Ricci Flow and Geometric Applications: Cetraro, Italy 2010 [1st ed.] 3319423509, 978-3-319-42350-0, 978-3-319-42351-7, 3319423517

Table of contents :
Front Matter....Pages i-xi
The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)....Pages 1-19
Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture....Pages 21-70
Singularities of Three-Dimensional Ricci Flows....Pages 71-104
Notes on Kähler-Ricci Flow....Pages 105-136
Back Matter....Pages 137-138

Citation preview

Lecture Notes in Mathematics 2166 CIME Foundation Subseries

Michel Boileau Gerard Besson Carlo Sinestrari Gang Tian

Ricci Flow and Geometric Applications Cetraro, Italy 2010

Riccardo Benedetti Carlo Mantegazza Editors

Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Catharina Stroppel, Bonn Anna Wienhard, Heidelberg

More information about this series at http://www.springer.com/series/304

2166

Michel Boileau • Gerard Besson • Carlo Sinestrari • Gang Tian

Ricci Flow and Geometric Applications Cetraro, Italy 2010 Riccardo Benedetti, Carlo Mantegazza Editors

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Authors Michel Boileau Aix-Marseille Université, CNRS, Central Marseille Institut de Mathematiques de Marseille Marseille, France Carlo Sinestrari Dip. di Ingegneria Civile e Ingegneria Informatica Università di Roma “Tor Vergata” Rome, Italy Editors Riccardo Benedetti Department of Mathematics University of Pisa Pisa, Italy

ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-42350-0 DOI 10.1007/978-3-319-42351-7

Gerard Besson Institut Fourier Université Grenoble Alpes Grenoble, France

Gang Tian Princeton University Princeton, NJ USA

Carlo Mantegazza Department of Mathematics University of Naples Naples, Italy

ISSN 1617-9692 (electronic) ISBN 978-3-319-42351-7 (eBook)

Library of Congress Control Number: 2016951889 Mathematics Subject Classification (2010): 53C44, 57M50, 57M40 © Springer International Publishing Switzerland 2016 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

Our aim in organizing this CIME course was to present to young students and researchers the impressive recent achievements in differential geometry and topology obtained by means of techniques based on the Ricci flow. We then invited some of the leading researchers in the field of geometric analysis and low-dimensional geometry/topology to introduce some of the central ideas in their work. Here is the list of speakers together with the titles of their lectures: • Gérard Besson (Grenoble) – The differentiable sphere theorem (after S. Brendle and R. Schoen) • Michel Boileau (Toulouse) – Thick/thin decomposition of three-manifolds and the geometrization conjecture • Carlo Sinestrari (Roma “Tor Vergata”) – Singularities of three-dimensional Ricci flows • Gang Tian (Princeton) – Kähler–Ricci flow and geometric applications. The summer school had around 50 international attendees (mostly PhD students and postdocs). Even though they were sometimes technically heavy, the lectures were followed by all the students with interest. The participants were very satisfied by the high quality of the courses. The not-so-intense scheduling of the lectures gave the students many opportunities to interact with the speakers, who were always very friendly and available for discussion. It should be mentioned that the wonderful location and the careful CIME organization were also greatly appreciated. We think that the fast-growing field of geometric flows and more generally of geometric analysis, which has always received great attention in the international community, but which is still relatively “young” in Italy, will benefit from its diffusion by this CIME course. We briefly describe the contents of the lectures collected in this volume. Gérard Besson presented the impact of the Ricci flow technique on the theory of positively curved manifolds, the central result being the differentiable 1/4-pinched sphere theorem, proved by Brendle and Schoen. It says that a complete, simply

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connected Riemannian manifold whose sectional curvature varies in .1=4; 1 is diffeomorphic to the standard sphere. The problem was first proposed by H. Hopf, and then in 1951, H.E. Rauch showed that a complete Riemannian manifold whose sectional curvature is positive and varies between two numbers whose ratio is close to 1 has a universal cover homeomorphic to a sphere. In the 1960s, M. Berger and W. Klingenberg obtained the optimal result: a simply connected Riemannian manifold which is strictly 1=4pinched is homeomorphic to the sphere. The analogous diffeomorphic conclusion remained open until S. Brendle and R. Schoen proved the following: Theorem (S. Brendle and R. Schoen, 2008) Let M be a pointwise strictly 1=4pinched Riemannian manifold of positive sectional curvature. Then M carries a metric of constant sectional curvature. Hence, it is diffeomorphic to the quotient of a sphere by a finite subgroup of O.n/. The proof relies on the use of the Ricci flow introduced by R. Hamilton and culminating in the work of G. Perelman. The idea is to construct a deformation of the Riemannian metric, evolving it by means of the Ricci flow toward a constant curvature metric. We recall that this was the method that R. Hamilton used in his seminal paper, proving the following theorem: Theorem (R. Hamilton, 1982) Let M be a closed 3-dimensional Riemannian manifold which carries a metric of positive Ricci curvature; then it also carries a metric of positive constant curvature. The lectures also focus on the extension to higher dimensions of the following result, due to C. Böhm and B. Wilking. Recall that a curvature operator is 2-positive if the sum of its two smallest eigenvalues is positive. Theorem (C. Böhm and B. Wilking, 2008) Let M be a closed Riemannian manifold whose curvature operator is 2-positive; then M carries a constant curvature metric. In the lectures, the connection between this method and the algebraic properties of the Riemann curvature operator is stressed, the main focus being the identification of those properties of the curvature operator which are preserved under the Ricci flow. In his lectures, Michel Boileau gave an introduction to the geometrization of 3-manifolds. Sections 2.1 and 2.2 cover Thurston’s classification of the eight 3dimensional geometries and the characterization of geometric (and Seifert) closed 3-manifolds in terms of basic topological properties. This follows by combining Thurston’s hyperbolization theorems (in particular the characterization of hyperbolic 3-manifolds that are fibered over S1 ), Perelman’s general geometrization theorem, and Agol’s recent (2013) proof of a deep conjecture of Thurston that closed hyperbolic 3-manifolds are “virtually fibered.” Section 2.3 discusses the following: (1) A central result of classical 3dimensional geometric topology, that is, the canonical decomposition of a

Preface

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3-manifold by splitting it along spheres and tori. (2) Thurston’s geometrization conjecture. This roughly says that every piece of a canonical decomposition is geometric together with a prediction on the geometry carried by the piece in terms of basic topological properties. It includes as a particular case the celebrated Poincaré conjecture. (3) Thurston’s fundamental hyperbolization theorem for Haken manifolds. Perelman’s proof of the general geometrization theorem deals with all of these topics and also allows us to recover, as a by-product, the canonical decomposition itself. This is done by completing the program based on the Ricci flow with surgeries, first proposed by R. Hamilton. This is the subject of Boileau’s notes from Sect. 2.4. Since the appearance of Perelman’s three celebrated preprints, several simplifications and variants of the original proofs have been developed by various authors. At the end of the day, we can say that the Poincaré conjecture (i.e., the case when the Ricci flow with surgery becomes extinct in finite time) is in a sense the “simplest” case. The general case (when the Ricci flow with surgery exists at all times, which includes the complete hyperbolization theorem) requires nontrivial extra arguments, in particular, to obtain a key non-collapsing theorem. In Perelman’s original work, these come from the theory of Alexandrov spaces. Bessières, Besson, Boileau, Maillot, and Porti developed instead an alternative approach where the basic tools are Thurston’s hyperbolization theorem for Haken manifolds and some well-established properties of Gromov’s simplicial volume, allowing one to bypass the need for the (somewhat more exotic) theory of Alexandrov spaces. Boileau’s notes are largely based on the monograph by L. Bessières, G. Besson, M. Boileau, S. Maillot, and J. Porti, Geometrisation of 3-Manifolds, EMS Tracts in Mathematics 13, 2010. In this tract, the authors developed a slightly different notion of surgery by defining the so-called Ricci flow with bubbling-off. Actually, one might roughly say that the Ricci flow with bubbling-off reduces the general hyperbolization theorem to Thurston’s hyperbolization theorem for Haken manifolds. Carlo Sinestrari provided an extensive introduction to the Ricci flow by first giving a survey of the basic results and examples, then concentrating on the analysis of the singularities of the flow in the three-dimensional case, which is needed in Hamilton and Perelman’s surgery construction. After reviewing the properties of the Ricci flow and the fundamental estimates of the theory, such as Hamilton’s Harnack differential inequality, the Hamilton–Ivey pinching estimate, and Perelman’s no collapsing result, he presented Perelman’s analysis of kappa-solutions and the canonical neighborhood property which gives a full description of the singular behavior of the solutions in dimension 3. All these results are central to the proof of the Poincaré and geometrization conjectures. The exposition is quite accessible to nonexperts. Indeed, the presentation is often informal, and the proofs are omitted except in some simple and significant cases, focusing more on the description of the results and their applications and consequences. A final detailed bibliographical section gives to the interested reader all the references needed for an advanced study of these topics.

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Gang Tian’s expository notes, based on his lectures, discuss some aspects of the Analytic Minimal Model Program through the Kähler–Ricci flow, developed in collaboration with other authors, particularly, J. Song and Z. Zhang. Very stimulating open problems and conjectures are also presented. Section 4.2 contains a detailed account of the sharp version of the Hamilton– DeTurck local existence theorem, which holds in the Kähler case: the maximal time Tmax 2 .0; C1 such that the flow exists on the interval Œ0; Tmax / is precisely determined in terms of a cohomological property of the initial Kähler metric. As a corollary, one deduces that Tmax D C1 for every initial metric on a compact Kähler manifold with a numerically positive canonical bundle. In Sect. 4.3, the limit singularities that can arise when t ! Tmax < C1 are analyzed. After having established a general convergence theorem (Theorem 4.3.1), one faces questions concerning regularity and the geometric properties of the limit. A combination of partial results (particularly in the case of projective varieties and when one can apply deep results of algebraic geometry) and well-motivated conjectures outlines a pregnant scenario. Sections 4.4 and 4.5 discuss the construction of a Kähler–Ricci flow with surgery (assuming the truth of a conjecture stated in Sect. 4.3) and its asymptotic behavior. Numerous conjectures arise throughout this discussion such as: the characterization (up to birational isomorphism) of “Fano-like” manifolds as those whose flow becomes extinct at a finite time; the characterization of uniruled manifolds (up to birational isomorphism) as those whose flow collapses in finite time; and the existence of a flow with surgery globally defined in time and with only finitely many surgery times. In Sect. 4.6, algebraic surfaces are considered, showing how most of the program is carried out in this case.

We are pleased to express our thanks to the speakers for their excellent lectures and to the participants for contributing with their enthusiasm to the success of the Summer School. The speakers, the participants, and the CIME organizers collectively created a stimulating, rich, pleasant, and friendly atmosphere at Cetraro. For this reason, we would finally like to thank the Scientific Committee of CIME and, in particular, Pietro Zecca and Elvira Mascolo. Pisa, Italy Naples, Italy

Riccardo Benedetti Carlo Mantegazza

Acknowledgements

CIME activity is carried out with the collaboration and financial support of: INdAM (Istituto Nazionale di Alta Matematica) and MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca).

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Contents

1

The Differentiable Sphere Theorem (After S. Brendle and R. Schoen).. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Gérard Besson

1

2 Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Michel Boileau

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3 Singularities of Three-Dimensional Ricci Flows . . . . . .. . . . . . . . . . . . . . . . . . . . Carlo Sinestrari

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4 Notes on Kähler-Ricci Flow .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 Gang Tian

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Chapter 1

The Differentiable Sphere Theorem (After S. Brendle and R. Schoen) Gérard Besson

Abstract In these notes we describe a major result obtained recently using the Ricci flow technique in the context of positive curvature. It is due to S. Brendle and R. Schoen and states that a strictly 1/4-pinched closed manifold carries a metric of constant (positive) sectional curvature. It relies on a technique developed by C. Böhm and B. Wilking who obtained the same conclusion assuming that the manifold has positive curvature operator. The maximum principle applied to the Ricci flow equation leads to studying an ordinary differential equation on the space of curvature operators.

1.1 Introduction In 1951, Rauch [38] showed that a complete Riemannian manifold whose sectional curvature is positive and varies between two numbers whose ratio is close to 1 has a universal cover homeomorphic to a sphere. Precisely, for ı > 0, we say that a closed Riemannian manifold M is pointwise ı-pinched if at each point the ratio between the smallest sectional curvature and the largest one is greater or equal to ı. We say that it is ı-pinched if this inequality is satisfied by the ratio between the global minimum (on M) of the sectional curvature and its global maximum. Finally, it is said to be strictly ı-pinched (pointwise or globally) if this inequality is strict. These notions make sense whether the curvature is positive or negative. In the sequel all Riemannian manifolds will be positively curved; for results in the negative setting the following articles [26, 40] can be consulted. Rauch’s Theorem then asserts that a ı-pinched simply connected Riemannian manifold, for an explicit value of ı close to 1, is homeomorphic to a sphere. On the other hand a standard computation (see [20, Sect. 3.D.2, p. 149]) shows that the complex projective spaces, of complex dimension greater than 1, have a sectional curvature varying between 1 and 4, for a suitable normalization; this is also true for the other closed Riemannian symmetric spaces of rank one. Rauch’s Theorem then cannot be true for a (non strictly) 1=4-pinched Riemannian manifold of positive curvature. It was Berger [1] G. Besson () Université Grenoble Alpes – Institut Fourier, CS40700, 38058 Grenoble Cedex 9, France e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Benedetti, C. Mantegazza (eds.), Ricci Flow and Geometric Applications, Lecture Notes in Mathematics 2166, DOI 10.1007/978-3-319-42351-7_1

1

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G. Besson

and Klingenberg [28], in the 1960’s, who got the optimal result: a simply connected Riemannian manifold which is strictly 1=4-pinched is homeomorphic to a sphere. The reader is referred to [4] (Sect. 12.2.2.1, p. 552) for a description of the technique and some historical notes. Let us remark that these results do not exclude the exotic spheres. It is indeed an interesting question to investigate whether an exotic sphere can carry a Riemannian metric of positive curvature. This question is still open. This text grew out of a series of lectures given at a CIME conference in Cetraro, Italy. The goal is to describe the following remarkable result due to S. Brendle and R. Schoen. Theorem 1.1 (Brendle and Schoen [9, 10]) Let M be a Riemannian n-manifold of positive sectional curvature and pointwise strictly 1=4-pinched, then M carries a metric of constant sectional curvature. Hence it is diffeomorphic to the quotient of a sphere by a finite subgroup of O.n/. In particular a consequence is that no exotic sphere can carry a strictly 1=4pinched metric. The method relies on the use of the Ricci flow introduced by Hamilton in [21]. The idea is to construct a deformation of the Riemannian metric which evolves towards a constant curvature metric. This ideal behavior only occurs when the metric one starts with has nice properties. Let us recall that the seminal article by R. Hamilton has had extraordinary developments which culminated in Perelman’s works ([34–36], and also [5]) proving the geometrization conjecture. In [21] Hamilton shows, using the same method, the following theorem. Theorem 1.2 (Hamilton [21]) Let M be a closed 3-dimensional Riemannian manifold which carries a metric of positive Ricci curvature, then it also carries a metric of positive constant curvature. Later on, the same technique extended to 4-manifolds was used to get the results proved in [22, 25]. It is a remarkable extension to higher dimensions which is at the core of the works we intend to describe and it is due to Böhm and Wilking [6]. We shall say that a curvature operator is 2-positive if the sum of its smallest eigenvalues is positive. This notion first appeared in the article [12] of Chen. Theorem 1.3 (Böhm and Wilking [6]) Let M be a closed Riemannian manifold whose curvature operator is 2-positive, then M carries a constant curvature metric. Using analytical methods to prove results such as Theorem 1.1 is not new. The theory of harmonic maps is used in [30] whereas the Ricci flow is used in [22, 27, 29, 33]. Note that in [29], Margerin studies a natural curvature condition and proves an optimal result. All this is part of the so-called Geometric Analysis and the maximum principle for parabolic systems is a crucial tool which allows to reduce the problem to an algebraic question and, in general, answers the question: what are the properties of the curvature operator which are preserved under the Ricci flow? In this text we shall limit ourselves to the dimensions greater than 3. After recalling the different curvature notions, we shall describe the main steps of the proof. The first one consists in using the above-mentioned maximum principle in

1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

3

order to reduce the problem to the study of an ordinary differential equation on the space of curvature endomorphisms. In the second step we shall exhibit inequalities satisfied by the curvature operator which are invariant under this dynamical system. Finally, standard geometric arguments allow to conclude. General references for the basics of Riemannian Geometry are [4, 20] for an overview and [14–16] for the Ricci flow. These last references are exhaustive and C. Böhm and B. Wilking’s method is described in details in [16, Chap. 11]. This text is then to be used as a guide for the reading of the original articles as well as these references. We certainly advise the reader to study the recent survey by S. Brendle and R. Schoen [11]. I wish to warmly thank C. Böhm, S. Brendle, L. Ni and H. Seshadri for their answers to my very naive questions as well as M. Berger, P. Bérard, L. Bessières, J-.P. Bourguignon, Z. Djadli, S. Gallot, H. Nguyen and T. Richard for fruitful discussions.

1.2 Basics of Riemannian Geometry In what follows .M; g/ is a closed Riemannian manifold. We shall denote the metric either by g or < :; : >. The Levi-Civita covariant derivative is denoted by r. Let X, Y, Z and T be vector fields, we define the .0; 4/-curvature tensor by (see [20]) R.X; Y; Z; T/ D< rY rX Z  rX rY Z  rŒY;X Z; T > : It is known that this tensor is skew symmetric in X and Y and in Z and T and satisfies R.X; Y; Z; T/ D R.Z; T; X; Y/. It defines a symmetric endomorphism of 2 .TM/ that we shall also denote by R and which is called the curvature operator. The convention used in this text are similar to those in [20], the reader is welcome to compare them to those in [16]. The curvature tensor also satisfies the first Bianchi identity, which is analogous to the Jacobi identity satisfied by the Lie bracket of a Lie algebra. Precisely, it reads R.X; Y; Z; T/ C R.Y; Z; X; T/ C R.Z; X; Y; T/ D 0 : The sectional curvature of a 2-plane P  Tm M tangent at m to M is K.P/ D< R.x ^ y/; x ^ y > ; where .x; y/ is an orthonormal basis of P. We note that K is the value of R computed on the decomposed vector whereas R is defined on the whole 2 .TM/ . If fei g is an orthonormal basis of Tm M, we endow 2 .Tm M/ with the scalar product such that fei ^ ej gi. The Ricci curvature at m is a symmetric bilinear form on Tm M that we also think of as a symmetric endomorphism of Rn

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(we shall give the same name to these two objects). It is defined by Ricij D Ric.ei ; ej / D< Ric.ei /; ej >D

X

Rikjk :

k

It is a metric-like tensor. Finally, the scalar curvature is the trace of the Ricci curvature, that is, for m 2 M X X scal.m/ D Ricii D Rikik D 2 trace R : i

i;k

With these conventions the curvature operator of the standard sphere is the identity, its sectional curvatures are all equal to 1, its Ricci curvature is .n  1/g and its scalar curvature is constant equal to n.n  1/.

1.2.1 Algebraic Curvature Operators For a point m 2 M the choice of an orthonormal basis fei g of Tm M gives an identification with the Euclidean space Rn . The scalar product also gives an identification of 2 .Rn / with the Lie algebra so.n/ by associating to the unit vector ei ^ ej the rank 2 endomorphism which is the rotation of angle =2 in the plane generated by ei and ej . Via this identification one has < A; B >D 1=2 trace.AB/, for A; B 2 2 .Rn /. The space of symmetric endomorphisms (which we identify with the symmetric bilinear forms) of 2 .Rn / is denoted by S 2 .so.n//. This space encodes the first three relations satisfied by the curvature tensor. We call algebraic curvature tensor an element in S 2 .so.n// which furthermore satisfies the first Bianchi identity; The space of algebraic curvature operators is denoted by SB2 .so.n// (see [16, p. 81]).

1.2.2 Algebraic Products on S 2 .so.n// Let A and B be two symmetric endomorphisms of Rn , we define a new symmetric endomorphism of so.n/ ' 2 .Rn / by (see [16, p. 74]) : .A ^ B/.v ^ w/ D

1 .A.v/ ^ B.w/ C B.v/ ^ A.w// : 2

It is easily checked that A ^ B 2 S 2 .so.n// and satisfies the following equality A ^ B D B ^ A.

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Let now R and S be two tensors in S 2 .so.n//, we define R]S 2 S 2 .so.n// by: < .R]S/.h/; h >D

1X < ŒR.!˛ /; S.!ˇ ; h > : < Œ!˛ ; !ˇ ; h > ; 2 ˛;ˇ

where h 2 so.n/ and f!˛ g is an orthonormal basis of so.n/. It is easily checked that this definition is independent of the choice of the basis and that R]S D S]R (see [6] and [16, p. 72]). Finally we shall denote by R] D R]R. This product which has been introduced by Hamilton (see for example [22]) will play a key role in the sequel.

1.2.3 Irreducible Components Under the Action of O.n/ The group O.n/ acts by changes of basis on the space of algebraic curvature operators. One can decompose the space SB2 .so.n// in three irreducible components under this action. For R a curvature operator we denote by Ric0 the traceless part of its Ricci tensor, that is < Ric.R/.ei /; ei >D

n X

Rikik

and

Ric0 .R/ D Ric.R/ 

kD1

scal.R/ I Rn ; n

where scal.R/ D 2 trace.R/. One then has SB2 .so.n// D RIS 2 ˚ < Ric0 > ˚ < W > : B

The first factor consists of algebraic curvature operators which are multiple of the identity, the second of multiple of operators of the type A^IRn where A is a traceless symmetric operator acting on Rn and the third is the kernel of the map R ! Ric.R/. This last space contains the Weyl curvature tensors, which among the components of R is the most difficult to understand. Let us recall that the Weyl component of the curvature tensor of a Riemannian metric vanishes, if and only if it is locally conformally flat. For a curvature tensor R we denote by RI , RRic0 and RW its various components and by I the identity in SB2 .so.n// as well as in Rn . We then have (see [16, p. 87] for the details), RD

scal.R/ 2 IC Ric0 .R/ ^ I C W : n.n  1/ n2

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1.2.4 A New Identity Böhm and Wilking establish in [6] a new identity satisfied by any algebraic curvature tensor. It is remarkably simple. Proposition 1.4 ([6]) For all R 2 SB2 .so.n// one has R C R]I D .n  1/RI C

n2 RRic0 D Ric ^I : 2

Below we shall deduce from it the decomposition along the irreducible components of the quadratic expression in R, which appears in the evolution equation (along the Ricci flow) of the curvature operator. It is remarkable that new simple identities on this well-studied space can still be discovered.

1.2.5

O.n/-invariant Endomorphisms of SB2 .so.n//

Let ` be a self adjoint linear map of SB2 .so.n// into itself which is invariant under the action of O.n/. It is diagonalizable and its eigenspaces are O.n/ invariant; it then preserves the above irreducible components. It is furthermore a multiple of the identity on each of them (see [16, p. 89]). In [6] the authors consider such maps which preserve the Weyl component. It can be written as follows, `a;b .R/ D RC2.n1/aRI C.n2/bRRic0 D .1C2.n1/a/RI C.1C.n2/b/RRic0 CRW ;

where a and b are real numbers.

1.3 The Ricci Flow It is an ordinary differential equation on the space of Riemannian metric introduced by Hamilton in the seminal article [21]. We consider a family g.t/ of smooth Riemannian metrics, depending smoothly in t, and solving the following Cauchy problem: 8 < @g D 2 Ric g.t/ @t : g.0/ D g0

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There is a normalised version for which the volume of the evolving metric is constant; indeed, it suffices to replace the first line of the above equation by: Z 2  @g 1 scal.x; t/dvg.t/ .x/ g.t/: D 2 Ricg.t/ C @t n vol.M; g.t// M The reader is referred to [5, 14] for details of this theory. Classical results show that the solutions exist for small time for any smooth initial data. The idea is now to show that under the hypothesis of Theorems 1.1 and 1.3 the normalized Ricci flow converges towards a constant curvature metric.

1.3.1 The Evolution Equations for the Curvatures A standard computation gives the evolution equation of the various curvature tensors. We shall limit ourselves to the scalar curvature and the curvature operator. We have, @ scal C g.t/ scal D 2j Ricg.t/ j2g.t/ @t were g.t/ is the Laplace operator acting on functions for the metric g.t/. The convention is here the one used in geometry (in dimension 1, it is d2 =dx2 ). The norm of the Ricci tensor is computed with the metric g.t/. Similarly one has @R C R D 2.R2 C R] /: @t We use here the rough Laplacian, that is the opposite of the trace of the second covariant derivative of the tensor R. The notation R2 denotes the square of the endomorphism R and R] the quadratic expression defined in the previous section. The evolution of the curvature tensor is written in this simple way thanks to a trick due to K. Uhlenbeck that we shall describe below.

1.3.2 The Maximum Principle It is the key tool for the study of the solutions of the heat equation. Here, we shall give a vector version that is adapted to our parabolic system. This is done by Hamilton in [21, 22]. The reader is referred to [14, 16]. Let us consider a PDE of the type @s=@t C t s D f .s/ and the ODE ds=dt D f .s/. The above PDE are so-called reaction-diffusion equations, the diffusion part is given

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by the Laplacian; if f  0, it is a heat equation which “spreads” the initial condition. The non-linear term f .s/ is the reaction term which, alone, leads to a blow up in finite time (convergence of certain norms towards C1). The main question is to know which term will win, reaction or diffusion. The maximum principle is a comparison between the solutions of the PDE and the solutions of the ODE. The equation satisfied by the scalar curvature can be written @R=@t C R  0. The simplest maximum principle leads to the fact that the minimum of the scalar curvature does not decrease along the Ricci flow. We now describe a vector version. Let M be endowed with a smooth family of metrics g.t/, for t 2 Œ0; T, and let  W E ! M be a vector bundle endowed with a fixed metric and a smooth family of compatible connections, rt . These datas allow to define a Laplace operator acting on sections of E , depending on t that we simply shall denote by . Let us now consider a smooth function f W E  Œ0; T ! E such that, for given t, f .:; t/ preserves the fibres. Let K be a closed subset of E that is assumed to be invariant under the parallel transport of rt , for all t 2 Œ0; T, and such that Km D K \  1 .m/ is closed and convex. The key hypothesis is a relation between K and the ODE du dt D f .u/, defined on each fibre Em of E ; we assume that any of its solution u such that u.0/ 2 Km remains in Km for all t 2 Œ0; T. Theorem 1.5 ([21, 22] or [14, Theorem 4.8]) Under the above hypothesis, let s.t/ be a solution of the PDE, @s C s D f .s/ @t such that s.o/ 2 K , then, for all t 2 Œ0; T, s.t/ 2 K . In order to apply this result to the curvature operator we notice that, although the metric on the bundle 2 .T  M/ depends on t, a trick, due to K. Uhlenbeck (see below and [14, Sect. 6.1]), allows to convert the situation to a fixed metric on a fixed bundle with however a time dependent connection. The set K is thought of as a geometric version of the inequalities satisfied by the curvature operator such as the positivity. Example 1.6 Let us recall the 3-dimensional situation, for which the maximum principle has played an important role in G. Perelman’s works; it indeed shows that the scalar curvature controls the full curvature tensor. At each point m 2 M and t 2 Œ0; T the endomorphism R diagonalises and has three eigenvalues denoted by .m; t/  .m; t/  .m; t/. In dimension 3 these numbers are sectional curvatures. One shows that the two tensors R2 and R] diagonalise in the same basis and have eigenvalues respectively equal to .2 ; 2 ;  2 / and .; ; /. The ordinary

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differential equation is thus, 8 d 2 ˆ ˆ < dt D  C  d D 2 C  dt ˆ ˆ : d D  2 C  dt

In general we have to consider the ODE dR D R2 C R] D Q.R/ ; dt defined on SB2 .so.n// (we omit the factor 2 which does not play any role).

1.3.3 Constructing K It is now useful to recall K. Uhlenbeck’s trick which allows to reduce to a fixed vector bundle. Let us consider a Ricci flow .M; g.t// defined on an interval Œ0; T/. One can construct a time dependent vector bundle isometry .t/ W .TM; g.0// ! .TM; g.t// solving the equation d .t/ D Ricg.t/ ı .t/ dt

and .0/ D id ;

where Ric is here to be understood as an endomorphism. It allows to pull back the whole situation to a fixed vector bundle endowed with a fixed metric. In particular, the pulled-back Levi-Civita connection depends on t. This isometry extends to all natural vector bundle such as the bundle of curvature tensor E whose typical fibre is isometric to SB2 .so.n//. Then, considering .t/ .Rg .t// we can write the evolution equation of the curvature operator on this bundle and then apply the above maximum principle. In order to construct K in E we proceed as follows. Let us consider a closed convex set F  SB2 .so.n//. We furthermore assume that F is O.n/-invariant. We can then transport F on each fibre Em by the identification given by the choice of an orthonormal basis of Tm M; the image set is denoted by Km . The O.n/invariance of F guaranties that this construction does not depend on the chosen basis. Similarly the parallel transport associated to the time dependent connection induces an isometries between the fibres of K ; the invariance of F by the O.n/ action ensures the invariance of K by the parallel transport of all these connections. If furthermore F is invariant by the ODE, the set K obtained satisfies the hypothesis

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of Theorem 1.5. Such a set transcribes curvature conditions which are preserved by the flow.

1.4 C. Böhm and B. Wilking’s Method The question is now reduced to the study of the quadratic term Q.R/, the right hand side of the ODE, which we view as the value at R of a vector field on SB2 .so.n//. A flow invariant curvature condition can be defined by a closed convex and O.n/invariant cone, for example the cone C of 2-nonnegative curvature operators (the sum of their two lowest eigenvalues is nonnegative). In order to show the invariance of the condition by the ODE we have to show that if a trajectory of this ODE starts inside the cone and reaches its boundary, the vector field pushes it back inside, or at least keep it on the boundary. We have then to show that this vector field Q.R/ is in the tangent cone TR C to C at each point of @C. Let C be such a cone, invariant by the ODE, one can built others by taking the image of C by the maps `a;b defined above, for suitable choices of a and b. The linear maps `a;b sends boundary into boundary and tangent cones into tangent cones, hence the set `a;b .C/ is invariant by the ODE, that is C is invariant by `1 a;b ı Q ı `a;b , if 2 ] Xa;b .R/ D `1 a;b ..`a;b .R// C .`a;b .R// / ;

is in the tangent cone TR C to C at each point of @C. As it is the case for Q.R/ it suffices, by convexity of C, to prove it for Da;b .R/ D Xa;b .R/  Q.R/. The maps `a;b preserves the Weyl component and consequently we show that Da;b .R/ does depend only on the Ricci component of the curvature operator R. The formulae then become much simpler and the computation is possible. Thanks to this enlightening idea C. Böhm and B. Wilking built families of convex cones C.s/s2Œ0;1 , O.n/-invariant, invariant by the ODE whose intersection is reduced to the curvature operator I, where  > 0, that is the curvature operator of the round sphere up to normalisation. If the theory works, the trajectories of the PDE are “trapped” in this family and constrained to converge towards the curvature operator of the round sphere. The underlying idea is to try to obtain Lyapunov functions associated to the ODE (projected on the unit sphere of SB2 .so.n//) whose minimum would be the curvature operator of a round sphere in order to prove that the trajectories converge towards this fixed point. Such a function is not really necessary; indeed, only its level sets play the key role. and they are not necessarily included in one another but may be used in the same way.

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1.4.1 Properties of Q.R/ and Da;b .R/ Below is a list of properties satisfied by these quantities; for more details the reader is referred to [6, 16]. For R 2 SB2 .so.n// one shows that Q.R/ 2 SB2 .so.n// (see [16, p. 88]). It was observed by G. Huisken that the ODE under consideration is the gradient flow of P.R/ D

1 trace.R3 C RR] / ; 3

which is unfortunately too difficult to study to play the role of a Lyapunov function. Finally, one can compute the components of Q.R/ in the decomposition in irreducible components. That is where the new identity is used. The following theorem is the key tool for proving the invariance of the cones `a;b .C/. Theorem 1.7 (Böhm and Wilking [6]) For a and b real numbers, Da;b .R/ D ..n  2/b2  2.a  b// Ric0 ^ Ric0 C2a Ric ^ Ric C2b2 Ric20 ^I C

trace.Ric20 / .nb2 .1  2b/  2.a  b/.1  2b C nb2 //I : n C 2n.n  1/a

The remarkable fact, to be expected, is that Da;b .R/ does not depend on the Weyl component of R. For example, one can compute its eigenvalues in terms of those of Ric. It suffices to choose and orthonormal basis of Rn diagonalising Ric, we then deduce an orthonormal basis of so.n/ in which Da;b .R/ diagonalises. Its eigenvalues then depend on the choice of a and b.

1.4.2 Construction of the Family of Cones p

2 Let us recall that n  4. We set bN D 2n.n2/C42 and a0 .b/ D b C .n2/ n.n2/ 2 b . These N The following propositions numbers are such that Da0 .b/;b is positive for b 20; b. are the starting point of the construction (see [6] and [16, p. 113]) :

Proposition 1.8 (Hamilton [24]) The cone C of 2-positive algebraic curvature operators is preserved by the ODE. This result is also true in dimension 3. In this case the hypothesis says that the Ricci curvature is positive (see [22]). Let f!i g be an orthonormal basis of so.n/ of eigenvectors R. In order to show that the cone is preserved we have to show that < Q.R/!i ; !i > C < Q.R/!j ; !j >  0 for all i < j such that < R!i ; !i > C < R!j ; !j >D 0I that is, that the vector field Q.R/ forces the trajectories back inside the cone. We here use the precise description of Q.R/.

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1.4.2.1 First Step Proposition 1.9 Let C be the cone of 2-nonnegative curvature operators for n  4 N then one has the following properties: and b 20; b, (i) Cb D `a0 .b/;b .C/ is preserved by the ODE. (ii) For b > 0, the vector field Q.R/ is transverse to the boundary of `a0 .b/;b .C/ at points R ¤ 0. (iii) `a0 .bN /;bN .C n f0g/ is included in the cone of positive curvature operators. We use the description of Da;b .R/ to prove this proposition following the scheme described above. We grosso modo show that Da;b .R/ is nonnegative. This produces a first family of cones that links C to a cone included in the set of positive curvature operators.

1.4.2.2 Second Step Following the same scheme and similar proofs we now construct the following family. For b 2 Œ0; 1=2 set .n  2/b2 C 2b 2 C 2.n  2/b2

a.b/ D

and p.b/ D

.n  2/b2 : 1 C .n  2/b2

Then, C0b D `a.b/;b

˚ trace.Ric/  R 2 SB2 .so.n//j R  0; Ric  p.b/ n

is a O.n/-invariant closed convex cone which is furthermore ODE-invariant [6, Lemma 3.4]. It links the cone of nonnegative operators to C01 ; 1 . 2 2

1.4.2.3 Third Step Finally, for b D 1=2 and s  0 we set a.s/ D

1Cs 2

and p.s/ D 1 

4 : n C 2 C 4s

We then define, ˚ trace.Ric/  R 2 SB2 .so.n//j R  0; Ric  p.s/ 2 n

C00s D `a.s/; 1

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which has the same properties than before. Let us notice that 1 `a.s/; 1 .R/ D 2.n  1/RI ; 2 s!C1 a.s/ lim

consequently the family C00s converges towards RC I when s goes to infinity. These two last steps consist in adding to the construction of convex invariant sets a pinching of the Ricci curvature. Indeed, the inequalities which define these families yield a control of the Ricci curvature when the curvature operator is normalised so that its largest eigenvalue is 1, for example. It then suffices to concatenate the three families: Cb , C0b \ CbN and C00s \ CbN to obtain, after a suitable change in the parameters, a collection of O.n/-invariant closed convex cones, invariant by the ODE which links the 2-nonnegative curvature operators to the multiples of the identity. We shall denote this family by C.s/s2Œ0;1 . Such a family is called a pinching family in [6].

1.4.3 Pinching Set The above construction is not really sufficient to obtain the desired conclusion, that is that the solution of the PDE converges, after rescaling, towards the curvature operator of the round sphere. Indeed, it could happen that the curvature operator goes to infinity in these cones without getting closer to the multiples of the identity. Somehow these cones are too widely opened at infinity. We then construct from C.s/ a set F which is called a pinching set. This notion was introduced by Hamilton in [22] and C. Böhm and B. Wilking describe a suitable generalisation. Theorem 1.10 (Böhm and Wilking [6]) For 20; 1Œ and h0 > 0, there exists a O.n/-invariant closed convex set F  SB2 .so.n// such that (i) F is preserved by the ODE, (ii) C. / \ fR W trace.R/  h0 g  F  C. /, (iii) the closure of F n C.s/ is compact for all s 2 Œ ; 1Œ. This theorem is shown by proving that the intersection of all O.n/-invariant closed convex sets which satisfy .i/ and .ii/ works. In [6] a more general result is proved which can be applied to a wide variety of families of cones. Intuitively one can think of the set F as a parabola that crosses all the cones centred at the origin; the asymptotic cone of such a set is reduced to the vertical axis. Going at infinity in this set thus forces to get closer and closer to it (after renormalisation). This set can now be “copied” in the fibers of the bundle of curvature operators on M as described before in order to built a set F invariant by parallel transport.

