The Ricci Flow: Techniques and Applications Part IV: Long-time Solutions and Related Topics 0821849913, 9780821849910, 978-0-8218-3946-1, 0-8218-3946-2, 9780821844298, 0821844296, 9780821846612, 0821846612

Table of contents :
Content: pt. 1. Geometric aspects. --
pt. 2 Analytic aspects.--
pt. 3 Geometric-analytic aspects.--
pt. 4 Long-time solutions and related topics.

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Mathematical Surveys and Monographs Volume 206

The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni

American Mathematical Society

The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics

Mathematical Surveys and Monographs Volume 206

The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman Michael I. Weinstein 2010 Mathematics Subject Classification. Primary 53C44, 53C21, 53C43, 58J35, 35K59, 35K05, 57Mxx, 57M50.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-206

Library of Congress Cataloging-in-Publication Data Chow, Bennett. The Ricci flow : techniques and applications / Bennett Chow. . . [et al.]. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 135) Includes bibliographical references and indexes. ISBN-13: 978-0-8218-3946-1 (pt. 1) ISBN-10: 0-8218-3946-2 (pt. 1) 1. Global differential geometry. 2. Ricci flow. 3. Riemannian manifolds. I. Title. QA670.R53 2007 516.362—dc22

2007275659

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by Bennett Chow. All rights reserved.  Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

20 19 18 17 16 15

Contents Preface

ix

Acknowledgments

xiii

Contents of Volume One and Parts I, II, and III of Volume Two Notation and Symbols

xv xvii

Chapter 27. Noncompact Gradient Ricci Solitons 1. Basic properties of gradient Ricci solitons 2. Estimates for potential functions of gradient solitons 3. Lower bounds for the scalar curvature of nonflat nonexpanding gradient Ricci solitons 4. Volume growth of shrinking gradient Ricci solitons 5. Logarithmic Sobolev inequality 6. Gradient shrinkers with nonnegative Ricci curvature 7. Notes and commentary

15 17 26 29 33

Chapter 28. Special Ancient Solutions 1. Local estimate for the scalar curvature under Ricci flow 2. Properties of singularity models 3. Noncompact 2-dimensional ancient solutions with finite width 4. Ancient solutions with positive curvature 5. Notes and commentary

35 35 40 49 63 66

Chapter 29. Compact 2-Dimensional Ancient Solutions 1. Statement of the classification result and outline of its proof 2. The Ricci flow equation on S 2 and some intuition 3. The King–Rosenau solution in the various coordinates 4. A priori estimates for the pressure function 5. The almost everywhere vanishing of R∞ 6. First properties of the backward limit v∞ 7. Isoperimetric constant of metrics on S 2 8. Characterizing round solutions 9. Classifying the backward pointwise limit 10. An unrescaled cigar backward Cheeger–Gromov limit 11. Irreducible components of ∇3 v 12. The heat-type equation satisfied by Q 13. That Q = 0 implies the solution is the King–Rosenau solution ¯ 14. The evolution equation for Q 15. The quantity Q must be identically zero

69 69 70 73 76 79 81 83 87 100 106 108 111 117 124 125

v

1 1 9

vi

CONTENTS

¯ 16. The equivalence of Q and Q 17. Notes and commentary

129 132

Chapter 30. Type I Singularities and Ancient Solutions 1. Reduced distance of Type A solutions 2. Reduced volume at the singular time for Type I solutions 3. Type I solutions have shrinker singularity models 4. Some results on Type I ancient solutions 5. Notes and commentary

133 133 145 154 159 169

Chapter 31. Hyperbolic Geometry and 3-Manifolds 1. Introduction to hyperbolic space 2. Topology and geometry of hyperbolic 3-manifolds 3. The Margulis lemma and hyperbolic cusps 4. Mostow rigidity 5. Seifert fibered manifolds and graph manifolds 6. Notes and commentary

171 171 178 185 192 193 194

Chapter 32. Nonsingular Solutions on Closed 3-Manifolds 1. Introduction 2. The main result on nonsingular solutions 3. The three cases of nonsingular solutions 4. The positive and zero cases of nonsingular solutions 5. The negative case—sequential limits must be hyperbolic 6. Notes and commentary

197 197 200 203 207 210 211

Chapter 33. Noncompact Hyperbolic Limits 1. Main results on hyperbolic pieces 2. Harmonic maps parametrizing almost hyperbolic pieces 3. Proof of the stability of hyperbolic limits 4. Incompressibility of boundary tori of hyperbolic pieces 5. Notes and commentary

213 214 219 226 237 254

Chapter 34. Constant Mean Curvature Surfaces and Harmonic Maps by IFT 1. Constant mean curvature surfaces 2. Harmonic maps near the identity of S n 3. Existence of harmonic maps near the identity of manifolds with negative Ricci curvature 4. Application of Mostow rigidity to the existence of isometries 5. Notes and commentary

266 273 278

Chapter 35. Stability of Ricci Flow 1. Linear stability of Ricci flow 2. Analytic semigroups and maximal regularity theory 3. Dynamic stability results obtained using linearization 4. Dynamic stability results obtained by other methods

279 280 287 296 304

257 257 260

CONTENTS

vii

Chapter 36. Type II Singularities and Degenerate Neckpinches 1. Numerical simulation of solutions with degenerate neckpinches 2. Matched asymptotic studies of degenerate neckpinches 3. Ricci flow solutions with degenerate neckpinch singularities 4. Concluding remarks

307 309 318 324 326

Appendix K. Implicit Function Theorem 1. The implicit function theorem 2. H¨ older spaces and Sobolev spaces on manifolds 3. Harmonic maps and their linearization 4. Spectrum of Δd on p-forms on S n 5. Notes and commentary

327 327 332 336 347 352

Bibliography

353

Index

371

Preface Keys to ignition, use at your discretion. – From “Starin’ Through My Rear View” by Tupac Shakur # This is Part IV (a.k.a. Rijk ), the sequel to Volume One ([75]; a.k.a. gij ) ∂ Rijk , ΔRijk , respectively) and Parts I, II, III ([69], [70], [71]; a.k.a. Rijk , ∂t of Volume Two on techniques and applications of the Ricci flow. For the reader’s convenience, we have included the titles of each chapter on the pages that follow. In this part we mainly discuss aspects of the long-time behavior of solutions to the Ricci flow, including the geometry of noncompact gradient Ricci solitons, ancient solutions, Hamilton’s classification of 3-dimensional nonsingular solutions, and the stability of the Ricci flow. Any theory about singularities of the Ricci flow requires an understanding of ancient solutions and, in particular, gradient Ricci solitons. Building on the success in dimensions at most 3, the study of higher-dimensional Ricci solitons is currently an active field; we discuss some of the progress in this direction. We also present recent progress on (1) the classification of ancient 2-dimensional solutions without the κ-noncollapsing hypothesis and (2) Type I ancient solutions and singularities. In a direction complementary to the study of singularities, we discuss 3-dimensional nonsingular solutions. These solutions underlie the Ricci flow approach to the geometrization conjecture; Hamilton’s work on this is a precursor to Perelman’s more general theory of immortal solutions to the Ricci flow with surgery. Finally, a largely unexplored direction in the Ricci flow concerns the sensitivity of solutions to their initial data; the study of stability of solutions represents an aspect of this. The choice of topics is based on our familiarity and taste. Due to the diversity of the field of Ricci flow, we have inevitably omitted many important works. We have also omitted some topics originally slated for this part, such as the linearized Ricci flow and the space-time formulation of the Ricci flow. We now give detailed descriptions of the chapter contents.

Chapter 27. This chapter is a continuation of Chapter 1 of Part I. Here we discuss some recent progress on the geometry of noncompact gradient Ricci solitons (GRS), including some qualitatively sharp estimates for the volume growth, potential functions, and scalar curvatures of GRS. We also discuss the logarithmic Sobolev inequality for shrinking GRS as well as shrinking GRS with nonnegative Ricci curvature. Chapter 28. This chapter complements the discussion in Part III on Perelman’s theory of 3-dimensional ancient κ-solutions. The topics discussed are a local lower bound for the scalar curvature under Ricci flow, some geometric properties of 3-dimensional singularity models, noncompact 2-dimensional ancient solutions ix

x

PREFACE

without the κ-noncollapsed condition, and classifying certain ancient solutions with positive curvature. Chapter 29. In this chapter we present the results of Daskalopoulos, Hamilton, and Sesum that any simply-connected ancient solution to the Ricci flow on a closed surface must be either a round shrinking 2-sphere or the rotationally symmetric King–Rosenau solution. The proof involves an eclectic collection of geometric and analytic methods. Monotonicity formulas that rely on being in dimension 2 are used. Chapter 30. This chapter is focused on the general study of Type I singularities and Type I ancient solutions. We study properties and applications of Perelman’s reduced distance and reduced volume based at the singular time for Type I singular solutions. We also discuss the result that Type I singular solutions have unbounded scalar curvature. Chapter 31. In the study of nonsingular solutions to the Ricci flow on closed 3-manifolds in the subsequent chapters, of vital importance are finite-volume hyperbolic limits. In this chapter we present some prerequisite knowledge on the geometry and topology of hyperbolic 3-manifolds. Key topics are the Margulis lemma (including the ends of finite-volume hyperbolic manifolds) and the Mostow rigidity theorem. Chapter 32. Hamilton’s celebrated result says that for solutions to the normalized Ricci flow on closed 3-manifolds which exist for all forward time and have uniformly bounded curvature, the underlying differentiable 3-manifold admits a geometric decomposition in the sense of Thurston. The proof of the main result requires an understanding of the asymptotic behavior of the solution as time tends to infinity. If collapse occurs in the sense of Cheeger and Gromov, then the underlying differentiable 3-manifold admits an F -structure and in particular admits a geometric decomposition. Otherwise, one may extract limits of noncollapsing sequences by the uniformly bounded curvature assumption. In the cases where these limits have nonnegative sectional curvature, we can topologically classify the original 3-manifolds. Chapter 33. In the cases where the limits do not have nonnegative sectional curvature, they must be hyperbolic 3-manifolds with finite volume, which may be either compact or noncompact. If these hyperbolic limits are compact, then they are diffeomorphic to the original 3-manifold. On the other hand, if these hyperbolic limits are noncompact, then the difficult result is that their truncated embeddings in the original 3-manifold are such that the boundary tori are incompressible in the complements. To establish this, one proves the stability of hyperbolic limits by the use of harmonic maps and Mostow rigidity. Then, assuming the compressibility of any boundary tori, one applies a minimal surface argument to obtain a contradiction. Chapter 34. The purpose of this chapter is to prove, by the implicit function theorem, two results used in the previous chapter. We first show that almost hyperbolic cusps are swept out by constant mean curvature tori. Second, for any metric g on a compact manifold with negative Ricci curvature and concave boundary and for any metric g˜ sufficiently close to g, we prove the existence of a harmonic diffeomorphism from g to g˜ near the identity map.

PREFACE

xi

Chapter 35. A potentially useful direction in Ricci flow is to study the perturbational aspects of the flow, in particular, stability of solutions, dependence on initial data, and properties of generic solutions and 1-parameter families of solutions. In this chapter we discuss the stability of solutions. The analysis of stability is partly dependent on understanding the Ricci flow coupled to the Lichnerowicz Laplacian heat equation for symmetric 2-tensors. Chapter 36. In this chapter we survey a numerical approach, due to Garfinkle and one of the authors, to modeling rotationally symmetric degenerate neckpinches including the reflectionally symmetric case of two Bryant solitons simultaneously forming as limits. We also survey the matched asymptotic analysis of rotationally symmetric degenerate neckpinches and the related Wa˙zewski retraction method. Appendix K. In this appendix we recall some concepts and results about the analysis on manifolds that are used in various places in the book. In particular, we discuss the implicit function theorem, H¨ older and Sobolev spaces of sections of bundles, formulas for harmonic maps, and the eigenvalues of the Hodge–de Rham Laplacian acting on differential forms on the round sphere.

Acknowledgments I didn’t think I never dreamed That I would be around to see it all come true. – From “Nineteen Hundred and Eighty-Five” by Paul McCartney and Wings

We would like to thank our colleagues, some of whom have been named in previous volumes, for their help, support, and encouragement. In addition, we would like to thank the following mathematicians for helpful discussions: Scot Adams, Jianguo Cao, Yu Ding, Patrick Eberlein, Joel Haas, Richard Hamilton, Emmanuel Hebey, Shengli Kong, John Lott, Chikako Mese, Kate Okikiolu, Anton Petrunin, Justin Roberts, Xiaochun Rong, Peter Scott, Jian Song, Peter Topping, Bing Wang, Deane Yang, and Jiaping Wang. We are especially grateful to John Lott for a number of corrections and suggestions and to Jiaping Wang for help on technical issues. We would like to especially thank Ed Dunne for his tireless efforts and patience in making the publication of our expository works on Ricci flow possible through the American Mathematical Society. Special thanks to Ina Mette and Sergei Gelfand for their continuing help and support. We would like to thank the editors of the Mathematical Surveys and Monographs series. We would like to thank Marcia Almeida for her assistance. Special thanks to Arlene O’Sean for her expert copy editing. We would like to thank Bo Yang and Shijin Zhang for proofreading parts of the manuscript. Ben would like to thank Peng Lu for his vast commitment and contribution to coauthoring this book series. Ben expresses extra special thanks to Classic Dimension for continued encouragement, support, guidance, understanding, patience, faith, forgiveness, and inspiration. Ben dedicates all of his expository works on Ricci flow and in particular this book to Classic Dimension. Sun-Chin Chu would like to thank Nai-Chung Leung and Wei-Ming Ni for their encouragement and help over the years. Sun-Chin would like to thank his parents for their love and support throughout his life and dedicates this book to his family. David Glickenstein would like to thank his wife, Tricia, and his parents, Helen and Harvey, for their love and support. Dave dedicates this book to his family. Christine Guenther would like to thank Jim Isenberg as a friend and colleague for his guidance and encouragement. She thanks her family, in particular Manuel, for their constant support and dedicates this book to them. Jim Isenberg would like to thank Mauro Carfora for introducing him to Ricci flow. He thanks Richard Hamilton for showing him how much fun it can be. He dedicates this book to Paul and Ruth Isenberg. xiii

xiv

ACKNOWLEDGMENTS

Tom Ivey would like to thank Robert Bryant and Andre Neves for helpful comments and suggestions. Dan Knopf thanks his colleagues and friends in mathematics, with whom he is privileged to work and study. He is especially grateful to Kevin McLeod, whose mentorship and guidance have been invaluable. On a personal level, he thanks his family and friends for their love, especially Dan and Penny, his parents, Frank and Mary Ann, and his wife, Stephanie. Peng Lu would like to thank the Simons Foundation for their support through Collaboration Grant 229727. Peng would like to take this opportunity to record: In memory of Professor Weiyue Ding (April 26, 1945–November 11, 2014). Feng Luo would like to thank the NSF for partial support. Lei Ni would like to thank Jiaxing Hong and Yuanlong Xin for initiating his interests in geometry and pde, Peter Li and Luen-Fai Tam for their teaching over the years and for collaborations. In particular, he would like to thank Richard Hamilton and Grisha Perelman, from whose papers he learned much of what he knows about Ricci flow. Bennett Chow, UC San Diego Sun-Chin Chu, National Chung Cheng University David Glickenstein, University of Arizona Christine Guenther, Pacific University Jim Isenberg, University of Oregon Tom Ivey, College of Charleston Dan Knopf, University of Texas, Austin Peng Lu, University of Oregon Feng Luo, Rutgers University Lei Ni, UC San Diego [email protected] May 19, 2015

Contents of Volume One and Parts I, II, and III of Volume Two Volume One: An Introduction 1. The Ricci flow of special geometries 2. Special and limit solutions 3. Short time existence 4. Maximum principles 5. The Ricci flow on surfaces 6. Three-manifolds of positive Ricci curvature 7. Derivative estimates 8. Singularities and the limits of their dilations 9. Type I singularities A. The Ricci calculus B. Some results in comparison geometry

Part I: Geometric Aspects 1. Ricci Solitons 2. K¨ ahler–Ricci Flow and K¨ ahler–Ricci Solitons 3. The Compactness Theorem for Ricci Flow 4. Proof of the Compactness Theorem 5. Energy, Monotonicity, and Breathers 6. Entropy and No Local Collapsing 7. The Reduced Distance 8. Applications of the Reduced Distance 9. Basic Topology of 3-Manifolds xv

xvi

CONTENTS OF VOLUME ONE AND PARTS I, II, AND III OF VOLUME TWO

A. Basic Ricci Flow Theory B. Other Aspects of Ricci Flow and Related Flows C. Glossary

Part II: Analytic Aspects 10. Weak Maximum Principles for Scalars, Tensors, and Systems 11. Closed Manifolds with Positive Curvature 12. Weak and Strong Maximum Principles on Noncompact Manifolds 13. Qualitative Behavior of Classes of Solutions 14. Local Derivative of Curvature Estimates 15. Differential Harnack Estimates of LYH-type 16. Perelman’s Differential Harnack Estimate D. An Overview of Aspects of Ricci Flow E. Aspects of Geometric Analysis Related to Ricci Flow F. Tensor Calculus on the Frame Bundle

Part III: Geometric-Analytic Aspects 17. Entropy, μ-invariant, and Finite Time Singularities 18. Geometric Tools and Point Picking Methods 19. Geometric Properties of κ-Solutions 20. Compactness of the Space of κ-Solutions 21. Perelman’s Pseudolocality Theorem 22. Tools Used in Proof of Pseudolocality 23. Heat Kernel for Static Metrics 24. Heat Kernel for Evolving Metrics 25. Estimates of the Heat Equation for Evolving Metrics 26. Bounds for the Heat Kernel for Evolving Metrics G. Elementary Aspects of Metric Geometry H. Convex Functions on Riemannian Manifolds I. Asymptotic Cones and Sharafutdinov Retraction J. Solutions to Selected Exercises

Notation and Symbols Doesn’t mean that much to me To mean that much to you. – From “Old Man” by Neil Young

The following is a list of some of the notation and symbols which we use in this book.

∇ covariant derivative adjoint heat operator ∗ Lichnerowicz Laplacian heat operator L adjoint Lichnerowicz Laplacian heat operator ∗L  defined to be equal to · dot product or multiplication ˜  Euclidean comparison angle 2 Hessian of f ∇ f  Kulkarni–Nomizu product # sharp operator symmetric tensor product ⊗S α dual vector field to the 1-form α tangential component of the vector W W normal component of the vector W W⊥ Area area of a surface or volume of a hypersurface ASCR asymptotic scalar curvature ratio AVR asymptotic volume ratio ball of radius r centered at p Bp (r) b Bianchi map the quadratic 4-tensor −Rapbq Rcpdq Babcd bounded curvature bounded sectional curvature (for time-dependent metrics, where the bound may depend on time) tangent cone at V of a convex set J ⊂ Rk CV J const constant xvii

xviii

NOTATION AND SYMBOLS

CK Conf d+ d− d+ d− dt , dt , dt , dt

vector space of conformal Killing vector fields group of conformal diffeomorphisms

a Dini time derivative d distance Gromov–Hausdorff distance dGH dμ volume form Euclidean volume form dμE dσ or dA volume form on boundary or hypersurface Laplacian, Lichnerowicz Laplacian, Δ, ΔL , Δd Hodge–de Rham Laplacian diam diameter div divergence Rn with the flat Euclidean metric En Minkowski (n + 1)-space En,1 heat ball of radius r based at (x, t) Er (x, t) exp exponential map F Perelman’s energy functional k Christoffel symbols Γij g (X, Y ) = X, Y  metric or inner product g (t) time-dependent metric, e.g., solution of the Ricci flow limit Riemannian metric or solution of Ricci flow g∞ or g∞ (t) GRS gradient Ricci soliton h or II second fundamental form H mean curvature HV J for V ∈ ∂J set of closed half-spaces H containing J ⊂ Rk with V ∈ ∂H Hess f Hessian of f (same as ∇2 f ) id identity Im imaginary part Inn (G) inner automorphism group int interior inj injectivity radius Isom group of isometries of a Riemannian manifold IVP initial-value problem J Jacobian of the exponential map bundle of k-jets of maps J k (M, N ) KV vector space of Killing vector fields L length

NOTATION AND SYMBOLS

left-hand side natural logarithm a time interval for the Ricci flow a time interval for the backward Ricci flow λ-invariant Perelman’s L-distance reduced distance or -function Lie derivative or L-length L-cut locus L-exponential map L-index form L-Jacobian linear trace Harnack quadratic static Riemannian manifold μ-invariant mean curvature flow space of Riemannian metrics on a manifold mean value property group of M¨ obius transformations multiplication, when a formula does not fit on one line ν ν-invariant or unit outward normal collection of n-dimensional κ-solutions Mn,κ n-dimensional κ-solutions with Harnack MHarn n,κ volume of the unit Euclidean (n − 1)-sphere nωn NRF normalized Ricci flow volume of the unit Euclidean n-ball ωn ode ordinary differential equation Out outer automorphism group PSL (n, C) projective complex special linear group the symmetric 3-tensor ∇i Rjk − ∇j Rik Pijk pde partial differential equation PIC positive isotropic curvature  Rm gm (opposite of Hamilton’s convention) Rijk m i ijk  i Rjk i Rijk = i, g Rijk (components of Ricci) a symmetric 2-tensor (Rjk = Rjk is a special case) Rjk RayM (O) space of rays emanating from O in M set of positive real numbers R>0 RF Ricci flow lhs log I J λ L  L L Cut L exp LI L JV L (v, X) (M, gˆ) μ MCF Met MVP M¨ ob ×

xix

xx

NOTATION AND SYMBOLS

rhs R, Rc, Rm Rm# R Rc (R) Rn SL (n, C) SO (n, R) 2 (so(n)) SB SΩT SV J for V ∈ ∂J sect Sn supp Tx M Tx∗ M τ (t) tr or trace V˜ Vˆ∞

right-hand side scalar, Ricci, and Riemann curvature tensors the quadratic Rm # Rm algebraic curvature operator a trace of R (of two indices) n-dimensional Euclidean space complex special linear group real orthogonal group space of algebraic curvature operators side boundary ∂Ω × (0, T ] set of support functions of J ⊂ Rk at V sectional curvature unit radius n-dimensional sphere support of a function tangent space of M at x cotangent space of M at x function satisfying dτ dt = −1 trace reduced volume

V Vol W W k,p

mock reduced volume vector bundle volume of a manifold Perelman’s entropy functional Sobolev space of functions with ≤ k weak derivatives in Lp

k,p Wloc WMP

space of functions locally in W k,p weak maximum principle

CHAPTER 27

Noncompact Gradient Ricci Solitons There is no elite Just take your place in the driver’s seat. – From “Driver’s Seat” by Sniff ’n’ the Tears

Gradient Ricci solitons (GRS), which were introduced and used effectively in the Ricci flow by Hamilton, are generalizations of Einstein metrics. A motivation for studying GRS is that they arise in the analysis of singular solutions. By Myers’s theorem, there are no noncompact Einstein solutions to the Ricci flow with positive scalar curvature (which would homothetically shrink under the Ricci flow). In view of this, regarding noncompact GRS, one may expect to obtain the most information in the shrinking case. As we shall see in this chapter, this appears to be true. A beautiful aspect of the study of GRS is the duality between the metric and the potential function (we use the term “duality” in a nontechnical way). On one hand, associated to the metric are geodesics and curvature. On the other hand, associated to the potential function are its gradient and Laplacian as well as its level sets and the integral curves of its gradient. In this chapter we shall see some of the interaction between quantities associated to the metric and to the potential function, which yields information about the geometry of GRS. In Chapter 1 of Part I we constructed the Bryant soliton and we discussed some basic equations holding for GRS, leading to a no nontrivial steady or expanding compact breathers result. In the present chapter we focus on the qualitative aspects of the geometry of noncompact GRS. In §1 we discuss a sharp lower bound for their scalar curvatures. In §2 we present estimates of the potential function and its gradient for GRS. In §3 we improve some lower bounds for the scalar curvatures of nontrivial GRS. In §4 we show that the volume growth of a shrinking GRS is at most Euclidean. If the scalar curvature has a positive lower bound, then one obtains a stronger estimate for the volume growth. In §5 we discuss the logarithmic Sobolev inequality on shrinking GRS. In §6 we prove that shrinking GRS with nonnegative Ricci curvature must have scalar curvature bounded below by a positive constant. Although much is known about GRS, there is still quite a lot that is unknown. In this chapter we include some problems and conjectures (often standard or folklore) which speculate about certain geometric properties of GRS. 1. Basic properties of gradient Ricci solitons The main topic of this section is a sharp lower bound for the scalar curvature of a complete GRS. As we shall see in §4 of this chapter, this is useful for the study the volume growth of GRS. The proof of the lower bound involves localizing the 1

2

27. NONCOMPACT GRADIENT RICCI SOLITONS

application of the maximum principle to the elliptic equation satisfied by the scalar curvature of a GRS. 1.1. Normalized GRS structure. A quadruple G = (Mn , g, f, ε), consisting of a connected manifold, Riemannian metric, real-valued function, and real constant, is called a gradient Ricci soliton (GRS) structure if ε (27.1) Rc +∇2 f + g = 0, 2 where Rc denotes the Ricci tensor and where ∇2 f denotes the Hessian of the potential function f . We say that G is complete if g is a complete metric. The GRS G is called shrinking, steady, or expanding if ε < 0, ε = 0, or ε > 0, respectively. Recall the following model cases:  (1) The Gaussian soliton on Rn , which is given by g = ni=1 dxi ⊗ dxi and f (x) = − 4ε |x|2 , where ε ∈ R. (2) An Einstein manifold (Mn , g, 0, ε): Rc + 2ε g = 0. (3) The Bryant soliton. This is the unique (up to homothety) complete rotationally symmetric steady GRS on Rn for n ≥ 3. It has positive curvature operator and its spherical sectional curvatures decay linearly, whereas its radial sectional curvatures decay quadratically. For the Gaussian soliton the curvature is as trivial as possible, whereas for an Einstein manifold the potential function is as trivial as possible. For this reason, one may expect such solutions to represent borderline cases for various geometric invariants of GRS. Let R denote the scalar curvature. We have the following standard formulas first derived by Hamilton (see Chapter 1 of Part I for example): nε (27.2) R + Δf = − , 2 (27.3)

2 Rc(∇f ) = ∇R,

(27.4)

ΔR + 2| Rc |2 + εR = ∇R, ∇f  ,

(27.5)

R + |∇f |2 + εf ≡ const,

where the constant const depends on g and f . If ε = 0, then we may take ε = ±1 and by adding a constant to the potential function f , we may assume that const = 0. If ε = 0 and g is nonflat, then by scaling the metric we may take const = 1. In these cases we say that G is normalized: (27.6a) (27.6b)

R + |∇f |2 + εf ≡ 0 for ε = ±1, R + |∇f |2 ≡ 1 for ε = 0.

A complete normalized shrinking, steady, or expanding GRS G is called a shrinker, a steady, or an expander, respectively. Let dμg denote the Riemannian measure of g. Given a metric measure space (Mn , g, e−f dμg ), define the f -Laplacian to be (27.7)

Δf  Δ − ∇f · ∇.

1. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS

3

Define the f -Ricci tensor to be Rcf  Rc +∇2 f ; this is also known as the Bakry– 2 Emery Ricci tensor. Define the f -scalar curvature to be Rf = R+2Δf −|∇f | . Given a measurable set X ⊂ M, define its f -volume to be  (27.8) Volf (X)  e−f dμ. X

The operator Δf is self-adjoint with respect to the L2 -inner product of functions using the measure e−f dμ. Exercise 27.1. Show that for any ϕ ∈ C ∞ (M) we have that     1 1 Δf − Rf ϕ = ef /2 Δ − R (e−f /2 ϕ). 4 4 Equations for normalized GRS may be conveniently expressed using Δf , Rcf , and Rf . First of all, ε Rcf + g = 0 2 by (27.1) and we also have Rf = ε (f − n) , Rf = −1,

for ε = ±1,

for ε = 0.

We may rewrite the difference of (27.2) and (27.6) as  n (27.9a) for ε = ±1, Δf f = ε f − 2 Δf f = −1 for ε = 0 (27.9b) and we may rewrite (27.4) as (27.10)

2 Δf R = −2| Rc |2 − εR ≤ − R2 − εR. n

1.2. Scalar curvature lower bound for a GRS structure. The following result of B.-L. Chen does not require any curvature bound in its hypothesis. In the case where M is compact, this follows directly from applying the maximum principle to equation (27.4). Theorem 27.2 (Lower bound for the scalar curvature of GRS). Let (Mn , g, f, ε), where ε = −1, 0, or 1, be a complete GRS structure. (1) If the GRS is shrinking or steady, then R ≥ 0. (2) If the GRS is expanding, then R ≥ − n2 . Remark 27.3. Note that the Gaussian soliton shows that part (1) is sharp, whereas the Einstein solutions with Rc = − 12 g show that part (2) is sharp. Proof of Theorem 27.2. We may assume that M is noncompact. The proof consists of localizing equation (27.4). ˜ ∈M Step 1. The Laplacian of R times a cutoff function η. Fix a point O ˜ and let r (x)  d(x, O). Let b ∈ [2, ∞) and define η : [0, ∞) → [0, 1] to be a C ∞ nonincreasing cutoff function with  2 1 for u ∈ [0, 1] , (η  ) (27.11) η (u) = and η  − 2 ≥ − const η 0 for u ∈ [1 + b, ∞)

4

27. NONCOMPACT GRADIENT RICCI SOLITONS

for some universal const < ∞. We define Φc : M → R by   r (x) (27.12) Φc (x) = η R (x) . c Throughout the proof, c ∈ [2, ∞); eventually we let c → ∞. Taking the f -Laplacian of (27.12), we have   r  η ◦c η  ◦ r 2   r Δf R + η ◦ ∇r, ∇R + Δf r + Δf Φc = η ◦ c c c c c2

r c

 |∇r|2 R.

2

Applying (27.10) and |∇r| = 1 to this, while dropping “ ◦ rc ” in our notation, we have (27.13)    η 2η  η  2 ∇r, ∇R + Δf r + 2 R Δf Φc = η(−2 |Rc| − εR) + c c c

2 2η  η 1 (η  ) 2  = η(−2 |Rc| − εR) + ∇r, ∇Φc  + (Δf r)R + 2 η − 2 R cη c c η at all points where η = 0. Step 2. Applying the maximum principle 1 to Φc . Now suppose that there exists xc ∈ M such that (27.14)

Φc (xc ) = min Φc < 0. M

Otherwise, we have R ≥ 0 in all of BO˜ (c). Applying the first and second derivative tests to (27.13), using |Rc|2 ≥ and dividing by R (xc ) < 0, we have that at xc ,

  2 2 1 (η  ) η  (27.15) 0 ≥ η − R − ε + Δf r + 2 η − 2 . n c c η

1 2 nR ,

We consider two cases, depending on the location of xc . Case (i): xc ∈ BO˜ (c). Then η ◦ rc ≡ 1 in a neighborhood of xc , so that (27.15) and (27.14) imply   r (x) 2 2 2 (27.16) 0 ≥ − R (xc ) − ε = − Φc (xc ) − ε ≥ − η R (x) − ε n n n c for all x ∈ M. This yields the estimate R (x) ≥ −

(27.17)

nε 2

for all x ∈ BO˜ (c) since η ◦ rc = 1 in BO˜ (c). Case (ii): xc ∈ / BO˜ (c). Regarding (27.15), since η  ≤ 0, we wish to estimate the term Δf r from above. Recall from Lemma 18.6 in Part III (or the original Lemma 8.3 in Perelman [312]) that  r(xc )   2 (n − 1) (ζ  ) (s) − ζ 2 Rc (γ  (s), γ  (s)) ds (27.18) Δr(xc ) ≤ 0 1 The distance function r (x) is in general only Lipschitz continuous. When applying the maximum principle, to address the possible nonsmoothness of r (x), one may use Calabi’s trick; see p. 395 of [77] or pp. 453–456 in Part I for example.

1. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS

5

˜ to xc and any for any unit speed minimal geodesic γ : [0, r (xc )] → M joining O continuous piecewise C ∞ function ζ : [0, r (xc )] → [0, 1] satisfying ζ (0) = 0 and ζ (r (xc )) = 1, where γ  (s)  dγ ds (s). We also have  r(xc ) d  ˜ (27.19) ∇f, ∇r ds − ∇f, ∇r (xc ) = −∇f (O), γ (0) − ds 0  r(xc ) ˜ γ  (0) − = −∇f (O), ∇2 f (γ  , γ  ) ds 0

since ∇r = γ  (s) and ∇γ  (s) ∇r = 0. Therefore applying (27.1) to (27.19) and combining with (27.18), we have  r(xc ) 2 ˜ γ  (0) + ε r (xc ) Δf r(xc ) ≤ (n − 1) (ζ  ) ds − ∇f (O), 2 0  r(xc )  1 − ζ 2 Rc (γ  , γ  ) ds. + 0

Let ζ (s) = s for 0 ≤ s ≤ 1 and let ζ (s) = 1 for 1 < s ≤ r (xc ). Then (27.20)

˜ γ  (0) + ε r (xc ) + 2 max Rc+ , Δf r(xc ) ≤ n − 1 − ∇f (O), 2 3 B¯O˜ (1)

where maxB¯O˜ (1) Rc+  maxV ∈Tx M, |V |=1, x∈B¯O˜ (1) {Rc (V, V ) , 0}. Applying (27.20) and (27.11) to (27.15) yields for all x ∈ BO˜ (c) (27.21)

2 2 R (x) ≥ Φc (xc ) n n   η  ( r(xc c ) ) ˜ γ  (0) + 2 max Rc+ ≥ n − 1 − ∇f (O), c 3 B¯O˜ (1) r(x )

  c η  ( c )r (xc ) r (xc ) const −η +ε − 2 , 2c c c

where const < ∞ is independent of b and c. Since xc ∈ BO˜ ((1 + b) c) − BO˜ (c) in Case (ii), we have (27.22)

1≤

r (xc ) < 1 + b. c

We now consider the nonexpanding and expanding cases separately, where we primarily need to deal with Case (ii). Step 3. Proof of the theorem when ε = 0 or −1. Here, Case (ii) must always hold since otherwise (27.16) contradicts assumption (27.14). We take b = 2. Then the inequality (27.21) and − const ≤ η  ≤ 0 imply that for all x ∈ BO˜ (c)   const 2 ˜ + 2 max Rc+ − const , R (x) ≥ − (27.23) n − 1 + |∇f | (O) n c 3 B¯O˜ (1) c2 where const is independent of c. By taking c → ∞, we conclude that R (x) ≥ 0 for all x ∈ M. This completes the proof of the theorem in the nonexpanding case.

6

27. NONCOMPACT GRADIENT RICCI SOLITONS

Step 4. Proof of the theorem when ε = 1. Define η : [0, ∞) → [0, 1] so as to satisfy the properties that η is C ∞ on (0, 1 + c), ⎧ ⎪ if u ∈ [0, 1] , ⎨ 1 1+c−u if u ∈ [2, 1 + c), (27.24) η (u)  c ⎪ ⎩ 0 if u ∈ [1 + c, ∞), and (27.25)

−

2 ≤ η  ≤ 0 and c

|η  | ≤ const

on (0, 1 + c) ,

where const < ∞ is independent of c. Note that η satisfies the prior conditions in (27.11) with b = c. Moreover, for the purposes below, the nondifferentiability of η at u = 1 + c shall not be an issue since η (1 + c) = 0. If Case (i) holds, i.e., xc ∈ BO˜ (c), then we have the estimate (27.17) in BO˜ (c). Now assume that we are in Case (ii). Then xc ∈ BO˜ ((1 + c) c) − BO˜ (c). and η  (u) = − 1c (ii)(a) If xc ∈ BO˜ ((1 + c) c)−BO˜ (2c), then since η (u) = 1+c−u c for u ∈ [2, 1 + c), from (27.21) we have that for all x ∈ BO˜ (c)   2 2 1 1 ˜ (27.26) R (x) ≥ − 2 n − 1 + |∇f | (O) + r (xc ) + max Rc+ n c 2 3 B¯O˜ (1) 1 + c − r(xc c ) 1 − 2 const c  c  1+c 2 1 1 ˜ ≥− − 2 n − 1 + |∇f | (O) + max Rc+ − 2 const . ¯ c c 3 BO˜ (1) c −

(ii)(b) Otherwise, if xc ∈ BO˜ (2c) − BO˜ (c), then since η ≤ 1, from (27.21) and from (27.25) we have that for all x ∈ BO˜ (c)   2 2 ˜ + c + 2 max Rc+ − 1 − 1 const . (27.27) R (x) ≥ − 2 n − 1 + |∇f | (O) n c 3 B¯O˜ (1) c2 From combining the estimates (27.26) and (27.27) for Case (ii) with the estimate (27.17) for Case (i), we have that for all x ∈ BO˜ (c)   2 1 2 1+c 1 (27.28) R (x) ≥ min − − 2 const, − 2 (const +c) − 1 − 2 const, −1 . n c c c c We then conclude when ε = 1 that, from taking c → ∞ in (27.28), n2 R (x) ≥ −1 for all x ∈ M. This completes the proof of Theorem 27.2 in the expanding case.  1.3. Characterizing completeness of GRS structures. Recall that the vector field ∇f is complete if for each p ∈ M, the integral curve γp to ∇f with γp (0) = p may be defined on all of R. In this case, ∇f generates a 1-parameter group of diffeomorphisms {ϕt }t∈R of M which is given by ϕt (p) = γp (t) for any p ∈ M and t ∈ R. We are interested in finding conditions to ensure that the vector field ∇f is complete on a GRS. First, by applying the lower bound for R in Theorem 27.2 to (27.6), we obtain the following result.

1. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS

7

Theorem 27.4 (Bounds for |∇f |). Suppose that (Mn , g, f, ε) is a complete ˜ ∈ M, we have the following: normalized GRS. Then, given any O (1) For a steady, |∇f | (x) ≤ 1

(27.29)

for all x ∈ M.

(2) For a shrinker we have f ≥ 0 and for all x ∈ M, ˜ ˜ + 1 d(x, O). (27.30) |∇f | (x) ≤ f 1/2 (x) ≤ f 1/2 (O) 2 (3) For an expander we have f ≤ n2 and for all x ∈ M, n 1/2  1/2 1 ˜ + n − f (O) ˜ (27.31) |∇f | (x) ≤ − f (x) ≤ d(x, O) . 2 2 2 Proof. (1) ε = 0. Since R ≥ 0, by (27.6b) we have 1 ≡ R + |∇f |2 ≥ |∇f |2 .

(27.32)

(2) ε = −1. Again since R ≥ 0, by (27.6a) we have |∇f |2 = −R + f ≤ f. ˜ → M be a unit speed minimal geodesic joining For any x ∈ M, let γ : [0, d(x, O)] ˜ to x. The function F (s)  f (γ (s)) satisfies O dF (s) = ∇f (γ (s)) · γ  (s) ≤ |∇f | (γ (s)) ≤ F 1/2 (s) . ds ˜ we obtain2 Integrating this over [0, d(x, O)], ˜ ˜ ≤ F 1/2 (0) + 1 d(x, O). F 1/2 (d(x, O)) 2 ˜ = x, this and (27.33) yield (27.30). Because γ(d(x, O)) (3) ε = 1. Since R ≥ − n2 , we have n (27.34) −f = R + |∇f |2 ≥ − + |∇f |2 . 2 Let the geodesic γ be as in part (2). The function G (s)  −f (γ (s)) + n2 ≥ 0 satisfies dG (s) = −∇f (γ (s)) · γ  (s) ≤ |∇f | (γ (s)) ≤ G1/2 (s) ds ˜ ≤ G1/2 (0) + 1 d(x, O), ˜ which implies (27.31). by (27.34). Therefore G1/2 (d(x, O)) 2  (27.33)

Remark 27.5. The example of the Gaussian soliton, where we have |∇f | (x) = |x| for x ∈ Rn , shows that the above upper bounds for |∇f | are qualitatively sharp. |ε| 2

The following is elementary. Lemma 27.6 (Criterion for completeness of a vector field). Suppose F : [0, ∞) → [0, ∞) is a locally Lipschitz function with the property that any solution u (t) to the ode du dt = F (u) , with u (0) ∈ [0, ∞), exists for all t ∈ [0, ∞). If a vector field X on ˜ satisfies |X| (x) ≤ F (d(x, O)) ˜ a pointed complete Riemannian manifold (Mn , g, O) for all x ∈ M, then X is complete. 2 For

each δ > 0 we have

d ((F ds

(s) + δ)1/2 ) ≤

1 . 2

8

27. NONCOMPACT GRADIENT RICCI SOLITONS

Combining Theorem 27.4 with Lemma 27.6 yields the following result of Z.-H. Zhang. Corollary 27.7 (Completeness of g implies completeness of ∇f ). If (Mn , g, f, ε) is a complete GRS, then ∇f is a complete vector field. 1.4. Equality case of the scalar curvature lower bound. We now return to Theorem 27.2 and consider the equality case. We have the following result of Pigola, Rimoldi, and Setti. Proposition 27.8. Let (Mn , g, f, ε) be a complete GRS. (1) If ε = 1 and if there exists x0 ∈ M such that R (x0 ) = − n2 , then Rc ≡ − 12 g. (2) If ε = 0 and if there exists x0 ∈ M such that R (x0 ) = 0, then Rc ≡ 0. (3) If ε = −1 and if there exists x0 ∈ M such that R (x0 ) = 0, then (M, g, f, −1) is a Gaussian soliton, so that (M, g) is isometric to Euclidean space. Proof. The idea is to apply the strong maximum principle. By Corollary 27.7 and by Theorem 4.1 in [77], we may extend the GRS structure (M, g, f, ε) canonically in time so that g (t) is a complete solution of the Ricci flow with g (0) = g and such that g (t) and f (t) satisfy ε g (t) = 0. Rcg(t) +∇2g(t) f (t) + 2 (εt + 1) Recall the standard equation ∂R = ΔR + 2 |Rc|2 . ∂t

(27.35) ˜ R+ We thus have that R

nε 2(εt+1)

satisfies  2   ˜  R  nε ∂R 2 ˜  ˜ ˜ R− R, = ΔR + 2 Rc − g  + ∂t n n εt + 1

(27.36) where we used

2     R  nε 2 ˜  ˜ 2 |Rc| − 2 = 2 Rc − n g  + n R − εt + 1 R. 2 (εt + 1) 2

nε2

Case i: ε = 0 or 1. By Theorem 27.2 we have nε R+ ≥ 0. 2 (εt + 1) ˜ (x0 , 0) = 0, by applying the parabolic strong Since there exists x0 ∈ M such that R ˜ (x, t) ≡ 0 for maximum principle (which is a local result) to (27.36), we have R ε x ∈ M and t ≤ 0 such that εt + 1 > 0. Thus, by (27.36), we have Rc ≡ R n g ≡ − 2 g. This proves parts (1) and (2). Case ii: ε = −1. By Theorem 27.2 we have R ≥ 0. Applying the parabolic strong maximum principle to (27.35), we have R ≡ 0, which in turn implies Rc ≡ 0. Since g is a shrinker, we then obtain (27.37)

∇2 f =

1 g > 0. 2

2. ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS

9

Thus f is uniformly convex and proper. In particular, f attains its infimum at a unique point O ∈ M and M is diffeomorphic to Rn . Part (3) of Proposition 27.8 is now a consequence of the following lemma.  Lemma 27.9 (A characterization of Euclidean space). Let (Mn , g) be a complete Riemannian manifold. If there exists a function f such that ∇2 f =

(27.38)

1 g, 2

then (M, g) is isometric to Euclidean space. In particular, any complete Ricci flat shrinking GRS must be isometric to Euclidean space. Proof. By (27.38), we have 2

∇i |∇f | = 2∇i ∇j f ∇j f = ∇i f, so that adding a suitable constant to f yields (27.39)

f = |∇f |2 ≥ 0,

√ which implies that inf M f = f (O) = 0. Hence, defining r  2 f , we have on M − {O} that  2 (27.40) ∇2 r 2 = 2g, |∇r| = 1. In particular, ∇∇r ∇r = 0, so that the integral  curves to ∇r are unit speed geodesics. Furthermore, by (27.40) we have that ∇ r 2 is a complete vector field which generates a 1-parameter group {ϕt }t∈R of homotheties of g. Since r : M → [0, ∞), where r 2 is C ∞ , proper, and the only critical point of r is at O with r (O) = 0, and since (M, g) is complete, by Morse theory we have that Sc  r −1 (c) is diffeomorphic to S n−1 for all c ∈ (0, ∞). Since |∇r| = 1, each homothety ϕt of g maps level sets of r to level sets of r. Hence g may be written as the warped product g = dr 2 + r 2 g˜, where g˜ = g|S1 . Since g is smooth at O, where r = 0, we have that (S1 , g˜) must  Sc = M − {O}, we conclude be isometric to the unit (n − 1)-sphere. Since c∈(0,∞)

that (Mn , g) is isometric to Euclidean space.



2. Estimates for potential functions of gradient solitons A good qualitative understanding of the potential function is crucial in understanding the geometry of GRS. In this section we study the potential function f of a complete GRS structure (Mn , g, f, ε). In particular, we shall obtain bounds for |∇f |, upper and lower bounds for f , and, as a consequence, upper bounds for the scalar curvature R, all depending on the distance to a fixed point. 2.1. Bounds for the potential function f . Immediate consequences of Theorem 27.4 are the following bounds for the potential functionof a GRS.

10

27. NONCOMPACT GRADIENT RICCI SOLITONS

Corollary 27.10 (Bounds on f for GRS). Let (Mn , g, f, ε) be a complete ˜ ∈ M. normalized GRS and fix O (1) For a steady, for all x ∈ M    ˜ ˜  ≤ d(x, O). (27.41) f (x) − f (O) (2) For a shrinker, for all x ∈ M 2  1 ˜ ˜ + 2 f (O) (27.42) f (x) ≤ . d(x, O) 4 If O is a minimum point of f , then f (O) ≤

(27.43)

n 2

and f (x) ≤

(27.44)

√ 1 (d (x, O) + 2n)2 . 4

(3) For an expander, for all x ∈ M   1/2 2 n 1 n ˜ ˜ (27.45) − + ≤ f (x) ≤ . d(x, O) + −4f (O) + 2n 4 2 2 Proof. (1) This follows from integrating inequality (27.29) along minimal geodesics. (2) Since f ≥ 0, this follows from squaring the right-hand inequality in (27.30). Now suppose that O is a minimum point of f . Since Δf (O) ≥ 0, by R+Δf = n2 we have R(O) ≤ n2 . Hence, by R + |∇f |2 = f and |∇f |2 (O) = 0, we conclude that f (O) ≤ n2 . Applying this to (27.42) yields (27.44). (3) Since f ≤

n 2,

this follows from squaring the right-hand inequality in (27.31). 

For shrinkers, the potential function is in fact uniformly equivalent to the distance squared. The elegant proof of this relies on the second variation of arc length formula and integration by parts. Let a+ = max {a, 0} for a ∈ R. We have the following originally due to H.-D. Cao and D. Zhou and later refined by Haslhofer and M¨ uller (with the sharper constants as presented here). Theorem 27.11 (Lower bound for f on shrinkers). Let (Mn , g, f, −1) be a ˜ ∈ M, we have complete normalized shrinking GRS. Given O   2  1 4 ˜ − 4n + ˜ − 2 f (O) (27.46) f (x) ≥ . d(x, O) 4 3 + If O is a minimum point of f , then   2 1 35 (27.47) f (x) ≥ . d (x, O) − n 4 8 +

2. ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS

11

Proof. Let x ∈ M − BO˜ (2) be any point and let γ : [0, r (x)] → M be a unit ˜ to x, where r(x)  d(x, O). ˜ Define ζ : [0, r (x)] → speed minimal geodesic joining O [0, 1] to be the piecewise linear function ⎧ ⎪ s if s ∈ [0, 1] , ⎨ 1 if s ∈ (1, r (x) − 1], (27.48) ζ (s) = ⎪ ⎩ r (x) − s if s ∈ (r (x) − 1, r (x)]. Let {E1 , . . . , En−1 , γ  (0)} be an orthonormal basis of TO˜ M. Define Ei (s) ∈ Tγ(s) M to be the parallel translation of Ei = Ei (0) along γ. Then the frame {E1 (s), . . . , En−1 (s), γ  (s)} forms an orthonormal basis of Tγ(s) M for s ∈ [0, r (x)]. Since γ is minimal, by the second variation of arc length formula, we have for each i  r(x)   2 2 (ζ ) − ζ 2 Rm (γ  , Ei ) Ei , γ   ds. L(γ) = 0 ≤ δζE i 0

Summing over i, we obtain (compare with (27.18))  r(x)  r(x) 2 (27.49) ζ 2 Rc (γ  , γ  ) ds ≤ (n − 1) (ζ  ) ds = 2(n − 1). 0

0

By applying the shrinker equation to Rc and integrating by parts, we obtain   r(x)  1  2 − (f ◦ γ) ds ζ 2(n − 1) ≥ (27.50) 2 0  1  r(x) 2 1   s (f ◦ γ) ds − 2 ζ (f ◦ γ) ds = r(x) − + 2 2 3 0 r(x)−1  1   2 1 ˜ + s ds s f (O) ≥ r(x) − − 2 2 3 2 0    r(x)  ζ (s) −2 ζ(s) f (x) + ds, 2 r(x)−1 where for the last inequality we used by (27.30) that   ˜ +s | (f ◦ γ) (s) | ≤ |∇f |(γ (s)) ≤ f (O) 2 √  ζ(s) for s ∈ [0, 1] and | (f ◦ γ) (s) | ≤ f (x) + 2 for s ∈ [r(x) − 1, r(x)]. Hence  4  ˜ 1 − f (x) . 2(n − 1) ≥ r(x) − − f (O) 2 3 That is,   ˜ − 2n + 2 . ˜ ≥ 1 d(x, O) (27.51) f (x) + f (O) 2 3 If O is a minimum point of f , then f (O) ≤ n2 by (27.43) and hence we have √ f (x) ≥ 12 r(x) − 35  16 n. Remark 27.12. Compare (27.51) with Lemma 19.46 in Part III of Volume Two. Corollary 27.13. For a complete normalized shrinker, if f attains its minimum at O1 and O2 , then √ 35 n + 2n. (27.52) d(O1 , O2 ) ≤ 8

12

27. NONCOMPACT GRADIENT RICCI SOLITONS

Proof. It follows from (27.47) that  35 d(x, O1 ) ≤ 2 f (x) + n for all x ∈ M. 8 Taking x = O2 and using f (O2 ) ≤ n2 yields (27.52).



Problem 27.14 (Potential functions of expanders). Regarding Corollary 27.10(3), what is the best qualitative estimate for f in the expanding case? Note that if an expander has Rc > 0, then the potential function is bounded from above ˜) d2 (x,O + C (see Lemma 9.51 in [77] for example). by − 4 2.2. Upper bounds for R. The estimates we proved in the previous subsections have various consequences, including bounds for the scalar curvature. For a shrinker, (27.6a) and (27.42) imply the following: Lemma 27.15 (Scalar curvature has at most quadratic growth). Let (Mn , g, ˜ ∈ M. Then f, −1) be a complete normalized noncompact shrinking GRS and let O for all x ∈ M, 2  1 ˜ + 2 f (O) ˜ (27.53) 0 ≤ R (x) = f (x) − |∇f |2 (x) ≤ d(x, O) . 4 By the work of Gromoll and Meyer [124], there exist complete noncompact Riemannian manifolds with positive Ricci curvature and infinite topological type. On the other hand, Theorem 27.11 implies the following for shrinking GRS assuming a growth condition on R. Corollary 27.16. Let (Mn , g, f, −1) be a complete normalized noncompact shrinking GRS. If its scalar curvature satisfies ˜ + C)2 (27.54) R (x) ≤ α(d(x, O) for some constants α < 14 and C < ∞, then M has finite topological type. In particular, any complete noncompact shrinking GRS with bounded scalar curvature has finite topological type. