1 ), z ∈ B . We shall show that g : (B , ρ) → (gB , kgB2 ) fails to be uniformly continuous. To this end, let xj = (ej − 1)/(ej + 1), j = 1, 2, . . . . It follows from (4.15) that ρ(0, xj ) = j and thus ρ(xj , xj +1 ) = 1. Let Y = B2 \ {0}. Since g(xj ) = exp(−ej ) we get by (5.4) and (4.27)     kY g(xj ), g(xj +1 ) ≥ jY g(xj ), g(xj +1 )     = log 1 + exp ej +1 exp(−ej ) − exp(−ej +1 )   = log 1 + exp(ej +1 − ej ) − 1 = ej +1 − ej → ∞ as j → ∞. In conclusion, g : (B2 , ρ) → (Y, kY ) cannot be uniformly continuous, because ρ(xj , xj +1 ) = 1. In this example ∂(gB2 ) consists of a point component {0} and the unit circle ∂B2 . We now show that if each boundary component of the image domain is nondegenerate, then the situation will be different if the image domain is uniform. Theorem 16.8 Let f : Bn → Rn be a non-constant quasiregular mapping, let E ⊂ n R \ f Bn be a non-degenerate continuum such that ∞ ∈ E, and let G = Rn \ E be a domain. (1) Then f : (Bn , ρ) → (G, jG ) is uniformly continuous. (2) If G is uniform, then f : (Bn , ρ) → (G, kG ) is uniformly continuous. Proof (1) The proof follows the same general pattern as the one in 16.2. The particular estimates needed for the present case are supplied by 9.29, 10.4, and (9.11). (2) The proof follows from (1) and the definition 6.1 of a uniform domain.   Lemma 16.9 Let G and G be proper subdomains of Rn , where G is uniform and n G has connected complement R \ G . If f : G → Rn is a quasiregular mapping with f G ⊂ G , then for all x, y ∈ G   jG f (x), f (y) ≤ a1 jG (x, y) + a2 , where a1 , a2 are positive numbers depending only on n, KI (f ), and the constant in the definition of a uniform domain.

16.1 The Schwarz Lemma and Quasiregular Maps

303

Proof By 10.8, 15.36(1), 10.7(2), and 6.1 we obtain     cn jG f (x), f (y) ≤ μG f (x), f (y) ≤ KI (f ) μG (x, y)   ≤ KI (f ) b1 kG (x, y) + b2 ≤ KI (f ) b1 A jG (x, y) + KI (f ) b2  

and the assertion follows. Rn

G

Exercise 16.10 Show that the hypothesis that \ be connected cannot be removed from 16.9 if n = 2 and G = Bn . [Hint: Show that the exponential function in 16.7 provides a counterexample in the present case, too. Recall that ρ ≈ jBn by 4.9(1).] Exercise 16.11 Apply 16.2(1) and (2) for a quasiregular mapping f : Bn → Hn to show that |f (x)| ≤ 22β |f (0)|

 1 + |x| β 1 − |x|

, β = KI (f ) ,

for x ∈ Bn . [Hint: Apply 16.2(2) for a quasiregular mapping of Bn into Hn and the inequality ρHn (x, y) ≥ | log(|x|/|y|)|, x, y ∈ Hn . The required inequality then follows from 4.9(2) and (4.30).] Exercise 16.12 Show that if f : Bn → Bn is K -quasiregular, then for all x ∈ Bn 1 − |f (x)| ≥ 2−2K (1 − |f (0)|)

 1 − |x| K 1 + |x|

.

[Hint: Observe that by 4.6(1) and 4.9(1) ρ(x, y) ≥ jBn (x, y) ≥ log

1 − |y| 1 − |x|

for all x, y ∈ Bn . Now apply 16.2(2) and (4.15).] Theorem 16.13 Suppose that f : G → Rn is a bounded quasiregular mapping and that F is a compact subset of G. Let α = KI (f )1/(1−n) and C = α λ1−α n d(f G)/d(F, ∂G) where λn is the Grötzsch constant in (9.6). Then f satisfies the Hölder condition |f (x) − f (y)| ≤ C |x − y|α for x ∈ F , y ∈ G.

(16.1)

304

16 Distortion Theory

Proof Set r = d(F, ∂G). Suppose first that |x − y| < r. Define g : Bn → Bn by g(z) =

f (x + rz) − f (x) . d(f G)

α Then g(0) = 0 and KI (g) = KI (f ) by 15.34(3). By 16.2 we get |g(z)| ≤ λ1−α n |z| . Setting z = (y − x)/r we obtain (16.1). Next assume that |x − y| ≥ r. Since λn ≥ 1 (in fact, λn ≥ 4, see 9.16) we have

|f (x) − f (y)| ≤ d(f G) ≤ r −α d(f G) |x − y|α ≤ C|x − y|α  

as desired. Theorem 16.14 Let f : Bn → Bn be a K -quasiregular mapping into Bn . Then   |f (x) − f (y)| ≤ bK th 12 ρ(x, y)

 3  2 (s) . The result is for all x, y ∈ Bn , where bK (s) = 2 ϕK,n (s) 1 + 1 − ϕK,n sharp if f in an element of M(Bn ) and f (x) = −f (y) .   Proof If we let t  = 12 ρ f (x), f (y) , it follows from (4.25) and 4.2(2) that |f (x) − f (y)| ≤ 2 th 12 t  =

2 th t   . 1 + 1 − th2 t 

The desired inequality follows now from 16.2(1). Since ϕ1,n (r) = r, the sharpness   assertion follows from the one in (4.25). Corollary 16.15 Under the assumptions of 16.14 3 (a) |f (x) − f (y)| ≤ ϕK,n (a) + ϕK,n

where a = th 12 ρ(x, y) for all x, y ∈ Bn . Proof The proof follows from the inequality 2 ≤ 1 + x2 , 0 ≤ x ≤ 1 , √ 2 1+ 1−x in Exercise 4.2(2) and Theorem 16.14.

 

Theorem 16.16 For n ≥ 2, r ∈ (0, 1), and K ∈ [1, ∞) there exists a number a(r) with limr→0 a(r) = 1 such that if f : Bn → Bn is a K -quasiregular mapping into Bn , then α |f (x) − f (y)| ≤ a(r) λ1−α n |x − y|

16.1 The Schwarz Lemma and Quasiregular Maps

305

for all x, y ∈ B n (r) where α = K 1/(1−n) . Proof Let r ∈ (0, 1) and x, y ∈ B n (r). Then th 12 ρ(x, y) ≤ th 12 ρ(−re1 , re1 ) =

2r 1 + r2

by 4.17. By the inequality in the proof of 16.15, by 16.14, 16.2(1), and 4.17 we obtain   |f (x) − f (y)| ≤ bK th 12 ρ(x, y) ≤

≤

2 λ1−α (th 12 ρ(x, y))α n 2 (2r/(1 + r 2 )) 1 + 1 − ϕK,n

λ1−α [1 + ϕK,n (2r/(1 + r 2 ))2 ] |x − y|α n " #α/2 . |x − y|2 + (1 − |x|2 )(1 − |y|2)

We may choose   2r 2  a(r) = 1 + ϕK,n (1 − r 2 )−α 1 + r2 and the assertion follows.

 

The following result is a generalization of Liouville’s theorem concerning the growth of entire analytic functions. Theorem 16.17 Suppose that f : Rn → Rn is a quasiregular mapping and that limx→∞ |x|−α |f (x)| = 0 where α = KI (f )1/(1−n) . Then f is a constant. Proof We can write |f (x)| ≤ |x|α (|x|) where (R) → 0 as R → ∞. Fix x ∈ Rn and choose R > |x|. Applying (16.1) for G = B n (R) and F = {0} we obtain −α 1−α |f (x) − f (0)| ≤ C|x|α where C = λ1−α n d(F, ∂G) d(f G) ≤ 2λn (R). Thus 1−α α |f (x) − f (0)| ≤ 2λn (R)|x| . Letting R → ∞ yields f (x) = f (0). Hence f is   a constant. Remark 16.18 The exponent α in 16.2(1) and 16.17 is best possible. As to 16.2(1), the function f : Bn → Bn , f (x) = x|x|α−1, x ∈ Bn , KI (f ) = K(f ) = α 1−n , is a desired example (see [524, 16.2] for the calculation of KI (f )). The same function, as a mapping of Rn onto Rn , shows that the condition limx→∞ |x|−α |f (x)| = 0 in 16.17 cannot be replaced by the requirement that |x|−α |f (x)| be bounded. For ϕ ∈ (0, 12 π) let C(ϕ) = {z ∈ Rn : z · en ≤ |z| cos ϕ}. We next give a formulation of 16.2 for maps into a cone or into an infinite cylinder.

306

16 Distortion Theory

Theorem 16.19 Let f : Bn → Rn be a non-constant quasiregular mapping. (1) If ϕ ∈ (0, 12 π) and f Bn ⊂ C(ϕ), then for all x ∈ Bn |f (x)| ≤ |f (0)| 4aϕ

 1 + |x| aϕ 1 − |x|

where a depends only on n and KI (f ). 2 < 1 }, then for all x, y ∈ Bn (2) If f Bn ⊂ { x ∈ Rn : x12 + · · · + xn−1 |f (x)| ≤ |f (y)| + A KI (f ) (ρ(x, y) + log 4) where A is a positive constant depending only on n. Proof (1) By 7.20, 15.36(1), 10.4(1), (9.12), and 9.17(2) we obtain   |f (x)| dn log ≤ μf Bn f (x), f (0) ≤ KI (f ) μBn (x, 0) ϕ |f (0)|  1 + |x|  ≤ KI (f ) 2n−1 cn log 4 . 1 − |x| The proof of (1) with a = 2n−1 cn KI (f )/dn follows. (2) Assume first that |f (x)| > |f (y)| + 1. From 7.20 we deduce that   μf Bn f (x), f (y) ≥ dn

|f(x)|

|f (y)|+1

2dn dr ≥ (|f (x)| − |f (y)| − 1) . ϕ(r)r π

Here ϕ(r) ∈ (0, 12 π) is such that 2 S n−1 (r) ∩ { x ∈ Rn : x12 + · · · + xn−1 < 1 } = S n−1 (r) ∩ C(ϕ(r))

for r > 1, i.e. ϕ(r) = arcsin(1/r) and rϕ(r) < 12 π. By 15.36(1) and (9.12) we obtain as in the proof of (1)   |f (x)| ≤ |f (y)| + 1 + T KI (f ) ρ(x, y) + log 4   ≤ |f (y)| + A KI (f ) ρ(x, y) + log 4 where T = 2n−2 cn π/dn and A = T + 1/ log 4. In the case |f (x)| ≤ |f (y)| + 1 the claim follows because A log 4 > 1 .  

16.2 Bounds for Moduli of Continuity

307

Remark 16.20 For small values of ρ(x, y) one can improve 16.19 by applying 9.17(1) instead of 9.17(2). Recall also 9.18(1).

16.2 Bounds for Moduli of Continuity Next we prove bounds for moduli of continuity for quasiconformal mappings. The next result is based on the tranformation rule for the conformal invariant λBn . Theorem 16.21 Let f : Bn → Bn be a quasiregular mapping with N(f, Bn ) = N < ∞. Then  β   th 14 ρ f (x), f (y) ≤ 2 th 14 ρ(x, y) holds for all x, y ∈ Bn where β = 1/(NKO (f )). Furthermore, if f (0) = 0, then for all x ∈ Bn  β |f (x)| |x|   ≤2 . 1 + 1 − |f (x)|2 1 + 1 − |x|2 Proof We may assume that f (x) = f (y). It follows from 10.4(2) and 10.5 that   λBn (x, y) = 12 τ sh2 21 ρ(x, y) ≥ −cn log th 14 ρ(x, y) .

(16.2)

Because f Bn ⊂ Bn , it follows from 10.2, 10.4(2), and 10.5 that     λf Bn f (x), f (y) ≤ λBn f (x), f (y) < cn log

2 th

1 4 ρ(f (x), f (y))

.

(16.3)

The proof now follows from (16.2), (16.3), and 15.36(2). If f (0) = 0, the assertion follows from the above inequality and (4.15), 4.2(2).   Exercise 16.22 Observe first that the proof of 16.21 yields the inequality sh2 b ≤ τ −1

 τ (sh2 a)  NKO (f )

  where a = 12 ρ(x, y) and b = 12 ρ f (x), f (y) . Next assume, in addition, that f (0) = 0 and N = 1. Exploiting the functional identity 7.31 and the definition (9.15) show that the above inequality with y = 0 yields 2 |f (x)|2 ≤ 1 − ϕ1/K,n



1 − |x|2



308

16 Distortion Theory

for all x ∈ Bn . (Compare this to the Schwarz lemma 16.3.) Exercise 16.23 Assume that f : Bn → Bn is K -quasiconformal with f (0) = 0 and f Bn = Bn . Show that    2 2 |f (x)|2 ≤ min ϕK,n , (|x|) , 1 − ϕ1/K,n 1 − |x|2    2 2 |f (x)|2 ≥ max ϕ1/K,n . (|x|) , 1 − ϕK,n 1 − |x|2 [Hint: Apply 16.22 and√16.3 also to f −1 .] Recall that in the case n = 2 we have 2 (r) = 1 − ϕ 2 2 ϕK,2 1/K,2 ( 1 − r ) for all K > 0 and 0 < r < 1 by 7.34(2) while the analogous relation fails to hold for n ≥ 3 by 9.39. Theorem 16.24 Let f : Bn → Rn \ {0} be a quasiregular mapping with N(f, Bn ) ≤ p < ∞. Then for x, y ∈ Bn    |f (x)| ≤ |f (y)| 1 + τ −1 A τ (sh2 21 ρ(x, y)) , where A = 1/(2pKO (f )). Proof If |f (x)| ≤ |f (y)| there is nothing to prove. Hence we may assume that |f (x)| > |f (y)|. By 7.18 and 10.2 we obtain       λf Bn f (x), f (y) ≤ λG f (x), f (y) ≤ M ( [0, f (y)], [f (x), ∞))   |f (x)| −1 ≤τ |f (y)| where G = Rn \ {0}. Next, by 10.4(2) λBn (x, y) =

1 2

  τ sh2 21 ρ(x, y)

and by 15.36(2)   λBn (x, y) ≤ p KO (f ) λf Bn f (x), f (y) . The desired bound follows from these relations.

 

We require the following important theorem of Martio, Rickman, and Väisälä [365, 2.3] on locally homeomorphic quasiregular maps of Bn , n ≥ 3. The proof of this theorem makes use of an ingenious method of V. A. Zorich [579]. The proof will be omitted. A similar result for quasimeromorphic mappings was proved by Martio and Srebro [372].It is not known whether the constant ψ(n, K) in the next theorem can be chosen independent of n .

16.2 Bounds for Moduli of Continuity

309

Theorem 16.25 For n ≥ 3 and K ≥ 1 there exists a number ψ = ψ(n, K) ∈ (0, 1) such that every locally homeomorphic K -quasiregular mapping f : Bn → Rn is injective in B n (x, (1 − |x|)ψ ) for all x ∈ Bn . K. Rajala [446] has proved the following quantitative local injectivity theorem for K-quasiregular mappings. Theorem 16.26 For n ≥ 3 there exists a constant εn > 0 such that if f : Bn → Rn is a non-constant K-quasiregular mapping with K < 1 + εn , then f is a local homeomorphism. Exercise 16.27 Applying (4.21) show that Bρ (x, M) ⊂ B n (x, T (1 − |x|)) where 3 T = (2 th 12 M) (1 − th 12 M), |x| < 1. Conversely show that Bρ (z, M) ⊂   B n (z, (1 − |z|)ψ ) where |z| < 1, ψ ∈ (0, 1), M = 2 arth ψ/(2 + ψ) . Theorem 16.28 Let f : Bn → Rn \ {0} be a locally homeomorphic quasiregular mapping and n ≥ 3. Then |f (x)| ≤ C|f (0)|

 1 + |x| a 1 − |x|

where C and a are positive numbers depending only on n and K(f ). Proof Let ψ = ψ(n, K(f )) be as in 16.25 and define gx (z) = fx (x + z(1 − |x|)ψ) for z ∈ Bn and x ∈ Bn . Then gx is injective and K -quasiconformal in Bn by 16.25. We are going to show first that |f (x)| satisfies the Harnack inequality (6.6) in Bn with s ∈ (0, 12 ψ] and Cs = 1 + τ −1 (A τ (16/9)) , A = 1/(2KO (f )) .

(16.4)

To this end let B n (z, r) ⊂ Bn and x1 , x2 ∈ B n (z, sr) , s ∈ (0, 12 ψ] . By 16.24 we obtain   |gz (y1 )| |f (x1 )| = ≤ 1 + τ −1 A τ (sh2 12 ρ(y1 , y2 )) |f (x2 )| |gz (y2 )| where yj = (xj − z)/((1 − |z|)ψ) ∈ Bn and A = 1/(2KO (f )) . Because |yj | ≤ for j = 1, 2 it follows from (4.15) that 2B = ρ(y1 , y2 ) ≤ 2 log 3 , and hence sh2 B ≤ 16/9 . We have thereby proved (16.4). By virtue of (16.4) and 6.23 we obtain |f (x)| ≤ Cs1+t |f (0)|

1 2

310

16 Distortion Theory

1+s where Cs is as in (16.4) and t = (log 1+|x| 1−|x| )/ log 1−s , s =

1 2ψ

. We have thus

proved the desired inequality with C = Cs and a = (log Cs )/ log 2+ψ 2−ψ . (Recall the relationship (15.5) between KO (f ) and K(f ) .)   Exercise 16.29 A counterexample to show that 16.25 is false for n = 2 is easily found. Denote fj (z) = exp(j z), j = 2, 3, . . . , z ∈ B2 . By considering the family {fj } we see that 16.25 is false for n = 2. Find a counterexample to show that 16.28, too, fails to hold for n = 2. Next we shall study the behavior of the function rG (x, y) =

|x − y| min{d(x), d(y)}

under quasiconformal mappings. Theorem 16.30 Let G and G be proper subdomains of Rn and let f : Rn → Rn be a K -quasiconformal homeomorphism such that f G = G . Then for all x, y ∈ G  1    τ (rG (x, y)) . rG f (x), f (y) ≤ τ −1 √ 2K Proof We may assume that d(f (x), ∂G ) ≤ d(f (y), ∂G ). Fix z ∈ ∂G such that |f (x) − z | = d(f (x), ∂G ) and z ∈ ∂G such that f (z) = z . Then by 15.36(2)   λD (x, y) ≤ K λD  f (x), f (y) , where D = Rn \ {z} and D  = f D = Rn \ {z }. By 10.18  |x − y| 

≥ τ (rG (x, y)) , |x − z|   √  |f (x) − f (y)|  √   = 2 τ rG (f (x), f (y)) . λD  f (x), f (y) ≤ 2 τ  |f (x) − z | λD (x, y) ≥ τ

The desired result follows immediately from the above inequalities. Applying Theorem 16.30 with G = Rn

Corollary 16.31 Let f : → 0 . Then for x, y ∈ Rn \ {0}

Rn

Rn

 

\ {0} yields the following result.

be a K -quasiconformal mapping with f (0) =

|f (x) − f (y)| ≤ τ −1 min{|f (x)|, |f (y)|}



 |x − y|  τ √ . min{|x|, |y|} 2K 1



Next we shall prove a result where f need not be defined on the whole space Rn as it was in 16.30 and 16.31.

16.2 Bounds for Moduli of Continuity

311

Theorem 16.32 Let G be a proper subdomain of Rn , suppose that G is c-QED , and let f : G → f G be K -quasiconformal with f G ⊂ Rn . Then  c    rf G f (x), f (y) ≤ τ −1 n τ (rG (x, y)) 2 K for all x, y ∈ G . Proof By 10.20 we obtain λG (x, y) ≥ c τ (r 2 + 2r) ≥ 21−n c τ (r) where r = rG (x, y). Next by 10.18 we get     λf G f (x), f (y) ≤ 2 τ rf G (f (x), f (y)) .  

The desired inequality now follows easily.

Example 16.33 We shall now show that the c-QED condition in 16.32 is necessary. Let G = B2 \ [0, e1 ) and let f : G → B2+ = B2 ∩ H2 be the conformal map √ f (z) = z, z ∈ G. Let xj = (1/2, 1/j ), yj = (1/2, −1/j ), j = 4, 5, . . . . Then rG (xj , yj ) = 2, while it is easy to see that   rf G f (xj ), f (yj ) −→ ∞   as j → ∞. In particular, rf G f (xj ), f (yj ) has no upper bound in terms of rG (xj , yj ). One can show that G is not a c-QED domain for any c > 0. The function rG (x, y) is invariant under similarities and, accordingly, the same n is true about 16.30 and 16.32. Next we shall give some M(R )-invariant results. n

n

Theorem 16.34 Let D ⊂ R be a c-QED domain with card(R \ D) ≥ 2 and n let f : D → f D ⊂ R be K -quasiconformal. Then for x, y ∈ D    mf D f (x), f (y) ≤ τ −1

c 2n+1 K

 τ (mD (x, y))

where mD is as in (5.17). Proof The proof follows from 10.22 and 15.36(2). n

 

→ R be a K -quasimeromorphic mapping, let Theorem 16.35 Let f : n a, d ∈ R \ f Bn be distinct and suppose that N(f, Bn ) ≤ p < ∞ . Then Bn

   q(a, d) q(f (x), f (y)) |x − y|2 1 ≤ τ −1 √ τ q(a, f (x)) q(f (y), d) (1 − |x|2)(1 − |y|2) 2 2Kp for all x, y ∈ Bn .

312

16 Distortion Theory

Proof By 10.4(2) 1 λBn (x, y) = τ 2



|x − y|2 (1 − |x|2)(1 − |y|2 )



n

for distinct x, y ∈ Bn . Let D = R \ {a, d} . By 10.2 λf Bn ≤ λD and thus by 10.21 we obtain     √   λf Bn f (x), f (y) ≤ λD f (x), f (y) ≤ 2 τ mD (f (x), f (y) √ ≤ 2 τ (| a, f (x), d, f (y) |) . The desired conclusion follows from the above inequalities and 15.36(2).   3 1 Exercise 16.36 Show that th 2 ρ(x, y) = |x − y| |x − y|2 + 4xn yn for x, y ∈ Hn . Show that if f : Hn → Hn is K -quasiregular with f (en ) = en then 

|f (x) − en | |f (x) − en |2 + 4f (x)n

& ≤

λ1−α n



|x − en |

'α

|x − en |2 + 4xn

for all x ∈ Hn . [Hint: Apply 16.2(1).] Assume next that f : Hn → Hn is K quasiconformal and f (en ) = en . Show that  1  |x − e |2  |f (x) − en |2 n τ . ≤ τ −1 4f (x)n K 4xn [Hint: Find first an expression for sh 12 ρ(x, y) and then apply 16.22.] Exercise 16.37 (1) Let f : Bn → Bn be K -quasiregular and α = K 1/(1−n) . From the proof of 16.16 derive the inequality |f (x) − f (y)| ≤ (2λn )1−α ρ(x, y)α for all x, y ∈ Bn . (2) Next extend this inequality to a K -quasimeromorphic map f : Bn → Bq (z, r) where Bq (z, r) is a ball in the chordal metric as defined in (3.8). Show that q(f (x), f (y)) ≤ (2λn )1−α ρ(x, y)α c(r) √ for all x, y ∈ Bn where c(r) = r/ 1 − r 2 . (3) Find a form of 16.16 where the majorant is independent of n .

16.3 The Schwarz Lemma in the Planar Case

313

(4) Let f : Bn → Rn be a K -quasiregular mapping with Bf = ∅ . Show that for n ≥ 3 there exists a number d(n, K) such that for all r ∈ (0, 1) N(f, B n (r)) ≤ d(n, K)(1 − r)1−n . [Hint: 6.28 and 16.25.] Exercise 16.38 Let f : Bn → Rn be K -quasiregular and assume that there are numbers M > 0 and A > 0 such that ρ(x, y) ≤ M implies |f (x) − f (y)| ≤ A . Show that |f (x) − f (y)| ≤ A λ1−α n

 th 1 ρ(x, y) α 2

th 12 M

for all x, y ∈ Bn with ρ(x, y) ≤ M where α = K 1/(1−n) . Next combine this inequality with 6.24 to obtain a bound valid for all x, y ∈ Bn .

16.3 The Schwarz Lemma in the Planar Case In the next few results we will refine the Schwarz Lemma for the case when n = 2 . This sharp result contains a constant c(K) determined in terms of the special function ϕK introduced in (9.13) Theorem 16.39 If f : B2 → B2 is a non-constant K -quasiregular mapping, then ρB2 (f (x), f (y)) ≤ c(K) max{ρB2 (x, y), ρB2 (x, y)1/K } for all x, y ∈ B2 where c(K) = 2arth(ϕK (th 12 )) and K ≤ u(K − 1) + 1 ≤ log(ch(Karch(e))) ≤ c(K) ≤ v(K − 1) + K  with u = arch(e)th(arch(e)) > 1.5412 and v = log(2(1+ 1 − 1/e2 )) < 1.3507 . In particular, c(1) = 1 . Our proof of Theorem 16.39 is based on Theorem 16.2. Before proving the result we need to introduce two lemmas to estimate the special function ϕK . To prove these estimates we need the next several results from [29]. For 0 < r < 1 , we define the complete elliptic integrals of the first and second kind as follows π/2 K(r) ≡ 0

dt

 = 1 − r 2 sin2 t

1  0

dt (1 − r 2 t 2 )(1 − t 2 )

dt.

314

16 Distortion Theory

and E(r) ≡

π/2

1



1 − r 2 sin2 t dt =

0

0

1 − r 2t 2 dt. 1 − t2

Lemma 16.40 ([29, Exercise 3.43 (13) (a)]) The function f (r) = 2E(r)−r 2 K(r) is increasing and log-convex from (0, 1) onto (π/2, 2) , and g(r) = r/f (r) is increasing and concave (hence log-concave) from (0, 1) onto (0, 1/2) . Lemma 16.41 ([29, Lemma 10.7 (3)]) For K > 1 , let s = ϕK (r) . Then the function f (r) = K(s)/K(r) is strictly increasing from (0, 1) onto (1, K) . Lemma 16.42 ( [29, Theorem 10.9 (3)]) Let f (r) = (arth ϕK (r))/ arth r . For K > 1 , f is a strictly decreasing function from (0, 1) onto (K, ∞) . For 0 < K < 1 , f is increasing from (0, 1) onto (0, K) . Lemma 16.43 ([29, Appendix E]) For s = ϕK (r) 1 ss 2 K(s)2 ss 2 K (s)2 ss 2 K(s)K (s) dϕK (r) = , = K 2  2 = 2 2 2 dr K rr K(r) rr K (r) rr K(r)K (r) and E(r) − r  2 K(r) dK(r) = . dr rr  2 Now we are ready to prove our estimates for ϕK . Lemma 16.44 For K > 1 , the function g(r) =

arth (ϕK (r)) (arth r)1/K

is strictly increasing on (0, 1) . Proof Let g1 (r) = rK(r)2 / arth r = g11 (r)/g12 (r) , where g11 (r) = rK(r)2 and g12 (r) = arth r . Then g11 (0) = g12 (0) = 0 , and  (r) g11 2  (r) = K(r)(2E(r) − r K(r)) , g12

which is strictly increasing by Lemma 16.40 and implies that g1 (r) is also strictly increasing by Lemma 12.1. Let s = ϕK (r) . Then we have that rK(r)2 sK(s)2 − >0 arth s arth r

(16.5)

16.3 The Schwarz Lemma in the Planar Case

315

since s > r for all K > 1 and 0 < r < 1 . By logarithmic differentiation, we get that   1 rK(r)2 g  (r) sK(s)2 1 = − > 0, g(r) K rr 2 K(r)2 arth s arth r which implies that for given K > 1 , the function g is strictly increasing on   (0, 1) . Lemma 16.45 For K > 1 the function t →

2arth(ϕK (th 2t )) , max{t, t 1/K }

is monotone increasing on (0, 1) and decreasing on (1, ∞) . Proof (1) Fix K > 1 and consider f (t) =

2arth(ϕK (th 2t )) , t

t > 0.

Let r = th 2t . Now t/2 = arthr , and t is an increasing function of r for 0 < r < 1 . Then f (t) =

2arth(ϕK (th 2t )) arth(ϕK (r)) = = F (r). t arthr

Then by Lemma 16.42, F (r) is strictly decreasing from (0, 1) onto (K, ∞) . Hence f (t) is strictly decreasing from (0, ∞) onto (K, ∞) . (2) Next consider g(t) =

2arth(ϕK (th 2t )) , t 1/K

and let r = th 2t . Then t = 2arthr and g(t) =

21−1/K arths 2arths = , 21/K (arthr)1/K (arthr)1/K

where s = ϕK (r) . We next apply Lemma 12.1. By calculus 1/(1 − r 2 ) .

d dr (arthr)

=

316

16 Distortion Theory

√ √ Write r  = 1 − r 2, s  = 1 − s 2 . In order to apply Lemma 12.1, we consider the quotient of the derivatives of the numerator and and the denominator of g(t) 21−1/K (1/(1 − s 2 )





r 2 1 ss 2 K(s)2 ds = 21−1/K K (arthr)1−1/K  2  1 1/K−1 (1/(1 − r 2 )) dr s K rr 2 K(r)2 K (arthr) )

= 21−1/K (arthr)1−1/K by Lemma 16.43. By Lemma 16.41, (arthr)1/K−1

K(s)2 K(r)2

sK(s)2 rK(r)2

is increasing, since K > 1 ,

is increasing. Finally, s/r is increasing by Lemmas 12.1 and 16.43. So g(t) is increasing in t on (0, ∞) by Lemma 12.1. (3) Fix K > 1 . Clearly t 1/K , for 0 ≤ t ≤ 1, t, for 1 ≤ t < ∞.

max{t, t 1/K } = Thus h(t) =

2arth(ϕK (th 2t )) , max{t, t 1/K }

increases on (0, 1) and decreases on (1, ∞) .

 

Proof of Theorem 16.39 The maximum value of the function considered in Lemma 16.45 is c(K) = 2arth(ϕK (th 12 )) . The inequality now follows from   Theorem 16.2.

16.3.1 Summary of Main Ideas We now recapitulate the main ideas of this chapter which yield a procedure for proving bounds for moduli of continuity. The key components are: 1/(1−n)

(1) Transformation rules of the two modulus metrics μG and λG under quasiconformal and quasiregular mappings, Theorem 15.36 and Corollary 15.38. 1/(1−n) in terms of geometric expressions and metrics. (2) Bounds for μG and λG In particular, the fundamental result, the quasiregular version of the Schwarz lemma follows immediately. Recall that there was also an alternative version of the Schwarz lemma for quasiconformal maps.

16.4 Further Results

317

In addition to the results proved in this chapter, there are many more results one could derive in the same fashion; one could combine the above procedure with the results from Part III. We now outline a few examples. Let f : G1 → G2 , G1 , G2 ∈ {Bn , Hn } , be a non-constant K -quasiregular mapping with f G1 ⊂ G2 . One could now combine Lemmas 11.12, 11.14, 11.18, Theorems 12.11, and 12.16 with the Schwarz lemma 16.2 and study the uniform continuity with respect to the following metric spaces • • • •

f f f f

: (G1 , vG1 ) → (G2 , vG2 ) , : (G1 , sG1 ) → (G2 , sG2 ) , ∗ ) → (G , j ∗ ) , : (G1 , jG 2 G2 1 ∗ ) → (G , ρ ∗ ) . : (G1 , ρG 2 G2 1

Some results of this type were proved by G. Wang [566] and P. Hariri [191]. Note that in the above list, we have the same metric in both domains G1 and G2 . But this is not necessary: the same procedure also yields bounds for moduli of continuity, for instance, between the metric spaces f : (G1 , sG1 ) → (G2 , vG2 ) . In conclusion, combining the results of Part IV with the Schwarz lemma one could prove many results; in the two-dimensional case we could use the refined version of the Schwarz lemma. It would make a good exercise for the interested reader to apply this procedure to prove some bounds for moduli of continuity. Perhaps a bit more challenging problem would be to study Ferrand’s problem 15.4.1 in this setup.

16.4 Further Results Next we shall survey some distortion theorems for quasiconformal mappings, which will not be proved in this book. In 1956 the following theorem of A. Mori [397] appeared. Theorem 16.46 Let f : B2 → B2 be a K -quasiconformal mapping with f (0) = 0 and f B2 = B2 . Then |f (x) − f (y)| ≤ 16 |x − y|1/K for all x, y ∈ B2 . Furthermore, the number 16 cannot be replaced by any smaller absolute constant. It has been conjectured that the best constant in place of 16 is 161−1/K [326, p. 68]. In 1997 S.L. Qiu [442, Thm 1] proved that the constant 461−1/K will do. In [134] R. Fehlmann and M. Vuorinen proved the following theorem for dimensions n ≥ 2.

318

16 Distortion Theory

Theorem 16.47 Let f : Bn → Bn be a K -quasiconformal mapping with f (0) = 0 and f Bn = Bn . Then |f (x) − f (y)| ≤ M1 (n, K) |x − y|α , α = K 1/(1−n) , for all x, y ∈ Bn , where the number M1 (n, K) has the following three properties (1) M1 (n, K) → 1 as K → 1 , uniformly in n . (2) M1 (n, K) remains bounded for fixed K and varying n . (3) M1 (n, K) ≤ 3λ2n for all K ≥ 1 . We remark that a multidimensional generalization of 16.46 (essentially part (3) of 16.47) follows if one extends Mori’s original argument to Rn . This fact was observed by B. V. Shabat in 1960 [488] (see also F. W. Gehring [151, p. 387] and K. Ikoma [252]). The point of 16.47 is that a quantitative constant is given which satisfies the property (1). See also G. D. Anderson and M. K. Vamanamurthy [23]. Some related results are given by R. Näkki and B. Palka [407] as well as by F. W. Gehring and O. Martio [162]. Remark 16.48 It is an open problem whether the constant M1 (n, K) in 16.47 can be chosen so that it remains bounded when both n → ∞ and K → ∞ . The following theorem was proved by P. Tukia and J. Väisälä [517, 528] and in its present improved dimension-independent form by G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen [24]. Theorem 16.49 For K ≥ 1 and s ∈ (0, 1) there exists a homeomorphism η : [0, ∞) → [0, ∞) with η(0) = 0 and with the following properties. If f : Bn → Rn , n ≥ 2 , is a K -quasiconformal mapping into Rn and x, y, z ∈ B n (s) with x = z , then  |x − y|  |f (x) − f (y)| ≤η . |f (x) − f (z)| |x − z| Exercise 16.50 Show that the inequalities η(1) ≥ 1 and $  |x − z|  |f (x) − f (y)| ≥1 η |f (x) − f (z)| |x − y| for all distinct x, y, z ∈ B n (s) follow from 16.49. Show also that η(1) yields a bound for the linear dilatation of the mapping f .

16.4 Further Results

319

16.4.1 An Open Problem For K ≥ 1 , n ≥ 2 , and r ∈ (0, 1) let ∗ ∗ ϕK,n (r) = ϕK (r) = sup{ |f (x)| : f ∈ QCK (Bn ), f (0) = 0, |x| ≤ r }

where QCK (Bn ) = { f : Bn → f Bn | f is K -quasiconformal and f Bn ⊂ Bn } . As shown in [326, p. 64] ∗ (r) = ϕK,2 (r) ≤ 41−1/K r 1/K ϕK,2

(16.6)

for each r ∈ (0, 1) and K ≥ 1 . By 16.3(1) ∗ ϕK,n (r) ≤ ϕK,n (r) ≤ λ1−α r α , α = K 1/(1−n) , n

(16.7)

for n ≥ 2 , K ≥ 1 , r ∈ (0, 1) . A. V. Sychev [509, p. 89] has conjectured that ∗ (r) ≤ 41−α r α ϕK,n

(16.8)

for all n ≥ 2 and K ≥ 1 . Because λ2 = 4 , (16.8) agrees with (16.6) for n = 2 . ∗ In [25] it is shown that ϕK,n ≡ ϕK,n for n ≥ 3 . It follows from 16.21 and 16.23 that ⎧ ⎨ϕ ∗ (r) ≤ 4 r 1/K , K,n (16.9) √ ⎩"ϕ ∗ (r)#2 ≤ 1 − ϕ 2 ( 1 − r2 ) . K,n

1/K,n

From (16.9) and (16.7) it follows, as shown in [26], that ∗ ϕK,n (r) ≤ 41−1/K r 1/K 2

(16.10)

holds for all n ≥ 2 , K ≥ 1 , r ∈ (0, 1) . Note that the right hand side of (16.10) is bounded when K → ∞ . Recall that λn → ∞ as n → ∞ by 9.16 and that λ1−α ≤ 21−1/K K by 9.34. Note that Sychev’s conjecture (16.8) still remains open. n Notes 16.51 Distortion theorems for quasiconformal and quasiregular mappings have been proved by many authors (see the bibliography of [94]). The Hölder continuity of plane quasiconformal mappings was proved by L. V. Ahlfors [8], and the Schwarz lemma by J. Hersch and A. Pfluger [232] and P. P. Belinski˘ı (see the references in [59, p. 13]). For n = 2 the explicit bound 41−1/K in 16.3(1) was found by C.-F. Wang [565] with the aid of a parametric method, and a simplified proof was given by O. Hübner [246]. See also O. Lehto and K. I. Virtanen [326, p. 65, (3.6.)] as well as P. P. Belinski˘ı [59, p. 15, formula (16’)]. The ndimensional form of the proof in [246] and [326] was given by G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen [24]. The Schwarz lemma 16.2(1) is from

320

16 Distortion Theory

[24, 1.15] and 16.2(2) occurs in [552, Theorem 3.3(2)]. This is not the end of the history: in the particular case n = 2 a statement similar to 16.2(2) was rediscovered by Epstein-Marden-Markovic [129, (5.1.8)] and Theorem 16.39 seems to give the latest result in this case. A future challenge for the general case n ≥ 2 is to solve the A.V. Sychev conjecture 16.8. The Hölder continuity of quasiregular mappings in Rn was proved by E. D. Callender [93], F. W. Gehring [151], Yu. G. Reshetnyak [453], [463, pp. 36–38]. A spatial form of the Schwarz lemma was found by B. V. Shabat [488] and O. Martio, S. Rickman, and J. Väisälä [364]. Most of these bounds depend essentially on n , with bounds that approach ∞ as n → ∞ . Dimension-free bounds (such as 16.3(1)) were given in [24] and [26]. For 16.28 see [552] and [390]. Both 16.34 and 16.35 were proved in [556]. For results similar to 16.34 and 16.35 see [153, p. 233] and [467]. Theorem 16.39 was proved in [73], but the proof contained a gap which was corrected in [567].

Chapter 17

Dimension-Free Theory

The present chapter is devoted to the study of uniform continuity properties of a quasiregular mapping f : G → f G as a mapping between the metric spaces (G, kG ) and (f G, kf G ) and to the study of its restrictions f |D : (D, kD ) −→ (f D, kf D ) whenever D is a subdomain of G. We shall consider the modulus of continuity of f (cf. (*))   ωf (t) = sup{ kf G f (x), f (y) : kG (x, y) ≤ t } . If f is a Möbius transformation, then ωf |D (t) ≤ 2t for all domains D in G by 5.12. In this chapter we shall prove an analogous result for quasiconformal maps. The situation for non-homeomorphic quasiregular mappings is entirely different, as Example 16.3 in the preceding chapter shows. However, under a natural additional condition, one can prove a positive result even for non-homeomorphic mappings. This additional condition is a necessary and sufficient condition for a quasiregular mapping f : (G, kG ) → (f G, kf G ) to be uniformly continuous. The condition requires that the function df defined by x −→ df (x) = d(f (x), ∂G ) satisfy the Harnack inequality (6.6) in G. Applying some results of Chap. 10 we shall show that this Harnack condition is satisfied if f is quasiconformal or if f is quasiregular and, in addition, N(f, G) < ∞. Furthermore, under mild restrictions on ∂f G the Harnack condition holds independently of N(f, G). For instance, it is sufficient to require ∂f G to be connected. Several results of this section were originally proved by G.D. Anderson, M.K. Vamanamurthy, and M. Vuorinen [24, 26]. Some of our results here have a special © Springer Nature Switzerland AG 2020 P. Hariri et al., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-32068-3_17

321

322

17 Dimension-Free Theory

feature: the constants involved do not depend on the dimension n . At the time of publication of these results, it was surprising that the results were independent of the dimension or dimension-free. These dimension-free properties follow from the “dimension-cancellation” property of the function ϕK,n (r); see Lemma 9.34 and Corollary 16.3. J. Väisälä [536] formulated questions asking to what extent one can develop dimension-free theory of quasiconformal maps in euclidean spaces. He also developed a theory of quasiconformal maps on spaces which may be infinite dimensional Banach spaces or even metric spaces [537].

17.1 Quasiregular Mappings and Harnack Functions We shall first prove some preliminary results. Lemma 17.1 Let R > 0, u, v ∈ Rn \ B n (R), u = v, and let F be a continuum with u, v ∈ F . Then   M (F, S n−1 (R)) ≥ γ (1 + a(u, v)) where a(u, v) =

2 min{ |v|(|u| − R) , |u|(|v| − R) } . R |u − v|

Proof Let h(x) = Rx/|x|2 , |x| > R. Then h(Rn \ B n (R)) = Bn . By (3.1) |h(u) − h(v)| =

|u − v| R. |u||v|

This together with the definition (4.27) yields   jBn h(u), h(v) = log(1 + 2/a(u, v)) . By conformal invariance 7.10, 9.20, and 4.9(1)      M (F, S n−1 (R)) = M (h(F ), S n−1 ) ≥ γ ≥γ Because th

1 2



1 th( 12 jBn (h(u), h(v)))



1



th 12 ρ(h(u), h(v)) .

 log(1 + s) = s/(2 + s), we obtain   M (F, S n−1 (R)) ≥ γ (1 + a(u, v))

as desired.

 

17.1 Quasiregular Mappings and Harnack Functions

323

Lemma 17.2 Let f : G → Rn be a quasiregular mapping, let G and f G be proper subdomains of Rn , x ∈ G, θ ∈ (0, 12 ), and let z ∈ ∂f G with df (x) = |f (x) − z| = d(f (x), ∂f G) . Assume that |x − y|