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1.4.4 Conclusion In order to finish the proof of Theorem 1.3 we proceed as follows. Let .M; g0 / be a closed Riemannian manifold whose curvature operator is (strictly) 2-positive and let g.t/ be the Ricci flow with initial datal g0 defined on a maximal interval t 2 Œ0; TŒ. Since M is closed, there exists 20; 1Œ and h0 > 0 such that the curvature operator of g0 is at each point m 2 M in C. / \ fR W trace.R/  h0 g, after identification of Tm M with Rn . Then there exists a pinching set F satisfying the above properties. Let F be the subset of the curvature operators bundle obtained. It follows from the maximum principle that the curvature tensor of the metric g.t/ is in F for all t  0. It is easy to check that the scalar curvature of a metric whose curvature operator is 2-positive is itself positive. It is then a classical fact [14] that the Ricci flow is singular in finite time, that is that T < C1 and that it exists .mi ; ti /, with ti ! T such that Qi D jR.mi ; ti /j D max jR.m; t/j ! C1 : MŒ0;ti 

We here denote by R.m; t/ the curvature operator of g.t/ at m, the norm being the endomorphism one. We then have a family of Ricci flow on M, gi .t/ D Qi g.ti C Q1 i t/. This transformation is called a parabolic renormalisation (see [5]) and produces a new Ricci flow. A compactness theorem due to Hamilton [23] shows that the sequence of flows converges for the pointed convergence towards a complete Ricci flow .M1 ; g1 .t/; m1 /. We can use here a result due to Perelman (see [34]) which shows that the injectivity radius of gi .0/ at mi is uniformly bounded below. By construction jR1 .m1 ; 0/j D 1, where R1 is the curvature operator of the limit space. Since F n C.s/ is compact and hence bounded for all s 2 Π; 1Πthen, for all s greater than , there exists i large enough so that R.mi ; ti / 2 C.s/. This implies that Q1 i R.mi ; ti / 2 C.s/. Taking the limit we then have R1 .m1 ; 0/ 2 \s2Π;1ΠC.s/, that is R1 .m1 ; 0/ 2 RC I. The sectional curvature of g1 .0/ is positive and constant at m1 (it does not depend on the 2-plane). By continuity, for p 2 M1 close to p1 , one has jR1 . p; 0/j > 1=2 and hence, for a sequence of points pi 2 M which converges towards p, jR. pi ; ti /j ! C1. The same argument applies at p showing that the sectional curvature of g1 .0/ at p is a positive constant. We then prove that at each point in M1 the sectional curvature is constant and positive (the constant may vary with the point); we deduce that M1 is closed and that the convergence is global, not pointed and hence M1 is diffeomorphic to M. In dimension greater than 3 the sectional curvature being constant at each point it is constant on M1 . It is possible to refine and show that the convergence is exponential in all Ck topologies (see [27, 29, 33]). The details of the proof are left to the reader (see [6] and [16, p. 130]).

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1.5 S. Brendle and R. Schoen’s Works The scheme briefly described is flexible enough so that it can be adapted to other situations. This approach is used in [9, 10]. The 2-positivity of the curvature tensor is replaced by the notion of positive isotropic curvature, usually denoted PIC. It first appeared in an article by Micallef and Moore [30]. Let us recall its definition. Let .M; g/ be a Riemannian manifold whose curvature endomorphism at m is denoted by R. The Riemannian metric can be extended to Tm M ˝ C as a complex bilinear form. Similarly R is extended as a complex linear map. For a 2-plane P  Tm M ˝ C, generated by fz; wg we define its complex sectional curvature K.P/ D

< R.z ^ w/; zN ^ w/ N : < z ^ w; zN ^ wN >

A subspace V  Tm M ˝ C is said to be totally isotropic if each z 2 V is isotropic, that is if < z; z >D 0. Definition 1.11 The manifold M is said to have positive isotropic curvature if K.P/ > 0 for all totally isotropic 2-plane P  Tm M ˝ C. This condition is non void for n  4 only. This notion appears in the study of the stability of minimal surfaces in a Riemannian manifold. It is exactly while studying the stability of harmonic and conformal 2-dimensional spheres in M that M. Micallef and J. Moore showed that a simply connected Riemannian manifold of dimension greater than 3 which is PIC is homeomorphic to a sphere. This results yields an alternative proof to M. Berger and W. Klingenberg’s pinching theorem thanks to the following remark: Proposition 1.12 A strictly 1=4-pinched Riemannian manifold is PIC. For a totally isotropic 2-plane P one can find four orthonormal vectors fe1 ; e2 ; e3 ; e4 g so that fe1 C ie2 ; e3 C ie4 g is a basis of P. An immediate computation now gives that the PIC condition reads R1313 C R1414 C R2323 C R2424  2R1234 > 0 : Now, if the manifold is strictly 1=4-pinched, for example if its sectional curvature varies in the interval 1=4; 1, all the quantities Rijij , which are sectional curvatures, are greater than 1=4. An inequality due to Berger (see [2, p. 69]) shows that jR1234 j < 1=2, which proves the above proposition. Let us remark that M. Micallef and J. Moore’s result requires the manifold to be simply connected. This restriction is necessary since the canonical metric on Sn1 S1 is PIC; this shows that one cannot even hope that being PIC implies that the universal cover is compact. An interesting question is to describe the fundamental group of PIC manifolds and the reader can check [10] for useful references. In [25], Hamilton studies the evolution of the Ricci flow when the initial data is PIC,

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in dimension 4, and shows that singularities should occur. He then introduces the notion of surgery extensively used afterwards in G. Perelman’s works; as in these works, singular points have neighbourhoods which could resemble open sets in S3  S1 . For these ideas the article [25] is of primary importance (see also [13]). For our purpose it shows that Böhm and Wilking’s method cannot work with the PIC condition as it i s stated; one has to take into account singularities and hence develop a surgery procedure comparable to the one used in Perelman’s works. It is a widely open question. In [10], a bright idea allows to circumvent this difficulty. Let .M; g0 / be a closed Riemannian manifold and denote by RQ the curvature tensor of M  R and RO the one of M  R2 . S. Brendle and R. Schoen prove the following properties: (1) the condition “R is PIC” is preserved by the Ricci flow. (2) The condition “RQ is PIC” is also preserved and is stronger than the previous one. Indeed, it implies, for example, that the Ricci curvature of R is positive. (3) The curvature operator RO cannot be PIC because of the 2-dimensional flat factor, as it can be easily checked. However, for > 0, let us consider R D R  I and its extension RO on M  R2 . Saying that RO has a nonnegative isotropic curvature (we shall write NIC) is a further restriction. It implies, for example, that R is PIC and has positive sectional curvature. The scheme develop by C. Böhm and B. Wilking can then be applied with these more restrictive notions. More precisely, one has Theorem 1.13 (Brendle and Schoen [10, Theorem 2]) Let .M; g0 / be a closed Riemannian manifold such that, for small enough RO is NIC. Then, the normalised Ricci flow with initial data g0 converges towards a metric with constant sectional curvature (in the smooth topology). The main Theorem 1.1 is then a consequence of Proposition 1.14 ([10]) Let .M; g0 / be a closed Riemannian manifold with positive sectional curvature and which is strictly 1=4-pinched then, for small enough, RO

is NIC. Here R denotes the curvature operator of .M; g0 /. Proof (Idea of the Proof) The proofs follow the scheme introduced by Böhm and Wilking. The first step is the most difficult one and consists in proving that the condition NIC is preserved by the flow. As before, using the above notations, it reduces to proving that if R1313 C R1414 C R2323 C R2424  2R1234 D 0 : then, Q.R/1313 C Q.R/1414 C Q.R/2323 C Q.R/2424  2Q.R/1234  0 : This tedious computation is clearly done and with several tricks in [10]. This result was independently obtained by H. Nguyen, in his thesis [32].

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The second step consists in constructing the family of convex cones which collapses to RC I. The starting point is ˚  CO D R W RO is NIC : The construction uses as before the maps `a;b for suitable choices of a and b. Finally the proposition is proved by computations of the type used in the proof of Proposition 1.12 although quite a bit more sophisticated. In a more recent work S. Brendle successfully develop the same formalism for a Riemannian metric such that .M; g0 /  R has a PIC curvature operator (that is for RQ with the above notations) (see [7]).

1.6 More Extensions In M. Berger and W. Klingenberg’s works the weak pinching is treated: a simply connected weakly 1=4-pinched (with positive sectional curvature) which is not homeomorphic to a sphere is diffeomorphic to a compact rank one symmetric space. The analogous version where we get a diffeomorphism with the sphere is proved by Brendle and Schoen in [9]. In [3], Berger shows that for each n, there exists a ".n/ with 0 < ".n/ < 14 such that any complete, simply connected n-dimensional manifold M which admits a ".n/-pinched metric is homeomorphic to Sn or diffeomorphic to a compact rank one Riemannian symmetric space. A version of that, in the spirit of the above works, is proved by Petersen and Tao in [37]. For more results on the geometric and topological consequences of the condition PIC, the reader is referred to: [8, 17–19, 31, 39] (and certainly many more). In this set up the new frontier is certainly to learn how to practice surgery on Ricci flows in dimension greater than 3.

1.7 Conclusion The quasi-coincidence between the works briefly described above and G. Perelman’s clearly shows that the Ricci flow technique has reached its full strength. We also note progresses in the understanding of the curvature tensor of a Riemannian manifold; indeed, it is remarkable that new identities can still be discovered as it is the case in Böhm and Wilking’s works. Finally, it is worth insisting on R. Hamilton’s work who constructed the building and opened most of the tracks.

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Acknowledgements The author is supported by ERC Advanced Grant 320939, “Geometry and Topology of Open Manifolds” (GETOM).

References 1. M. Berger, Les variétés Riemanniennes .1=4/-pincées. Ann. Scuola Norm. Sup. Pisa (3) 14, 161–170 (1960) 2. M. Berger, Sur quelques variétés riemanniennes suffisamment pincées. Bull. Soc. Math. France 88, 57–71 (1960) 3. M. Berger, Sur les variétés riemanniennes pincées juste au-dessous de 1=4. Ann. Inst. Fourier (Grenoble) 33(2), 135–150 (loose errata) (1983) 4. M. Berger, A Panoramic View of Riemannian Geometry (Springer, Berlin, 2003) 5. G. Besson, Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci (d’après Perelman), in Séminaire Bourbaki 2004–2005. Astérisque, vol. 307 (Société mathématiques de France, Paris, 2006), pp. 309–348 6. C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008) 7. S. Brendle, A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145(3), 585–601 (2008) 8. S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151(1), 1–21 (2010) 9. S. Brendle, R. Schoen, Classification of manifolds with weakly 1=4-pinched curvatures. Acta Math. 200(1), 1–13 (2008) 10. S. Brendle, R. Schoen, Manifolds with 1=4-pinched curvature are space forms. J. Am. Math. Soc. 22(1), 287–307 (2009) 11. S. Brendle, R. Schoen, Sphere theorems in geometry, in Handbook of Geometric Analysis, No. 3, Advanced Lectures in Mathematics (ALM), vol. 14 (International Press, Somerville, MA, 2010), pp. 41–75 12. H. Chen, Pointwise 14 -pinched 4-manifolds. Ann. Global Anal. Geom. 9(2), 161–176 (1991) 13. B.-L. Chen, X.-P. Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature. J. Differ. Geom. 74(2), 177–264 (2006) 14. B. Chow, D. Knopf, The Ricci Flow : An Introduction. Mathematical Surveys and Monographs, vol. 110 (American Mathematical Society, Providence, RI, 2004) 15. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part I. Mathematical Surveys and Monographs, vol. 135 (American Mathematical Society, Providence, RI, 2007). Geometric Aspects 16. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part II. Mathematical Surveys and Monographs, vol. 144 (American Mathematical Society, Providence, RI, 2008). Analytic Aspects 17. A. Fraser, Fundamental groups of manifolds with positive isotropic curvature. Ann. Math. (2) 158(1), 345–354 (2003) 18. A. Fraser, J. Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups. Duke Math. J. 133(2), 325–334 (2006) 19. S. Gadgil, H. Seshadri, On the topology of manifolds with positive isotropic curvature. Proc. Am. Math. Soc. 137(5), 1807–1811 (2009) 20. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Universitext, 3rd edn. (Springer, Berlin, 2004) 21. R. Hamilton, Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

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22. R. Hamilton, Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986) 23. R. Hamilton, A compactness property for solutions of the Ricci flow. Am. J. Math. 117(3), 545–572 (1995) 24. R. Hamilton, The formation of singularities in the Ricci flow, in Surveys in Differential Geometry, vol. II (International Press, Cambridge, MA, 1995), pp. 7–136 25. R. Hamilton, Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 1, 1–92 (1997) 26. L. Hernández, Kähler manifolds and 1=4-pinching. Duke Math. J. 62(3), 601–611 (1991) 27. G. Huisken, Ricci deformation of the metric on a Riemannian manifold. J. Diff. Geom. 21(1), 47–62 (1985) 28. W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv. 35, 47–54 (1961) 29. C. Margerin, A sharp characterization of the smooth 4-sphere in curvature terms. Commun. Anal. Geom. 6(1), 21–65 (1998) 30. M. Micallef, J. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. Math. 127, 199–227 (1988) 31. M. Micallef, McK. Wang, Metrics with nonnegative isotropic curvature. Duke Math. J. 72(3), 649–672 (1993) 32. H. Nguyen, Isotropic curvature and the Ricci flow. Int. Math. Res. Not. IMRN 2010(3), 536– 558 (2010) 33. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, in Geometric Measure Theory and the Calculus of Variations (Arcata, Calif., 1984). Proceedings of Symposia in Pure Mathematics, vol. 44 (American Mathematical Society, Providence, RI, 1986) 34. G. Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159, November 2002 35. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. arXiv:math.DG/0307245, July 2003 36. G. Perelman, Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109, March 2003 37. P. Petersen, T. Tao, Classification of almost quarter-pinched manifolds. Proc. Am. Math. Soc. 137(7), 2437–2440 (2009) 38. H.E. Rauch, A contribution to differential geometry in the large. Ann. Math. (2) 54, 38–55 (1951) 39. H. Seshadri, Manifolds with nonnegative isotropic curvature. Commun. Anal. Geom. 17(4), 621–635 (2009) 40. S.-T. Yau, F. Zheng, Negatively 14 -pinched Riemannian metric on a compact Kähler manifold. Invent. Math. 103(3), 527–535 (1991)

Chapter 2

Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture Michel Boileau

Abstract These notes are intended to be an introduction to the geometrisation of 3-manifold. The goal is not to give detailed proofs of the results presented here, but mainly to emphasize geometric properties of 3-manifolds and to illustrate some basic ideas or methods underlying Perelman’s proof of the geometrisation conjecture. The material is largely based on the monographs (Bessière et al., EMS Tracts Math 13, 2010) and (Boileau et al., Monographie, Panorama et Synthèse 15:167 pp, 2003). The author wants to thank the organizers of the CIME Summer School in Cetraro 2010 for their patience whilst these notes were completed.

2.1 The 3-Dimensional Geometries Basic references for this section are the articles [14, 82], the books [3, 78, 95] and the monograph [12]. A Riemannian manifold X is homogeneous if its isometry group Isom.X/ acts transitively. It is unimodular if it has a quotient of finite volume. We call geometry a homogeneous, simply-connected, unimodular Riemannian manifold, and say that a manifold is geometric if it is diffeomorphic to the quotient of a geometry by a discrete subgroup of its isometry group. Let X be a geometry. If  is a discrete subgroup of Isom.X/ acting freely, then the quotient space X= is a smooth manifold with a natural Riemannian metric which is locally isometric to X: we say that it admits a X-geometry. The notion of Geometry appeared in B. Riemann’s habilitation in 1854, where he introduced the notion of manifolds with constant curvature and stated the classification of the corresponding geometries. In 1891 W. Killing raised the problem of finding all the closed n-dimensional Riemannian manifolds with constant sectional curvature and showed that such spaces can be obtained as the quotient of the round n-sphere Sn , the Euclidean n-space En , or the hyperbolic n-space H2 by a free and cocompact action of a discrete group of isometries.

M. Boileau () Institut de Mathématiques de Marseille, Aix-Marseille Université, CNRS, Central Marseille, Technopôle Château–Gombert, Rue F. Joliot Curie 39, 13453 Marseille Cedex 13, France e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Benedetti, C. Mantegazza (eds.), Ricci Flow and Geometric Applications, Lecture Notes in Mathematics 2166, DOI 10.1007/978-3-319-42351-7_2

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The general notion of X-geometry has been introduced in 1931 by O. Veblen and J.H.C. Whitehead, after E. Cartan’s article on locally homogeneous Riemannian manifolds.

2.1.1 2-Dimensional Geometries In dimension 2 the situation is rather special since a geometry is always isotropic, hence has a constant curvature. So there are only three possible models (after rescaling): S2 ; E2 ; H2 with corresponding constant sectional curvature K  1, 0, 1. Theorem 2.1 (Riemann Uniformization Theorem) Any compact surface admits one and only one of these three geometries. The fact that a surface belongs to a unique geometric type follows from the Gauss-Bonnet formula Z K ds: .M/ D M

One cannot hope a similar result in dimension 3. The product manifold S1  S cannot carry a Riemannian metric with constant curvature since its universal covering S2  R has two ends while S3 , E3 and H3 have only one end. 2

2.1.2 3-Dimensional Geometries For a geometry X, let Isom.X/0  Isom.X/ be the component of the identity. The classification is up to equivalence, where X  X 0 if and only if there is a diffeomorphism W X ! X 0 conjugating the actions of the groups Isom.X/0 and Isom.X 0 /0 . In the 70s W. Thurston observed that, up to equivalence, there are only eight 3-dimensional geometries which are maximal in the sense that there is no Isom.X/-invariant Riemannian metric on X whose isometry group is strictly larger than Isom.X/: Theorem 2.2 (Classification of 3-Dimensional Geometries) Up to equivalence there are exactly eight maximal geometries X in dimension 3: • Three isotropic geometries of constant curvature S3 , E3 , and H3 ;

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• Four anisotropic geometries with isotropy subgroup SO.2/, namely: S2 E1 ; H2  Q E1 ; Nil and SL.2; R/. Each of these geometries has a natural fibration by geodesic lines or circles over S2 ; E2 or H2 • The geometry Sol with trivial isotropy subgroup, based on the only simplyconnected 3-dimensional Lie group which is solvable but not nilpotent. According to major results of Perelman [34–36] and Agol [1], it is possible now to give a precise description of geometric 3-manifolds from a topological point of view, up to passing to a finite cover. 2.1.2.1 The Spherical Geometry S3 Spherical 3-manifolds include S3 and the Lens spaces S3 =Zn . All spherical 3manifolds are closed orientable and finitely covered by S3 . Moreover, due to the proof of the geometrisation conjecture by Grigori Perelman in 2002 (see Sect. 2.3.2, Theorem 2.25), the following equivalent statements holds for a spherical 3-manifold: Theorem 2.3 (Perelman [71]) The following properties are equivalent for a closed 3-manifold M: (i) M is spherical (i.e. carries a complete Riemannian metric of constant sectional curvature equal to C1). (ii) M is finitely covered by the 3-sphere S3 . (iii) 1 .M/ is finite. The classification of Spherical 3-manifolds started with H. Hopf in 1926, and has been completed by H. Seifert and W. Threlfall in 1932 [90], see also [67, 82, 88]. 2.1.2.2 The Euclidean Geometry E3 From Bieberbach’s theorem, each closed Euclidean 3-manifold is the quotient of the 3-torus T 3 D S1  S1  S1 by a finite group of isometries (see for example [78]). Theorem 2.4 The following properties are equivalent for a closed 3-manifold M: (i) M is Euclidean (i.e. carries a complete Riemannian metric of constant sectional curvature equal to 0) (ii) M is finitely covered by the 3-torus T 3 . (iii) 1 .M/ contains an Abelian subgroup of rank 3. Such a subgroup is of finite index. W. Nowaski classified Euclidean 3-manifolds in 1934 by checking among the list of the 17 3-dimensional crystallographic groups the ones acting freely on the Euclidean space E3 (i.e. without torsion). In this case, up to conjugacy, the crystallographic groups preserves a direction in E3 (see for example [82]) and thus

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the Euclidean manifold carries a Seifert fibration. A more conceptual proof has been given independently in 1935 by W. Hantzsche and H. Wendt. There are only six Euclidean orientable closed 3-manifolds. They correspond to torus bundles over S1 with periodic monodromies. 2.1.2.3 The Hyperbolic Geometry H3 In 1977 W. Thurston proved a topological criterion for the existence of hyperbolic structures on a large class of compact 3-manifolds (the hyperbolisation theorem for Haken 3-manifolds, see Sect. 2.3.2, Theorem 2.24) and conjectured that this criterion holds true in general. This conjecture has been proved by Perelman in 2002, as part of his proof of the geometrisation conjecture (see Sect. 2.3.2, Theorem 2.25). Here is a special case of Thurston’s hyperbolisation theorem (see [70, 94]) which appears to be generic after passing to some finite cover by I. Agol’s work [1]. Definition 2.5 Let F be a closed orientable surface of genus g.F/ > 1, a diffeomorphism W F ! F is pseudo-Anosov if for every element  2 1 F n f1g and every integer n n . / is not conjugated to  , where  is the induced map on 1 . Theorem 2.6 (Hyperbolisation of Surface Bundles) A mapping torus .F; / WD F  Œ0; 1=f.x; 0/  . .x/; 1/g has a complete hyperbolic structure of finite volume if and only if W F ! F is a pseudo-Anosov diffeomorphism. A recent result by Agol [1] answered a deep conjecture of Thurston by showing that any closed hyperbolic 3-manifold is virtually a mapping torus. This result combines with Perelman’s hyperbolisation theorem to give the following characterization of closed hyperbolic 3-manifolds: Theorem 2.7 (Perelman, Agol) The following properties are equivalent for a closed 3-manifold M: (i) M is hyperbolic. (ii) 1 .M/ has one end and does not contain a subgroup isomorphic to Z2 . (iii) M is finitely covered by a pseudo-Anosov mapping torus. Poincaré’s theorem allows to build examples of hyperbolic 3-manifolds by identifying the faces of a convex polyhedron of the hyperbolic space H3 . In 1912 Gieseking constructed the first example of a complete 3-dimensional hyperbolic manifold with finite volume by identifying the faces of a ideal regular geodesic tetrahedron. This gives a non-orientable 3-manifold whose orientable twofold cover is the complement of the figure eight knot. The hyperbolic structure on the figure eight knot complement has been rediscovered in a different way by R. Riley in 1975. It is only in 1931 that Löbell [55] constructed the first example of a closed hyperbolic 3-manifold by gluing along their faces eight copies of a convex

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polyhedron with 14 faces and right diedral angles. In 1933 Seifert and Weber [99] built their hyperbolic dodecahedral space by identifying the faces of a regular convex dodecahedron with diedral angles 2=5. Other examples comes out from arithmetic geometry (see [56]). 2.1.2.4 The Product Geometry S2  E1 There are only two orientable examples: S2  S1 and RP3 ] RP3 . Proposition 2.8 The following properties are equivalent for a closed 3-manifold M: (i) (ii) (iii) (iv)

M carries a product geometry S2  E1 . M is finitely covered by S2  S1 . 1 .M/ contains a finite index subgroup isomorphic to Z. 1 .M/ has two ends

2.1.2.5 The Product Geometry H2  E1 A 3-manifolds carrying this geometry is finitely covered by a trivial circle bundle over a hyperbolic surface. More precisely: Proposition 2.9 The following properties are equivalent for a closed 3manifold M: (i) M carries a product geometry H2  E1 . (ii) M is finitely covered by F 2  S1 , with .F 2 / < 0. (iii) 1 .M/ is virtually a non Abelian product. An orientable 3-manifolds carrying this geometry is the mapping torus of a periodic homeomorphism of a hyperbolic surface.

2.1.2.6 The Nilpotent Geometry Nil Nil is the nilpotent Lie group of dimension 3 (Heisenberg Matrix group) 0 1x f@0 1 00

1 z yA W x; y; z 2 Rg 1

One has the extension: R ! Nil ! R2

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A nilpotent 3-manifold is closed, orientable and finitely covered by a non trivial circle bundle over a torus. More precisely [82]: Proposition 2.10 The following properties are equivalent for a closed 3manifold M: (i) (ii) (iii) (iv)

M carries the geometry Nil M is finitely covered by a S1 -bundle over T 2 with non-zero Euler class. M is finitely covered by a torus bundle with a parabolic monodromy. 1 .M/ is infinite, nilpotent, but not virtually Abelian.

B

2.1.2.7 The SL2 .R/ Geometry

B

On can identify PSL2 .R/ with the unit tangent bundle T1 H2 , and thus SL2 .R/ Š T1 H2 . In particular a manifold carrying the SL2 .R/ geometry is closed, orientable and finitely covered by a non-trivial circle bundle over a hyperbolic surface. More precisely [82]:

A

B

Proposition 2.11 The following properties are equivalent for a closed 3manifold M:

B

(i) M carries the SL2 .R/ geometry. (ii) M is finitely covered by a S1 -bundle over a surface F 2 , with .F 2 / < 0 and a non-trivial Euler class. (iii) 1 .M/ contains an infinite cyclic normal subgroup and is not nilpotent, nor virtually a product. Example 2.12 Let   PSL2 .R/ be a discrete cocompact subgroup.  can act on T1 H2 as follows:  W .x; u/ 7! ..x/; dx .u//. Then T1 H2 = has a SL2 .R/ geometry.

B

2.1.2.8 The Geometry Sol Sol is the 3-dimensional solvable Lie group given by the split extension R2 ! Sol ! R; given by: t:.x; y/ D

   t x e 0 0 et y

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Proposition 2.13 The following properties are equivalent for a closed 3-manifold M [82]: (i) M carries the Sol geometry. (ii) M is finitely covered by a torus bundles, with an Anosov monodromy W T 2 ! T 2 , that is conjugated to    0 ; > 1 0 1= (iii) 1 .M/ is infinite, solvable but not nilpotent. Remark 2.14 It follows from the descriptions above that except for hyperbolic geometry, all other compact geometric orientable 3-manifolds admit a foliation by circles or by tori. The Tits alternative holds for the fundamental group of a geometric 3-manifold: either it contains a free group (i.e. is modelled on the geometry H3 , SL2 .R/ or H2  E1 ), or it is virtually solvable. The fact that the Tits alternative holds for the fundamental group of every compact 3-manifolds is a consequence of Perelman’s proof of the geometrisation conjecture.

B

2.2 Seifert 3-Manifolds The notion of Seifert fibration generalizes the notion of circle bundle [85], see also [46, 62, 67, 82, 89]. This notion appeared in the classification of spherical 3-manifolds: every finite subgroup of SO(4) acting freely and orthogonally on S3 , commutes with the action of a subgroup SO.2/ in SO.4/. This SO.2/-action induces a circle foliation on the manifold, such that each leaf has a saturated tubular neighborhood. This phenomenon led Seifert to study in 1933 the 3-manifolds carrying such generalized fibrations by circles. which are called Seifert fibrations and play an important role in 3-manifold topology. Definition 2.15 A Seifert fibration on a compact orientable 3-manifold M is a partition by circles (called fibers), such that each circle admits a foliated tubular neighborhood. If such a foliation exists, M is called a Seifert (fibered) 3-manifold. One can show that a compact orientable 3-manifold is a Seifert 3-manifold if and only if it admits a geometry modelled on one of the following six models: S3 , E3 , Nil, H2  E1 , SL2 .R/ or S2  E1 , see for example [12, Chap. 3]. The fibration of a saturated tubular neighborhood of a fiber f may not be the product one. Define T.˛; ˇ/ to be the quotient of the solid torus T.1; 0/ D D2  S1 (with the product fibration) by the finite cyclic group of order ˛ generated by the

B

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following transformation: D2  S 1 ! D2  S 1 .z; x/ 7! .e2iˇ=˛ z; e2i˛ x/: We choose the normalisation ˛  1. The integer ˛ can be interpreted as card.@D2 t fiber/. The fiber f is said exceptional if its saturated tubular neighborhood is fiber preserving diffeomorphic to T.˛; ˇ/ with ˛ > 1. The fraction ˇ=˛ 2 .Q=Z/? is called the type of the exceptional fiber. Fact 1 If M is compact and Seifert fibered, there are only finitely many exceptional fibers. Fact 2 If M is not covered by S3 , the base of the fibration (the leaf space of the foliation) is a 2-dimensional geometric orbifold whose geometry is modelled on S2 , E2 or H2 . Fact 3 Fact 2 is not true for spherical 3-manifolds. The only exceptions are S3 and the Lens spaces L. p; q/ which have a Seifert fibration with basis a “bad” orbifold, i.e. a 2-sphere with exactly one cone point (a teardrop) or the 2-sphere with two cone points of different order (a spindle) (see Scott [82, p. 425]). Fact 4 Let M be a compact orientable Seifert fibered 3-manifold. One can associate to the Seifert fibration of M a set of invariants which determines M up to homeomorphism by [68] (a) The genus g of the underlying space of the basis (with g  0 if the basis is orientable, and g if the basis is non-orientable) (b) the rational Euler class e0 2 Q, (c) the invariant of the exceptional fibers ˇ1 =˛1 ; : : : ; ˇn =˛n with ˇi =˛i 2 .Q=Z/? , ˛i > 1 The rational Euler class e0 is an obstruction for the existence of an essential horizontal surface in M (i.e. a ramified section to the projection on the basis). Here is a definition of e0 : Let f1 ; : : : ; fr be the singular fibers. Pull out regular saturated neighborhoods Ti of these fibers and one saturated solid torus T0 around a regular fiber f0 . The manifold M n int.T0 [ : : : [ Tr / is a S1 -bundle over a surface F with boundary. It is a product fibration because the Euler class vanishes. Choose sections s0 ; s1 ; : : : ; sr to the Seifert fibration on each torus @Ti , i D 0; : : : r. Let e 2 Z be the obstruction to find a horizontal surface in F  S1 with boundary the chosen sections s0 ; s1 ; : : : ; sr . It is clear that e depends on the choice of the sections (hence depends on ˛ and ˇ), but e0 , as defined below, is well defined: e0 WD e 

r X iD1

ˇi =˛i

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The following characterization of Seifert manifolds has been first obtained for aspherical 3-manifold, as a combination of works of Casson and Jungreis [17], Gabai[29], Mess [60] (see also [57, 58]), Scott [81, 83], Tukia [96] and [98]. It follows from Perelman’s proof of the geometrisation conjecture, see Sect. 2.6.2, Theorem 2.43. Theorem 2.16 A compact, orientable 3-manifold M admits a Seifert fibration if and only if 1 .M/ is finite or admits an infinite cyclic normal subgroup.

2.3 The Geometric Decomposition It is easy to see that not every 3-manifold is geometric. However, it was W. Thurston’s groundbreaking conjecture that any compact 3-manifold should be decomposable along a finite collection of disjoint embedded surfaces into canonical geometric pieces. Conjecture 2.17 (Thurston’s Geometrisation Conjecture [92]) The interior of any compact, orientable 3-manifold splits along a finite collection of essential, pairwise disjoint, embedded spheres and tori into a canonical collection of geometric 3manifolds after capping off all boundary spheres by 3-balls. Such a decomposition will be called a geometric decomposition. A closed, connected, orientable surface F in a compact orientable 3-manifold M is essential if its fundamental group 1 .F/ injects in 1 .M/ and F does not bound a 3-ball nor cobound a product region in M with a connected component of @M. The Geometrisation Conjecture reduces many problem on 3-manifolds to combination theorems and understanding the case of geometric manifolds. In particular it implies that the homeomorphism problem for closed orientable 3-manifolds is solvable, see [84].

2.3.1 Canonical Decomposition of a 3-Manifold In this section we present the topological splitting underlying Thurston’s Geometrisation Conjecture: it is a splitting along spheres and tori into canonical pieces, which is a central result in the topology of 3-manifolds. The main references are [43, 46– 48, 66], see also [12]. An orientable 3-manifold M is irreducible if any embedding of the 2-sphere into M extends to an embedding of the 3-ball into M. The connected sum of two orientable 3-manifolds is the orientable 3-manifold obtained by pulling out the interior of a 3-ball in each manifold and gluing the remaining parts together along the boundary spheres.

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The first stage of the decomposition is due to Kneser [53], see also Milnor for the uniqueness [61]: Theorem 2.18 (Kneser’s Decomposition) Every compact, orientable 3-manifold is a connected sum of 3-manifolds that are either homeomorphic to S1  S2 or irreducible. Moreover, the connected summands are unique up to ordering and orientation-preserving homeomorphism. This result reduces the proof of the Geometrisation Conjecture to the case of irreducible manifolds, since each reducible summand S1  S2 is geometric. The second stage of the decomposition involves some specific embedded, essential tori in the 3-manifold. A compact orientable 3-manifold M is atoroidal if 1 .M/ is not virtually Abelian and any subgroup Z ˚ Z  1 .M/ is peripheral (i.e. conjugated to some 1 .P/, where P  @M is a connected component). The manifold M is said topologically atoroidal if it contains no essential embedded torus and is not homeomorphic to a product T 2  Œ0; 1 or a twisted IQ 1 over a Klein bottle. bundle K 2 Œ0; An atoroidal manifold is topologically atoroidal, while the converse is not true: there are topologically atoroidal Seifert manifolds which admit finite regular coverings containing essential tori, and thus which are not atoroidal. Example 2.19 Let   PSL2 .R/ be a triangle group, i.e.  WD T. p; q; r/ WD p q h1 ; 2 ; 3 j 1 D 2 D .1 2 /r D 1i. The quotient T1 H2 = is a geometric SL2.R/manifold which is topologically atoroidal but not atoroidal. It is finitely covered by a quotient T1 H2 = 0 , where  0   is a torsion–free subgroup of finite index. This last quotient is a S1 -bundle over a surface of genus  2, which always contains essential tori.

B

This is the only possibility for an irreducible 3-manifold thank to the following consequence of Theorem 2.16. Corollary 2.20 (Torus Theorem) A compact orientable irreducible 3-manifold is either Seifert fibered or atoroidal. A compact, orientable 3-manifold is Haken if it is irreducible and if its boundary is not empty, or if it contains a closed, orientable, essential surface. We can now state the theorem which gives a canonical splitting of a compact orientable, irreducible 3-manifold along a finite collection of essential tori. This result is due to Jaco and Shalen [47] and independently to Johannson [48] for the case of Haken 3-manifolds. The Torus theorem (Corollary 2.20) allows to extend the torus decomposition to any irreducible orientable 3-manifold. The following useful notion has been introduced by Neumann and Swarup [66]: an embedded torus in a compact, orientable, irreducible 3-manifold M is canonical if it is essential and can be isotoped off any incompressible embedded torus. Theorem 2.21 (JSJ-Splitting) A maximal (possibly empty) collection of disjoint, non-parallel, canonical tori in a compact, orientable, irreducible 3-manifold M is

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finite and unique up to isotopy. It splits M into 3-submanifolds that are atoroidal or Seifert fibered. Remark 2.22 If M is a torus bundle over S1 , it is either an Euclidean or a Sol geometric manifold. Hence, in this case the minimal family of splitting tori is empty. A crucial step for the existence of a finite splitting along spheres and canonical tori is given by Haken-Kneser finiteness theorem [36, 37, 53], see [46]. Theorem 2.23 (Haken-Kneser Finiteness Theorem) Let M be a compact orientable 3-manifold. There is an integer h.M/ such that for any family of n disjoint embedded essential surfaces in M either n < h.M/ or at least two surfaces are parallel. Haken-Kneser finiteness theorem is a fundamental result for finiteness properties of compact 3-manifolds.

2.3.2 The Geometrisation Conjecture The previous Sect. 2.3.1 shows that the interior of any compact, orientable 3manifold can be split along a finite collection of essential, pairwise disjoint, embedded 2-spheres and tori into a canonical collection of Seifert fibered, Solv or irreducible atoroidal 3-manifolds, after capping off all boundary spheres by 3-balls. The Seifert fibered and Solv pieces carry a geometric structure. So the content of Thurston’s Geometrisation Conjecture is that the same holds for the interior of compact, irreducible and atoroidal pieces, that is to say that they carry geometric structures modeled on S3 or H3 . In the mid 70s W. Thurston proved his Geometrisation Conjecture for Haken 3manifolds [92–94], see McMullen [59], Otal [69, 70], Kapovich [49], and also [8] for a survey. Theorem 2.24 (Thurston’s Hyperbolisation Theorem) The interior of a Haken 3-manifold M carries a hyperbolic structure if and only if M is atoroidal. The general case has been solved by Perelman at the beginning of 2000s [71– 73], see Kleiner and Lott [51, 52], Morgan and Tian [63, 64], Cao and Zhu [16], and also [5]. Theorem 2.25 (G. Perelman) Let M be a closed, orientable irreducible 3manifold. Then: (i) M is spherical if and only if 1 M is finite. (ii) M is hyperbolic if and only if 1 M is infinite and M is atoroidal. Theorems 2.24 and 2.25 together imply: Corollary 2.26 (Geometrisation Theorem) The Geometrisation Conjecture is true for all compact orientable 3-manifolds.

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Perelman’s proof deals in fact with all cases and allows to recover the geometric splitting of the manifold along spheres and tori. The main ingredient of the proof is the Ricci flow, which is an evolution equation introduced by R. Hamilton. In the next Sect. 2.4, we review general facts about this equation, and construct an object called Ricci flow with bubbling-off, defined in [5] and which is a variation of Perelman’s Ricci flow with surgery.

2.4 Ricci Flow with Surgery Let M be a closed, orientable, irreducible 3-manifold. If g is a Riemannian metric on M, we denote by Rmin .g/ the minimum of its scalar curvature, by Ricg its Ricci tensor, and by vol.g/ its volume.

2.4.1 Ricci Flow In early 1980, R. Hamilton proposed to study solutions of the following evolution equation, called Ricci flow equation: dg D 2 Ricg.t/ ; dt

(2.1)

A solution is an evolving metric fg.t/gt2I , i.e. a 1-parameter family of Riemannian metrics on M defined on an interval I  R, see [24]. R. Hamilton proved in [38] the following fundamental results: (1) The existence and uniqueness of a short time solution: for any metric g0 on M, there exists " > 0 such that Eq. (2.1) has a unique solution defined on Œ0; "/ with initial condition g.0/ D g0 (2) The solution can be extended as long as the norm of the curvature tensor does not blow up at some point: there exists T 2 .0; C1 such that the solution to (2.1) with initial condition g0 is defined on the maximal interval Œ0; T/. In general the metrics tends to become more homogeneous along the flow, but the curvature can blow up in finite time T at some points of the manifold. One says that the flow has a singularity at time T. A first goal is to understand the geometry of M when t tends to T. Using the Ricci flow, Hamilton [38–40, 42] obtained some results towards the Poincaré Conjecture and the Geometrisation Conjecture: Theorem 2.27 ([38]) On a closed, orientable 3-manifold M with initial Riemannian metric g0 of positive Ricci curvature, the volume-rescaled Ricci flow vol.g.t//2=3 g.t/ converges (modulo diffeomorphisms) to a metric of positive

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constant sectional curvature as t tends to a finite time T < 1. In particular, M is spherical. Theorem 2.28 ([42]) Let M be a closed, orientable 3-manifold and g.t/ be a Ricci flow on M defined on Œ0; C1/. If the sectional curvature of the time-rescaled flow gQ .t/ WD .4t/1 g.t/ is bounded independently from t, then one of the following assertions holds: 1. As t tends to infinity, gQ . / converges (modulo diffeomorphisms) to a hyperbolic metric. 2. As t tends to infinity, gQ . / collapses with bounded sectional curvature. 3. M contains an incompressible torus. In the general case, however, it may happen that T < 1 while the behavior of g.t/ as t tends to T does not allow to determine the topology of M.

2.4.2 Ricci Flow with Surgery The heart of Perelman’s proof is the construction of a “discontinuous piecewise smooth solution” of the Ricci flow equation called Ricci flow with surgery, we refer to [51] and [63] for a detailed exposition of Perelman’s construction. Here we briefly describe the construction of a variant of the Ricci flow with surgery, called the Ricci flow with bubbling-off, we refer to [5] for the details.The most significant difference between Ricci flow with bubbling-off and Perelman’s Ricci flow with surgery is that surgery occurs before the Ricci flow becomes singular, rather than at the singular time. The construction of Ricci flow with bubbling-off is in this respect closer to the surgery process envisioned by Hamilton [41]. From now on we suppose that M is a closed, orientable and irreducible 3-manifold which is not RP3 . In particular M does not contain any surface diffeomorphic to RP2 . The appearance of singularities is one of the main difficulties in the Ricci flow approach to geometrisation. Maximum principle arguments show that singularities in a 3-dimensional Ricci flow only occur when the scalar curvature tends to C1 somewhere (see e.g. [42]). One of Perelman’s major breakthroughs was to give a precise description of the geometry of neighborhoods at points of large scalar curvature: he showed that such a point has a so-called canonical neighborhood. Such a canonical neighborhood can be: 1. The whole manifold M, which is thus spherical. As t ! T, .M; g.t// shrinks to a point and the volume-rescaled metric vol.g.t//2=3 g.t/ converges, up to diffeomorphism, to a metric of positive constant sectional curvature. This is the case of Hamilton’s Theorem 2.27, when one starts with a metric with Ricci curvature > 0. 2. An "-neck, which is almost homothetic to the product of the round 2-sphere of unit radius with an interval of length 2"1 . 3. An "-cap, which is a 3-ball such that a collar neighborhood of @U is an "-neck.

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Let fix a large number , which plays the role of a curvature threshold. As long as the maximum of the scalar curvature Rmax .g.t/ < , the Ricci flow is defined. If it reaches  at some time t0 , then: Either the minimum Rmin .g.t0 / of the scalar curvature at the time-t0 is large enough so that every point has a canonical neighborhood, and M is spherical by the next Lemma 2.29. Lemma 2.29 A closed, orientable, irreducible Riemannian 3-manifold .M; g/ such that every point is the center of a "-neck or a "-cap is diffeomorphic to S3 . Otherwise one can modify the metric g.t0 / so that the maximum of the scalar curvature of the new metric, denoted by gC .t0 /, is at most =2. This modification, called metric surgery, consists in replacing the metric in some 3-balls containing regions of high curvature by a standard type of "-caps (see [5, Chap. II] fore the details). Since M is irreducible this surgery does not change the topology of M. Then one starts again the Ricci flow with initial metric gC .t0 /. Estimates on the Ricci flow show that it takes a certain amount of time for the maximum of the scalar curvature to double its value. Hence once  is fixed, the surgery times cannot accumulate, and the procedure can be repeated as many times as necessary. One main technical issue is to choose  in order to perform metric surgeries and to iterate this construction. To do so, one fixes two parameters r; ı > 0 such that a point x of scalar curvature R.x/  r2 has a canonical neighborhood, while ı describes the precision of surgery. The number  is then determined by the parameters r and ı. The construction gives an evolving metric fg.t/gt , which is piecewise C 1 -smooth with respect to t, and which is left-continuous and has a right-limit gC .t0 / at each singular time t0 , where the map t 7! g.t/ is not C 1 . It is called Ricci flow with .r; ı/-bubbling-off. A precise construction of a Ricci flow with .r; ı/-bubbling-off is outside the scope of this lectures, we refer to [5, Chap. II] fore the details. For these lectures, we use a simpler definition which retains the essential properties needed to understand the proof of the geometrisation conjecture. Definition 2.30 We say that an evolving metric fg.t/gt is a Ricci flow with surgery defined on a compact interval [0, T] if: 1. There is a finite subset of singular times. where the function t ! g.t/ is not continuous. 2. Equation (2.1) is satisfied at all non singular times. 3. At every singular time t0 , the function t ! g.t/ is left-continuous and has a right-limit gC .t0 / such that: (i) Rmin .gC .t0 // > Rmin .g.t0 //, and (ii) gC .t0 / 6 g.t0 /. The following results of Perelman show that the behavior of the Ricci flow with surgery depends on the topology of the 3-manifold. It is a main step for the proof of the geometrisation conjecture:

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Theorem 2.31 Let M be a closed, orientable, irreducible 3-manifold and let g0 be a Riemannian metric on M. Then: (1) Extinction If j1 .M/j < 1, there is a time T0 > 0 such that any Ricci flow with surgery on M, with initial condition g.0/ D g0 , cannot be defined for time T  T0 . (2) Long time existence If j1 .M/j D 1, there exists a Ricci flow with surgery on M, with initial condition g.0/ D g0 , which is defined on Œ0; C1/. The first statement corresponds to the case where the Ricci flow with surgery becomes extinct in a finite time (cf. Sect. 2.5). It means that the minimum of the scalar curvature Rmin become large enough at every point of the manifold M in a finite time. Therefore every point has a canonical neighborhood, and thus M is spherical by Lemma 2.29. The second statement corresponds to the case where M is aspherical, which means that the higher homotopy groups 2 .M/ and 3 .M/ vanish. However in order to conclude the proof of the hyperbolisation conjecture, some further work is still needed. (see Sect. 2.6).

2.5 Elliptisation In this section we sketch the proof of the extinction case of Theorem 2.31, which with Lemma 2.29 implies Theorem 2.25(i). The argument is based on a result of Colding and Minicozzi [25, 26]. Proof (Outline of the Proof) Let M be a closed, orientable, irreducible 3-manifold with finite fundamental group, and g0 be a Riemannian metric on M. Since 1 M Q of M is compact, hence H3 .M; Q Z/ Š Z. Since M is is finite, the universal cover M Q D 0. Therefore 1 .M/ Q D H1 .M; Q Z/ D 0 and 2 .M/ Q D irreducible, 2 .M/ D 2 .M/ Q Z/ D 0. By Hurewicz theorem, 3 .M/ D 3 .M/ Q D H3 .M; Q Z/ Š Z. H2 .M; Let ˝ be the space of smooth maps f W S2  Œ0; 1 ! M such that f .S2  f0g/ and f .S2 f1g/ are points. Since 3 .M/ Š Z, there exists f0 2 ˝ which is not homotopic to a constant map. Let  be the homotopy class of f0 and set W.g/ WD inf max E. f . ; s//; f 2 s2Œ0;1

where E denotes the energy 1 E. f . ; s// D 2

Z S2

jrx f .x; s/j2 dx:

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M. Boileau

Let fg.t/gt2Œ0;T be a Ricci flow with surgery such that g.0/ D g0 . At regular times the function t 7! Rmin .t/ is continuous and satisfies: 2 dC Rmin  R2min .t/; dt 3

(2.2)

C

where ddt denotes the limsup of forward difference quotients. If t0 is a singular time, properties 3 and 4 in the definition of a Ricci flow with surgery imply that the left limit of Rmin at t0 remains smaller or equal to its right limit. So Rmin .t/ satisfies the following inequality: Rmin .t/ 

Rmin .0/ : 1  2tRmin .0/=3

(2.3)

At regular times, the function t 7! W.g.t// is continuous, and since  is nontrivial, by Colding and Minicozzi [25, 26] it satisfies 1 dC W.g.t//  4  Rmin .t/W.g.t//; dt 2 Rmin .0/  4 C 4tR .0/ W.g.t//: min  2 3

(2.4) (2.5)

If t0 is a singular time, it follows again, from the definition of a Ricci flow with surgery, that the left limit of W.g.t// at t0 remains greater or equal to its right limit. Since W.g.t// cannot become negative, this implies an upper bound on T depending only on Rmin .0/ and W.g0 /, and completes this outline of proof of the extinction case of Theorem 2.31. Here is a useful consequence of the proof: Corollary 2.32 Let M be a closed, orientable, irreducible 3-manifold. If M has a metric of positive scalar curvature, then M is spherical.

2.6 Aspherical 3-Manifolds Let M be a closed, orientable, aspherical 3-manifold (i.e. irreducible with infinite fundamental group). The goal of this section is to geometrise these manifolds (part (ii) of Theorem 2.25). Namely we will prove: Theorem 2.33 Let Mbe a closed, orientable, aspherical 3-manifold. Then M is hyperbolic or Seifert fibered or contains an incompressible torus. In particular M admits a geometric decomposition.

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37

2.6.1 Long Time Behavior of the Ricci Flow with Surgery Let M be a closed orientable and aspherical 3-manifold. Since M has infinite fundamental group, it is not spherical, hence does not carry any metric with positive scalar curvature by Corollary 2.32. Moreover, in the following all hyperbolic 3manifolds will be complete and of finite volume. For any metric g on M we consider the two following scale invariant quantities: O R.g/ WD Rmin .g/ vol.g/2=3 6 0; and O V.g/ D



Rmin .g/ 6

3=2

3=2 O vol.g/ D 63=2 .R.g// > 0:

Those quantities are monotonic along the Ricci flow on a closed manifold, as long as Rmin remain non positive and this remains true in presence of surgeries by conditions (3) and (4) of Definition 2.30. Proposition 2.34 Let fg.t/g be a Ricci flow with surgeries on M. Then: O (i) The function t 7! R.g.t// is non decreasing; O (ii) The function t 7! V.g.t// is non increasing. For a Ricci flow, a consequence of the maximum principle is that the scalar curvature R.x; t/ and the volume V.t/ satisfy : dV D dt

Z R dV

(2.6)

dR 2 D R C 2j Ric0 j2 C R2 dt 3

(2.7)

M

Then the following lemma implies Proposition 2.34 Lemma 2.35 dV 6 Rmin V; dt dRmin 2 > R2min ; dt 3 Z 2 O 1 d RO > RV .Rmin  R/ dV: dt 3 M

(2.8) (2.9) (2.10)

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M. Boileau

Proof The following calculation shows the inequality (2.8): Z d d V.t/ D dvg.t/ dt dt M Z 1 dg trg . / dvg.t/ D 2 dt Z D R dv 6 Rmin V.t/ The inequality (2.9) follows from (2.7) and the maximum principle. Finally we prove the last inequality (2.10): 2 d O 2 R D R0min V 3 C Rmin dt 3

Z

1

R dvV  3

Z 2 1 2 2 2 3 > Rmin V C Rmin R dvV  3 3 3   Z Z 2 2 D Rmin V 3 Rmin V 1 dv C R dvV 1 3 M Z  2 2 D Rmin V 3 V 1 .Rmin  R/ dv 3 M

We will use the following standard notion of convergence. For more general notions of convergence of Riemannian manifolds we refer to the Appendix. Definition 2.36 A sequence of pointed Riemannian manifolds .Mn ; gn ; xn / converges to a pointed Riemannian manifold .M1 ; g1 ; x1 / in the C k -topology if there exists a sequence of numbers "n > 0 tending to zero, and a sequence of C k -diffeomorphisms 'n from the metric ball B.x1 ; "1 n /  M1 to the metric ball  k B.xn ; "1 /  M such that ' .g /  g has C -norm less then "n everywhere. We n n 1 n n say that the sequence subconverges if it has a convergent subsequence. According to Perelman, we introduce the following definitions: Definition 2.37 Let .M; g/ be a Riemannian manifold and " > 0 a real number. A point x 2 M is "-thin with respect to g if there exists a radius  2 .0; 1 such that the ball B.x; / has the following two properties: 1. All sectional curvatures on this ball are bounded below by 2 , and 2. The volume of this ball is less than "3 . Otherwise, x is said "-thick with respect to g.

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When g is a Riemannian metric with bounded sectional curvature, then Definition 2.37 is equivalent to the one given in the Appendix using injectivity radius. We state now Perelman’s result on the long time behavior of the Ricci flow with surgery (cf. [36, Sects. 6 and 7], [51], [5, Chap. III]). Theorem 2.38 (Long Time Behavior) Let M be a closed, orientable, aspherical 3-manifold. For every Riemannian metric g0 on M, there exists a Ricci flow with surgery fg.t/gt defined on Œ0; C1/ with the following properties: (i) g.0/ D g0 . (ii) The volume of the time-rescaled metric gQ .t/ D .4t/1 g.t/ is bounded independently from t. (iii) For every " > 0 and every sequence .xn ; tn / 2 M  Œ0; C1/, if tn tends to C1 and xn is "-thick with respect to gQ .tn / for each n, then there exists a complete hyperbolic 3-manifold H with finite volume and a basepoint 2 H such that the sequence .M; gQ .tn /; xn / subconverges to .H; ghyp ; / in the C 2 -topology. (iv) For every sequence tn ! 1, the sequence gQ .tn / has local curvature bound (LCB). When t goes to C1 if there does not exist a sequence of uniformly thick basepoints, then the conclusion .iii/ is vacuous and one says that the Riemannian manifolds .M; gQ .t/ collapse, see Sect. 2.7.2. The conclusion .iv/, i.e. the local curvature bound (LCB), is a technical property introduced by Perelman, which is weaker than a global two-sided curvature bound, but sufficient to carry some limiting arguments. For a precise definition see Definition 2.57 in Sect. 2.7.2, where this notion will be used. O For a complete hyperbolic manifold H with finite volume, let V.H/ D vol H D O hyp /, and R.H/ O O hyp /, where ghyp is the hyperbolic metric. A useful V.g D R.g corollary of Theorem 2.38 is: Corollary 2.39 Let M be a closed, orientable, aspherical 3-manifold. For every Riemannian metric g0 on M, there exists an infinite sequence of Riemannian metrics g1 ; : : : ; gn ; : : : with the following properties: (i) The sequence .vol.gn //n0 is bounded. (ii) For every " > 0 and every sequence xn 2 M, if xn is "-thick with respect to gn for each n, then the sequence .M; gn ; xn / subconverges in the pointed C 2 topology to a pointed complete hyperbolic 3-manifold .H; ghyp ; / with finite O O 0 /. volume, where vol H D V.H/ 6 V.g (iii) The sequence gn has local curvature bound (LCB). Proof The existence of a sequence gn follows from Theorem 2.38, by picking any sequence tn ! C1 of non-singular times and setting gn WD gQ .tn / for n  1. Then assertions (i), (iii) and the first part of (ii) are direct consequences of Theorem 2.38. O O 0 / one uses that Rmin .ghyp / > lim inf Rmin .gn / To show the inequality V.H/ 6 V.g O O n / 6 V.g O 0 / since by and vol.H/ 6 lim inf vol.gn /. Hence V.H/ 6 lim inf V.g

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O n //n0 is non Proposition 2.34 and the scale invariance of VO the sequence .V.g increasing.

2.6.2 Geometrisation A link L in a closed 3-manifolds M is a (possibly empty, possibly disconnected) closed 1-submanifold of M. A link is hyperbolic if its complement is a hyperbolic 3-manifold. Definition 2.40 For a closed orientable, irreducible 3-manifold M, let V0 .M/ be the infimum of the volumes of all hyperbolic links in M. This quantity is well defined since any closed orientable 3-manifold contains a hyperbolic link [65]. Moreover this infimum is always realized by some hyperbolic submanifold H0 because the set of volumes of hyperbolic 3-manifolds is wellordered by Thurston-Jørgensen theory (see e.g. [32]). In particular, it is positive and M is hyperbolic if and only if H0 D M (see e.g. [4].) Let introduce another volume invariant for a closed, orientable, aspherical 3manifold: Definition 2.41 For a closed, orientable and aspherical 3-manifold M, let define n o O V.M/ WD inf V.g/ j g metric on M D inf fvol.g/ j Rmin .g/ > 6g : Remark 2.42 Since M is orientable and aspherical, it can not carry any metric with positive scalar curvature by Corollary 2.32 and thus the invariant V.M/ is welldefined. Since an aspherical Seifert fibered 3-manifold is never atoroidal, the next theorem completes the proof of the hyperbolisation case .ii/ in Perelman’s Theorem 2.25 Theorem 2.43 Let M be a closed, orientable, irreducible 3-manifold with infinite fundamental group. (a) If V.M/ > V0 .M/, then M is hyperbolic and there is equality. (b) If V.M/ < V0 .M/, then M is a Seifert fibered manifold or contains an essential torus. Proof Assertion (a) follows from Anderson [2, pp. 21–23]. We indicate an alternative proof in Sect. 2.6.3, following Salgueiro [80] (cf. also [6, Sect. 6].) O 0 / < V0 .M/, which exists by the To prove (b), take a metric g0 on M such that V.g hypothesis V.M/ < V0 .M/. Applying Corollary 2.39 with initial metric g0 , gives a sequence of Riemannian metrics gn fulfilling exactly the assumptions of the weak collapsing Theorem 2.44 below. Indeed, if H is a hyperbolic limit appearing in .2/, O O 0 / < V0 .M/, and thus the we have by Corollary 2.39 .ii/, vol H D V.H/ 6 V.g conclusion follows from Theorem 2.44.

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41

The weak collapsing theorem is the last bit of the proof of Perelman’s geometrisation theorem, which will be explained in Sect. 2.7.3: Theorem 2.44 (Weak Collapsing Theorem) Let M be a closed, orientable, aspherical 3-manifold. Suppose that there exists a sequence gn of Riemannian metrics on M satisfying: (1) The sequence vol.gn / is bounded. (2) For every " > 0 and every sequence xn 2 M, if xn is "-thick with respect to gn for each n, then the sequence .M; gn ; xn / subconverges in the pointed C 2 -topology to a pointed hyperbolic 3-manifold .H; ghyp ; /, where volH < V0 .M/. (3) The sequence gn has local curvature bound (LCB). Then M is Seifert fibered or contains an essential torus. This theorem is a strengthened version of Perelman’s collapsing theorem (see Sect. 2.7.2, Theorem 2.58). A graph 3-manifold is obtained by gluing along some boundary components finitely many elementary pieces homeomorphic to a solid torus S1  D2 or a composite space S1  {punctured disk}. Therefore a graph manifold is obtained by gluing together geometric pieces which are not hyperbolic, hence it has a geometric decomposition with no hyperbolic pieces. In particular any Seifert fibered manifold is a graph manifold. The proof of the weak collapsing theorem relies deeply on the notion of graph manifolds and in particular on the fact that an aspherical graph manifold is Seifert fibered or contains an essential torus. The two cases are not exclusive. For instance, if F is a closed, orientable surface of genus at least 1, then F  S1 is a Seifert fibered manifold and contains essential tori. If M is a 3-manifold satisfying the hypotheses of Theorem 2.43, then V.M/ is always less than or equal to V0 .M/, with equality if and only if M is hyperbolic. This result can be refined using an invariant V00 .M/, defined as the minimum of the volumes of all hyperbolic submanifolds H  M having the property that either H is the complement of a link in M or @H has at least one component which is incompressible in M. By definition, we always have 0 < V00 .M/  V0 .M/.Then we obtain the following characterization of hyperbolic and graph manifolds among closed, orientable and aspherical 3-manifolds: Corollary 2.45 Let M be a closed, orientable, aspherical 3-manifold. Then, one of the following occurs: (i) 0 < V.M/ D V00 .M/ D V0 .M/ if M is hyperbolic. (ii) 0 < V00 .M/  V.M/ < V0 .M/ if M contains an incompressible torus and has some hyperbolic pieces in the JSJ decomposition. (iii) 0 D V.M/ < V00 .M/  V0 .M/ if M is a graph manifold.

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2.6.3 A Sufficient Condition for Hyperbolicity The following proposition implies Assertion (a) of Theorem 2.43: Proposition 2.46 Let M be a closed, orientable, aspherical 3-manifold. Suppose that the inequality V.M/  V0 .M/ holds. Then equality holds, M is hyperbolic, and V.M/ is realized by the hyperbolic metric. Proof Let H0 be a hyperbolic manifold homeomorphic to the complement of a link L0 in M and whose volume realizes V0 .M/. Claim 2.47 below shows that V.M/  vol.M; g" / < vol.H0 / D V0 .M/, which contradicts the hypothesis. The link L0 is thus empty and M D H0 is hyperbolic. Claim 2.47 If L0 ¤ ;, then M carries a metric g" such that vol.M; g" / < V0 .M/ and Rmin .g" /  6. This claim can be proved by a direct construction as in [2]. We give here a different argument relying on Thurston’s hyperbolic Dehn filling theorem as in [80] and [5]. Proof Since L0 ¤ ;, we can consider the orbifold O with underlying space M, singular locus L0 and branching index n along L0 . For n > 1 sufficiently large the orbifold carries a hyperbolic structure, by the hyperbolic Dehn filling theorem [91] (cf. [3], [10, Appendix B]). Then, according to [80], see also [5], it is possible to desingularize the hyperbolic conical metric on M corresponding to the orbifold structure, by a radial deformation in a tubular neighborhood of L0 , in such way that for each " > 0 there exists a Riemannian metric g" on M with sectional curvature bounded below by 1 and such that vol.M; g" / < .1 C "/ vol.O/. More precisely, let g be the hyperbolic cone metric on M induced by the hyperbolic orbifold structure O. Let N  O be a tubular neighborhood of radius r0 > 0 around the singular locus L0 . In N the local expression of the singular metric g in cylindrical coordinates .r; ; h/ is: ds2 D dr2 C



1 sinh.r/ n

2

d 2 C cosh2 .r/dh2 ;

where r 2 .0; r0 / is the distance to L0 , h is the linear coordinate of the projection to L0 , and  2 .0; 2/ is the rescaled angle parameter. The deformation depends only on the radial parameter r and consists in replacing N with the metric g by a smaller cylinder N 0 with a smooth metric g0 of the form ds2 D dr2 C 2 .r/d 2 C

2

.r/dh2 ;

where for some ı D ı."/ > 0 sufficiently small the functions

;

W Œ0; r0  ı ! Œ0; C1/

2 Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture

43

are smooth and chosen to verify the following properties: (1) (2) (3) (4)

In a neighborhood of 0, .r/ D r and .r/ is constant. In a neighborhood of r0  ı, .r/ D 1n sinh.r C ı/ and .r/ D cosh.r C ı/. 00 .r/ 00 .r/

0 .r/ 0 .r/  1 C "; 8r 2 .0; r0  ı/;

.r/ .r/  1 C " and .r/ .r/  1 C ": As ı ! 0, vol.M; g0 / ! vol.O/.

By .1/ the new metric is Euclidean near L0 , and thus non-singular. It matches the original metric away from N by .2/ and has sectional curvature  1  " by (3). Moreover by (4) for small, vol.M; g0 /  .1 C "/ vol.O/. Then the pı sufficiently 0 rescaled metric g" D 1 C " g on M has sectional curvature  1 and volume 5 vol.M; g" /  .1 C "/ 2 vol.O/. Since vol.O/ < vol.H0 / [12, 91], for " > 0 sufficiently small we obtain a Riemannian metric on M such that vol.M; g" / < vol.H0 / D V0 .M/ and Rmin .g" /   6.

2.7 Collapsing Sequences This section is devoted to the proof of the weak collapsing Theorem 2.44. We start in Sect. 2.7.1 by quoting the notion of simplicial volume and the statement of Gromov vanishing theorem. In Sect. 2.7.2 we sketch the proof of the weak collapsing theorem, under the stronger assumption that the sequence of Riemannian metrics collapses (Theorem 2.58, see also [6]): in particular the assumption (2) of Theorem 2.44 is then vacuous, and the assumption (1) is not needed. Then in Sect. 2.7.3 we explain the proof of Theorem 2.44.

2.7.1 Simplicial Volume The purpose of this section is to review the notion of Gromov’s simplicial volume. The main references for this section are [34] and [45], see also [3]. Definition 2.48 The simplicial volume of a compact, orientable n-manifold M (possibly with boundary) is defined as: 8 9 k ˆ ˇ P   is a cycle representing a fundamental > ˆX > k < = i i ˇ ˇ kMk WD inf ji j ˇ iD1 ˆ ˇ class in Hn .M; @MI R/; where i W n ! M > ˆ > : iD1 ; is a singular simplex and i 2 R; i D 1; : : : ; k: It follows from the definition that if f W .M; @M/ ! .N; @N/ is a proper map, then kMk  jdeg. f /j kNk: Moreover, if f is a covering then the equality holds.

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Hence if a manifold admits a self-map of degree d with jdj > 1, then its simplicial volume vanishes. So, for the n-sphere and the n-torus, kSn k D kT n k D 0. The same argument shows that the simplicial volume of spherical or Euclidean manifolds is zero. A covering fUi gi by open sets of a topological space X has dimension  k if every point of X belongs to at most k C 1 open sets of the covering. The dimension of the covering is the minimum of such k and is equal to the dimension of the nerve of the covering. A connected subset U  M is amenable if the image of its fundamental group 1 .U/ ! 1 .M/ is amenable. A (possibly disconnected) subset is amenable if all its connected components are amenable. The following vanishing theorem of Gromov will be useful for the proof of Perelman’s collapsing theorem.. Theorem 2.49 (Gromov’s Vanishing Theorem) Let M be a closed n-manifold. If M admits a covering of dimension at most n  1 by amenable open sets, then kMk D 0. This theorem is proved in [34, 45] using bounded cohomology, a notion dual to simplicial volume. Here is a corollary which shows that the simplicial volume of a Seifert fibered manifold vanishes, see also [100]. Corollary 2.50 The simplicial volume of a compact, connected, orientable smooth manifold M which admits a nontrivial smooth S1 -action is zero. The following theorem gives a non-vanishing result [34, 91]. We let vn denote the volume of the regular ideal hyperbolic n-simplex. Theorem 2.51 (Gromov-Thurston) If M is a complete hyperbolic n-manifold with finite volume, then kMk D

1 vol.M/: vn

The additivity property below allows to compute the simplicial volume of a 3manifold from its geometric decomposition [34, 54, 87]: Theorem 2.52 The simplicial volume of compact, orientable, 3-manifolds is additive by connected sums and gluing along incompressible tori. Perelman geometrisation Theorem (Corollary 2.26), combined with Proposition 2.50 and Theorems 2.51, 2.52 implies the following computation: Corollary 2.53 Let M be a compact, orientable irreducible 3-manifold with vanishing Euler characteristic, then: kMk D

k 1 X vol.Hi /; v3 iD1

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45

where H1 ; : : : ; Hk are the hyperbolic pieces of the geometric decomposition of M. In particular kMk D 0 if and only if M is a graph manifold. The connection between collapsing and simplicial volume is that the simplicial volume vanishes for a manifold which admits a collapsing sequence of Riemannian metrics. The model for such a result is Gromov’s isolation theorem, see [34]. The following result can be seen as a first version of Perelman’s collapsing Theorem 2.58 with à priori a weaker result. Theorem 2.54 (Vanishing of Simplicial Volume) Let M be a closed, orientable 3-manifold. If M admits a collapsing sequence of Riemannian metrics with local curvature bound (LCB), then kMk D 0. Proof (Outline of Proof) Let M be a closed, orientable 3-manifold and gn be a sequence of Riemannian metrics on M which collapses with local curvature bound (LCB). For n large enough, because of the local curvature bound, each point x 2 M has a neighborhood Ux in .M; gn / which is close to a metric ball in some manifold of nonnegative sectional curvature, whose volume is small compared to the cube of the radius. These neighborhoods are called local models and they have virtually Abelian, hence amenable fundamental groups (see for example Sect. 2.7.2.2). Adapting the arguments of Gromov’s isolation theorem [34], one can extract from this open cover a finite cover U1 ; : : : ; Up , of dimension at most 2 and thus applies Gromov’s vanishing theorem. First using the fact that these open sets are close to small metrics balls, one obtains a covering of dimension 3. The crucial step then is to go down to dimension 2, which is possible because of the upper bound on the volumes of the Ui ’s given by the collapsing hypothesis (see Sect. 2.7.2.3). The argument outlined above is not sufficient to prove Perelman’s collapsing Theorem 2.58, since we do not know that the manifold M has a geometric decomposition to use Corollary 2.53. To overcome this difficulty we will translate the original problem into one about irreducible manifolds with non-empty boundary, where Thurston’s geometrisation theorem (Theorem 2.24) can be applied to use Corollary 2.53 since these manifolds are Haken. The next corollary is a consequence of Theorems 2.51 and 2.54. It implies Proposition 2.77 of Sect. 2.7.4 Corollary 2.55 A closed, orientable, hyperbolic 3-manifold cannot admit a collapsing sequence of Riemannian metrics with local curvature bound (LCB).

2.7.2 Collapsing Theorem Definition 2.56 Let M be a 3-manifold and gn be a sequence of Riemannian metrics on M. The sequence gn collapses if there is a sequence "n ! 0 such that for all x 2 Mn3 : there exists 0 < n .x/ < 1 satisfying:

46

M. Boileau 1 on n .x/2 vol.B.x;n .x/// n .x/3

(i) sec 

B.x; n .x//

(ii) and

 "n

The number n .x/ is the collapsing scale These are scale invariant inequalities. If we normalize by an homothety to have n .x/ D 1, then we get vol.B.x; 1//  "n . Definition 2.57 (LCB) A sequence .Mn ; gn / has local curvature bound if for n large enough the following properties hold for the metric gn : 8ı > 0 9Nr.ı/ > 0, K0 D K0 .ı/ and K1 D K1 .ı/ such that: If 0 < r < rN .ı/, vol.B.x; r// > ır3 and sec   r12 on B.x; r/ then jRmj < K0 r2 and jrRmj < K1 r3 The constants rN .ı/, K0 .ı/ and K1 .ı/ are independent of n. 1 When r ! 0, vol.B.x;r// ! 4 3 and r2 ! 1. Hence such an r always exist, but r3 r  n .x/ is small compared to the collapsing scale. The local curvature bound is a technical property, which is weaker than a global two-sided curvature bound, but suffices for some limiting arguments. The goal of this section is to explain the proof of the following collapsing theorem, stated by Perelman in [36] without proof, see [6] for a detailed proof along the lines given here, and also [52, 64, 86] for a different proof along the lines given by Perelman. Theorem 2.58 Let .Mn ; gn /n be a sequence of closed, irreducible, aspherical and orientable 3-manifolds such that: (i) gn collapses. (ii) gn has local curvature bounds (LCB). Then Mn is a graph manifold for n large. We first give a short outline of the proof of Theorem 2.58, then we give some details on each step of the proof. For a complete and detailed account we refer to [6] and [5].

2.7.2.1 Outline of the Proof of Theorem 2.58 Step 1: We find local models for the metric balls B.x; x / with the following properties: 1. Their volume is collapsed:

vol.B.x;x // x3 3 1

< "n ! 0.

Q 2. They are homeomorphic to B , S  D2 , S2  I, T 2  I or K 2 I. 3. For one of the local models V0 D B.x0 ; x0 / the image of 1 .V0 / ! .Mn / is nontrivial. The last property insures that Mn n V0 is Haken, and thus geometric, by Thurston hyperbolisation theorem for Haken 3-manifolds.

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47

Step 2: We construct a covering of Mn n V0 by local models B.x; x / such that: 1. The covering has dimension 2 on the interior: every point belongs to at most three open subsets. 2. The covering has dimension 0 on @.Mn n V0 /: each component of @.Mn n V0 / belongs to a unique open subset. Then the simplicial volume kMn n V0 k D 0, by a relative version of Gromov’s vanishing Theorem 2.49 (see [11]) Thus Mn n V is a graph manifold by Corollary 2.53, since it admits a geometric decomposition, and so is Mn by Dehn filling.

2.7.2.2 The Local Models Let ı be a positive real number. Recall that a diffeomorphism f W X ! Y is .1 C ı/bi-Lipschitz if both f and f 1 are .1 C ı/-Lipschitz. We say that two Riemannian manifolds X; Y are ı-close if there exists a .1 C ı/-bi-Lipschitz diffeomorphism between them. A .1Cı/-bi-Lipschitz embedding is a map f W X ! Y which is a .1C ı/-bi-Lipschitz diffeomorphism onto its image (see Section “Lipschitz Topology” in Appendix). First we describe the Cheeger-Gromoll soul theorem and some of its consequences. Let M be a Riemannian n-manifold. A subset S  M is totally convex if every geodesic segment M with endpoints in S is contained in S. We say that S is a soul of M if S is a closed, totally convex submanifold of M. Theorem 2.59 ([20]) Every Riemannian manifold M of non negative sectional curvature has a soul S. When M is closed, then S D M. The following lemma gives some strong topological consequences in the case of open manifolds: Lemma 2.60 ([20]) Let M be an open Riemannian manifold of non negative sectional curvature, and let S  M be a soul. Then M is diffeomorphic to the normal bundle of S in M. In addition, for every r > 0, the tubular neighborhood Nr .S/ D fx 2 M j d.x; S/ < rg is diffeomorphic to the disc subbundle of the normal bundle. The fact that the soul S itself is a closed manifold of nonnegative curvature, allows to classify the diffeomorphism type of low dimensional manifolds with nonnegative curvature, [20]: Corollary 2.61 (Classification of Open 3-Manifolds of Curvature  0) Let M be an open, orientable 3-manifold M of sectional curvature  0. Then exactly one of the following possibilities occurs: 1. If dim S D 0, then S is a point, and M Š R3 . 2. If dim S D 1, then S is a circle, and M Š S1  R2 .

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3. If dim S D 2, then S is diffeomorphic to S2 , T 2 , RP2 , or K 2 , and M is Q or K 2 R. Q diffeomorphic to one of the line bundles S2  R, T 2  R, RP2 R, Q All these manifolds are irreducible except S2  R and RP2 R. In the closed 3-dimensional case, the classification has been obtained by R. Hamilton using Ricci flow [38, 39]: Theorem 2.62 (Classification of Closed 3-Manifolds of Curvature  0) Every closed, orientable 3-manifold of nonnegative sectional curvature is spherical, Euclidean, or diffeomorphic to S2 S1 or RP3 #RP3 . In particular, any such manifold is a graph manifold. A group is virtually Abelian if it contains an Abelian subgroup of finite index. Here is a straightforward consequence of the classification Theorem 2.62 and Corollary 2.61: Corollary 2.63 Let M be an orientable 3-manifold of nonnegative sectional curvature. Then 1 M is virtually Abelian. The description of the metric balls B.x; x / in a collapsing sequence of Riemannian 3-manifolds is given by the following lemma, whose proof uses the Cheeger-Gromoll soul theorem (Theorem 2.59 and its consequences Lemma 2.60, Corollary 2.61). Lemma 2.64 Let .Mn ; gn /n be a sequence of closed, aspherical and orientable 3manifolds which collapses with local curvature bound (LCB). Then for all D > 1 there exists n0 .D/ with the property that for n > n0 .D/: (a) Mn is D1 -close to some closed Riemannian 3-manifold X 3 with sec  0, (b) or for all x 2 Mn there exists a radius x 2 .0; n .x// such that: (i) B.x; x / is D1 -close to Nx .Sx /  Xx3 , (ii) Xx3 is a non compact Riemannian 3-manifold with sec  0 and soul Sx , such that diam.Sx /  (iii) vol.B.x; .x/// 

1 3 D x

x D

and secjB.x;x /   12 . x

The fact that the Mn are orientable and aspherical implies: • In the compact case .a/ X 3 is Euclidean by Hamilton’s Theorem 2.62, and thus a graph manifold. • In the non compact case .b/ by Cheeger-Gromoll soul theorem (Theorem 2.59), Xx3 is the normal bundle of its soul S D , S1 , S2 , T 2 , K 2 . Hence B.x; x / is Q diffeomorphic to B3 , S1  D2 , S2  I, T 2  I, K 2 I. The proof of Lemma 2.64 is by contradiction. Let assume that there is a constant D0 and a sequence of counterexamples xn 2 Mn3 .

2 Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture

If we set n D inffr j following claim:

vol.B.xn ;r/ r3



1 D0 g,

49

then the contradiction is reached by the

Claim 2.65 Up to taking a subsequence 1n B.xn ; n .xn // converges in the C 1 topology to a Riemannian 3-manifold X 3 with sec  0 Proof Since vol.Mn / < C, one has n < 1. Moreover n > 0 because vol.B.xn ;n // lim vol.B.x;r// D 4 D D10 . 3 . By continuity, r3 3

r!0

n

By hypothesis sec  0, it follows that

n .xn / n

1 n .xn /2

on B.xn ; n .xn // and

vol.B.xn ;n .xn /// n .xn /3

 "n . Since "n !

! 1. 2

1 n B.xn ; n .xn // 3

has curvature   n .xnn /2 ! 0 and converges to a comHence plete metric space X in the Hausdorff-Gromov topology (see Section “GromovHausdorff Distance Between Metric Spaces” in Appendix, for more details about the Hausdorff-Gromov topology). Now, by the choice of n , the local curvature bound (LCB) applies and jRmj, jrRmj are uniformly bounded on balls of bounded radius in 1n B.xn ; n .xn //. This implies that X 3 is a Riemannian manifold and the convergence is C1 . Since closed manifolds with sectional curvature  0 are graph manifolds, we may assume that only non compact models B.x; x / occur. Therefore Mn is covered by local models B.x; x / 1 Nx .Sx /  Xx3 where Xx3 is noncompact with sec  0 and

vol.B.x;x / x3


= > ;

By definition, triv.x/  x since we assume 1 .B.x; x // ! 1 .Mn / trivial. 1 Let define r.x/ D minf1; 11 triv.x/g. If B.x; r.x// \ B.y; r.y// ¤ ;, then:  43 . 1. 34  r.x/ r.y/ 2. B.x; r.x//  B.x; 4r.y// Let choose a maximal pairwise disjoint collection of balls B.x1 ; 14 r1 /; : : : ; B.xk ; 14 rk /. Then, by maximality B.x1 ; 23 r1 /; : : : ; B.xk ; 23 rk / cover Mn : if y 2 Mn , there exists 1  i  q such that B.y; 14 r.y// \ B.xi ; 14 r.xi // ¤ ;, so y 2 B.xi ; 23 r.xi //. The goal is to shrink this covering to get a covering of dimension 2. The collection of balls fB.x1 ; r1 /; : : : ; B.xq ; rq /g defines a covering of Mn with nerve K. By Bishop-Gromov inequality the dimension of K is uniformly bounded. Let define a characteristic map f D P1 i . 1 ; : : : ; q / W Mn ! K  p1 with

i W B.xi ; ri / ! Œ0; 1 a test function such that: • i jB.xi; 2 ri / Š 1, 3 • i [email protected];ri / D 0, • jr i j  r4i Then jrf j  Cri on B.xi ; ri / for a uniform constant C, since Mn D [B.x; 23 ri /. The strategy now is to show that f can be retracted into the 2-skeleton K .2/ to a function f .2/ W Mn ! K .2/ and to take the covering of Mn given by the preimages of the open stars of the vertices of K .2/ , i.e. by the open subsets Vi D . f .2/ /1 (open star of the i-th vertex of K .2/ ). This gives a covering of M of dimension 2 by homotopically trivial open subsets. Claim 2.68 One can inductively deform f by homotopy into the 3-skeleton K .3/ of the nerve K, while keeping the local Lipschitz constant under control. Proof (Sketch of Proof) The idea is to find a constant  D .d; L/ > 0 such that each d-simplex   K (d  4) contains a point z whose distance to @ and to the image of f is  . Then one composes f on  with the radial projection from z to push f into the .d  1/-skeleton. This construction increases the Lipschitz constant by a multiplicative factor bounded from above by a function of .d; L/. and decreases the inverse image of the open stars of the vertices.

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Let assume now that f W Mn ! K .3/ . Then: vol.Im. f / \ 3 /  vol. f .B.xi ; ri ///  jrf jvol.B.xi ; ri // r3

 . rCi /3 Di 

C3 D

< vol.3 / :

As a consequence for D large enough f does not fill any 3 simplex 3 . Thus one can again define f .2/ W Mn ! K .2/ by composing f with the radial projection 3 n f?g ! @3 . This finishes the proof of Claim 2.66. In the next Sect. 2.7.2.4 we complete the proof of Theorem 2.58.

2.7.2.4 The Vanishing of Simplicial Volume By the previous Sects. 2.7.2.2 and 2.7.2.3 there exists a local model V0 D B.x0 ; x0 / such that: 1. Im.1 .V0 / ! 1 .B.x0 ; x0 /// ¤ f1g Q 2. V0 Š S1  D2 , T 2  I or K I. 3. Mn n V0 is a Haken manifold with @ Š T 2 or T 2 [ T 2 . The fact that Mn nV0 is a Haken manifold follows from the facts that Mn is irreducible and that V0 can not be contained in a ball. Claim 2.69 The simplicial volume of Mn n V0 vanishes. Proof As before, we find a covering U1 ; : : : ; Uk of Mn n V0 such that: • the dimension of the covering is 2 on the interior of Mn n V0 and 0 on @.Mn n V0 /. • im.1 .Ui / ! 1 .Mn3 n V0 // is virtually Abelian. The construction of the covering follows as in the previous Sect. 2.7.2.3 by playing an analogous game with a radius of Abelianity: 8 ˆ < ab.x/ D sup r ˆ :

ˇ ˇ B.x; r/ is virtually abelian relatively to V 0 ˇ ˇ ˇ and B.x; r/ is contained in a ball B.x0 ; r0 / ˇ ˇ with curvature   .r10 /2

9 > = > ;

and by taking: r.x/ D minf

1 ab.x/; 1g: 11

Then one applies, the following relative version of Gromov’s vanishing theorem, obtained by applying the absolute version (Theorem 2.49) to all closed 3-manifold obtained by Dehn filling along the boundary @.Mn n V0 /, (see [11]).

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Theorem 2.70 If a compact orientable irreducible 3-manifold N with boundary an union of tori admits a covering fUi g of dimension 2 on its interior and of dimension 0 on @N by open amenable subsets, then the simplicial volume of N vanishes: kNk D 0.

2.7.3 Weak Collapse The goal of this section is to explain the proof of the weak collapsing Theorem 2.44 which is a strengthened version of Perelman’s collapsing Theorem 2.58. For convenience we recall the statement: Theorem 2.71 (Weak Collapsing Theorem) Let M be a closed, orientable, aspherical 3-manifold. Suppose that there exists a sequence gn of Riemannian metrics on M satisfying: (1) The sequence vol.gn / is bounded. (2) For every " > 0 and every sequence xn 2 M, if xn is "-thick with respect to gn for each n, then the sequence .M; gn ; xn / subconverges in the pointed C 2 -topology to a pointed hyperbolic 3-manifold .H; ghyp ; /, where volH < V0 .M/. (3) The sequence gn has local curvature bound (LCB). Then M is Seifert fibered or contains an essential torus. A few more arguments are needed in order to carry over the previous arguments of Sect. 2.7.2 to this case since the thick part may be non-empty, and needs to be taken in account. We will assume that M contains no essential torus, and show that M is a graph manifold., and so is Seifert fibered. Let .M; gn / be a sequence satisfying the hypotheses of Theorem 2.44. The first N ni step in Sect. 2.7.3.1 consists to cover the thick part by compact submanifolds H which approximate compact cores of the limiting hyperbolic manifolds given by hypothesis (2). The uniform bound on the volume [hypothesis (1)] is used to control the number of such “quasi-hyperbolic” thick pieces. Their boundaries consist of tori and no such thick piece can be the exterior of a link in M because of hypothesis (2). Then the assumption that M contains no essential torus allows to embed these thick pieces into solid tori, by using the following result which is classical: Lemma 2.72 An embedded torus T in a compact orientable irreducible 3-manifold M either is 1 -injective or bounds a solid torus or is embedded in a ball. In Sect. 2.7.3.2 we use the solid tori containing the thick pieces and the local models around points of the thin part to build, as in Sect. 2.7.2.3, a covering of M by open subsets such that at least one of the open subsets is homotopically nontrivial in M. Then, like in Sect. 2.7.2.4, relatively to this homotopically nontrivial subset we build a covering of dimension 2 by virtually Abelian subset and apply the relative version of Gromov’s vanishing theorem, Theorem 2.70, to conclude that M is a graph manifold.

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2.7.3.1 Embedding Thick Pieces in Solid Tori Under the assumptions of the weak collapsing Theorem 2.44, we have the following description of the thick part: Proposition 2.73 Up to taking a subsequence of .M; gn /, there exists a finite (possibly empty) collection of pointed hyperbolic manifolds .H 1 ; 1 /; : : : ; .H m ; m / and for every 1  i  m a sequence xin 2 M satisfying: (i) lim .M; gn ; xin / D .H i ; i / in the C 2 topology. n!1

(ii) For all sufficiently small " > 0, there exist S n0 ."/ and C."/ such that for all n  n0 ."/ the "-thick part .M; gn /C ."/  i B.xin ; C."//. Proof By assumption, the sequence vol.M; gn / is bounded above. Let 0 > 0 be a Margulis constant such that any hyperbolic manifold has volume at least 0 . If for all " > 0 we have .M; gn /C ."/ D ; for n large enough, then Proposition 2.73 is certainly true. In fact, this would correspond to the case considered in Sect. 2.7.2, where the sequence of Riemannian manifolds .M; gn / is collapsing. Otherwise, we use Hypothesis (2) of Theorem 2.44: up to taking a subsequence of .M; gn /, there exists "1 > 0 and a sequence of points x1n 2 .M; gn /C ."1 / such that .M; gn ; x1n / converges to a pointed hyperbolic manifold .H 1 ; 1 /. If for all " > 0 there exists C."/ such that, for n large enough, .M; gn /C ."/ is included in B.x1n ; C."//, Proposition 2.73 is proved. Otherwise there exists "2 > 0 and a sequence x2n 2 .M; gn /C ."2 / such that d.x1n ; x2n / ! 1. Again by Hypothesis (2) of Theorem 2.44, after taking a subsequence, the sequence .M; gn ; x2n / converges to a pointed hyperbolic manifold .H 2 ; 2 /. The argument now is to show that this construction has to stop after finitely many steps, and thus that eventually condition (ii) of Proposition 2.73 is satisfied. For each step i, and for n sufficiently large, .M; gn / contains a submanifold C 2 -close to some compact core of a hyperbolic manifold Hi , and hence of volume greater than or equal to say 0 =2. Moreover, for n fixed and large, these submanifolds are pairwise disjoint. The hypothesis .1/ of Theorem 2.44 that the sequence vol.M; gn / is bounded above implies that the construction has to stop. A consequence of Proposition 2.73 is that, up to subsequence, one can choose sequences "n ! 0 and rn ! 1 such that the ball B.xin ; rn / is arbitrarily Sclose to a metric ball B. i ; rn /  H i , for i D 1; : : : ; m, and every point of .M; gn /X i B.xin ; rn / is "n -thin. For the remaining of the section let fix a sequence of positive real numbers "n ! N i denote a compact core of the hyperbolic limits H i 0. For each i D 1; ; m let H N ni  given by the previous Proposition 2.73, and for each n let fix a submanifold H i i i N N .M; gn / and an approximation nSW Hn ! H . Up to renumbering, one can further N ni  .M; gn / ."n /, and that the submanifolds assume that for all n .M; gn / X H i N Hn ’s are disjoint. The definition of the volume V0 and the hypothesis (2) of Theorem 2.44 that the volume of each hyperbolic limit H i is less than V0 imply that for n sufficiently large

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N ni is homeomorphic to the exterior of a link in M. Since we assume no submanifold H N ni are not essential, and by that M is atoroidal, all boundary components of each H i N n is contained in a 3-ball or a solid torus. More precisely, for all Lemma 2.72 each H sufficiently large n and each i 2 f1; : : : ; mg, there exists a connected submanifold N ni , whose boundary is a boundary component of H N ni , and which Yni  M containing H is a solid torus, or contained in a 3-ball. In the sequel we pass to a subsequence, so that this holds for all n. If for two indices i; j M D Yni [ Ynj , then M would be a Lens space, which is not the case, since M is aspherical. Therefore for all i ¤ j, the submanifolds Yni and i Ynj are disjoint or one contains S i the other. By taking Wn to be the union of the Yn , N n  Wn . Since each connected component of Wn is we obtain for each n that H i homeomorphic to one of the YS n , the boundary of each connected component of Wn is a connected component of i @HN ni . Moreover, each connected component of Wn is a solid torus, or contained in a 3-ball embedded in M. We show now that all components of Wn can be assumed to be solid tori. If there exists a component X of some Wn which is not a solid torus, X is a knot exterior contained in a 3-ball B  M, and then it is possible to replace X by a solid torus Y without changing the topology of M, since the change takes place in a ball. From the metric point of view, we can endow M with a Riemannian metric g0n , equal to gn away from Y and such that an arbitrarily large collar neighborhood of @Y in Y is isometric to a collar neighborhood of @X in X, because the end of the interior of X is close to a hyperbolic cusp. For n large, this neighborhood is thus almost isometric to a long piece of a hyperbolic cusp. This last geometric property is sufficient to carry on the covering arguments of Sect. 2.7.2.3 when both the thick part and the local models for the thin part are involved. By repeating this construction for each connected component of Wn which is not a solid torus, we obtain a sequence of Riemannian manifolds .M; g0n / together with submanifolds Wn0  M satisfying the following properties: (i) (ii) (iii) (iv)

The metrics g0n D gn on the set M X Wn0 D M X Wn . Mn0 X Wn0 D Mn X Wn is "n -thin. Each component of Wn0 is a solid torus. When n goes to infinity, there exists a collar neighborhood of @Wn0 in Wn0 of arbitrarily large diameter isometric to the corresponding neighborhood in Wn .

For simplicity, we keep using the notation gn , Wn instead of g0n and Wn0 . This construction allows to assume that under the hypothesis of Theorem 2.44 all components of Wn are solid tori. In particular, for each n, the thick part is amenable.

2.7.3.2 Coverings by Abelian Subsets From now on the proof of Theorem 2.44 follows the same scheme as in Sect. 2.7.2.3, starting with the covering of M by solid tori containing the thick pieces and the local models around points of the thin part.

2 Thick/Thin Decomposition of Three-Manifolds and the Geometrisation Conjecture

55

The first step is to show that at least one of the open subsets is homotopically nontrivial in M. It is given by the following lemma, where the constant n0 .D/ and the local models B.x; .x// are given by Lemma 2.64: Lemma 2.74 There exists D0 > 1 such that for D > D0 and n  n0 .D/ either some connected component of Wn is not homotopically trivial, or there exists x 2 .M; gn / X Wn with a not homotopically trivial local model B.x; .x// A proof by contradiction of this lemma is analogous to the one given in Sect. 2.7.2.3, using the fact that M is not simply connected and Theorem 2.67. With the notation above, we may assume that for arbitrarily large D there exists n  n0 .D/ such that each component of Wn is homotopically trivial in M as well as the local models B.x; .x// for x 2 .M; gn /. Then like in Sect. 2.7.2.3 one introduces the following radius of triviality: For all x 2 .M; gn / X Int.Wn / we set: 8 ˆ < triv.x/ D sup r ˆ :

ˇ ˇ  .B.x; r// !  .M / is trivial and 1 n ˇ 1 ˇ B.x; r/ is contained in B.x0 ; r0 / with ˇ ˇ ˇ curvature   .r10 /2

9 > = > ;

Notice that triv.x/  .x/, by Lemma 2.64. For x 2 .M; gn / X Wn we set r.x/ WD min

n1 o triv.x/; 1 : 11

The proof follows then by contradiction from this claim: Claim 2.75 There exists a 2-dimensional covering of .M; gn / by open sets U1 ; : : : ; Up such that each Ui is either contained in some local model B.xi ; triv.xi // or in a subset that retracts by deformation to a component of Int.Wn /. In particular, Ui is homotopically trivial in M. To prove Claim 2.75, one plays an analogous game as in Sect. 2.7.2.3, using the radius r.x/ defined above. For n large enough, let first pick points x1 ; : : : ; xq 2 @Wn such that: (i) A tubular neighborhood of each connected component of the boundary of Wn contains precisely one of the xj ’s (ii) The balls B.xj ; 1/ are disjoint, have volume  D1 and sectional curvature close to 1. (iii) Each B.xj ; 1/ is contained in some submanifold Wn0 which contains Wn and can be retracted by deformation onto it. (iv) Each B.xj ; 23 r.xj // contains an almost horospherical torus corresponding to a boundary component of Wn . In particular, Wn0 and B.xj ; 1/ are homotopically trivial, since one has assumed that Wn is. This implies that triv.xj / is close to 1.

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Then one completes the previous finite sequence to a sequence of points x1 ; x2 ; : : : in M X Int.Wn / with the property that the balls B.x1 ; 14 r.x1 //, B.x2 ; 14 r.x2 //; : : : are pairwise disjoint. Such a sequence is finite, since M is compact. Let choose a maximal finite sequence x1 ; : : : ; xp with this property, then the balls B.x1 ; 23 r.x1 //; : : : ; B.xp ; 23 r.xp // cover M X Int.Wn /. Set ri WD r.xi / and let Wn;1 ; : : : ; Wn;q be the components of Wn such that the almost horospherical torus @Wn;i  B.xi ; 23 ri /, for i D 1; : : : ; q. Then we define: • Vi WD B.xi ; ri / [ Wn;i , for i D 1; : : : ; q. • Vi WD B.xi ; ri /, for i D q C 1; : : : ; p. 1 triv.xi /, the open sets Vi , i D 1; : : : ; p, are homotopically trivial, Since ri  11 and cover M. Furthermore each connected component of Wn;i can be retracted to miss B.xj ; 23 rj / when j ¤ i. Let K be the nerve of the cover fVi g. As in Sect. 2.7.2.3, by the Bishop-Gromov inequality the dimension of K is bounded above by a uniform constant. Then one plays the same game as in Sect. 2.7.2.3 to extract from the cover fVi g a 2dimensional cover of M by null-homotopic open subsets. Having proved the existence of a non-homotopically trivial subset V0 in the covering of .M; gn / by solid tori containing the thick pieces and by the local models around points of the thin part, for D > D0 and n  n0 .D/, the argument follows now the lines of Sect. 2.7.2.4. Namely, we build, as above, a cover of M n V0 by virtually Abelian subset which has dimension 2 on the interior of M n V0 and 0 on @.Mn n V0 / and then apply the relative version of Gromov’s vanishing theorem (Theorem 2.70) to conclude that M is a graph manifold, see [5, Chap. IV] for the complete details.

2.7.4 Thick/Thin Decomposition of a Closed Orientable 3-Manifold For a compact orientable irreducible 3-manifold with boundary a (possibly empty) collection of tori, the geometric decomposition can be interpreted as a topological version of Margulis thick/thin decomposition for a hyperbolic manifold: Theorem 2.76 (Thick/Thin Decomposition) Every compact orientable irreducible 3-manifold M splits into two (possibly empty or not connected) compact 3-submanifolds: M D H [ G such that: 1. int.H/ admits a complete hyperbolic structure with finite volume. 2. G is a graph 3-manifold. 3. @H D @G is a collection (maybe empty) of 1 -injective tori. Moreover this splitting is unique up to isotopy.

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57

In the decomposition given by Theorem 2.76, a component of the graph part G can be homeomorphic to a product T 2  Œ0; 1 between to boundary components of H. The following proposition is a consequence of the notion of Gromov’s simplicial volume and Corollary 2.55 (see Sect. 2.7.1). Proposition 2.77 There is no Riemannian metric with a lower bound on the curvature and arbitrarily small volume on the interior of the union H of the hyperbolic pieces. The submanifold H is called the thick part of M. In contrast, if M is a graph manifold, then by Cheeger-Gromov collapsing theory [21] one can construct Riemannian metrics on M with sectional curvature pinched between 1 and 1 whose volume is arbitrarily small, see the Appendix. Proposition 2.78 There are Riemannian metrics on the union of Seifert fibered pieces G with bounded curvature and arbitrarily small volume. The submanifold G is called the thin part of M. The following result is a consequence of Perelman’s geometrisation theorem (Corollary 2.26) and previous works of Besson et al. [7] and Boyland et al. [13] about the minimal volume of a Riemannian manifold with lower curvature bound. Theorem 2.79 Any sequence gn of Riemannian metrics on M, with sectional curvature Kgn  1 and which is volume minimizing, converges to a complete hyperbolic metric with constant curvature 1 on the thick part H and collapses on G. In particular vol.H/ is the minimal volume Minvol1 .M/ over all Riemannian metrics on M, with sectional curvature bounded below by 1.

Appendix: Sequences of Riemannian Manifolds The goal of this appendix is to present some fundamental tools and results in the theory of Gromov-Hausdorff convergence of Riemannian manifolds. Details of the proofs of the results presented here can be found in the following basic references: [15, 27, 28, 33, 35, 77].

Gromov-Hausdorff Distance Between Metric Spaces In the 1980s Gromov extended the classical Hausdorff distance between compact subspaces of a metric space to a distance between abstract metric spaces, called the Gromov-Hausdorff distance (G-H distance for short). However two metric spaces which are close for this distance generally can be topologically different. All metric spaces in this section will be separable metric space .X; d/.

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For a subset A  X the "-neighborhood around A is B.A; "/ D fx 2 X W d.x; A/ < "g. The classical Hausdorff distance between two subsets A; B in a metric space X is: dHX .A; B/ D inff" W A  B.B; "/ and B  B.A; "/g This metric is only a pseudo-metric since dHX .A; B/ D 0 implies only that the closures A D B. However for closed subspaces of X it is a metric. Gromov extended this notion to the setting of abstract metric spaces by getting ride of the role of the ambient space and found important applications to differential geometry. Definition 2.80 (Gromov-Hausdorff Distance) Two metric spaces X and Y are "near in the Gromov-Hausdorff topology if there is a metric on the disjoint union X t Y which extends the metrics on X and Y such that dHXtY .X; Y/  ". Then define : dGH .X; Y/ D inff" such that X and Y are "-nearg: From the definition it follows that dGH .X; Y/ D dGH .Y; X/ and that dGH .X; Y/ is finite if X and Y are compact. We will show below that dGH is a distance on the set of compact metric spaces. First we give some basic examples which show that often one can give upper bound for the G-H distance, even if it is usually very hard to compute it exactly. Example 2.81 Let X and Y be compact metric spaces with diam.X/  D and diam.Y/  D. Then dGH .X; Y/  D=2. Example 2.82 Let .X D fx1 ; : : : ; xk g; d/ and .Y D fy1 ; : : : ; yk g; d/ be finite metric spaces with k points. If jd.xi ; xj /  d.yi ; yj /j < " for all 1  i; j  k, then dGH .X; Y/  " Example 2.83 A map f W X ! Y between two compact metric spaces is called a "-Hausdorff approximation if the following holds: (i) Y is the "-tube around f .X/. (ii) 8u; v 2 X; jd. f .u/; f .v//  d.u; v/j < " If f W X ! Y is a "-Hausdorff approximation then dGH .X; Y/  3". The following examples show that the Hausdorff dimension is not continuous with respect to the Gromov-Hausdorff topology. Example 2.84 Let X be a compact space with a metric d. Then .X; d/ converges to a point for the Gromov-Hausdorff distance when  ! 0. Example 2.85 Consider the unit sphere S3  C2 with the standard S1 -action induced by C . The quotient  W S3 ! S3 =S1 D S2 is the Hopf fibration, where S2  R3 is the standard sphere with curvature 4. The finite cyclic subgroup Zn  S1 of order n acts freely and orthogonally on S3 . One denotes by S3 =Zn D Ln the lens space obtained. As n ! 1, the subgroup Zn fills up S1 and the 3-dimensional lens spaces Ln converge for the Gromov-Hausdorff distance to the 2-dimensional base S2

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59

of the Hopf fibration. This phenomenon is called a collapse because the dimension of the limit space is smaller than the dimension of the spaces in the sequence. We will come back to this phenomenon latter on. Example 2.86 Let Xn D f n1 . p; q/ W p 2 Z; q 2 Zg, then Xn with the induced metric from R2 converges for the Hausdorff-Gromov metric to R2 . In the unit cube Œ0; 13  R3 consider the subspace Xn D f.x; y; z/ 2 Œ0; 13 g, where at least two coordinates are of the form pn ; p 2 Z. Then Yn D @Xn is a surface in R3 which fills up Œ0; 13 as n ! 1. This phenomenon is called an explosion, since the limit space has larger dimension than the spaces in the sequence. Let M be the set of isometry classes of compact metric spaces, then dGH is a distance on M and : Theorem 2.87 .M ; dGH / is a metric space which is separable and complete.

Gromov’s Precompactness Theorem In order to state Gromov’s precompactness criterion we need the following definitions: Definition 2.88 Let X be a compact metric space and " > 0 a real number: A "-net is a finite set of points Z " in X such that X D B.Z; "/. Define Cov.X; "/ has the minimal number of points of a "-net in X. Lemma 2.89 Let X; Y 2 M such that dGH .X; Y/  ı, then Cov.X; " C 2ı/  Cov.Y; "/. The following precompactness criterion for subset C in M is important and very useful : Theorem 2.90 A subset C  M is precompact for the Gromov-Hausdorff topology iff there is a function N W .0; ˇ/ ! .0; 1/ such that : 8 " > 0 and 8 X 2 C one has Cov.X; "/  N."/. Example 2.91 Let N W .0; ˇ/ ! .0; 1/ be a function and let CN  M be the class of compact metric spaces X such that Cov.X; "/  N."/; 8 " 2 .0; ˇ/. Then CN is compact. We present now two important applications of Gromov’s precompactness criterion to Riemannian geometry.

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Riemannian Manifolds with a Lower Bound on the Injectivity Radius The injectivity radius inj.M; x/ of a Riemannian manifold M at a point x is the maximal radius r so that the exponential expx W B.0; r/  Tx M ! M is an embedding. The injectivity radius of M is inj.M/ D infx2M inj.M; x/. Denote by R.n; ı; v/ the set of closed, connected Riemannian manifolds of dimension n with injectivity radius inj.M/  ı > 0 and volume vol.M/  v. The following result is a consequence of Croke’s isoperimetric inequality, see [18, Sect. 6.6]: Proposition 2.92 Let M be a closed Riemannian n-manifold. If inj.M/  ı, there is a constant c.n/ depending only on the dimension n such that vol.B.x; r/  c.n/rn for any 0 < r  ı=2 and any x 2 M. Given a Riemannian manifold M 2 R.n; ı; v/ one chooses a maximal set fB.xi ; "=2/g of disjoint balls in M. Then the set of balls fB.xi ; "/g covers M and 2n v n one gets that Cov.M; "/  c.n/ " . Thus Gromov’s criterion applies to show: Corollary 2.93 The set R.n; ı; v/ is precompact in M for the Gromov-Hausdorff topology.

Riemannian Manifolds with a Lower Bound on the Ricci Curvature For our second application of Gromov’s precompactness criterion we consider the set R.n; k; D/ of closed, connected Riemannian manifolds of dimension n with Ricci curvature RicM  k.n  1/ and diameter diam.M/  D. The Ricci curvature reflects important information on the growth of the volume of the balls in M. Theorem 2.94 (Bishop-Gromov) Let M be a complete Riemannian n-manifold with RicM  k.n1/. Then for every point x 2 M the quantity vol.B.x;r// is decreasing vk .r/ with respect to r, where vk .n; r/ denotes the volume of a geodesic ball in the space form of constant sectional curvature k and of dimension n. In particular : p R n1 . jkjt/dt vk .n; D/ vol.M/ Œ0;r sinh   R p 8 0 < r < D: n1 . jkjt/dt vol.B.x; r// vk .n; r/ Œ0;D sinh

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Therefore as above Cov.M; "/  once more to show:

vk .n;D/ vk .n;r/

61

 c.n; k; D/"n . Gromov’s criterion applies

Corollary 2.95 The set R.n; k; D/ is precompact in M for the Gromov-Hausdorff topology.

Dimension of the Limit Space We have shown that every sequence in R.n; ı; v/ or R.n; k; D/ subconverges in M , but a priori the limit space may not be a manifold and may have a dimension different from n. Here we show that in both cases the dimension of the limit space stays  n, so no explosion can occur like in Example 2.86. However the Example 2.85 shows that collapses may occur in R.n; k; D/, even with pinched sectional curvature. We will show that no collapse can occur in R.n; ı; v/, this points out the importance of controlling the injectivity radius. We first recall the definition of the (covering) dimension of a topological space: Definition 2.96 The (covering) dimension of a topological space X is  n if every locally finite, open covering of X admits a refinement such that no point in X belongs to more than n C 1 open subsets. The dimension dim.X/ is the smallest integer n such that X has dimension  n. The (covering) dimension of a n-dimensional manifold is n. For a metric space there is another concept of dimension which has a more metric flavor. Both concept coincides for a compact n-dimensional manifold Definition 2.97 For a compact metric space X the Hausdorff dimension is: dimH .X/ D limsup"!0

log.Cov.X; "//  log."/

This metric dimension can take non integral values for Cantor sets. It is related to the usual (covering) dimension by the following inequality [44] for a compact metric space X, dim.X/  dimH .X/. Proposition 2.98 Let fMi g be a sequence of closed Riemannian n-manifolds which converges in the Gromov-Hausdorff topology to a compact metric space X. (1) If fMi g  R.n; ı; v/ then dim.X/ D dimH .X/ D n. So no collapse, nor explosion occur. (2) If fMi g  R.n; k; D/ then dim.X/  dimH .X/  n. So no explosion occur, but collapses are possible. The fact that in both cases (1) and (2) dimH  n follows immediately from the bound Cov.M; "/  c"n , where the constant c depends only of the dimension n

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and the bounds given on the injectivity and the volume, or on the Ricci curvature and the diameter. The fact that in case (1) the dimension cannot decrease is more subtle: it uses the notion of .n  1/-diameter and a local contractibility argument, see [76]. As a consequence of Perelman’s stability theorem for Alexandrov spaces with lower curvature bound (see [15, 50]) one obtains: Corollary 2.99 The set of closed, connected Riemannian n-manifolds M with KM  1, inj.M/  ı > 0 and vol.M/  v contains only finitely many homeomorphism types. Instead of working in the general class of compact metric spaces we could have work in the smaller class of length spaces: Definition 2.100 Let X be a compact metric space. A continuous map ` W Œ0; a ! X is a minimizing geodesic if d.`.u/`.v// D juvj holds for each 0  u  v  a. The space X is a length space if two points in X can be joined by a minimizing geodesic. An easy application of Ascoli-Arzela ’s theorem shows that a Gromov-Hausdorff limit of length spaces is a length space.

Lipschitz Topology We introduce a new topology on the set of metric spaces called the Lipschitz topology. Definition 2.101 Let X and Y be two metric spaces. .x/;f .y// For a map f W X ! Y let dil. f / D supx6Dy d. fd.x;y/ denote the dilatation of f . It is finite for a Lipschitz map. A homeomorphism f W X ! Y is said bi-Lipschitz if both dil. f / and dil. f 1 / are finite

Example 2.102 Let f W M ! N be a C 1 -map between two compact Riemannian manifolds, then dil. f / D supx2M kdf .x/k. Definition 2.103 (Lipschitz Distance) Let X and Y be metric spaces. The Lipschitz distance dL .X; Y/ between X and Y is defined as: • dL .X; Y/ D inffj log.dil. f //j C log.dil. f 1 /j over all bi-Lipschitz homeomorphism f W X ! Y • dL .X; Y/ D 1 if X and Y are not bi-Lipschitz homeomorphic. Proposition 2.104 Let M be the set of isometry classes of compact metric spaces, then dL is a distance on M .

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The next proposition shows that Lipschitz convergence implies GromovHausdorff convergence. Hence the Gromov-Hausdorff topology is weaker than the Lipschitz topology on M . Proposition 2.105 Let fXn gn2N and X be compact metric spaces in M . (i) Assume that limn!0 dGH .Xn ; X/ D 0. Then given " > 0 and  > 0, for any -discrete "-net Z "  X there is a sequence of "n -nets Zn"n  Xn such that limn!0 dL .Zn"n ; Z " / D 0 with 0  "n  " ! 0. (ii) Conversely assume that 8" > 0 there is a "-net Z "  X and a sequence of "-nets Zn"  Xn such that limn!0 dL .Zn" ; Z " / D 0. Then limn!0 dGH .Xn ; X/ D 0. Part (ii) of the above proposition immediately implies the following: Corollary 2.106 Let fXn gn2N and X be compact metric spaces in M . If limn!0 dL .Xn ; X/ D 0, then limn!0 dGH .Xn ; X/ D 0.

Rigidity Theorem In general the Lipschitz topology is stronger than the Gromov-Hausdorff topology. However in the setting of Riemannian manifolds with pinched sectional curvature both topology coincides. More precisely, let R1 .n; ı; v/ be the set of Riemannian n-manifolds M with a pinched sectional curvature jKM j  1, a lower bound on the injectivity radius inj.M/  ı > 0 and an upper bound on the volume vol.M/  v. Then two Riemannian manifolds in R1 .n; ı; v/ which are sufficiently nearby in the GromovHausdorff topology are in fact bi-Lipschitz homeomorphic: this the content of the following result due to Gromov [33, 35]. Theorem 2.107 (Rigidity Theorem) Given " > 0 there is a constant .n; ı; v; "/ > 0 such that if dGH .M; N/   for M and N in R1 .n; ı; v/, then dL .M; N/  ". The proof of this theorem goes back in fact to Cheeger’s finiteness Theorem [19, 74] which now is a consequence of it and of Gromov’s precompacness theorem: Corollary 2.108 (Finiteness Theorem) Up to diffeomorphism there are only finitely many manifolds in R1 .n; ı; v/. Another important corollary is the following convergence theorem due to Gromov [33, 35, 75]: Corollary 2.109 (Gromov Convergence Theorem) Every sequence fMk gk2N in R1 .n; ı; v/ subconverges in the Lipschitz topology to a smooth n-dimensional Riemannian manifold .M1 ; g1 / where the Riemannian metric g1 is of class C 1;˛ with 0 < ˛ < 1. Moreover M is diffeomorphic to Mk for k sufficiently large.

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It is in fact shown in [31] that the convergence takes place in the C 1;˛ . More precisely, there exists diffeomorphisms fk from M1 to Mk such that fk .gk / converges to g1 in the C 1;˛ norm (cf. [31, pp. 139–140]).

Pointed Topologies Gromov-Hausdorff or Lipschitz convergences, as defined above, are too restrictive, because one may be interested in sequences Xn with diamXn ! 1. Such a sequence cannot converge to a compact space in any reasonable sense. For instance, intuitively, a sequence of round 2-spheres of radius n should converge to E2 . But if Xn is obtained by gluing a round 2-sphere of radius n to a round 3-sphere of radius n (the union occurring at a single point), then what should the limit lim Xn be: E2 or E3 ? This problem is solved by considering sequences of pointed spaces, i.e. pairs .X; x/ where X is a metric space and x is a point of X. This works well when the spaces considered are proper (which means that metric balls are compact). Definition 2.110 Let .Xn ; xn / be a sequence of pointed proper metric spaces and .X; x/ be a pointed proper metric space. Then .Xn ; xn / converges to .X; x/ for the pointed Gromov-Hausdorff topology if for every R > 0 lim dGH .B.xn ; R/; B.x; R// D 0:

n!1

If the limit exists, it is unique up to isometry. The next example illustrates the importance of the choice of basepoint in a hyperbolic context. Example 2.111 Let M be a noncompact hyperbolic manifold. Set Xn D M and choose xn 2 M. • When the sequence xn stays in a compact subset of M, .Xn ; xn / subconverges to some .X1 ; x1 / with X1 isometric to M. • When xn goes to infinity in a cusp of maximal rank, .Xn ; xn / converges to a line. The cusp is a warped product of a compact Euclidean manifold with a line, and the diameter of the Euclidean manifold containing xn converges to zero as xn goes to infinity. • When xn goes to infinity in a geometrically finite end of infinite volume, .Xn ; xn / converges to a hyperbolic space of dimension dim M. This holds true because one can find metric round balls BRn .xn / with Rn ! 1. Here is the version of Gromov’s precompactness criterion for pointed metric spaces: For a metric space X and for constants R > " > 0, let Cov.X; R; "/ denote the maximal number of disjoints balls of radius " in a ball of radius R in X.

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Theorem 2.112 (Precompactness Criterion) A sequence of pointed metric geodesic spaces .Xn ; xn / is precompact for the pointed Hausdorff-Gromov topology if and only if, for every " > 0 and R > 0, Cov.Xn ; R; "/ is bounded on n. In an analogous way there is a notion of Lipschitz convergence for pointed proper metric spaces: Definition 2.113 (Pointed Lipschitz Convergence) A sequence of pointed proper metric spaces .Xn ; xn / converges to a proper metric space .X; x/ for the pointed Lipschitz topology if for every R > 0 lim dL .B.xn ; R/; B.x; R// D 0:

n!1

Remark 2.114 When .Xn ; xn / ! .X; x/ for the pointed Lipschitz topology, if the limit X is compact, then for n large enough Xn is bi-Lipschitz homeomorphic to X. Then one has the following compactness theorem: Theorem 2.115 (Compactness Theorem) The set R1 .n; ı/ of complete Riemannian n-manifolds M with bounded sectional curvature jKM j  1 and lower bound on the injectivity radius inj.M/  ı > 0 is compact for the Lipschitz topology.

Collapses We say that a family of Riemannian metrics on a manifold collapses with bounded geometry if all the sectional curvatures remain bounded while the injectivity radius goes uniformly everywhere to zero. Collapsing phenomena have received much attention in all dimensions. Example 2.116 (Berger Spheres) Let  W S3 ! S2 be the Hopf fibration and g the standard metric on S3 . Let g" be the metric obtained after rescaling by " the metric g in the direction tangent to the fibers. It means that for a tangent vector v 2 Tx S3 , g" .x/.v; v/ D "g.x/.v; v/ if dx .v/ D 0, while g" .x/.v; v/ D g.x/.v; v/ if v is orthogonal to the Hopf fiber. Moreover sup.0;1 jKg" j  1. If we put on S2 a Riemannian metric with constant curvature equal to 4, then lim"!0 dGH ..S3 ; g" /; S2 / D 0. This example can be generalized to any isometric locally free S1 -action on a closed Riemannian manifolds. For example any flat torus T n collapses to any smaller dimensional torus T k with k < n by rescaling the metric on some of the S1 factors. These examples turn out to be basic. Cheeger and Gromov [21] have proved that a necessary and sufficient condition for a manifold to have such a collapse with bounded geometry is the existence of a “generalized torus action” which they call an F-structure, where F stands for “flat” in this terminology. Intuitively an F-structure corresponds to different tori of varying

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dimension acting locally on finite coverings of open subsets of the manifold. Certain compatibility conditions on these local actions on intersections of these open subsets will insure that the manifold is partitioned into disjoint orbits of positive dimension. Definition 2.117 A pure F-structure F of positive rank k > 0 on a manifold M is a partition of M into compact submanifolds (leaves of variable dimension) which support an affine flat structure. Moreover M has an open covering fU˛ g such that f˛ ! U˛ coincides with the partition induced on some regular finite covering ˛ W U f˛ . the orbits of a smooth affine action of the k-dimensional torus Tk on U Two pure F-structures F1 and F2 are compatible if either F1  F2 (i.e. every leaf of F1 is an affine submanifold of a leaf of F2 ) or F2  F1 . Definition 2.118 A F-structure F on a manifold M is an open covering f.U˛ ; F˛ /g of M where F˛ is a pure F-structure on U˛ such that F˛ and Fˇ are compatible on U˛ \ Uˇ . The rank of F is the minimum rank of the local F-structures F ˛. A more precise definition of an F-structure can be given using the notion of sheaf of local groups actions. A compact orientable 3-manifold M with an F-structure admits a partition into orbits which are circles and tori, such that each orbit has a saturated subset. It follows from the definition of F-structure that such a partition corresponds to a graph structure on M (see [79, 97]). Another description of the family of all graph manifolds is that they are precisely those compact three manifolds which can be obtained, starting with the family of compact geometric non-hyperbolic three-manifolds, by the operations of connect sum and of gluing boundary tori together. Thus they arise naturally in both the Geometrisation of 3-manifolds and in Riemannian geometry. The following theorem is a version of Cheeger-Gromov’s thick/thin decomposition (see [23, Theorems 1.3 and 1.7], see also [9]). In this setting the "-thin part of a Riemannian n-manifold .M; g/ is the set of points F ."/ D fx 2 M ; inj.x; g/ < "g Theorem 2.119 For each n, there is a constant n , depending only on the dimension n, such that for any 0 < "  n and any complete Riemannian nmanifold .M; g/ with jKg j  1, there exists a Riemannian metric g" on M such that: (1) The "-thin part F ."/ of .M; g / admits an F-structure compatible with the metric g" , whose orbits are all compact tori of dimension  1 and with diameter < ". (2) The Riemannian metric g" is "-quasi-isometric to g and has bounded covariant derivatives of curvature, i.e. it verifies the following properties: • e" g"  g  e" g" . • kr g  r g" k  ", where r and r g" are the Levi-Civita connections of g and g" respectively.

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• k.r g" /k Rg" k  C.n; k; "/, where the constant C depends only on ", the dimension n and the order of derivative k. Using Cheeger-Gromov’s chopping theorem [22] one can prove the following proposition which is the analogue in bounded curvature of Jørgensen’s finiteness theorem [32, 91], which states that all complete hyperbolic 3-manifolds of bounded volume can be obtained by surgery on a finite number of cusped hyperbolic 3manifolds. The finiteness of hyperbolic manifolds with volume bounded above and injectivity radius bounded below is a particular case of Cheeger finiteness theorem, while the Margulis lemma takes the place of the Cheeger-Gromov thick/thin decomposition [21]. Proposition 2.120 Let M be a closed Riemannian n-manifold with jKM j  1 and vol.M/  v. Then M has a decomposition M D N [ G into two compact nsubmanifolds such that: • G admits an F-structure such that @N D @G is an union of orbits. • N belongs, up to diffeomorphism, to a finite set N .n; v/ of smooth, compact, orientable n-manifolds.

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72. G. Perelman, Ricci flow with surgery on three-manifolds. ArXiv: math.DG/0303109 (2003) 73. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. ArXiv: math.DG/0307245 (2003) 74. S. Peters, Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds. J. Reine Angrew. Math. 394, 77–82 (1984) 75. S. Peters, Convergence of Riemannian manifolds. Compos. Math. 62, 3–16 (1987) 76. P. Petersen, Gromov-Hausdorff convergence of metric spaces, in Proceedings Symposia in Pure Mathematics, Part 3, vol. 54 (1993), pp. 489–504 77. P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, vol. 171 (Springer, Berlin, 1997) 78. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149 (Springer, New York, 1994) 79. X. Rong, The limiting eta invariant of collapsed 3-manifolds. J. Differ. Geom. 37, 535–568 (1993) 80. A. Salgueiro, Orders and actions of branched coverings of hyperbolic links. Topol. Appl. 156, 1703–1710 (2009) 81. P. Scott, A new proof of the annulus and torus theorems. Am. J. Math. 102, 241–277 (1980) 82. P. Scott, The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983) 83. P. Scott, There are no fake Seifert fibre spaces with infinite 1 . Ann. Math. (2) 117(1), 35–70 (1983) 84. P. Scott, H. Short, The homeomorphism problem for closed 3-manifolds. Algedr. Geom. Topol. 14, 2431–2444 (2014) 85. H. Seifert, Topologie der dreidimensionaler gefaserter Raüme. Acta Math. 60, 147–288 (1933) 86. T. Shioya, T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound. Math. Ann. 333(1), 131–155 (2005) 87. T. Soma, The Gromov invariant of links. Invent. Math. 64 445–454 (1981) 88. C.B. Thomas, Elliptic Structures on 3-Manifolds. London Mathematical Society Lecture Notes Series, vol. 104 (Cambridge University press, Cambridge, 1986) 89. W. Threlfall, Quelques résultats récents de la topologie des variétés. Enseign. Math. 35, 242–255 (1936) 90. W. Threlfall, H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes I and II. Math. Ann. 104, 1–70 (1931); 107, 543–586 (1932) 91. W.P. Thurston, The Geometry and Topology of 3-Manifolds (Princeton University, Princeton, 1979) 92. W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. 6, 357–381 (1982) 93. W.P. Thurston, Hyperbolic structures on 3-manifolds, I: deformations of acylindrical manifolds. Ann. Math. (2) 124, 203–246 (1986) 94. W.P. Thurston, Hyperbolic structures on 3-manifolds, II: surface groups and manifolds which fiber over S1 . Preprint (1986) 95. W.P. Thurston, Three-Dimensional Geometry and Topology, vol. 1 (Princeton Mathematical Press, Princeton, 1997) 96. P. Tukia, Homeomorphic conjugates of Fuchsian groups. J. Reine Angew. Math. 391, 1–54 (1988) 97. F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. Invent. Math. 3–4, 308–333, 87–117 (1967) 98. F. Waldhausen, Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten. Topology 6, 505–517 (1967) 99. C. Weber, H. Seifert, Die beiden Dodekaederräume. Math. Z. 37, 237–253 (1933) 100. K. Yano, Gromov invariant and S1 -actions. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 29(3), 493–501 (1982)

Chapter 3

Singularities of Three-Dimensional Ricci Flows Carlo Sinestrari

Abstract The Ricci flow is an evolution of a Riemannian metric driven by a parabolic PDEs and was introduced by Hamilton in 1982. It has been the fundamental tool for some important achievements in geometry in the early 2000s, such as Perelman’s proof of the geometrization conjecture and Brendle–Schoen’s proof of the differentiable sphere theorem. In these notes we provide an introduction to the Ricci flow, by giving a survey of the basic results and examples. In particular, we focus our attention on the analysis of the singularities of the flow in the threedimensional case which is needed in the surgery construction by Hamilton and Perelman.

3.1 The Ricci Flow Let M be an n-dimensional Riemannian manifold with a metric g0 . The Ricci flow, also called Hamilton–Ricci flow of .M ; g0 / is a time-dependent family g.t/ (with t  0) of metrics on M satisfying g.0/ D g0 and evolving according to the equation @ g.t/ D 2Ricg.t/ @t where Ricg.t/ is the Ricci curvature tensor associated with the metric g.t/. The Ricci flow is a parabolic system of partial differential equations which has a unique solution at least in some finite time interval t 2 Œ0; T/ if M is compact. The Ricci flow was introduced by R. Hamilton in [30]. The motivation was to define an evolution of the metric tensor analogous to the evolution of functions defined by the heat equation. An earlier example of the use of parabolic PDEs in geometric problems was the paper by Eels and Sampson [27], who considered the heat flow of a map between two Riemannian manifolds, in order to obtain a harmonic mapping as the long time limit of the solution. Hamilton expected that

C. Sinestrari () Dip. di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, Via Politecnico 1, 00133 Roma, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Benedetti, C. Mantegazza (eds.), Ricci Flow and Geometric Applications, Lecture Notes in Mathematics 2166, DOI 10.1007/978-3-319-42351-7_3

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the Ricci flow should enjoy similar properties and exhibit convergence to stationary states in many cases. On the other hand, it was also clear that a complete analogy with the heat equation could not be expected. In fact, in the Ricci flow the evolution equation of the curvature contains additional reaction terms which may induce a singular behavior in finite time. To illustrate the powerful applications of the flow, let us start by describing the first important result obtained by Hamilton in [30]. Theorem 3.1 Any closed three-dimensional Riemannian manifold with positive Ricci curvature is diffeomorphic to a quotient of the sphere S3 under a finite group of isometries. To prove this result, Hamilton considered the evolution of the metric under the Ricci flow and showed that it converges to a metric of constant positive sectional curvature. More precisely, there is a finite time T > 0 at which the flow becomes singular and the manifold “shrinks to a point”: that is, the metric tends to zero and the curvature becomes unbounded everywhere. However, by choosing an appropriate rescaling factor .t/, the normalized metric .t/g.t/ converges, as t ! T, to a metric of positive constant sectional curvature. On the other hand, it is known that a manifold with such a metric must be S3 or one of its quotients. After that seminal paper, a rich variety of studies of the Ricci flow followed through the years to obtain several geometric applications. Other classes of Riemannian manifolds were found which converge under the Ricci flow to a limit of constant curvature after rescaling. For instance, this holds for all closed twodimensional manifolds, a property which provides an alternative proof of the uniformization theorem. Moreover, manifolds with positive curvature operator also converge to space forms under the Ricci flow. This was proved by Hamilton [31] in dimension 4 and by Böhm and Wilking [7] in the general case. A further spectacular result in this direction is the recent proof of the differentiable sphere theorem by Brendle and Schoen [10], which is treated in G. Besson’s contribution in this volume. On the other hand, it was soon clear that in many cases the Ricci flow can develop local singularities where no global information on the manifold is available. During the 1990s, Hamilton proposed a strategy to study these cases, whose main goal was the proof of the Thurston geometrization conjecture, which provides a complete classification of the closed three-dimensional manifolds, and includes in particular the Poincaré Conjecture Every closed simply connected three-dimensional manifold is homeomorphic to the sphere S3 . Hamilton’s idea to handle local singularities is to define a flow with surgeries. The Ricci flow is stopped shortly before the singular time, the regions with large curvature are removed by a surgery and replaced by more regular ones, and the flow is restarted. Hamilton conjectured that, after a finite number of surgeries, each component of the manifold converges to one of the structures described by Thurston, with the consequence that the initial manifold admits the desired decomposition.

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Hamilton was able to perform various important steps of his program [32, 33, 35, 36] but some crucial parts remained unsolved. Then, in 2002 and 2003 G. Perelman posted on the web three papers [49–51] which introduced several new ideas and gave a more detailed understanding of the Ricci flow. In particular, these new results allowed Perelman to finally prove the geometrization conjecture. A central part in Hamilton’s and Perelman’s surgery construction is the study of the possible singular profiles of 3-manifolds under Ricci flow. This allows a description of the regions with large curvature enough detailed to perform a surgery which preserves the relevant curvature estimates and changes the topology of the manifold in a controlled way. In these notes we will present the basic properties and techniques in the study of the Ricci flow, and the main results about the analysis of singularities which are used in the proof of the geometrization conjecture. In order to make the exposition easily accessible to non experts, the presentation will be often informal and the proofs will be omitted except in some simple and significant cases. A final bibliographical section will give to the interested reader the references for a detailed study of these topics. These notes describe the content of the lectures given at a CIME Summer Course in 2010. The author wishes to thank CIME and the organizers of the course for the invitation and their patience while these notes were written.

3.2 Notation, Examples and Special Solutions We consider an n-dimensional Riemannian manifold M and denote by g D .gij / its metric. We assume that the reader has some familiarity with the basic notions of Riemannian geometry, and refer to the notes by G. Besson in this volume for more details. The Riemann curvature tensor associated with the metric will be denoted by Rm D .Rijkl /, the Ricci curvature by Ric D .Rij / and the scalar curvature by R. Associated to the metric there is the Levi-Civita connection, which induces a covariant differentiation r on tangent vector fields and on tensor fields of arbitrary type. Because of its symmetries, the Riemann curvature tensor can also be interpreted as a symmetric bilinear map on 2 .Tp M /, the algebra of 2-forms on Tp M ; such a map is called the curvature operator of the Riemannian manifold. In these notes, we are mainly interested in three-dimensional manifolds, where the curvature quantities admit a simpler representation than in the general case. At a given point p 2 M 3 , let e1 ; e2 ; e3 be an orthonormal basis of Tp M which diagonalizes Ric. Let ; ;  be the sectional curvatures at p associated with the planes orthogonal to e1 ; e2 ; e3 respectively. Then the Ricci tensor has the form 0

1 C 0 0 Ric. p/ D @ 0  C  0 A : 0 0 C

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The scalar curvature is given by R. p/ D 2. C  C /. Thus, positive sectional curvature implies positive Ricci curvature, which in turn implies positive scalar curvature, but the reverse implications do not hold. On the other hand, it is easily seen that ; ;  are also the eigenvalues of the curvature operator. Therefore, positive sectional curvature is equivalent to positive curvature operator, a property which fails in higher dimension. As we said at the beginning, we say that a time-dependent family of metrics g.t/ on M is a solution of the Ricci flow if it satisfies @ g.t/ D 2Ricg.t/ : @t

(3.1)

Usually, an initial data g.0/ D g0 is given and the problem is studied for t > 0. The choice of the sign in the right-hand side is essential to ensure the parabolic character of the equation and the well-posedness for positive times, while the factor 2 is only a matter of convenience which simplifies some later formulas. Before giving general results about the existence and uniqueness, it is interesting to consider special solutions of the Ricci flow. There are very few cases where the solutions can be described explicitly, see e.g. [33, Sect. 2] or [17, Sect. 2]; here we present some of them. Trivial examples are Ricci-flat spaces, which are constant solutions. Other easy examples are provided by Einstein manifolds, which give rise to homothetic solutions. In fact, if g0 satisfies Ricg0 D cg0 for some constant c, then it is easy to check that the metric g.t/ D .1  2ct/g0 is a solution to the Ricci flow. Observe that the flow is defined for t 2 .1; .2c/1 / if c > 0 and for t 2 ..2c/1 ; C1/ if c < 0. Thus, for instance, if .M ; g0 / is a Euclidean sphere of radius r0 and dimension n  2, which corresponds to c D .n  1/r02 , its evolution at time t is a sphere of radius q r.t/ D r02  2.n  1/t; which shrinks to a point as t approaches the maximal time T D r02 =2.n  1/. As t ! T, the sectional curvature blows up like .T  t/1 . On the other hand, the flow starting from a manifold with constant negative curvature is defined for all positive times and is homothetically expanding with a curvature decay of order t1 . When we have a product metric, each factor evolves independently under Ricci flow. For example, given a cylinder of the form M D Srk0  Rnk where Srk0 is a k-dimensional sphere of radius r0 , the spherical factor shrinks homothetically while the flat factor remains unchanged. Thus, the global evolution is given by a cylinder with shrinking radius. At an intuitive level, it is often useful to picture the evolution of a metric under the Ricci flow as if the manifold were immersed in an Euclidean case with a shape which

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changes in time, contracting or expanding in the regions of positive or negative curvature respectively. This description corresponds to the exact behavior of the flow in the case of the homothetic solutions described above, but it should only be regarded as a heuristic tool in general. In addition to the exact solutions described above, there are some examples of “intuitive solutions”, described in [33, Sect. 3], which are very important to understand the possible singular behavior of the flow. Consider a dumbbell-shaped manifold, of dimension at least three, consisting approximately of two big spheres SRn joined by a thin tube (or “neck”) close to an Srn1  Œa; b. Both the spheres and the cylinder shrink under the flow; however, if r is enough small compared to R, we expect the cylinder to shrink faster. Therefore, the neck should pinch in its central part before the two spheres have become singular. Such a behavior is called neckpinch. It is natural to expect that a similar behavior should occur for much more general shapes, whenever there is a thin cylindrical region connecting large regions with lower curvature. A solution developing a neckpinch singularity has been later constructed rigorously in [4, 53], see also [17, Sect. 2.5]. By considering variants of the dumbbell example above, a further interesting possible behavior can be detected. We can observe that, if we take a dumbbell with r only slightly smaller than R, then the manifold has positive Ricci curvature and it will shrink with an asymptotically spherical profile as described in Theorem 3.1. Intuitively speaking, the two spheres “catch up” with the cylinder while they shrink, and the three parts merge in a shape which becomes more and more round. Therefore, there must be a threshold value of r; R where a limiting behavior occurs, and the cylinder pinches off at the same time as the two spheres collapse. Such a behavior, again conjectured in [33, Sect. 3], is called “degenerate neckpinch”. A rigorous construction of solutions exhibiting these properties has been performed in [5]. An important class of solutions to the Ricci flow is provided by the so-called solitons. A steady Ricci soliton is a manifold M (not necessarily closed) with a metric gQ which is a constant solution of the Ricci flow up to a diffeomorphism. By this we mean that there exists a time-dependent family of diffeomorphisms t of M such that, if we set g.t/ D t .Qg/, i.e. the pull-back metric under t , then g.t/ solves the Ricci flow. More generally, a shrinking (resp. expanding) Ricci soliton is a homothetically shrinking (resp. expanding) solution of the flow up to a diffeomorphism. That is, the evolving metric g.t/ has the form g.t/ D .t/ t .Qg/, where gQ is a fixed metric, .t/ is a scalar function and t a family of diffeomorphisms. It can be proved [17, Lemma 2.4] that the scaling factor is necessarily of the form .t/ D 1 C 2t for some  2 R. The case  D 0 corresponds to the steady solitons, while  < 0 and  > 0 yield a shrinking or expanding metric respectively. As in the case of Einstein metrics, such solutions are defined in an unbounded time interval of the form .1; .2/1 / or ..2/1 ; C1/, depending on the sign of . If the family of diffeomorphisms is generated by the gradient of a function f , these solutions are

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called gradient solitons, and f is characterized by the equation RicgQ C r 2 f D gQ :

(3.2)

An explicit example of steady gradient soliton in dimension 2 is the so-called cigar, which is R2 endowed with the metric gij D

ıij ; 1 C x2 C y2

see e.g. [17, Sect. 2.2] or [8, Sect. 2.1] for the details. To understand how the metric looks like, we can take a generic circle centered at the origin r .t/ D .r cos t; r sin t/; its length in the above metric is 2r.1 C r2 /1=2 , and thus it tends to 2 as r ! 1. Intuitively speaking, the manifold looks like a one-ended infinite cylinder for r large, and it closes with a round cap for r close to zero. It is also easy to see that the curvature decays very rapidly away from the origin, while the injectivity radius is close to the value  of a cylinder. It follows in particular inf inj.P/R.P/ D 0:

P2R2

(3.3)

These properties are important in connection to the non-collapsing property which will be introduced later in these notes. It can be proved that the cigar is the unique rotationally symmetric nonflat steady gradient soliton on R2 . Similarly, on any Rn with n  3 there exists a unique rotationally symmetric steady gradient soliton with positive sectional curvature, as shown by Bryant [11], see also [19, Sect. 1.4]. In contrast to the cigar soliton, these higher dimensional solutions look like a paraboloid in RnC1 rather than a cylinder. In addition, they satisfy inf inj.P/jRmj.P/ > 0:

P2R2

(3.4)

In general, a solution which is defined on a time interval of the form t 2 .1; T/ for some finite T > 0 is called an ancient solution; if it is defined for t 2 .1; 1/, it is called an eternal solution. Examples are given by the shrinking and steady Ricci solitons respectively, and other such solutions exist which are not solitons. Solutions defined for all negative times are very special, since the Ricci flow in general ill-posed backward in time. However, they are of great importance since they describe the possible profile of general solutions near a singularity, as we will see in the following. To conclude this section, it is interesting to mention another geometric flow which has many similarities with the Ricci flow. A hypersurface in Euclidean space, or in a general Riemannian manifold, is said to evolve by the mean curvature flow if every point moves with normal speed given by the opposite of the mean curvature. The signs are chosen in such a way that a closed hypersurface with positive mean

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curvature is contracting under the flow. While the Ricci flow applies to abstract Riemannian manifolds, this evolution deals with immersed ones. In many cases, however, the two flows exhibit striking analogies: for instance, the examples of the sphere, of the cylinder, of the standard and degenerate neckpinch occur in the mean curvature flow with almost identical properties. The analogy between the two flows can be useful at a heuristic level, because results and techniques can often be exported from one problem to the other. In addition, since immersed manifolds are sometimes easier to visualize, some examples in the Ricci flow can be better understood in connection with their mean curvature flow analogue. It should be pointed out that the two evolutions are not equivalent, and that there are some different properties as well. For example, in the mean curvature flow any closed hypersurface of the Euclidean space develops singularity in finite time. However, most of the relevant results on the Ricci flow treated on these notes have some analogues for the mean curvature flow, although possibly with some substantial difference in the hypotheses or in the method of proof. For example, a counterpart of Theorem 3.1 was obtained by G. Huisken [39], who proved that every closed convex hypersurface evolving by mean curvature flow in Euclidean space converges to a sphere after rescaling. We will mention in the following the other main correspondences and differences between the two flows.

3.3 Short Time Existence and Singularity Formation When written in coordinates, the Ricci flow is a parabolic system of partial differential equations for the components of the metric. There exists a standard theory giving short time existence of solutions for systems which are strictly parabolic; however, the Ricci flow does not completely fit into this framework since for this system the parabolicity is not strict. Nevertheless, using the special structure of the equations, Hamilton was able to prove short time existence for the Ricci flow, as stated in the next result [30]. Theorem 3.2 Given a closed manifold and a smooth initial metric, the Ricci flow has a unique smooth solution in a time interval Œ0; t0 / for some t0 > 0 . The original proof by Hamilton [30] was rather difficult and used a sophisticated version of the implicit function theorem due to Nash and Moser. Shortly afterwards, De Turck [24] gave a simpler proof, which exploited an equivalent formulation of the flow where the parabolicity becomes strict. For more details about these matters, one can consult [30, Sects. 4–6], [33, Sect. 6], [17, Sects. 3.1–3.4]. A typical feature of parabolic problems is that boundedness of the solution implies boundedness of its derivatives of any order. A property of this kind holds also for the Ricci flow and was first proved by W.X. Shi [52]. Theorem 3.3 Let g.t/ be a solution of the Ricci flow on a compact manifold M , defined for t 2 Œ0; t0 . Suppose that the associated Riemann tensor Rm is bounded

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on M , uniformly for t 2 Œ0; t0  . Then any derivative r k Rm, with k  1, is also bounded on M uniformly for t 2 Œ0; t0 . Using the above estimates, a standard continuation argument allows to prove that, whenever the maximal time of a solution is finite, the curvature necessarily becomes unbounded, as shown by the next theorem [33, Sect. 8]. Theorem 3.4 Each solution of the Ricci flow on a compact manifold can be extended to a maximal time interval Œ0; T/, with T  C1. If T is finite, then necessarily lim sup M.t/ D C1; t!T

where M.t/ is the maximum of the norm of the Riemann curvature tensor at time t. We describe the above behavior by saying that the flow becomes singular at time T. Such a behavior is very frequent on compact manifolds, as it can be seen for instance from Hamilton’s Theorem 3.1 in the positive Ricci case. However, there are also examples where the Ricci flow is defined for all positive times, like compact quotients of the hyperbolic space which give rise to homothetically expanding solutions. The examples of the previous sections show that the behavior of the flow as a singularity is approached can be very different depending on the cases considered. In a shrinking soliton, and in a general compact solution with positive Ricci curvature, see Theorem 3.1, the metric tends to zero and the curvature becomes unbounded everywhere, so that we can say that the whole manifold becomes singular and collapses to a point. In the case of a neckpinch singularity, instead, the curvature becomes unbounded only in a part of the manifold, while on the rest remains regular even at the singular time.

3.4 Evolution of Curvature, Preservation of Positivity As the metric on a manifold evolves by Ricci flow, the Riemann curvature tensor also evolves and satisfies an equation which can be computed explicitly and has the form @ Rm D  Rm C Q.Rm/: @t

(3.5)

Here  D g.t/ is the Laplace operator associated to the evolving metric g.t/, while Q.Rm/ is a tensor which is a quadratic function of Rm. Its explicit expression can be found for instance in [30] or in the notes of G. Besson in this volume. From the evolution equation for the Riemann tensor one can easily derive equations satisfied by the Ricci tensor and other quantities. The equation satisfied by the scalar

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curvature has the following particularly simple expression @R D R C 2jRicj2 : @t

(3.6)

Using the parabolic maximum principle one can obtain several invariance results for the positivity of curvature under Ricci flow, see [17, Sect. 6], [33, Sect. 5]. For example, Eq. (3.6) immediately implies that the minimum of the scalar curvature on a compact manifold evolving by Ricci flow is nondecreasing in time. As a consequence, if the initial metric has positive scalar curvature, then the same holds at all positive times. More precisely, the strong maximum principle says that, if the initial metric has nonnegative scalar curvature and is not Ricci flat, then the solution has strictly positive scalar curvature everywhere for all positive times. In addition to the usual maximum principle for scalar functions evolving by parabolic equations, see e.g. [28, Sect. 7.1.4], there are maximum principles for systems of reaction-diffusion equations which ensure the invariance of sets satisfying suitable conditions. Some of these statements, which are due to Hamilton, are particularly useful for the study of geometric flows. A first version, see [30, Theorem 9.1], gave a criterion for the preservation of the positivity of a 2-tensor. We state here a more general formulation which was proved in [31], see also [20, Chap. 10]. Theorem 3.5 Let .M ; g.t// be a Riemannian manifold evolving by Ricci flow and let F be a time dependent section of a tensor bundle E on M . Suppose that F satisfies the system @F D F C ˚.F/ @t

(3.7)

for some function ˚ mapping each fiber of E into itself. Let Z be a closed subset of E which is invariant under parallel translation and such that its intersection with each fiber is convex. If Z is invariant in each fiber under the ordinary differential system dZ=dt D ˚.Z/, then Z is also invariant for system (3.7). That is, if F belongs to Z at a given time, it also belongs to Z for all later times. The above maximum principle for tensors is a fundamental tool for the analysis of the Ricci flow, and its application relies on the study of the ordinary differential equation dtd Rm D Q.Rm/ associated to (3.5). In the three dimensional case, it is enough to study the corresponding equation for the Ricci tensor, which can be obtained by taking the trace of (3.5). It can be proved, see [30] or [8, Chap. 6], that if the Ricci tensor is diagonal at the initial time with respect to a given basis, it remains so along the evolution by the ODE. In addition, the sectional curvatures

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; ;  in the principal directions evolve according to the system 8 ˆ ˆ d D 2 C  ˆ ˆ ˆ < dt d D 2 C  ˆ dt ˆ ˆ ˆ d ˆ : D  2 C : dt

(3.8)

The following result then follows as an easy application of Theorem 3.5. Theorem 3.6 Let g.t/ be a solution to the Ricci flow on a closed three-dimensional manifold M . (i) If the initial metric has positive sectional curvature, then the same holds at any positive time. (ii) If the initial metric has positive Ricci curvature, then the same holds at any positive time. Proof Both properties describe a convex cone in the space of 2-tensors which is invariant under parallel translations. Positive sectional curvature corresponds to  > 0;  > 0;  > 0, a property which is preserved by the ODE (3.8) since the expressions in the right hand side are positive. Positive Ricci means instead  C  > 0;  C  > 0;  C  > 0. From (3.8) we deduce d . C / D 2 C 2 C . C /  . C /: dt Thus, if  C  is positive, it remains so for all later times. The other two expressions are treated similarly. Thus the maximum principle for tensors can be applied to deduce the desired properties. It should be pointed out that the above invariance properties are peculiar of the three-dimensional case. In higher dimensions, neither positive Ricci nor positive sectional curvature is preserved. Properties valid in any dimension are the preservation of the positivity for the scalar curvature and for the curvature operator. It is important to study the limiting cases of Theorem 3.6. Using a strong version of the maximum principle for tensors [31, Sects. 8–9], Hamilton proves the following statement: Theorem 3.7 Let g.t/, with t 2 Œ0; T/, be a solution to the Ricci flow on a complete, connected, not necessarily compact, three-dimensional manifold M . If .M ; g.t// has nonnegative sectional curvature, and there is at least a point P 2 M where one sectional curvature vanishes at a positive time t, then M splits as a product with a flat factor, that is .M ; g.t// D .M0 .MQ; gQ .t//, where M0 is a flat one-dimensional factor and .MQ; gQ .t// is a two-dimensional solution to the Ricci flow with strictly positive sectional curvature for all t > 0. The same result holds if .M ; g.t// has

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nonnegative Ricci curvature, and the Ricci curvature vanishes at some point and positive time.

3.5 Curvature Properties Which Improve In addition to the invariance properties described in the previous section, there are some remarkable results showing that when the flow becomes singular some curvature properties improve. A property of this kind is the crucial step in the proof of Hamilton’s first result [30] on manifolds with positive Ricci curvature. To show this, we follow the simpler approach of the paper [31, Sect. 5], see also [8, Chap. 6]. Theorem 3.8 Let g.t/ be a solution to the Ricci flow on a closed three-dimensional manifold M with positive Ricci curvature. Denote by      the sectional curvatures in an orthonormal frame which diagonalizes Ricci, and let ı; C be positive constants such that the initial metric satisfies at every point  C   2ı;

.  /1Cı  C. C /:

(3.9)

Then the same inequalities hold on .M ; g.t// for all t > 0 such that the flow exists. Proof Let us check that the two inequalities in (3.9) define a set which satisfies the hypotheses of the maximum principle for tensors. In general, any set defined by inequalities on the sectional curvatures ; ;  is invariant under parallel translation. We therefore only need to check the convexity and the invariance with respect to the ODE (3.8). Recall that, if A is a symmetric matrix, then its smallest and largest eigenvalues ;  are given by  D min hAv; vi; jjvjjD1

 D max hAv; vi: jjvjjD1

It follows that  is a concave function of A, being the infimum of linear functions. Similarly,  is a convex function of A. In addition, the trace  C  C  is a linear function of A since it coincides with the sum of the elements on the diagonal. Thus, we also obtain that  C  D . C  C /   is a concave function of A. These properties show that each of the two inequalities in (3.9) defines a convex set, and so do the two together. Let us now show that the first inequality  C   2ı, with ı > 0, defines an invariant set under the ODE. This is equivalent to the property  C  D 2ı

H)

d d . C /  2ı: dt dt

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We therefore suppose that  C  D 2ı. Observe that this is only possible if ı  1. We compute from (3.8) d . C   2ı/ D 2 C 2  2ı C . C   2ı/ dt D 2 C 2  2ı D .1  ı/.2 C 2 / C ı.  /2  0: Thus, the set defined by the first inequality in (3.9) is invariant. Again from (3.8), we find d 2 C 2 1 ln. C / D C   . C / C   .1 C ı/: dt C 2 In addition, d  2 C   2   ln.  / D D  C     : dt  From this we deduce d Œ.1 C ı/ ln.  /  ln. C /  0: dt This shows that the ratio .  /1Cı =. C / is decreasing in time, and that the set defined by (3.9) is invariant under (3.8). The assertion follows. Observe that condition (3.9) is always satisfied on the initial metric for suitable constants ı; C, by the positivity of the Ricci tensor and by compactness. However, when the singular time is approached and the curvature becomes unbounded, the second inequality in (3.9) has important consequences. We see that the difference   is only allowed to grow at a lower rate than the sum C. This implies that, if the three curvatures become unbounded, their ratio must tend to one. After justifying that the curvatures blow up everywhere on M as the singular time is approached, and that a smooth limit of the rescaled flow exists, Hamilton obtained in this way that the limit has  D  D  at each point, a property which implies constant curvature on M . This gives an outline of the strategy of the proof of Theorem 3.1. Using the maximum principle one can prove further estimates which yield the improvement of some curvature properties near the singular time. An important example is given by the next result, usually called Hamilton-Ivey pinching estimate and which was proved in [33, Theorem 24.4] and [43]. We follow here the presentation of [8, Sect. 6.2]. Theorem 3.9 Let g.t/ be a solution of the Ricci flow on a closed three-manifold M and let R0 be the minimum of the scalar curvature at time 0. Then there exists a function W ŒR0 ; C1/ ! .0; 1/ such that .r/=r ! 0 as r ! C1 and such that

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the smallest sectional curvature  at any point and time satisfies    .R/:

(3.10)

Proof Up to a homothety in our solution, we can assume that the initial metric satisfies R D 2. C  C /  1. Then this inequality also holds for positive times, as it follows from (3.6) and the maximum principle. Let us now consider the function f .x/ D x ln x  x, whose derivative is f 0 .x/ D ln x. We have that f is convex and strictly increasing for x 2 .1; C1/, and therefore one-to-one from .1; C1/ to .1; C1/. Let us denote by W .1; C1/ ! .1; C1/ its inverse. Then is increasing, concave and satisfies

0 .y/ D

1 : ln. .y//

(3.11)

In addition, lim

y!1

.y/ x D lim D 0: x!1 f .x/ y

Let us consider the set of 3  3 symmetric matrices defined by the inequalities 

1  C  C    ;    . C  C / 2

 (3.12)

on their eigenvalues     . As observed in the proof of the previous theorem,  is a concave function while  C  C  is linear. From the concavity of we deduce that the set defined above is convex. We claim that it is also invariant under the ODE (3.8). The first condition corresponds to R  1, which we already know to hold. To check the invariance of the second inequality, assume that 1 CC  ; 2

 D  . C  C /:

Then  < 1 and  C  D f ./   D  ln./ > 0: We also have ln. . C  C // D ln./ D 

C : 

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It follows, using (3.8), (3.11) d  d d ΠC . C  C / D   . C  C / dt dt  C  dt D 2 C  

.2 C 2 C  2 C  C  C / C

D .  /  

2 C 2 : C

We recall that  < 1 and  C  > 0, which also implies   12 . C / > 0. Thus the above expression is positive, showing the invariance of the set defined by (3.12). Intuitively speaking, the previous theorem says that when the scalar curvature becomes large (that is, when the singular time is approached) the negative sectional curvatures, if there are any, become negligible compared to the other ones. In fact, if we consider a sequence of points approaching the singular time such that  < 0 and R D 2. C  C / ! C1, we have 2 2

.R/ jj D  4 ! 0:  2 CC R Thus, even if the sign of the curvature at the initial time is completely arbitrary, the asymptotic profile near the singularity necessarily has nonnegative curvature. This property will be stated in a more precise way later in these notes, when we introduce the rescaling of a solution near a singularity. Invariance properties for positive curvature as the ones of the previous section also hold for the mean curvature flow described in Sect. 3.2. For instance, convexity or positive mean curvature are invariant properties in all dimensions. There is also an analogue of the Hamilton-Ivey estimate for the mean curvature flow, which was proved in [41] and [56]. Unlike the Ricci flow case, the result for the mean curvature flow holds for general dimension, but requires the positivity of the mean curvature. Under this hypothesis, the smallest principal curvature satisfies a lower bound similar to (3.10), which implies that its negative part becomes negligible near a singularity.

3.6 Differential Harnack Inequality The classical Harnack inequality for elliptic equations is an estimate controlling the oscillation of positive solutions. We recall the statement in the case of the Laplace equation, see e.g. [29, Theorem 2.5].

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Theorem 3.10 Let A  Rn be open and let ˝ be any bounded set such that ˝  A. Then there exists a constant C > 0, depending only of ˝ and A, with the following property: given any nonnegative function u 2 C2 .A/ such that u D 0 in A, we have sup  C inf u: ˝

˝

Similar estimates hold for parabolic equations; in this case, however, the supremum of the solution in a spatial domain at a given time is estimated above by a multiple of the infimum in the same domain at a later time, (see e.g. [28, Sect. 7.1.4b]). In [45] P. Li and S.-T. Yau introduced an alternative approach to Harnack inequalities in the parabolic case, showing that in certain cases they can be obtained from suitable estimates involving derivatives. Since this approach has been of fundamental importance in the study of the Ricci flow afterwards, we illustrate the main ideas in the “toy model” provided by the heat equation in Rn ; to avoid technicalities, we assume some a priori bound on the derivatives of the solutions which are stronger than needed for the validity of the result. Our exposition follows [13, Chap. 2]. Proposition 3.11 Let w 2 C2 .Rn  Œ0; T/ satisfy @w D w: @t Suppose in addition that w  0 and that its first and second derivatives are bounded. Then w satisfies D2 w C

Dw ˝ Dw w I  0I 2t w

nw jDwj2 @w C   0: @t 2t w

(3.13) (3.14)

Here D2 w denotes the Hessian matrix of w with respect to the space variables and I the identity matrix; inequality (3.13) means that the matrix at the left-hand side is positive semi-definite. Proof It is not restrictive to assume that w is greater than some positive constant; if this is not the case, we can replace w by w C " and then let " ! 0C . Let us set u.x; t/ D  ln.w.x; t//. Then it is easily checked that u is a solution of equation @u C jDuj2 D u: @t

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In addition, u is bounded together with its first and second derivatives. Given any @2 u unit vector  2 Rn , let us then set h.x; t/ D t @ 2 .x; t/. We have ˇ  ˇ2 ˇ @u ˇˇ h @h D C h  2hD.jDuj2/; Dhi  2t ˇˇD @t t @ ˇ 

h .1  2h/ C h  2hD.jDuj2/; Dhi: t

Since h. ; t/ ! 0 uniformly as t ! 0, the maximum principle implies that 2h.x; t/  1 for all .x; t/ 2 Rn  Œ0; T. By the definition of h and the arbitrariness of , this means D2 u 

1 I: 2t

On the other hand, an easy computation shows that D2 u D 

Dw ˝ Dw D2 w C w w2

and this proves (3.13). Taking the trace of the left-hand side of (3.13), we obtain w C

nw jDwj2   0; 2t w

which implies (3.14), since w solves the heat equation. Inequalities of the form (3.13)–(3.14) are often called differential Harnack inequalities. The connection with the classical Harnack inequality is explained by the following result. Corollary 3.12 Let w be as in the previous proposition. Then w satisfies   n=2  t1 jy  xj2 w.y; t2 /  w.x; t1 / exp  t2 4.t2  t1 / for all x; y 2 Rn , t2 > t1 > 0. Therefore, given any bounded set ˝  Rn and t2 > t1 > 0, we have max w. ; t1 /  C min w.t2 ; / ˝

˝

(3.15)

for some constant C > 0 depending on t1 ; t2 and the diameter of ˝ but not on w.

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Proof Let us set .s/ D x C

s  t1 .y  x/; t2  t1

s 2 Œt1 ; t2 :

Then we have, using (3.14), d w..s/; s/ D @t w C Dw P ds  @t w  

jDwj2 wjP j2  w 4

nw wjP j2  : 2t 4

It follows that  ln

w.y; t2 / w.x; t1 /



Z

t2

d ln w.s; .s// ds ds  Z t2  jy  xj2 n    ds 2t 4.t2  t1 /2 t1   t2 jy  xj2 n ;  D  ln 2 t1 4.t2  t1 /

D

t1

which proves our statement. After the work [45] by Li and Yau, Hamilton developed extensively this approach for various geometric evolution equations. In particular, for the Ricci flow he obtained the following result [32]. Theorem 3.13 Let .M ; g.t// be a solution to the Ricci flow, defined for t 2 Œ0; T/, which is either closed or complete with bounded curvature, and has nonnegative curvature operator. Then, for any vector field V and any time t 2 Œ0; T/ we have 1 @R C R C 2hrR; Vi C 2Ric.V; V/  0: @t t

(3.16)

The above result is sometimes called “trace differential Harnack inequality” for the Ricci flow, because it is obtained taking the trace of a more general tensor inequality, similarly to what we have described above for the heat equation in Rn . Integrating along a suitable path in space time as in Corollary 3.12, one obtains the following result.

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Corollary 3.14 Under the same hypotheses as in the previous theorem, given any P1 ; P2 2 M and 0 < t1 < t2 , we have R.P2 ; t2 / 

2 t1  d R.P1 ; t1 / e 2.t2 t1 / t2

where d is the distance between P1 and P2 at time t1 . Differential Harnack estimates for geometric flows are elegant and mysterious at the same time. They are strictly related to special solutions of the flow, where they hold identically as equalities, similarly to the case of the heat equation in Rn , where they hold as equalities on the heat kernel. Some geometric interpretations that give a deeper insight in these inequalities can be found in [14]. A detailed general exposition can be found in [48]. Harnack estimates have also been obtained for the mean curvature flow [34] and for more general curvature flows of immersed manifolds [1].

3.7 The Intuitive Picture of the Flow with Surgeries To motivate the analysis of the following sections, we give here an intuitive description of the strategy of proof of the Poincaré and Thurston conjecture using the Ricci flow with surgeries. We follow Hamilton’s original approach [31], where surgeries are performed shortly before the singular time, which in our opinion is slightly easier to picture than Perelman’s one [50], where surgeries are performed exactly at the singular time. Let us consider an arbitrary closed three-dimensional manifold M , and suppose that we want to study its possible topology. It is not restrictive to assume that M has a differentiable structure, and some smooth Riemannian metric g0 chosen arbitrarily. Then, we let the metric g0 evolve by the Ricci flow and study the behavior of the solution. The aim is to show that the metric g.t/ eventually converges, up to rescaling, to some limit that can be explicitly described. We then obtain that the initial manifold M has to be diffeomorphic to one of the possible limits of the flow. This strategy works very well in the case of Hamilton’s first result [30]. Here we have the additional assumption that the initial metric has positive Ricci curvature, and the only possible limits are the sphere and its quotients. However, for more general initial metrics, neckpinch singularities can occur and the smooth Ricci flow does no longer give a global information on the manifold. In a neckpinch singularity, in fact, we can hope to describe the structure of the region were the curvature becomes unbounded, but we have no knowledge of the remaining part of the manifold, which can have arbitrary topology. To overcome the difficulty, Hamilton proposed a way to continue the flow after singularities using a surgery procedure. Consider a three-manifold which develops a neckpinch singularity, where the region with the largest curvature looks like a

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portion of a cylinder with radius shrinking to zero as the singular time approaches. Such a cylindrical region is usually called a neck. The strategy is to stop the flow shortly before the singular time, remove the neck and fill smoothly the two remaining holes with two 3-balls. Such a modification is called a surgery, and in general it can change the topological type of the manifold. However, the surgery defined in this way is the reverse of a standard operation in algebraic topology called connected sum. In particular, the possible topological changes of our manifold under surgery can be precisely described. After the surgery, the flow can be restarted until the next singularity occurs, and then other surgeries are performed. At each surgery time, we are allowed to discard connected components of the manifold of known topology. Now suppose that we can show that, after a finite number of surgeries, all the remaining components of the manifold are diffeomorphic to one of the eight model manifolds allowed by Thurston conjecture. This implies that the initial manifold can be obtained performing a finite number of connected sums on these model geometries, and proves the validity of the conjecture. In particular the Poincaré conjecture is obtained, since the only simply connected manifold allowed by Thurston conjecture is the sphere. In order to implement this program, one needs to prove that the singular behavior of the Ricci flow in three dimension must be, roughly speaking, of one of two kinds described above, namely: (i) Either the curvature becomes large on the whole manifold, and the manifold converges to a sphere up to rescaling, or (ii) the curvature becomes unbounded only in a part of the manifold, which becomes asymptotically close to a portion of a shrinking cylinder. In case (i) the flow is stopped, while the other connected components of the manifold, if any, continue their evolution. In case (ii) the flow is stopped shortly before the singular time, the neck is removed by a surgery, and the flow is restarted. Therefore, a crucial part of the implementation of the program is the analysis of the possible asymptotic profile of the singularities. This will be the object of the remainder of these notes. The other part of the program, consisting of showing that after finitely many surgeries we are left with components which satisfy Thurston conjecture, is outlined in M. Boileau’s notes in this volume. Before passing to more precise statements in the next sections, let us add two important details to the intuitive picture given above. A first observation is that, in order to really increase the lifespan of the solution, we must perform the surgery in such a way that the maximum of the curvature on the manifold is substantially decreased after removing the necks. However, if our necks are close to a portion of a cylinder, as in the crude description above, the curvature on the boundary of the neck is comparable to the one in the interior, and we cannot claim that the surgery decreases the curvature. Instead, our necks should be only diffeomorphic to a cylinder, but the curvature on the boundary should be much smaller than the one in the middle part. Intuitively speaking, we should think of them as long tubes, with a very small radius in the middle region which becomes

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slowly larger as we move towards the ends. To obtain such a picture, one needs to prove that the points where the curvature is large enough, but not necessarily close to the maximum value, possess a neighborhood almost isometric to a portion o f a cylinder. By gluing together all these neighborhoods, we obtain a neck with the desired properties which covers a whole region with large curvature but has much smaller curvature on its boundary. The other caveat is that there is a third possible singular behavior in addition to (i) and (ii) described above. In fact, in the case of the degenerate neckpinch described in Sect. 3.2, the region with the largest curvature is not cylindrical, but it is instead diffeomorphic to a ball. However, the surgery procedure can be adapted to this case too. In fact, it is possible to show that the spherical region with the largest curvature is surrounded by a neck along which the curvature gradually decays. Then we can remove these two regions, which together are diffeomorphic to a ball, and fill the remaining hole with another ball with smaller curvature. In this case the surgery is topologically trivial, but it again reduces the maximum of the curvature and it allows to restart the flow to obtain a solution defined in a longer time interval. The above description should be kept in mind in the following sections, to understand the goal of the analysis of the singularities.

3.8 Rescaling Around Singularities To study the behavior of the solutions of the Ricci flow when the curvature becomes unbounded one can use rescaling procedures which are common also for other kinds of PDEs. We will describe the technique in an informal way because the rigorous statements are rather technical, see [33, Sect. 16]. Let us first observe that the Ricci flow is invariant under parabolic rescalings, that is, if we dilate a solution by a factor  > 0 in space and 2 in time, we obtain another solution of the flow, which has the norm of the curvature jRmj reduced by a factor 2 . Suppose now that we have a solution .M ; g.t// of the Ricci flow which becomes singular as t ! T. We can consider a sequence of rescalings with larger and larger factors near the singular time and then take a limit which describes, intuitively speaking, the singular profile of the original solution. More precisely, let us take a sequence of points Pj 2 M and times tj such that tj " T and in addition jRm.P; t/j  CjRm.Pj ; tj /j

8P 2 M ; t 2 Œ0; tj 

for some constant C  1 independent of j. For any j  1 we now rescale our flow p by a factor j , where j D jRm.Pj ; tj /j. In addition, we take Pj to be the origin of the rescaled flow and we translate the time so that tj becomes zero. Then the j-th flow is defined for t 2 Œ2j tj ; .T  tj /2j . Observe that the initial endpoint of the time interval tends to 1 at j ! 1; the final endpoint is positive, and it can be proved that it stays bounded away from zero for all j. By construction, each rescaled flow satisfies jRmj  C everywhere at all times t  0. It is possible to show that

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this curvature bound ensures the existence of a converging subsequence, provided the rescaled flows also satisfy an injectivity radius bound. Theorem 3.15 Let .M ; g.t// a solution of the Ricci flow which becomes singular as t ! T, and let us consider a family of rescaled flows defined as above. Suppose in addition that the injectivity radius of our manifold satisfies the estimate c inj.P; t/  p ; maxM jRmj. ; t/

8P 2 M ; t 2 Œ0; T/

(3.17)

for some c > 0. Then a subsequence of the rescaled flows converges uniformly on O ; gO .t//, which is a solution to the Ricci flow and is defined compact sets to a limit .M in an interval of the form .1; T  /, with T  > 0 (possibly infinite). If n D 3 then the limit flow has nonnegative sectional curvature at every point and satisfies the improved differential Harnack estimate @R C 2hDR; Vi C 2Ric.V; V/  0: @t

(3.18)

For the proof of the first part of this statement, see [33, Sect. 16]. The assertion concerning the sectional curvature can be obtained from Theorem 3.9; in fact, the right-hand side of (3.10) disappears in the rescaling procedure due to the sublinearity of . Observe also that in three dimensions positive sectional curvature is equivalent to positive curvature operator. Thus the limit flow satisfies the Harnack inequality (3.16) where the R=t term can be replaced by R=.t  t0 / with t0 arbitrarily small since the solution is defined in .1; T  /. Thus, letting t0 ! 1, this term vanishes and we obtain the improved inequality (3.18). In [33, Sect. 16] one can also find a precise definition of the “convergence on compact sets” mentioned in the statement. In particular, even if the rescaled flows are all compact, their diameter can go to C1, and thus the limit flow can be noncompact. The typical example is the neckpinch of Sect. 3.2, where the limit flow is an infinite cylinder S2  R. Hamilton then proved the following classification results of the possible structure of the limit flow in dimension 3. Theorem 3.16 Let g.t/ be a solution of the Ricci flow on a closed three-manifold M . Suppose that the flow becomes singular as t ! T and that we have an injectivity radius estimate of the form (3.17). Then it is possible to choose the sequence .Pj ; tj / in the above construction in such a way that the limit flow is one of the following (or a quotient under a finite group of isometries) (i) the shrinking sphere S3 , or (ii) the shrinking cylinder S2  R, or (iii) ˙  R, where ˙ is the “cigar” soliton described in Sect. 3.2. The above theorem is given at the end of the paper [33] and the proof uses all the properties of the limit flow which we have mentioned before. Although such a result

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already gave strong restrictions on the possible structure of the singularities, there remained two unsatisfactory aspects. One was the lack of a general argument which could provide the injectivity radius estimate needed in the theorem. The second problem regarded case (iii): if such a limit could occur then it would represent a fatal obstruction to Hamilton’s program, because there is no clear way to do surgery on a singularity which exhibits such a profile. Hamilton conjectured that case (iii) cannot occur, but did not succeed in proving this. We will see in the next sections how Perelman’s new results have solved both of these difficulties.

3.9 Perelman’s Monotonicity Formula In [49, Sect. 3] Perelman introduced the following functional. Let M be a closed n-dimensional manifold. Given a metric g on M , a function f W M ! R and a positive number , consider Z W .g; f ; / D

M

Œ.jrf j2 C R/ C f  n.4/n=2 ef dg ;

known as Perelman’s W -entropy functional. Define also, for fixed g and ,  Z .g; / D inf W .g; f ; / W f such that

M

.4/

n=2 f



e dg D 1 :

Then the following result holds. Theorem 3.17 Let g.t/ be a solution of the Ricci flow for t 2 Œt0 ; t1  on a closed manifold M , and let .t/ D Nt  t for some Nt > t1 . Let f W M  Œt1 ; t2  ! R satisfy @f n D f C jrf j2  R C : @t 2 Then dW D dt

Z M

ˇ ˇ ˇ 1 ˇˇ2 n ˇ 2 ˇRij C ri rj f  gij ˇ .4/ 2 ef d: 2

(3.19)

In addition, the quantity .g.t/; .t// is nondecreasing in t for t 2 Œt0 ; t1 . The above result, known as Perelman’s entropy monotonicity formula, has important applications to the analysis of singularities of the Ricci flow. Let us introduce the notion of local collapsing. Definition 3.18 Let .M ; g.t// be a solution of the Ricci flow for t 2 Œ0; T/, with T finite. We say that the solution is locally collapsing at time T if there exists a sequence of times tk " T, of points Pk 2 M and of radii rk > 0 such that frk g is

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bounded and such that, if we denote by Bk the ball of center Pk and radius rk with respect to the metric g.tk /, we have that jRmj.P; tk /  rk2 for all P 2 Bk and that Vol .Bk /=rkn ! 0 as k ! 1. Roughly speaking, collapsing means that we can find metric balls where the volume ratio is arbitrarily small. This is due to an increasingly smaller injectivity radius, which implies that the balls actually cover many times the same tiny portion of the manifold. The hypothesis that jRmj.P; tk /  rk2 on Bk shows that the smallness of the injectivity radius is not due to the size of the curvature. Therefore, for instance, a family of shrinking n-dimensional spheres, with n > 1, is not collapsing. Instead, a family of manifolds of the form M D S1r.t/  M 0 , where S1r.t/ is a one-dimensional circle with radius r.t/ ! 0 and M 0 is any fixed .n  1/dimensional manifold, is collapsing: the shrinking one-dimensional factor is flat and does not influence the curvature, but the injectivity radius and the volume ratio go to zero as r.t/ ! 0. The intuitive expectation is that such a behavior should not occur in the Ricci flow, since a flat factor would stay constant and not shrink. Indeed, Perelman shows that the monotonicity of W prevents the collapsing described above, and he obtains from Theorem 3.17 the following crucial result [49, Sect. 4]. Theorem 3.19 If g.t/ is a solution of the Ricci flow for t 2 Œ0; T/ on a closed manifold M , then .M ; g.t// is not locally collapsing at time T. To prove this result, Perelman shows that if the flow is collapsing at time T, then .g.tk /; rk2 / ! 1, by plugging suitable functions f in the functional W . On the other hand, by Theorem 3.17, .g.tk /; rk2 /  .g.0/; tk C rk2 /, which cannot be arbitrarily small, and this gives a contradiction. As mentioned before, the collapsing behavior is related to the smallness of the injectivity radius at the points .Pk ; tk /, see e.g. [55, Sect. 8.4]. In particular, if the solution is not locally collapsing, then it also satisfies the injectivity radius estimate required in Theorem 3.16. Thus, Perelman’s result ensures that the injectivity radius estimate is always satisfied. Theorem 3.19 also allows to exclude that the cigar ˙  R is obtained as limit of rescaled flows. In fact, using (3.3) one can check that the metric on the cigar ˙ is locally collapsing. Since the collapsing property is invariant under rescaling, ˙  R cannot occur as the limit of the rescalings of a noncollapsed solution. The noncollapsing property of the Ricci flow allows to exclude case (iii) of Hamilton’s Theorem 3.16. This result, however, does not suffice yet to define a flow with surgeries. For this purpose, we need to know that, when the singular time is approached, all points of the manifold with curvature larger than a certain threshold lie in regions that can be removed by the surgeries. In this way, we know that the manifold after surgeries has bounded curvature, so that the flow can be restarted and exists for some given time before possible new singularities occur. We therefore need to obtain a more detailed description of the singular regions than the one provided by Theorem 3.16, which only describes the behavior around suitable

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sequences of points with large curvature. We will see this in the remaining part of these notes.

3.10 L -Distance and Reduced Volume In [49, Sect. 7] Perelman introduces some geometric quantities which provide further powerful results for the analysis of the singularities of the Ricci flow. Let .M ; g.t// be a solution of the Ricci flow, for t 2 Œ0; T. For the purposes of this section, it is convenient to reverse the time direction, and consider the variable  WD T  t. The metric then satisfies the backward Ricci flow dd g./ D 2 Ric g./. Given a curve  W Œ1 ; 2  ! M , with 0  1 < 2  T, we define the L -length of  as Z L . / D

2 1

p 2 ŒR..// C j./j P  d:

(3.20)

2 Here R..// and j./j P are computed with respect to the metric g./. Functionals of this type are classical in Calculus of Variations, see e.g. [13, Chap. 6] and the references therein. The L -length can be regarded as a generalization of the usual energy functional for curves on a Riemannian manifold, see e.g. [25, Sect. 9.2], whose minimizers are geodesics, and it shares some basic properties. Given any pair of points p; q 2 M , we can minimize the L -length over all the curves such that .1 / D p, .2 / D q. By standard methods, it can be proved that the minimum exists, and it is called the L -distance between . p; 1 / and .q; 2 /. The curve  achieving the minimum is not necessarily unique; any such curve is called an L -geodesic and satisfies a suitable ordinary differential equation. It should be noted that the L -distance is not necessarily positive unless the evolving metric has positive scalar curvature. It is convenient to fix the initial endpoint p 2 M and 1 D 0 and analyze the properties of the L -distance as a function of the final endpoint. Namely, we define L.q; / to be the L -distance between . p; 0/ and .q; / for any given q 2 M and  > 0. In [49, Sect. 7], several identities and inequalities relating the derivatives of L and the minimizers are derived, which are inspired by the corresponding first and second order conditions satisfied by the classical geodesics. As in the case of the ordinary distance from a given point, the function L is in general Lipschitz continuous but not differentiable everywhere; more precisely, it is not differentiable at those points .q; / for which the minimizing geodesic from . p; 0/ is not unique, see e.g. [13, Corollary 6.4.10]. Therefore, the differential inequalities satisfied by L must be understood in a suitable weak sense, for instance in the sense of barriers introduced by Calabi in [12], or the viscosity sense [23]. We recall two important inequalities derived by Perelman, see formulas (7.13) and (7.15) in [49].

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Theorem 3.20 (i) If we set l.q; / WD

1 p L.q; /, 2 

then l satisfies

n @l  l  jrlj2 C R  : @ 2

(3.21)

p N / WD 2  L.q; /, then LN satisfies (ii) If we set L.q; @LN C LN  2n: @

(3.22)

As an immediate consequence, we obtain Corollary 3.21 (i) Let us define Z V./ WD M

 n=2 el.q; / dg. / :

(3.23)

Then V is a nonincreasing function of . (ii) We have minM l. ; /  n2 for all  > 0. Proof We argue as if the function L were smooth, but the computations can be justified also in the case where the inequalities of the previous theorem only hold in a weak sense. To prove (i), let us set w.q; t/ D  n=2 el.q; / . We find, using (3.21),   n @l @w  w C wR D w   jrlj2 C l  C R  0: @ 2 @ When we compute the derivative of V./, we must take into account that the volume element on a solution of the backward Ricci flow evolves according to @ dg. / D R dg. / ; @ see e.g. [17, Lemma 3.9]. We conclude 0

V ./ D



Z M

 Z @w C Rw dg. /  w dg. / D 0; @ M

which proves the monotonicity of V. To obtain (ii), let us first estimate l.q; / in the case where  is small and q D p. If we compute the L -distance using the constant path   p and use the local boundedness of R, we easily obtain an upper bound of the form L. p; /  C 3=2 for a suitable C > 0. It follows that minM l. ; / < n2 for  enough small. Now, if apply

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the maximum principle backward in time to (3.22), we deduce that the minimum of N /  2n is a nonincreasing function of . On the other hand, by definition, L. ; o n ˚  N /  2n D 4 min l. ; /  n ; min L. ; M M 2 which is negative for small  > 0. Therefore, the above minimum stays negative for all  > 0, which proves (ii). The function V defined in (3.23) is called the reduced volume by Perelman. The monotonicity of the reduced volume gives an alternative argument to prove the noncollapsing property for the solutions of the Ricci flow. Roughly speaking, one can consider a sequence of points as in Definition 3.18 and consider for each k the reduced volume Vk obtained setting p D pk and  D k  t. Then, using the smallness of the standard volume of suitable balls around pk given by the collapsing assumption, it can be proved that Vk ./ becomes close to zero for suitably small . On the other hand, one can show that Vk .tk / is bounded away from zero, since  D tk corresponds to t D 0 and thus Vk .tk / can be estimated in terms of the behavior of the manifold in a neighborhood of a fixed regular time. The two properties together are in contradiction with the monotonicity of the reduced volume proved above, and this allows to prove a result similar to Theorem 3.19. It is interesting to remark that the monotonicity of the reduced volume stated in Corollary 3.21 has some similarity with a well-known result for the mean curvature flow, called Huisken’s monotonicity formula [40]. The proof of the two results, however, are not related, and the applications to the analysis of singularities are rather different in the two cases. The entropy monotonicity and the noncollapsing property described in the previous section have also some analogues in the mean curvature flow, see [2, 26].

3.11 Properties of -Solutions In the rest of these notes, we restrict ourselves three-dimensional manifolds. In Sect. 3.8 we have seen that the profile of the solution of the Ricci flow near a singularity can be studied by rescaling techniques. In particular, by taking the limit of a sequence of flows rescaled around points where the curvature becomes unbounded, one obtains an ancient solution which describes the singular profile and enjoys some special properties. The study of such ancient solutions is the first step to a more detailed analysis of the singularities which will enable the surgery construction. The results in this section are taken from [49, Sects. 11 and 12], [50, Sects. 1 and 3] We start with a definition. Definition 3.22 Given  > 0, a nonflat solution of the Ricci flow on a (possibly noncompact) three-dimensional manifold M is called a -solution if it satisfies the

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following properties (i) It is ancient, i.e., it is defined for t 2 .1; T/ for some T > 0. (ii) It has bounded curvature and nonnegative curvature operator at each fixed time t. (iii) The solution is -noncollapsed, in the sense that for any time t and any ball B of radius r in .M ; g.t// which satisfies jRmj. p; t/  r2 for all p 2 B, we have that Vol .B/  r3 . As we have observed in Sect. 3.8, any limit obtained by rescaling a given solution near a singularity satisfies properties (i) and (ii) above. Property (iii) follows from Theorem 3.19 and from the fact that the noncollapsing property is scaleinvariant. In addition, a -solution satisfies the stronger form of Hamilton’s Harnack inequality (3.18). Choosing V D 0 in that inequality, we obtain in particular that @R @t  0. This means that the scalar curvature is pointwise nondecreasing, and therefore any bound on R at a certain time holds also for all previous times. Examples of -solutions are the shrinking sphere, the shrinking cylinder or the Bryant soliton mentioned in Sect. 3.2. The product ˙  R, where ˙ is the cigar soliton, is not a -solution because it does not satisfy the noncollapsing property (iii). A product S2  S1 , where S2 is homothetically shrinking while S1 remains constant because it has no curvature, also violates property (iii), as it is seen by considering arbitrarily large negative times. There exist also more elaborate -solutions which are not solitons. In [50, Sect. 1.4], Perelman describes a compact -solution for t 2 .1; 0/, which is close to a shrinking sphere as t ! 0, while as t ! 1 it resembles a more and more eccentric oval. However, the analysis is simplified by the next result, see [49, Sect. 11.2], which associates to any -solutions a gradient shrinking soliton. Theorem 3.23 Let .M ; g.t// be a -solution of the Ricci flow, let p 2 M , t0 2 .1; T/ be fixed arbitrarily, and let l.q; / be the reduced length centered at . p; t0 / defined in Sect. 3.10. For any  > 0, let q./ 2 M be a point such that l.q; /  n=2, whose existence follows from Corollary 3.21. Then the rescalings of the metrics g.t0  / around the point q./ with factor  1 converge along a subsequence k ! 1 to a nonflat gradient shrinking soliton. The soliton obtained from the above theorem is called an asymptotic soliton of the -solution. It can be proved that the only possibilities for such a soliton are the following, up to quotients: (i) the asymptotic soliton is a shrinking sphere; (ii) the asymptotic soliton is a shrinking cylinder S2  R. In addition, case (i) only occurs if the -solution is itself a sphere. This also shows that the asymptotic soliton is unique, because the two possibilities are incompatible.

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An interesting property of noncompact -solutions concerns the asymptotic volume ratio, defined by Vol B. p0 ; r/ ; r!C1 r3

V D lim

where p0 is any fixed point in M and B. p0 ; r/ is the metric ball of radius r around p0 . Then, it can be proved that on a -solution we have V D 0 at each time. Such a result is not in contrast with the noncollapsing property; in fact, for a fixed p0 and arbitrarily large r, we cannot have the property jRmj  r2 in the ball B. p0 ; r/, and therefore the lower bound on the volume ratio in the noncollapsing property does not apply on such a ball. Intuitively speaking, the property that V D 0 implies that a noncompact -solutions cannot open up asymptotically like a cone, but rather like a paraboloid, as in the case of the Bryant soliton. A fundamental step in Perelman’s analysis of the singularities is the following compactness result modulo scaling for -solutions [49, Sect. 11.7]. Theorem 3.24 (i) For any r > 0, there exists a universal constant M D M.r/ such that, given any point and time . p; t/ in a -solution such that R. p; t/ D 1 and any other point q such that dg.t/ . p; q/ < r, we have R.q; t/  M. (ii) Given a sequence of -solutions f.Mk ; gk .t//g and points pk 2 Mk such that R. pk ; 0/ D 1, there exists a subsequence centered at . pk ; t/ which converges smoothly to a limit which is also -solution. (iii) There exists a constant C > 0 such that, on any  solution, we have the following derivative estimates: jrR. p; t/j2  CjR. p; t/j3 ;

j@t R. p; t/j  CjR. p; t/j2 ;

p 2 M ; t 2 .1; T/:

The above result is the main step towards a precise description of the structure of -solutions. We first give a formal definition of the notion of “almost cylindrical region” inside a manifold evolving by Ricci flow. Definition 3.25 Given a solution of the Ricci flow and " > 0, we say that . p0 ; t0 / is the center of an "-neck if, after setting Q0 D R. p0 ; t0 /, the parabolic neighborhood  . p; t/ W t0 

1 1 2  t  t0 ; dg.t . p; p0 /  0/ "Q0 "Q0



is "-close, after scaling with the factor Q, to a subset of a shrinking round cylinder. Here “"-close” means that the metric, and its derivatives up to a suitable order, differ from the corresponding ones of the cylinder by no more than ". The following result [49, 11.8], see also [44, Sect. 48] or [46, Theorem 9.93], states that on any -solution the points either lie at the center of a neck, or they belong to a compact region whose diameter satisfies an apriori bound.

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Theorem 3.26 For any " > 0 enough small, there exists C D C."/ > 0 with the following property. Take any -solution of the Ricci flow and denote by M" the points which are not at the center of an "-neck at some given time t0 . Then M" is compact. In addition, M" can be written as the union of at most two components M1 ; M2 which have the following properties, after setting Qi D R. pi ; t0 / for an arbitrary pi 2 Mi : 1=2

(i) the diameter of Mi is at most CQi , (ii) we have C1 Qi  R.q; t0 /  CQi , for any q 2 Mi . Examples of the possible structure of M" can be obtained by looking at the solutions described at the beginning of the section. The set M" is clearly empty on a shrinking cylinder. On a sphere, M" is instead the whole manifold if " is small enough, and thus it consists of single component, which satisfies (i) and (ii). The Bryant soliton is a more interesting example. As mentioned in Sect. 3.2, it consists of a rotationally symmetric metric on R3 where the curvature decreases as the distance from the origin increases. It turns out that any point sufficiently far from the origin lies at the center of an "-neck, so that M" is a ball of radius R" , with R" becoming large if " becomes small. Therefore, M" consists of a single compact component, which satisfies properties (i) and (ii). The other -solution described at the beginning, which is compact and becomes more and more oval as t ! 1, gives instead an example where M" consists of two components: they are the two opposite ends of the solution, while the central part consists of points which are all centers of a neck.

3.12 Canonical Neighborhoods and the Structure of Singularities The results of the previous section give an accurate description of the structure of -solutions. The next fundamental result [49, Sect. 12.1] shows that the same description extends to the regions with large curvature of an arbitrary solution of the Ricci flow in three dimensions. Theorem 3.27 Let .M ; g.t//, with t 2 Œ0; T/, be a solution of the Ricci flow on a closed three-dimensional manifold M . For any " > 0 there exists r0 > 0, only depending on " and the initial data, with the following property. Let . p0 ; t0 / be any point with t0  1 and R. p0 ; t0 /  r02 . Then, if we set Q D R. p0 ; t0 /, we have that the parabolic neighborhood n

o 2 1 . p; t/ W t0  ."Q/1  t  t0 ; dg.t . p; p /  ."Q/ ; 0 / 0

rescaled by a factor Q, is "-close to a suitable space-time subset of a -solution.

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The power of the above result lies in the fact that the curvature R. p0 ; t0 / is only required to be larger than some given threshold, but we do not need to assume, for instance, that R. p0 ; t0 / is the maximum of R at time t0 or is comparable with the maximum. Thus we have no apriori bound on the curvature in the parabolic neighborhood under consideration, which would easily yield a proof by compactness. Combining Theorems 3.26 and 3.27 one obtains a precise description of the possible structure of the solution near the points with large curvature. We introduce a further terminology. We say that a parabolic neighborhood inside a solution of the Ricci flow is an "-cap if each point lies at the center of an "-neck outside of a compact set satisfying properties (i) and (ii) of Theorem 3.26. Then we have the following result. Theorem 3.28 Let .M ; g.t// be a three dimensional solution of the Ricci flow. Then, for any " > 0 there exists r0 > 0, only depending on " and the initial data, such that each point . p0 ; t0 / with R. p0 ; t0 /  r02 has a parabolic neighborhood P satisfying one of the following properties (i) P is an "-neck. (ii) P is an "-cap. (iii) P has positive curvature and coincides with the whole manifold. A neighborhood satisfying one of the properties (i)–(iii) above is called a canonical neighborhood. The above result allows to define surgeries at the first singular time, as outlined in Sect. 3.7. After restarting the flow, in order to do the following surgeries, one must then repeat the previous analysis at the subsequent singularities. This is a highly nontrivial part of the procedure, because the estimates needed in this study are derived in the case of a smooth solution and one should justify them also in the case of a flow which has been modified by surgeries. Sections 4 and 5 of [50] are devoted to this delicate issue. The rest of [50] then studies the long time behavior of the flow with surgeries, leading to the proof of the Thurston conjecture. Some important aspects of this part are described in the notes of M. Boileau in this volume. Let us finally mention that a surgery procedure has also been defined for the mean curvature flow of suitable classes of hypersurfaces. More precisely, in [42] a flow with surgeries was constructed for hypersurfaces in RnC1 , with n  3, which are 2convex, i.e. the sum of the two smallest principal curvatures is positive everywhere. The procedure of [42] is inspired by Hamilton’s original approach [35] and the main part consists of the analysis of the singularities, which yields a similar picture to the one described here for the three-dimensional Ricci flow. The flow with surgeries allows to prove that any closed 2-convex hypersurface is diffeomorphic to a sphere or to a connected sum of Sn1  S1 . More recently, the procedure was extended to the case of surfaces in R3 with positive mean curvature [9]. An alternative derivation of these results, which uses techniques closer to Perelman’s ones for the Ricci flow, has been given in [37, 38].

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3.13 Bibliographical Notes A landmark date in the literature on the Ricci flow are the years 2002–2003 when Perelman’s papers appeared. While now many expository books and survey articles can be found, before that date the only references in the field were the original papers, especially Hamilton’s ones. A notable exception was the survey [14] which gave a nice overview of Hamilton’s program for geometrization and its progress at that time. Despite the vast literature appeared in the last decade, Hamilton’s original works still represent a fundamental reference in the field. In particular, the long paper [33], which gives a survey of the previous results and presents many original ones, can be recommended to anyone interested in the Ricci flow for the richness of ideas and the beauty of the exposition. A useful reference for the results on the Ricci flow before 2002 is the volume [16], which collects all the most relevant papers appeared until that time. Perelman’s papers [49–51] are famous not only for the historical relevance of the results, but also for the difficulty of their mathematical content. Even the experts in the field have required a long time of careful analysis before working out all the details of the arguments. After some time, three different detailed expositions of Perelman’s papers have appeared, namely the notes by Kleiner and Lott [44], the paper by Cao and Zhu [15], and the book by Morgan and Tian [46]. In particular, the notes [44] have had a great influence on the understanding of Perelman’s papers, because preliminary versions were posted on the web while the work was in progress. The references [15, 46] are self-contained, while the notes [44] are meant as a complement to the Perelman’s papers, to be read along with them. All these three references are a valuable source for Perelman’s results, and anyone who wants to learn these topics in detail is strongly advised to look at least at one of them. Nevertheless, interested readers should also absolutely read the original papers by Perelman; although the proofs are in most of the cases very difficult to understand, the main ideas are often easy to follow and the beauty of the results is stunning even without following all the details. While the three above mentioned references are mainly focussed on the proof of Poincaré conjecture, the two later books [6] and [47] treat in more detail the long time behavior of the flow with surgery and the proof of the full Thurston conjecture. We also mention the interesting commentary on Perelman’s proof by T. Tao [54]. In addition, there are other books on the Ricci flow which are not aimed at a presentation of the full Hamilton-Perelman’s theory, but which are excellent references for the basic results as well as some parts of that theory. A particularly rich and detailed source is provided by the series of books by B. Chow and many coauthors [17–22]. On the other opposite, the books by S. Brendle [8], by P. Topping [55] give short and clear expositions of some basic important aspects of the theory and are definitely recommendable for a beginner. The book by B. Andrews and C. Hopper [3] is another interesting reference for the various developments of the Ricci flow.

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References 1. B. Andrews, Harnack inequalities for evolving hypersurfaces. Math. Z. 217, 179–197 (1994) 2. B. Andrews, Noncollapsing in mean-convex mean curvature flow. Geom. Topol. 16, 1413– 1418 (2012) 3. B. Andrews, C. Hopper, The Ricci Flow in Riemannian Geometry (Springer, Berlin/Heidelberg, 2011) 4. S. Angenent, D. Knopf, An example of neckpinching for Ricci flow on SnC1 . Math. Res. Lett. 11, 493–518 (2004) 5. S. Angenent, J. Isenberg, D. Knopf, Degenerate neckpinches in Ricci flow. J. Reine Angew. Math. 709, 81–117 (2015) 6. L. Bessières, G. Besson, S. Maillot, M. Boileau, J. Porti, Geometrisation of 3-Manifolds (European Mathematical Society, Zürich, 2010) 7. C. Böhm, B. Wilking, Manifolds with positive curvature operator are space forms. Ann. Math. 167, 1079–1097 (2008) 8. S. Brendle, Ricci Flow and the Sphere Theorem (American Mathematical Society, Providence, 2010) 9. S. Brendle, G. Huisken, Mean curvature flow with surgery of mean convex surfaces in R3 . Invent. Math. 203, 615–654 (2016) 10. S. Brendle, R. Schoen, Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc. 22, 287–307 (2009) 11. R.L. Bryant, Ricci flow solitons in dimension three with SO(3)-symmetries. Available at www. math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf 12. E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J. 25, 45–56 (1958) 13. P. Cannarsa, C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control (Birkhäuser, Boston, 2004) 14. H.D. Cao, B. Chow, Recent developments on the Ricci flow. Bull. Am. Math. Soc. 36, 59–74 (1999) 15. H.D. Cao, X. Zhu, A Complete Proof of the Poincaré and geometrization conjectures application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10, 165–492 (2006) 16. H.D. Cao, B. Chow, S.C. Chu, S.T. Yau (eds.), Collected Papers on Ricci Flow (International Press, Boston, 2003) 17. B. Chow, D. Knopf, The Ricci Flow: An Introduction (American Mathematical Society, Providence, 2004) 18. B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow (American Mathematical Society/Science Press, Providence/New York, 2006) 19. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part I: Geometric Aspects (American Mathematical Society, Providence, 2007) 20. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part II: Analytic Aspects (American Mathematical Society, Providence, 2008) 21. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part III: Geometric-Analytic Aspects (American Mathematical Society, Providence, 2010) 22. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part IV: Long-Time Solutions and Related Topics (American Mathematical Society, Providence, 2015) 23. M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

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24. D.M. DeTurck, Deforming metrics in the direction of their Ricci tensors. J. Differ. Geom. 18, 157–162 (1983) 25. M. Do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992) 26. K. Ecker, A formula relating entropy monotonicity to Harnack inequalities. Commun. Anal. Geom. 15, 1025–1061 (2008) 27. J. Eels, J.H. Sampson, Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109– 160 (1994) 28. L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, 1998) 29. D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin/Heidelberg, 1983) 30. R.S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982) 31. R.S. Hamilton, Four–manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986) 32. R.S. Hamilton, The Harnack estimate for the Ricci flow. J. Differ. Geom. 37, 225–243 (1993) 33. R.S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in Differential Geometry (Cambridge, MA, 1993), vol. II (International Press, Cambridge, 1995), pp. 7–136 34. R.S. Hamilton, The Harnack estimate for the mean curvature flow. J. Differ. Geom. 41, 215– 226 (1995) 35. R.S. Hamilton, Four–manifolds with positive isotropic curvature. Commun. Anal. Geom. 5, 1–92 (1997) 36. R.S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7, 695–729 (1999) 37. R. Haslhofer, B. Kleiner, Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. (2013). Preprint 38. R. Haslhofer, B. Kleiner, Mean curvature flow with surgery (2014). Preprint, arXiv:1404.2332 39. G. Huisken, Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984) 40. G. Huisken, Asymptotic behavior of singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990) 41. G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999) 42. G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175, 137–221 (2009) 43. T. Ivey, Ricci solitons on compact three-manifolds. Differ. Geom. Appl. 3, 301–307 (1993) 44. B. Kleiner, J. Lott, Notes on Perelman’s papers. Geom. Topol. 12, 2587–2855 (2008) 45. P. Li, S.T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) 46. J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture (American Mathematical Society/Clay Mathematics Institute, Providence/Cambridge, 2007) 47. J. Morgan, G. Tian, The Geometrization Conjecture (American Mathematical Society/Clay Mathematics Institute, Providence/Cambridge, 2014) 48. R. Müller, Differential Harnack Inequalities and the Ricci Flow (European Mathematical Society, Zürich, 2006) 49. G. Perelman, The entropy formula for the Ricci flow and its geometric applications (2002). Preprint, arXiv:math/0211159 50. G. Perelman, Ricci flow with surgeries on three-manifolds (2003). Preprint, arXiv:math/0303109 51. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds (2003). Preprint, arXiv:math/0307245 52. W.X. Shi, Deforming the metric on complete Riemannian manifolds. J. Differ. Geom. 30, 223– 301 (1989)

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Chapter 4

Notes on Kähler-Ricci Flow Gang Tian

Abstract In these notes some aspects of the Analytic Minimal Model Program through Kähler-Ricci flow which was initiated by J. Song and the author are discussed. Some open problems will be also presented.

4.1 Introduction These notes are based on my CIME lectures ar Cetraro, 2010. In these notes, I will discuss some aspects of my joint works with others, particularly, J. Song and Z. Zhang on the Kähler-Ricci flow. We will also present some open problems on the Kähler-Ricci flow. Because of time and length, we are unable to include all the results on Kähler-Ricci flow. We refer the readers to other survey papers, e.g., [19], on some works which may not be covered in these notes. Let X be an n-dimensional compact Kähler manifold. We denote a Kähler metric by its Kähler form !, in local complex coordinates z1 ; ; zn , !D

p 1 giNj dzi ^ dNzj ;

where we use the standard convention for summation and .giNj / is the positive Hermitian matrix valued function given by giNj D g

@ @

: ; @zi @Nzj

The Kähler-Ricci flow was given as follows: @!Q t D  Ric.!Q t /; @t

!Q 0 D !0 ;

(4.1)

G. Tian () Beijing Normal University, Beijing, China Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Benedetti, C. Mantegazza (eds.), Ricci Flow and Geometric Applications, Lecture Notes in Mathematics 2166, DOI 10.1007/978-3-319-42351-7_4

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where !0 is any given Kähler metric and Ric.!/ denotes the Ricci form of !, i.e., in the complex coordinates above, Ric.!/ D

p 1 RiNj dzi ^ dNzj ;

where .RiNj / is the Ricci tensor of !. Many results in this paper are taken from my joint works with people mentioned above. I would like to thank them for sharing their thoughts and time with me in last few years.

4.2 A Sharp Local Existence for Kähler-Ricci Flow By the well-known local existence theorem of Hamilton-DeTurck, we know that there is a unique solution g.t/ for the Ricci flow with any given initial metric g0 . As shown first by R. Hamilton in the case of curvature tensor and by N. Sesum in general, such a solution exists so long as the Ricci curvature is bounded, that is, one can have g.t/ on an interval Œ0; T/ such that either T D 1 or limt!T jjRic.g.t//jjg.t/ D 1, where Ric.g.t// denotes the Ricci curvature of g.t/ and the norm is given with respect to g.t/. However, it is hard to determine T explicitly and T may even get close to 0 when one varies initial metrics. However, in the Kähler case, as shown by Z. Zhang and myself, the maximal interval for solution depends on the Kähler class of the initial Kähler metric !0 , which is a cohomological condition. In this section, we present this local existence theorem and outline its proof. The following theorem is due to Zhang and myself (see [27], also [25]). Theorem 4.2.1 Let X be a compact Kähler manifold. Then for any initial Kähler metric !0 , the flow (4.1) has a maximal solution !Q t on X  Œ0; Tmax /, where Tmax D supft j Œ!0   t c1 .X/ > 0g:1 This theorem was known in many special cases. In [1], Cao proved this theorem in the case that c1 .X/ is definite and proportional to the initial Kähler class. In the case that KX is nef, i.e., numerically effective, and the initial metric !0 is sufficiently positive, Tsuji proved in [29] the above theorem, that is, (4.1) has a global solution !Q t . Also in [2], the authors claimed a proof of a similar local existence result under certain extra technical assumptions.

1 This means that Œ!0   tc1 .X/ > 0 represents a Kähler class, where c1 .X/ denotes the 2 multiple of the first Chern class of X in the usual terminology.

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Corollary 4.2.2 Let X be a compact Kähler manifold with numerically positive canonical bundle KX . Then for any initial Kähler metric !0 , the flow (4.1) has a global solution !Q t for all t > 0. Proof Since KX is numerically positive, Œ!0   tc1 .X/ D Œ!0  C tc1 .KX / is ample for all t  0. Then this corollary follows from Theorem 4.2.1. For readers’ convenience, we outline a proof for Theorem 4.2.1 by essentially following the arguments in the proof of Proposition 1.1 in [27].2 For simplicity, we write T for Tmax . First we reduce (4.1) to a scalar flow: Choose a real closed .1; 1/ form representing c1 .X/ and a smooth volume form ˝ such that Ric.˝/ D . This ˝ is unique up to multiplication by a positive constant. p Write !t D !0  t for t 2 Œ0; T/ and !Q t D !t C 1 @@N u. Then one can easily show that !Q t satisfies (4.1) if u.0; / D 0 and u.t; / satisfies @u !Q n D log t ; @t ˝

!t C

p 1@@N u > 0:

(4.2)

Thus our main theorem follows from the following lemma. Lemma 4.2.3 For any smooth function u0 , there is a unique smooth solution u of (4.2) on X  Œ0; T/ with u.0; / D u0 and the following estimates: u  C .1 C t log.1 C t// and

@u  C.t C t1 /; @t

where C is a uniform constant depending only on !0 ,

(4.3)

and u0 .

Proof First we observe that for t sufficiently small, !t is a Kähler metric, so by the standard theory on parabolic equations, there is a unique solution of (4.2) for t small. Now we assume that (4.2) has a maximal solution u for t 2 Œ0; T 0 / for some 0 T  T. Let us establish the estimates in (4.3) for u. Set Z v.t/ D sup u.t; /  F.t/; X

t

F.t/ D 0

f .s/ ds;

where f .t/ is defined by ˇ  ˇ  ˇ .!0  t /n ˇ ˇ .x/ : ˇ f .t/ D log sup 1 C ˇ ˇ ˝ x2X

Then v.0/ D supX u and .!0  t /n @v  log  f .t/  0; @t ˝ 2

The flow equation in [27] is not the same as, but equivalent to, (4.1).

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so v.t/ is non-increasing and consequently, u.t; /  sup u0 C F.t/: X

Put A D 1 C max

0kn

 ˇ nk ˇ !0 ^ ˇ ˇ ˝

ˇ ˇ : ˇ

kˇ

Then by a direct computation, we have f .t/  log .1 C A .1 C t/n /  log.1 C A/ C n log.1 C t/:

It implies that F.t/  C.1 C t log.1 C t// for C D log 2n .1 C A/2 C n. This establishes the first estimate in (4.3). To prove the second estimate, we take t-derivative of (4.2) to get @ @u @u . / D !Qt . /  h!Q t ; i; @t @t @t where ! denotes the Laplacian of a Kähler metric ! and h!; i means the trace of

with respect to ! for a real .1; 1/-form . It follows



@u @ @u  u D !Qt t  u C n  h!Q t ; !0 i: t @t @t @t

(4.4)

Since h!Q t ; !0 i  0, we have



@u @ @u t  u  n t  !Qt t u nt @t @t @t By this and the Maximum Principle, we see that the maximum of t non-increasing, so we have t

@u  u  n t   inf u0 : X @t

@u @t

 u  nt is

(4.5)

Therefore, we have @u  C .t C t1 /: @t If T 0 D T, we have finished the proof of the main theorem of this section. So we may assume that T 0 < T  1, then we want to prove that u can extend across T 0 . For this purpose, we need to establish a prior estimates for u and its derivatives.

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This will be achieved by considering (4.2) with a good representative of c1 .X/ as follows: Choose > 0 such that T 0 C < T. By our definition of T, there is a

representing c1 .X/ such that !0  .T 0 C / > 0. Then for any t 2 Œ0; T 0 C , T0 C  t t !0 C 0 .!0  .T 0 C / / > 0: 0 T C

T C

p N for some real function f over X. Since Ric.˝/ D We can write D C 1 @@f , we have Ric.˝ / D , where ˝ D ef ˝. Now set !0  t



D

! ;t D !0  t



D !t C t

Then v D u  tf satisfies (4.2) on Œ0; T 0 / with

p N : 1 @@f

replaced by

:

 n p !Q t @v D log ; where !Q t D ! ;t C 1 @@N v > 0: @t ˝

(4.6)

Note that v.0; / D u0 . Clearly, it suffices to show that v extends across T 0 . We have seen that v and @v @t are uniformly bounded from above. Since ! ;t  c !0 > 0 on X  Œ0; T 0 C  for some constant c > 0, by applying the Maximum Principle to (4.6), we can easily see that jvj  C for some uniform constant C which may depend on . Taking t-derivative on (4.6), we get @v @ @v . / D !Qt . /  h!Q t ; @t @t @t

i:

It follows @F

D !Qt F C h!Q t ; !0  .T 0 C / @t

i;

(4.7)

where F D .T 0 C  t/ @v @t C v C nt. Since h!Q t ; !0  .T 0 C / i  0, it follows from the Maximum Principle that infX F is an non-decreasing function of t. Hence, .T 0 C  t/

@u ˇˇ @v C v C nt  .T 0 C / inf C inf u0 ; X @t tD0 X @t

and consequently, we obtain @v > C : @t

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Now we have gotten the C0 -estimates for both v and @v . If we have uniform @t higher order estimates for v, then v converges to a smooth limit v.T 0 ; /, by the local existence of parabolic equations, one can extend v, and consequently u, across T 0 . This would complete the proof of Lemma 4.2.3. The higher order estimates for v can be proved by using the standard arguments (cf. [27]). For the readers’ convenience, we outline a proof here. As usual, we first derive a Laplacian estimate for v by using a parabolic version of the generalized N Note that it was done Schwartz lemma [15]. This will imply the L1 -bound on @@v. differently in [27] in which the Aubin-Yau type estimate was used as Cao in [1]. For simplicity, we write !Q for !Q t and ! for ! ;t in the following. In any local holomorphic coordinates z1 ; ; zn , we denote by .QgiNj / and .giNj / the metric tensors of !Q and !, respectively, that is, !Q D

p

1

n X

gQ iNj dzi ^ dNzj and ! D

n X p 1 giNj dzi ^ dNzj :

i;jD1

i;jD1

Then N

h!; Q !i D gQ ij giNj ; where .QgiNj / is the inverse of .QgiNj /. Lemma 4.2.4 Let Q be the Laplacian operator of the metric !Q and K be the upper bound of the bisectional curvature of !. Then   @ Q  h!; Q !i  h!; Q @t

i

N

N

 K h!; Q !i2 C gQ ij gQ kl gpNq viNlp vNjmNq ;

(4.8)

where viNlp ’s are the third derivatives of v with respect to !. Q p Proof Since !Q D ! C 1 @@N v, by a direct computation, we have   N N N N N N N Q gQ ij giNj D RQ kNl gQ il gQ kj giNj  gQ ij gQ kl RiNjkNl C gQ ij gQ kl gpNq viNlp vNjmNq ; where RQ iNj denote the Ricci curvature of !Q and RiNjkNl denote the bisectional curvature of !. Since !Q satisfies (4.1), we deduce from the above   @  iNj  Q gQ giNj D h!; Q  @t

i

N

N

N

N

 gQ ij gQ kl RiNjkNl C gQ ij gQ kl gpNq viNlp vNjmNq :

Then we get (4.8). By our choice of , we have   C ! for some constant C which may depend on and T 0 . So it follows from (4.8)

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  @ Q .log h!; Q !i/   C  K h!;  Q !i C I; @t

(4.9)

where I D

gQ iNj gQ kNl gpNq viNlp vNjmNq h!; Q !i



Q !; jrh Q !ij2 ; h!; Q !i2

where rQ denotes the covariant derivative of !. Q Using the Cauchy-Schwartz inequality, one can show that I  0. Then (4.9) implies   @ Q  Q !i: .log h!; Q !i  Av/   .A n C C / C .A  K/ h!; @t

(4.10)

Taking A D KC1 and using the fact that v is bounded on Œ0; T 0 C , say inf v   a, we deduce from (4.10)   @ Q Q .log h!; Q !i  Av/   .A n C C / C ea A elog h!;!iAv  : @t Then the Maximum Principle implies that the function logh!; Q !i  Av is bounded from above by a uniform constant which may depend on , so we have for some c > 0 c !t  !Q t :

(4.11)

is uniformly bounded on Œ0; T 0 C , we have the uniform equivalence Since @v @t of two volume forms !Q tn and ˝ , so by (4.11), we have3 c !t  !Q t  c0 !t :

(4.12)

This means that !Q t is uniformly bounded by !t . Next we prove the third order estimate on derivatives of v. This is the same as an estimate on the 1st derivative of Kähler metric !Q t and can be done in several ways, such as using a parabolic version of the Krylov-Evans estimate or the method I showed in [20]. Here we follow more common method in Kähler geometry, that is, using a parabolic version of Calabi’s third derivative estimate. Such an estimate was previously used in [31] for elliptic Monge-Ampère equations and adapted in [1] for Kähler-Ricci flow. Our flow (4.1) is more general than the one studied in [1], but the computation there can be extended here without difficulty. Let us indicate how to carry this out.

3

We always use C , c0 etc. to denote some uniform constants which may depend on and T 0 .

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Let r be the covariant derivative of !. We define S D jr !j Q 2!Q D jr 3 vj2!Q ;

(4.13)

Q where the norm j j!Q is taken with respect to !. By a direct computation and using Eq. (4.1), we can show   @ Q  S  jr 2 !j Q 2!Q  C1 S  C2 @t where C˛ (˛ D 1; 2; ) are positive constants which may depend on c ; c0 above. It follows from (4.8)   @ Q  h!; Q !i  C3 C C4 S: @t

(4.14)

,

! and

(4.15)

Hence, if we choose A1 such that A1 C4 D C1 C 1, then   @ .S C A1 h!; Q  Q !i/  .S C A1 h!; Q !i/  C5 : @t Applying the Maximum Principle to this, we get a bound on S, thus the third derivatives of v are uniformly bounded. We can follow this line to get higher order estimates on v. We denote by r .k;l/ the covariant derivative r k rN l and define X Sm D jr k;l !j Q 2!Q : kClDm

Clearly, S1 D 2 S. We observe: (1) If we already bound all the covariant derivatives of !Q up to order m  1, then in order to bound all the derivatives of !Q up to order m C 1, we only need to bound SmC1 ; (2) If we bound all the derivatives of !, Q then we bound all the derivatives of v. So it suffices to bound Sm by induction. Suppose that we have bounded all the derivatives of !Q up to order m  1. Then by a straightforward, though tedious, computation, we have   X

@ N k;l !j Q  Sm  jrr k;l !j Q 2!Q C jrr Q 2!Q  CmC5 : @t kClDm This implies   @ 0 Sm  SmC1  CmC5 Q  : @t

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On the other hand, a similar computation shows   @ 00 Q  SmC1  cmC5 SmC1  CmC5 ; @t 00 depends only on the derivatives of !Q up to order m. Hence, if we choose where CmC5 AmC1 D cmC5 C 1, we deduce from the above estimates

  @ Q .SmC1 C AmC1 Sm /  .SmC1 C AmC1 Sm /  CQ mC6 :  @t Thus we can bound SmC1 by using the Maximum Principle. To summarize the above discussions, we have that for any m  1, there is a uniform constant C ;m such that jjvjjCm  C ;m :

(4.16)

It follows from (4.16) and (4.6) jj

@v 0 jjCm  C ;m : @t

Therefore, as t tends to T 0 , v converges to a unique smooth function vT 0 in the C1 -topology, consequently, !Q t converge to a smooth metric !Q T 0 . Then (4.6) has a solution v on Œ0; T 0 C 0  for a sufficiently small 0 > 0 which coincides with original v on Œ0; T 0 /. Consequently, we get a solution u of (4.2) on Œ0; T 0 C 0 . This is a contradiction to our choice of T 0 . Our lemma is proved. Remark 4.2.5 Note that !t may not be a Kähler metric for t close to T. However, the solution u of (4.2) exists for all t < T and !Q t is a Kähler metric.

4.3 Finite-Time Singularity In this section, we assume that T D Tmax < 1, that is, the Kähler-Ricci flow develops singularity at finite time T. We want to examine the limiting behavior of !Q t given by Theorem 4.2.1 p as t tends to T. We shall adopt the notations in the last N t and u.t; / D ut . / solves (4.2) with u0 D 0. section, e.g., !Q t D !t C 1@@u First we have Theorem 4.3.1 The solution !Q t converges to a unique positive (1,1)-current !Q T as t tends to T. Moreover, ut converge to a unique uT in the L1 -topology. Proof This theorem was first speculated in [25] and its proof was first given by Zhang in [33]. Here we will give a simplified proof by using an idea of Zhang.

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We observe !Q t D !0  t

C

Z

p 1 @@N ut  0;

X

!Q t ^ !0n1  CT ;

where CT depends only on T, Œ!0  and c1 .X/. Therefore, there is some p > 0 such that Z e p .ut sup ut / !0n  CT0 ; (4.17) X

where CT0 may depend on T, !0 and . This follows from the same arguments as we did for estimating the ˛-invariant in [22]. It follows from (4.5) t

@u  u  n t  0: @t

This implies that t1 u  n log t is non-increasing in t. Hence, ut D u.t; / converges to a unique limit uT which may take value 1. First we assume that uT is not identically 1. Then sup ut is uniformly bounded. Integrating by parts, we get Z ˇ ˇ2 n ˇ ˇ ˇr.1 C sup ut  ut / 2.nC1/ ˇ !0n X

1 D 4 1  2

 

n nC1 n nC1

3 Z  3

X

1 1 C sup ut  ut

1  nC1

.!t  !Q t / ^ !0n1

CT < 1:

Combining this with the Sobolev embedding theorem and (4.17), we can show that n 2 ut converge to uT in the Lq -topology for any 1  q  n1 . It follows that !Q t p N T, converge to a positive (1,1)-current !Q T D !T C 1@@u It remains to prove that uT cannot be identically 1. Here we use an argument due to Zhang in [33]. By the definition of T, we know that for any > 0, .1 C

/Œ!0   Tc1 .X/ is a Kähler class, so there is a smooth ' satisfying: sup ' D 0 and .1 C / !0  T X

C

p

1 @@N ' > 0:

(4.18)

It follows that 0 '   A0 for 2 .0; 1/, where 0 denotes the Laplacian of !0 and A0 D 2n C T supX supX jh!0 ; ij. If G0 denotes the positive Green function of !0 , then we have the Green formula Z Z 1 n ' !0  sup G0 .x; y/0 ' .y/ !0n .y/; 0 D sup ' D V X x2X X

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where V D Œ!0 n . This implies for some uniform CN depending on A0 and !0 Z N .' / !0n  C: (4.19) X

Using the same computation as we did for (4.4), we can derive     @ @u @u .T  t/ C u D !Qt .T  t/ C u  n C h!Q t ; !0  T @t @t @t

i:

(4.20)

Subtracting -multiple of (4.4) to (4.20), we get   @ @u .T  .1 C / t/ C .1 C / u @t @t   @u D !Q t .T  .1 C / t/ C .1 C / u  .1 C / n C h!Q t ; .1 C / !0  T @t

i:

Set F D .T  .1 C / t/ @u @t C .1 C / u  ' C .1 C / n t. Then it follows from the above and (4.18)   p @  !Qt F D h!Q t ; .1 C / !0  T C 1 @@N ' i  0: @t Applying the Maximum Principle to this, we see that the minimum of F is nondecreasing. So .T  .1 C / t/

@u @u C .1 C / u  ' C .1 C / n t  T inf .0; x/: x2X @t @t

For any t < T, choose sufficiently small so that T  .1 C / t > 0. Then we deduce from the above, (4.19) and (4.3) sup ut   C0 ; X 0

where C > 0 is some uniform constant. Thus uT can not be identically 1, so our theorem is proved. If .Œ!0   T c1 .X//n > 0, we have a simpler proof for that uT is not identically 1. Taking t-derivative of (4.2), we have @2 u D h!Q t ;  @t2

  p @u N i: C 1 @@ @t

Taking t-derivative again, we get @ @t



@2 u @t2



ˇ  2  ˇ @ u Q D   ˇˇ 2 @t

 ˇ2 p @u ˇˇ N : C 1 @@ @t ˇ

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Applying the Maximum Principle to this, we see that the maximum of increasing, consequently, @2 u  c0 ; @t2

@2 u @t2

is non-

(4.21)

@u where c0 depends only on !0 and ˝. It implies that @u @t  c0 t is non-increasing, so @t converge to a unique limit, say v, which is bounded from above. However, we have

Z

Z 0 < .Œ!0   T c1 .X// D lim n

t!T

X

!Q tn

ev ˝:

D X

So v, and consequently uT , can not be identically 1. A Natural Question Arises How regular is this limit !Q T and what are its geometric properties? It is reasonable to expect that !Q T is bounded and smooth on a Zariski open subset of X. We also expect to have a understanding of the behavior along the subvariety of its singularity in a suitable sense. A sufficient good knowledge of the limit !Q T will enable us to understand how finite-time singularity forms and how the Kähler-Ricci flow extends across the singular time T. Note that the positive current !Q T represents Œ!0 Tc1 .X/ in H 2 .M; R/. We expect Conjecture 4.3.2 For any compact Kähler manifold .X; !0 /, Œ!0   Tc1 .X/ can be represented by a semi-positive (1,1)-form. This conjecture is closely related to the Abundance conjecture in algebraic geometry and its preliminary significance can be seen in the following. Proposition 4.3.3 If Œ!0   Tc1 .X/ can be represented by a semi-positive, smooth (1,1)-form, then !Q T has bounded local potentials, i.e., for any p 2 X, there is a N for some bounded '. neighborhood over which !Q T D @@' Proof By the assumption, we can choose such that !T D !0  T is a semipositive (1,1)-form represents Œ!0   Tc1 .X/. Then it follows from (4.17)     @u @u @ .T  t/ C u C nt  !Qt .T  t/ C u C nt : @t @t @t Applying the Maximum Principle, we have .T  t/ Since

@u @t

@u @u C u C nt  T inf .0; /: X @t @t

is bounded from above, u is uniformly bounded from below.

Remark 4.3.4 It is plausible that uT is continuous and ut converge to uT in the C0 topology. This actually follows if one can show that ut converges to uT in the L1 topology. In the non-collapsing case, i.e., .Œ!0   T c1 .X//n > 0, this follows from

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extensions of Kolodziej’s continuity and stability theorems for degenerate complex Monge-Ampere equations by Dinew and Zhang (see [7, 32]). If X is a projective algebraic manifold and Œ!0  2 H 2 .X; Q/, i.e., it is a rational class, we can say much more about !Q T , including affirmation of Conjecture 4.3.2, because of certain deep theorems in algebraic geometry. In this case, there is an integer m such that mŒ!0  D c1 .L/ for some ample line bundle over X. First we observe Lemma 4.3.5 The maximal time T D Tmax is a rational number. Proof It follows from the well-known rationality theorem which states: For any ample line bundle L on X, the number T 0 D supft0 j L C t0 KX is nef g is either 1 or contained in Q, where KX denotes the canonical bundle of X. For 0 any t0 < T 0 , L C t0 KX is ample, so Œ!0   mt c1 .X/ is a Kähler class. It follows that 0 T D mT and T is rational. Of course, the rationality follows if there is a holomorphic curve along which Œ!0   T c1 .X/ vanishes. It will be desirable to have an analytic proof of the existence of such holomorphic curves. Lemma 4.3.6 If T 0 D mT < 1, then L C T 0 KX is semi-ample, that is, for ` sufficiently large, H 0 .X; `.L C T 0 KX // is free of base points or equivalently, `.L C mTKX / is generated by its global sections. Proof Put H D k.L C T 0 KX / for k large such that kT 0 > 1. Clearly, H is nef. Moreover, for any a  1, aH  KX D

1 b1 H C ..a  1/ k C 0 / L b T

is ample, where b D kT 0 > 1. Then a theorem of Kawamata [11, Theorem 6.1] implies that H is semi-ample. Lemma 4.3.6 implies that there is a fibration T W X 7! Y  CPN generated by global sections of `.L C T 0 KX /, where ` is sufficiently large, such that `.L C T 0 KX / D T O.1/. Note that Y is a variety possibly with singularities. Furthermore, 1 Œ!0   Tc1 .X/ can be represented by a semi-positive form m` T !FS , where !FS N denotes the Fubini-Study metric on CP . In particular, it follows that uT is bounded. Let us first discuss briefly some special cases: If !Q T D 0, then T contracts X to a point Y and the corresponding flow becomes extinct at T. In this case, c1 .X/ D T1 Œ!0  is positive, i.e., X is a Fano manifold. Conversely, if X is a Fano manifold, then Œ!0   tc1 .X/ is negative for t sufficiently big, so we expect that the Kähler-Ricci flow becomes extinct at finite time (cf. Conjecture 4.4.5 in Sect. 4).

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If dim Y < dim X, then we say the Kähler-Ricci flow (4.1) collapses at time T. Then one can easily show that the generic fibers of T are Fano manifolds. The simplest example is X D ˙  S2 , where ˙ is a Riemann surface of genus g  1. Write Œ!0  D a1 Œ!1 Cb2 Œ!2 , where i is the natural projection onto its ith factor and !i is the positive generator of the ith factor. Then Œ!t  > 0 if and only if 2t < b since c1 .X/ D .2g  2/Œ!1  C 2Œ!2 , so T D b=2. One can prove that !Q T descends to a metric on ˙ (possibly in the sense of currents). The Kähler-Ricci flow then continues on ˙. In general, X is uni-ruled in this case. Conversely, if X is uni-ruled, we expect that the Kähler-Ricci flow collapses at finite time (cf. Conjecture 4.4.6 in Sect. 4). If dim Y D dim X, we say that the Kähler-Ricci flow (4.1) is non-collapsing at time T. As we will show soon, this is always the case when X has non-negative Kodaira dimension and T < 1. Now let us state our first main conjecture in our program which concerns the formation of finite-time singularity for the Kähler-Ricci flow. Conjecture 4.3.7 Let .X; !0 / be a compact Kähler manifold with T < 1, then .X; !Q t / converges to a metric space .XT ; dT / in the Gromov-Hausdorff topology: (1) XT is a compact Kähler space and there is a holomorphic map Q T W X 7! XT ; (2) The current !Q T descends to a current !N T on XT , i.e., !Q T D Q T !N T , which is a smooth Kähler metric on XT0  XT and induces the distance function dT , where XT0 denotes the set of regular values of Q T . (3) .X; !Q t / converge to .X; !Q T / in the C1 -topology on any compact subsets contained in Q T1 .XT0 /. Moreover, the solution ut converges to uT in the C1 topology on Q T1 .XT0 /. This conjecture is a slightly refined version of corresponding ones in [15, 25]. If X is projective and Œ!0  is rational, then Q T W X 7! XT should be naturally identified with the holomorphic map W X 7! CPN induced by a basis of H 0 .X; `.LCmTKX // for a sufficiently large `. Such a holomorphic map is assured by Lemma 4.3.6, that is, Kawamata’s theorem on base locus. To see how to identify XT with .X/, we use the approach by the partial C0 -estimate originated by myself: Let fHt g (t 2 .0; T) be a smooth family of Hermitian metrics on L C mTKX with curvature form R.Ht / D Qt D et Put H

T

ut

T T t !t  !0 : t t

Ht , then its curvature form is given by Q t/ D R.H

T T t !Q t  !0 : t t

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For each t 2 .0; T/ and `, we have an induced Hermitian inner product h ; it on Q t . As in [21, 24], we define H 0 .X; `.L C mTKX // by !Q t and H t;` D

N X

hSi ; Si it ;

(4.22)

iD0

where fSi g is any orthonormal basis of H 0 .X; `.L C mTKX // with respect to h ; it . For any t, such a basis fSi g defines a holomorphic map t from X onto a subvariety

t .X/  CPN which coincides with t . .X// for some automorphism t of CPN . We expect that t converge to an automorphism T and t converge to a holomorphic map T from XT onto the subvariety T1 . .X//, moreover, the limiting map T should be Lipschitz with respect to dT and a homeomorphism. Thus, we can easily identify XT with .X/. Now we discuss some results on the first singularity formation in the noncollapsing case. The following lemma shows that the Kähler-Ricci flow does not collapse at any finite time for most projective manifolds. Lemma 4.3.8 Assume that the Kodaira dimension of X is non-negative. Then for any rational T < 1, Œ!0   Tc1 .X/ is big, i.e., Z X

!Tn > 0:

Proof We will adopt the notations from Lemma 4.3.6. Since the Kodaira dimension is non-negative, for m sufficiently large, mTKX admits a holomorphic section S. It follows that Sk S0 is a global section of k.L C mTKX / for any section S0 of kL, so dim H 0 .X; k.L C mTKX //  ckn for some c > 0. Hence, .L C mTKX /n > 0, i.e., it is big. The following lemma is well-known and referred as the Kodaira lemma (cf. [10]). Lemma 4.3.9 Let E be a divisor in a projective manifold X. If E is nef and big, then there is an effective R-divisor D such that E  D > 0. The proof follows essentially from the openness of the big cone of X which clearly contains the positive cone and the fact that E is in the closure of the positive cone. Recall that the non-ample locus BC .E/ of E is defined to be the intersection of Supp.D/, where D ranges over all effective R-divisor such that E  D is ample. The above lemma implies that BC .E/ is a subvariety whenever E is big. We will apply this to L C mKX in the following theorem. Recall that !Q t of (4.1) converges to a unique current !Q T as t ! T. In [27], the following regularity theorem is proved.4 4 In [27], KX is assumed to be big. It is clear from the arguments in the proof that this assumption was not used.

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Theorem 4.3.10 Let X, L C mTKX , !Q t and !Q T be as above. Then we have (1) !Q T is a smooth Kähler metric outside the non-ample locus5 BC .L C mKX /  X which is a subvariety; (2) !Q t converges to !Q T on any compact subset outside BC .L C mKX / in the C1 topology. Proof For the readers’ convenience, we outline a proof of this theorem following [27]. Since L C mTKX is semi-positive and big, by Proposition 4.3.3, we know that the limiting current !Q T exists with locally bounded potential and cohomology class Œ!0   Tc1 .X/. To prove (1) and (2), we need only establish uniform high derivative estimates for u.t; / outside BC .L C mTKX /. Recall that BC .L C mTKX / is the intersection of Supp.D/ for all effective Rdivisors suchPthat L C mKX  D is ample. Let D be any such effective R-divisor. Write D D i ˛i Di, where Di are effective divisors and ˛i > 0. For each i, let i be the defining section of Di and choose P a Hermitian metric hi on the holomorphic line bundles ŒDi  induced by Di . Then i ˛i log hi .i ; i / is a well-defined function outside Supp.D/  X. Since L C mTKX  D is ample, we can choose hi such that !T C

p X 1 ˛i @@N log hi .i ; i / > 0: i

For simplicity, we will write log jj2 D

X

˛i log hi .i ; i /:

i

Formally, we regard  as a defining section of D and j j a norm for ŒD. First we derive a second order estimate. Set !t;D D !t C

p

1@@N log jj2 :

Then there is a ı D ı.D/, which may depend on D, such that !t;D is a smooth Kähler metric for any t 2 ŒT  ı; T C ı, in particular, there is a uniform bound on their curvature when t 2 ŒT  ı; T C ı. Note that this bound may depend on D. In order to derive the second order estimate, we need a lower bound on @u @t for any t 2 ŒT  ı; T. Using the same arguments in deriving (4.7), we can deduce @w D !Qt w  n C h!Q t ; !TCı;D i; @t

5

In [25], it was misnamed as the stable base locus.

(4.23)

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where w D .T C ı  t/

@u C u  log jj2 : @t

Since h!Q t ; !TCı;D i  0, by the Maximum Principle, we can show that the minimum of w C nt is non-decreasing. Since u is bounded for t 2 ŒT  ı; T, we conclude from this 1 @u  log jj2  Cı ; @t ı

(4.24)

where Cı is a uniform constant which may depend on ı. Now we write !Q t D !t;D C

p N  log jj2 /: 1@@.u

Note that the function v D u  log jj2 is only defined outside D. On X n D, we can rewrite (4.2) as .!t;D C

p N n D e @u@t ˝: 1@@v/

Note that Ric.˝/ D . Computing as in [1, 29, 31] and using bounds on one can derive the following inequality

@u @t

and the curvature of !t;D ,

@ Cv / e h!t;D ; !Q t i @t @u n > C0 C .C  C0 /h!t;D ; !Q t i C C0 h!t;D ; !Q t i n1 @t n C > C0 C . log jj2  C0 /h!t;D ; !Q t i C C0 h!t;D ; !Q t i n1 : ı

eCv .!Qt 

(4.25)

Here C, C0 are constants which may depend on D. For instance, we need to choose C such that C C infM Rm.!t;D /  1 for t 2 ŒT  ı; T C ı, where Rm.! 0 / denotes the bisectional curvature tensor of ! 0 . 2 Clearly, eC.ulog j j / h!t;D ; !Q t i attains its maximum in X n f D 0g. At such a maximum point, we have n

0 > C0 C .C00 log jj2  C0 /h!t;D ; !Q t i C C0 h!t;D ; !Q t i n1

1 D C0 C C0 h!t;D ; !Q t i h!t;D ; !Q t i n1 C C00 log jj2  C0 :

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Here C00 D C=ı. Since jj is bounded, we deduce from the above h!t;D ; !Q t i  .C0  C00 log jj2 /n1 : Hence, at this maximum point, eCv h!t;D ; !Q t i  .C0  C00 log jj2 /n1 eCv  C1 .1  log jj2 /jj2C : Here we have used that fact that u is uniformly bounded and C1 is a constant which may depend on D. Then we obtain the second order estimate: h!0 ; !Q t i 6 C2 jj2C :

(4.26)

On the other hand, we observe that there is a uniform bound on compact subset outside D. It implies the volume estimate:

@u @t

on any given

!Q tn > C3 jj2 !0 n : Hence, for t 2 ŒT  ı; T, c1 Q t  cF !0 as Kähler metrics on compact subset F !0  ! F in X n f D 0g. The higher order derivative estimates for u.t; / (t 2 ŒT  ı; T) outside f D 0g follow from the standard theory on Monge-Ampere equations [8, etc.] or Calabi’s third order estimates as shown in [31]. We have shown that u.T; / D limt!T u.t; / exists. The above estimates pass to u.T; /, so u.T; / is smooth and defines a smooth Kähler metric !T outside D. Moreover, we have .!T C

p N T /n D e @u@tt jT ˝; 1@@u

on X n D:

(4.27)

where ut . / D u.t; /. Since D can be any effective R-divisor defining BC .L C mTKX /, we have shown that !Q T is a smooth Kähler metric and satisfies the above equation outside BC .L C mTKX /. Thus (1) is proved. To prove (2), we only need to notice that ut converges to uT in any Lp -norm and the kth derivatives of ut are uniformly bounded on any compact subset F  X n BC .L C mTKX / by a constant cF;k depending only on F and k, so the convergence outside the non-ample locus is in the C1 -topology. The theorem is proved. Theorem 4.3.10 can be generalized to the case of Kähler manifolds. All we need is a generalization of the Kodaira lemma to Kähler manifolds. This can be proved by methods in potential theory. Once we have such a generalization, we can argue in the same way as we did above to prove a version of Theorem 4.3.10 for Kähler manifolds. We refer the readers to [5] for more details.

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It follows from Theorem 4.3.10 or its generalization to Kähler manifolds that the solution !Q t extends to a Kähler metric !Q T outside the subvariety BC  X. However, the limit !Q T does have singularity along BC . This singularity is because of the metric’s either blowing-up or failing to be non-degenerate. Conjecture 4.3.7 gives a description of how such a singularity may behave. In fact, in the situation given in Theorem 4.3.10, we can formulate the conjecture in a simpler and more precise way as we discussed in [15, 25]. Conjecture 4.3.11 Let X and BC be as above. Then XT in Conjecture 4.3.7 is the metric completion of XnBC with respect to the distance dT on XnBC induced by !Q T . If X is projective and Œ!0  D mc1 .L/, then BC D BC .L C mTKX / and XC can be obtained from X by flips or algebraic surgeries of certain “standard” type. This is indeed the case if X is also of general type (see [17] and papers in its list of references for more details). Moreover, L C mTKX yields a Q-bundle LC over XC such that LC C KXC is nef and big for > 0 sufficiently small. If X has the Kodaira dimension 1, then L C mTKX may not be big and consequently, .X; !Q t / may collapse to a lower dimensional space XT as described in Conjecture 4.3.7. What about the metric !Q T D limt!T !Q t in this collapsing case? We do not have as many results as in the non-collapsing case. By examining the limit in the best scenario and using some evidences we have, we can make the following speculation: Let Q T W X 7! XT be the fibration given by the line bundle L C mTKX , then generic fibers are Fano manifolds. First we discuss how !Q t may behave along fibers. It is plausible that for a generic fiber F of Q T , the restriction !Q t jF shrinks to a point at the scale T  t, more precisely, inspired by the Hamilton-Tian conjecture for Fano manifolds, we conjecture that the scaled metric .T  t/1 !Q t jF converges to a Kähler-Ricci soliton on the Fano manifold F as t goes to T. If this conjecture holds, then .T  t/1 !Q t converges to Kähler-Ricci solitons along fibers, at least away from singular fibers where the limiting metrics may have singularities. Next we discuss how !Q t may behave horizontally as t goes to T. We have shown that as currents, !Q t converges to a (1,1)-current  .!Q T / on XT with bounded local potentials. Then we can consider the Kähler-Ricci flow on a certain resolution XC of XT with initial metric given by the pull-back of .!Q T /. The solution of this flow on XC can be regarded as the extension of !Q t across T. We have the following speculation on !Q T : Let ˝ be the volume form with !Q n Ric.˝/ D given in (4.2), then the leading term of ˝t is of the form: .T  t/ !N Tn ^ ˝XjY C o..T  t/ / modulo constants; ˝ where Y D XT and ˝XjY is a relative volume form whose curvature is the KählerRicci soliton with Kähler class c1 .F/. If !Q T has sufficiently regularity, then we can deduce from this speculation lim

t!T

@!Q t D Ric.!N T / C f  !WP ; @t

(4.28)

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where !N T is the pull-forward Q T !Q T on Y and f W Y 7! M is the developing map (possibly with singularities6 which maps generic point of Y to the corresponding fiber in the moduli space M of Kähler-Ricci solitons on generic fibers, and !WP is the L2 -metric, i.e., the Weil-Peterson metric, on the moduli M. Obviously, it remains to prove that (4.28) is the limit of the Ricci flow !Q t (t < T) in a suitable sense. One can use this modified Ricci flow to extend !Q t across T. One of advantages for using this flow instead of the Kähler-Ricci flow is that it encodes more information of the original X through the developing map. For simplicity, we will just use the Ricci flow to extend solution across the singular time T in our program. It should not be difficult to prove a local existence theorem on the stationary solutions of (4.28). This is amount to solving a complex Monge-Ampere equation and analogous to the generalized Kähler-Einstein metrics studied in [15, 16]. In fact, our discussions are inspired by those joint works.

4.4 Extending Kähler-Ricci Flow Across Singular Time In this section, we discuss how to extend the Kähler-Ricci flow !Q t across the singular time T, assuming that we have solved Conjecture 4.3.7 proposed in last section. Then we have a Kähler variety XT and a Gromov-Hausdorff limit !N T on XT which is smooth outside a subvariety B  XT . A natural question is how to continue the Kähler-Ricci flow on XT starting at !N T . There are two difficulties: 1. XT may not be smooth, so we need to study the Kähler-Ricci flow on a singular space. It is not even clear whether or not we can make sense of the Kähler-Ricci flow in a useful way. 2. Even if XT is smooth, !N T may not be smooth, so we need to solve the KählerRicci flow with weak initial value !N T . In short, we need a local existence theorem for (4.1) when the underlying space may be singular or initial Kähler metric is not smooth. First we assume that XT is smooth. Then we can choose the Ricci representative in setting up (4.2) such that !T is the pull-back of a smooth Kähler metric on XT , still denoted by !T for simplicity. As we did in Sect. p 4.2, we can reduce (4.1) N uT for some Kähler on XT to a scalar flow: First we write !N T D !T0 C 1 @@N 0 0 0 metric !T . Choose a (1,1)-form D Ric.˝ / for a volume form ˝ 0 on XT . Define 0 0 !t D !T  .t  T/ for t  T, then (4.1) is reduced to a variant of (4.2): p N n @u !tCT C 1 @@u D log ; lim u.t; / D uN T : t!0 @t ˝0

(4.29)

6 The singularities are caused by those singular fibers, but the developing map should be meromorphic.

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If (4.29) has a solution u. ; t/ for t 2 .0; T 0 /, then !Q TCt extends previous KählerRicci flow !Q t (t 2 Œ0; T/) to .T; T C T 0 /, where 0 !Q TCt D !TCt C

p N 1 @@u. ; t/:

Moreover, we have limt!T !Q t D !Q T and limt!TC !Q t D !N T in the sense of distribution. It follows p from the potential theory that for any bounded function ' such that N  0 as currents, one can have an associated volume form !' D !00 C 1 @@' n !' . In particular, we have a volume form !N Tn . A direct computation shows that the function !N n =˝ 0 is Lp -integrable for some p > 1. Thus, the solvability of (4.29) for some T 0 > 0 is provided by the following. 0 Theorem 4.4.1 ([17]) Let X be a compact Kähler manifold and p !t be given as 0 N 0  0 as above. Assume that u0 is a bounded function such that !0 C 1 @@u n 0 p currents and !u0 =˝ lies in L for some p > 1. Then there is a unique smooth solution u. ; t/ of (4.29) on XT  .0; T 0 / such that limt!0 u. ; t/ D u0 .

This extends Theorem 4.2.1 to the case of weak initial metrics. Previously, in [4], a weak version of theorem was proved under the condition that p > 3. If the Kodaira dimension of X is non-negative, then L0 C aKX is nef and big on XT and dimC XT D n. According to Conjecture 4.3.7, if XT is smooth, then !N T extends to be a Kähler class on XT . Since @Q@tut is uniformly bounded from above for t 2 .0; T/, we can show that the assumptions in the above theorem are satisfied. Then one can extend (4.1) across T and continue the flow on XT until T2 > T when Œ!N T   .t  T/c1 .XT / fails to be a Kähler class. If T2 is finite, one can proceed as we did for !Q t at T. However, in general, the resulting variety XT from the surgery at T may not be smooth.7 Nevertheless, we expect Conjecture 4.4.2 There is a Q-factorial variety8 X 0 with a holomorphic map T0 W X 0 7! XT , referred as a flip of X, satisfying: 1. X 0 has only log-terminal singularities, that is, there is a smooth resolution  W XQ 7! X 0 such that KXQ D   KX 0 C

k X

a i Ei ;

(4.30)

iD1

where Ei are exceptional divisors of the resolution  and ai > 1 are rational numbers. 2. For > 0 sufficiently small, Π0  !N T  C KX 0 > 0. 7

It will be interesting to construct an explicit example of such a singular XT , even though no one doubts its existence. 8 A projective variety is Q-factorial if it is normal and any Q-Weil divisor is Q-Cartier.

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Clearly, our description of flip X 0 does not require that X is smooth. If we can affirm Conjecture 4.4.2, then we need to extend Kähler-Ricci flow to X 0 above with initial metric  0  !N T . First we carry out such an extension by establishing a scalar flow on potential functions in the way as we did from (4.1) to (4.2). Since X 0 is Q-factorial, there is an m > 0 such that mKX 0 is a line bundle, that is, locally, it is of the form C ˛ for some local holomorphic section ˛. We can further assume that X 0 is covered by finitely many open subsets fUi g satisfying: a. mKX 0 jUi D C ˛i for a local holomorphic section over Ui . b. For each i, there is a holomorphic embedding X 0 \ Ui 7! CN for some large N.9 Then we can choose a smooth volume form ˝ 0 on the regular part Reg.X 0 / of X 0 such that on each Reg.X 0/ \ Ui , it is of the form 1

˝ 0 jReg.X 0/\Ui D fi .˛i ^ ˛N i / m ;

(4.31)

where fi is a smooth function on a neighborhood of Ui in CN . Put 0 D p N 1 @@ log ˝ 0 and !t0 D  0  !N T C .t  T/ 0 . Though not really needed, we can even choose ˝ 0 such that !t0 > 0 on X 0 for t > T and t  T sufficiently small. Now we introduce a scalar flow on X 0 : p .!t0 C 1 @@N u/n ˝ 0 @u D log on Reg.X 0 / and u 2 L1 .X 0 /: @t

(4.32)

The following theorem was proved in [17] and extends Theorem 4.4.1 to Qfactorial varieties which satisfy 1 and 2. Theorem 4.4.3 ([17]) Let X 0 be a variety which arises from Conjecture 4.4.2. Define 

T1 D supft j Œ 0 !N T  C t KX 0  0g: Then (4.29) has a unique solution u.t/ on X 0  Œ0; T1 / such that (1) u is smooth on p 0 0 Reg.X /  .0; T1 /; (2) !t C 1 @@N u > 0 on Reg.X 0 /; (3) u is continuous on X 0 . This theorem provides a solution !Q t to Kähler-Ricci flow on X 0 with initial data  !N T in the sense of distribution, where T  t < T C T1 . Thus we extend KählerRicci flow !Q t on X  Œ0; T/ across T to a Kähler-Ricci flow on X 0  ŒT; T C T1 /. So we get a solution !Q t with surgery for (4.1) for t 2 Œ0; T C T1 / satisfying: As usual, we call T a surgery time. Assuming that Conjectures 4.3.7 and 4.4.2 are confirmed, we can repeat the above process to continue the flow beyond T C T1 and so on. Thus, we can construct a Kähler-Ricci flow with surgery on X for all t  0. By a Kähler-Ricci flow with 0

9

Without loss of generality, we may assume that N is independent of i.

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surgery on X with initial metric !0 , we mean a sequence of Kähler-Ricci flows !i .t/ on Xi  ŒTi1 ; Ti /, where i D 0; 1; (possibly only finitely many), satisfying: KR1. T1 D 0 < T1 < and for any T < 1, there are only finitely many Ti ’s which are less than T; KR2. X0 D X and each Xi for i  1 is either an empty set or a Q-factorial Kähler variety with at most log-terminal singularities and which is obtained from Xi1 as described in Conjecture 4.4.2, that is, Xi is a flip of Xi1 ; KR3. For each i, !.t/ is a Kähler-Ricci flow on Xi  .Ti ; TiC1 / or Xi  .Ti ; 1/ if TiC1 does not exist, defined by the corresponding scalar flow either (4.2) or (4.29); KR4. For each i  0, as t ! Ti , .Xi ; !i .t// converge to .XN i ; !N i / as described in Conjecture 4.3.7. Let iC1 W XiC1 7! Xi be the projection T0 given by  Conjecture 4.4.2, then limt!Ti C !iC1 .t/ D iC1 !N i . We denote by f.Xi ; !i .t//g such a Kähler-Ricci flow with surgery. From the definition, a Kähler-Ricci flow with surgery may have infinite surgery times. However, we expect that there should be only finitely many surgery times, that is, Conjecture 4.4.4 After finitely many surgeries at T1 < T2 < < TN1 < 1, we arrive at XN which is either an empty set or a Q-factorial Kähler variety Xmin with nef canonical bundle KXmin  0. If XN D ;, we say that (4.1) becomes extinct at T D TN1 . At each Ti (i D 1; ; N  1), we do surgery along some “rational” components along which KXi integrates negatively. In particular, Xi is birational to X for all i. This leads us to expect Conjecture 4.4.5 The Kähler-Ricci flow (4.1) becomes extinct at finite time if and only if X is birational to a Fano-like manifold.10 Partial progress on this conjecture has been made by Jian Song. In particular, he proved that if X is Fano manifold, then (4.1) becomes extinct at finite time, If X is an uni-rule manifold, then one can show the flow (4.1) collapses at some finite time. We expect the converse holds, too. Conjecture 4.4.6 The Kähler-Ricci flow (4.1) collapsed at finite time if and only if X is birational to an uni-ruled manifold. There are strong supporting evidences for validity of both Conjectures 4.3.7 and 4.4.2. In [17], using deep results in algebraic geometry, we have established these two conjectures for X being a projective manifold of general type and !0 having rational Kähler class.

10

It is likely that such a Fano-like manifold is actually Fano. This is indeed the case if the dimension is not greater than 3.

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4.5 Asymptotic Behavior of Kähler-Ricci Flow In last two sections, we have discussed results and speculations on singularity formation of the Kähler-Ricci flow at finite time. We also conjectured that there is always a global solution .Xt ; !Q t / with surgery of (4.1) with only finitely many surgery times. This generalized solution with surgery becomes an usual solution !Q t of (4.1) on a variety with nef canonical bundle when t is sufficiently large. In this section, we study the asymptotic behavior of !Q t as t goes to 1. For simplicity, we assume that X is a compact Kähler manifold with KX nef. The general case can be dealt with in a similar approach. We refer the readers to [17] for a more detailed discussion in case of possible singular varieties. It is known that (4.1) has a global solution !Q t for any given initial metric. Set t D es  1 and !.s/ Q D es !Q t , then !.s/ Q is a solution of the following normalized Kähler-Ricci flow: @!.s/ Q D Ric.!.s// Q  !.s/; Q @s

!.0/ Q D !0 :

(4.33)

The advantage of doing this is that Œ!.s/ Q D es Œ!0   .1  es /c1 .X/, which converges to c1 .X/ as s ! 1. We also assume that there is a (1,1)-form  0 representing c1 .X/. This is of course the case if KX is semi-positive or equivalently, for m sufficiently large, H 0 .X; KXm / is free of base points. The Abundance conjecture in algebraic geometry claims that it is true for any X with KX nef. Since H 0 .X; KXm / is base-point free, any basis of it induces a holomorphic map

W X 7! CPN for some N > 0 so that  OCPN .1/ D KXm . The dimension of ’s image is just the Kodaira dimension  D .X/ of X. If .X/ D 0, then c1 .X/ D 0 and by the result in [1], the global solution !Q t of (4.1) converges to a Calabi-Yau metric on X. This is still true even if X is a Qfactorial variety with only log-terminal singularities. If .X/ D dim X D n, then X is minimal and of general type. It follows from [27, 29] that !.s/ Q converges to the unique (possibly singular along a subvariety) Kähler-Einstein metric with scalar curvature n on X as s tends to 1. The more tricky cases are for those X with 1  .X/  n  1. If X is such a manifold, one can not expect the existence of any Kähler-Einstein metrics (even with possibly singular along a subvariety) on X since  6D 0 and KXn D 0. Hence, the first problem is to find what limiting metrics for !.s/ Q one supposes to have as s tends to 1. To solve this problem, we introduced a class of new canonical metrics which we call generalized Kähler-Einstein metrics in [15]11 and [16]. Let us briefly describe them.

11

Reference [15] is mainly for complex surfaces, but the part on limiting metrics works for any dimensions.

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Since we assume that KX is semi-ample, the canonical ring R.X/ D ˚m0 H 0 .X; KXm / is finitely generated, so there is a canonical model Xcan of X (possibly singular). Let  W X 7! Xcan be the natural map from X onto its canonical model Xcan . Then generic fibers of  are Calabi-Yau manifolds of dimension n  , and consequently, there is 0 0 a holomorphic map f W Xcan 7! MCY which assigns p 2 Xcan to the fiber  1 . p/ in 0 1 the moduli MCY , where Xcan consists of all p such that  . p/ is smooth. The moduli MCY admits a canonical metric, the Weil-Petersson metric. Let us recall its definition. Let X ! MCY be a universal family of Calabi-Yau manifolds. Let .UI t1 ; : : : ; t` / be a local holomorphic coordinate chart of MCY , where ` D dim M. Then each @t@i corresponds to an element . @t@i / 2 H 1 .Xt ; TXt / through the Kodaira-Spencer map . The Weil-Petersson metric is defined by the L2 -inner product of harmonic forms representing classes in H 1 .Xt ; TXt /. In the case of Calabi-Yau manifolds, as shown in [20], it has the following simple expression: Let  be a nonzero holomorphic .n  ; 0/-form on the fiber Xt and  y . @t@i / be the contraction of  and @t@i . Then the Weil-Petersson metric is given by 

@ @ ; @ti @tNj

R

 D !WP

Xt

 y . @t@i / ^  y . @t@i / R : Xt  ^ 

(4.34)

Now we can introduce the generalized Kähler-Einstein metrics. Definition 2 Let X, Xcan etc. be as above. A closed positive .1; 1/-current ! on Xcan is called a generalized Kähler-Einstein metric if it satisfies the following. 1. f  ! 2  c1 .X/; 0 12 2. ! is smooth p on Xcan ; 0 3. Ric.!/ D  1 @@ log !  lifts to a well-defined current on X and on Xcan Ric.!/ D ! C f  !WP :

(4.35)

If  D n, then it is just the equation for Kähler-Einstein metrics with negative scalar curvature.

In fact, one can prove that .  !/ ^  extends to a continuous function on X, where  is the (n-, n-)-form which restricts to polarized flat volume form on each smooth fiber (see [16, p. 15]).

12

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Remark 4.5.1 More generally, one can consider the generalized Kähler-Einstein equation: Ric.!/ D  ! C f  !WP ; where  is a constant. In [16], the following theorem was proved. Theorem 4.5.2 Let X be an n-dimensional projective manifold with semi-ample canonical bundle KX . Suppose that 0 < .X/  n. There exists a unique generalized Kähler-Einstein metric on Xcan . To prove this theorem, we reduce (4.35) to a complex Monge-Ampere equation as in the proof of the Aubin-Yau theorem. First we introduce a function which will appear in such a complex MongeAmpere equation. Since KX is semi-ample, there is a semi-ample form   representing c1 .X/, where is defined in the following way: Xcan can be embedded into some projective space CPN by using any basis of H 0 .X; KXm / for a sufficiently large m, then D

1 !FS jXcan : m

Let ˝ be a volume form on X satisfying: p 1 @@ log ˝ D : We push forward ˝ to get a current  ˝, where  W X ! Xcan as above, as follows: For any continuous function on Xcan Z

Z

.  / ˝:

 ˝ D Xcan

X

0 It is easy to see that for any x 2 Xcan , we have

Z  ˝.x/ D

 1 .x/

˝:

Definition 3 We define a function F on Xcan by F  D  ˝:

(4.36)

There is another way of defining F: Choose any Kähler class ˇ on X, by using 0 the Hodge theory, one can find a flat relative volume form  on X 0 D  1 .Xcan / n n in the cohomology class ˇ , this means a .n  ; n  /-form  in ˇ whose

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131

0 restriction to each fiber  1 .x/ for x 2 Xcan is Ricci-flat, that is,

@@ log j 1 .x/ D 0: This is possible because c1 .X/ vanishes along each smooth fiber. One can show 



c F D

˝  ^   

 ;

(4.37)

where c is a constant determined by Z c

 1 .x/

ˇ n D 1;

0 . For simplicity, assume that c D 1. In particular, it where x is any point in Xcan   follows that  ^  can be extended to X as a current. Furthermore, one can show (see [20])

f  !WP D

p p 1@@ log. ^  /  1@@ log  :

The function F may not extend smoothly to Xcan , but we have some controls on 0 it along the subvariety Xcan nXcan . 0 Lemma 4.5.3 F is smooth on Xcan and is in L1C .Xcan / for some > 0, where the p L -norm is defined by using the metric corresponding to .

To prove it, we notice Z

F 1C  D Xcan

Z

  F 1C    ^  D X

Z

  F ˝:

X

Furthermore, one can show that if W Y ! Xcan is any resolution of Xcan , then  F has at worst pole singularities on Y. The proof is a bit technical and we refer the readers to [16] for details. Consequently,   F is integrable for sufficiently small

> 0 (see [16], Proposition 3.2). Consider . C

p 1@@'/ D Fe'  :

(4.38)

p If ' is a bounded solution for (4.38), then ! D C 1@@' is a generalized Kähler-Einstein metric. To see this, we first observe that Œ  ! D Œ   D c1 .X/. Next we observe p p p p Ric.!/ D  1 @@ log !  D  1@@ log   1@@ log F  1@@'

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is a well-defined current on Xcan . A direct computation shows p p p 1@@ log  C 1@@ log F C 1@@'   p p ˝  C! D 1@@ log C 1@@ log  ^   p  D ! C 1 @@ log. ^  / C @@ log  D !  f  !WP : Therefore Ric.!/ D ! C f  !WP : Thus, in order to prove Theorem 4.5.2, we only need to prove the following 0 Theoremp 4.5.4 There exists a unique solution ' 2 C0 .Xcan / \ C1 .Xcan / for (4.38) with C 1@@'  0.

This is proved by using the continuity method and establishing an a priori C3 estimate for solutions of (4.38). We refer the readers to [16] for its proof.  We would like to point out that   !  ^  D ˝ ef ' is continuous since both   ' and ˝ are continuous on X. There are still unknown about the generalized Kähler-Einstein metric !can constructed above, for instance, we do not know if it has finite diameter in general. Conjecture 4.5.5 Let Xcan and !can be as above, then the metric completion of 0 .Xcan ; !can / coincides with Xcan . In particular, the diameter of !can is finite. Recently, J. Song made some progress on this problem, especially, in the case of projective manifolds of general type. Now we can discuss the limit of !.s/ Q in (4.33) as s tends to 1. The following theorem was proved in [16] (also see [15] for complex surfaces). Theorem 4.5.6 Let X be a projective manifold with semi-ample canonical bundle KX . So X admits an algebraic fibration  W X ! Xcan over its canonical model Xcan . Suppose 0 < dim Xcan D  < dim X D n. Then for any initial Kähler metric !0 , the solution !.s/ Q for (4.33) converges to   !can as currents, where !can is the unique generalized Kähler-Einstein metric on Xcan . Moreover, for any compact subset K  0 Xcan , there is a constant CK such that .n/s jjR.!.s//jj Q sup jj!.s/ Q n j 1 .x/ jjL1 . 1 .x//  CK ; L1 . 1 .K// C e x2K

where R.!.s// Q denotes the scalar curvature of !.s/. Q

(4.39)

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If n D 2, then the above implies the convergence in the C1;˛ -topology for any 0 ˛ 2 .0; 1/ on any compact subset in Xcan . We believe that the same can be proved in any dimensions. Moreover, we also expect Conjecture 4.5.7 The solution !.s/ Q converges to the unique limit   !GKE in the Gromov-Hausdorff topology and the convergence is in the smooth topology in 0  1 .Xcan /. This is even open for complex surfaces. In the above, we assume that X has semi-ample KX . This is indeed true if the Abundance conjecture holds. If KX is nef, (4.33) still has a global solution !.s/. Q Clearly, it will be extremely interesting to study the asymptotic behavior of !.s/ Q without assuming the Abundance Conjecture, namely, give a differential geometric proof of the convergence of !.s/. Q The success of such a direct approach will yield many deep applications to studying the structures of Kähler manifolds. To solve the above conjecture or succeed in the above direct approach, we may need to develop a theory of compactness for Kähler metrics with bounded scalar curvature. For Kähler surfaces, a compactness theorem of this sort was proved in [26]. Also note that the scalar curvature is uniformly bounded along (4.33) on any compact projective manifold with semi-positive canonical bundle (see [18]). We also refer the readers to [13, 14] for corresponding scalar curvature estimate for Kähler-Ricci flow on Fano manifolds.

4.6 The Case of Algebraic Surfaces In this section, we will carry out the program described above for complex surfaces. Almost all the results in this section are taken from [27] (for surfaces of general type) and [15] (for elliptic surfaces). We just make a few simple observations in order to deduce the program from those previous works. For simplicity, we assume that X is an algebraic surface. The general case for Kähler surfaces can be done in a similar way. As before, let !Q t be a maximal solution of (4.1) on X  Œ0; T. If T < 1, then Œ!0   Tc1 .X/ is nef. There are three possibilities: Q D .1  Tt /1 !Q t , 1. If Œ!0   Tc1 .X/ D 0, then X is a Del-Pezzo surface and !.s/ t where s D T log.1  T /, converges to a Kähler-Ricci soliton as s ! 1 or equivalently, t ! T (cf. [21, 28, 30]). 2. If Œ!0 Tc1 .X/ 6D 0 but .Œ!0 Tc1 .X//2 D 0, then there is a fibration  W X 7! ˙ with rational curves as fibers (possibly with finitely many singular fibers) such that Œ!0   Tc1 .X/ D   Œ!˙  for some Kähler metric !˙ on ˙. It follows that p as t ! T, !Q t converges to a positive current of the form   .!˙ C 1@@uT / for some bounded function uT on ˙. To extend (4.1) across T, one needs to

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solve (4.2) on ˙ with uT as the initial value. This is the same as solving the following for t  T, !˙  .t  T/ ˙ C @u D log @t ˝˙

! p 1@@u

; u.T; / D uT ;

(4.40)

where ˝˙ is a volume form on ˙ with Ric.˝˙ / D ˙ . One can solve this flow by using the standard potential theory in complex dimension 1. Let !Q t be the resulting maximal solution of (4.40) (t  T). If the genus g.˙/ of ˙ is zero, then !Q t becomes extinct at some finite time T2 > T or after appropriate scaling, these metrics converge to the standard round metric on ˙ D S2 as t ! T2 . Hence, it verifies Conjecture 4.4.5 in case of algebraic surfaces. If g.˙/ D 1, then !Q t exists for all t  T and converges to a flat metric as t ! 1. If g.˙/ > 1, then !Q t exists for all t  T and after scaling, converges to a hyperbolic metric as t ! 1. 3. If .Œ!0   Tc1 .X//2 > 0, then Œ!0   Tc1 .X/ is semi-ample, so it can vanish only along a divisor. It is easy to see that for each irreducible component D of this divisor, KX D < 0. Moreover, D2 < 0. By the Adjunction Formula, D is a rational curve of self-intersection 1, so the divisor is made of finite disjoint (-1) rational curves and consequently, we can blow down them to get a new algebraic surface XT . Moreover, the limit !Q T descends to a positive current with continuous potential and well-defined bounded volume form. By Theorem 4.4.1, one can extend (4.1) across T. Notice that the extension !Q t for t > T is smooth. Either KXT is nef and there is a global solution on XT ,or !Q t develops finite-time singularity at some T2 > T. In the later case, one can repeat the above steps 1, 2 and 3. Since H2 .X; Z/ is finite, after finitely many surgeries, we will arrive at a minimal algebraic surface XN , that is, KXN is nef. Then (4.1) has a global solution, denoted again by !Q t , on XN . Let us study its asymptotic behavior. There are three possibilities according to the Kodaira dimension .X/ of X: 1. If .X/ D 0, then c1 .X/R D 0 or a finite cover of X is either a K3 surface or an Abelian surface. In this case, the solution !Q t on XN converges to a Ricci flat Kähler metric. In other two cases, we better use the normalized Kähler-Ricci flow (4.33) on XN : @!.s/ Q D Ric.!.s// Q  !.s/; Q @s

!.0/ Q D !0 ;

where t D es  1 and !.s/ Q D es !Q t . 2. If .X/ D 1, then XN is a minimal elliptic surface:  W XN 7! ˙. It was proved in [15] that as s ! 1, !.s/ Q converges to a positive current of the form   .!Q 1 / and the convergence is in the C1;1 -topology on any compact subset

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outside singular fibers Fp1 ; ; Fpk , where p1 ; ; pk 2 ˙. Furthermore, !Q 1 satisfies the generalized Kähler-Einstein equation: Ric.!Q 1 / D !Q 1 C f  !WP ;

on ˙nfp1 ; :pk g;

where f is the induced holomorphic map from ˙nfp1 ; :pk g into the moduli of elliptic curves. 3. If .X/ D 2, then XN is a surface of general type and its canonical model Xcan is a Kähler orbifold with possibly finitely many rational double points and ample canonical bundle. By the version of the Aubin-Yau Theorem for orbifolds, there is an unique Kähler-Einstein metric !Q 1 on Xcan with scalar curvature 2. It was proved in [27] that as s ! 1, !.s/ Q converges to !Q 1 and converges in the C1 topology outside those rational curves over the rational double points. This shows that our program indeed works for algebraic surfaces except that we did not check if the blown-down surfaces coincide with the metric completions described in Conjecture 4.3.7. Acknowledgements This work was partially supported by NSF grants.

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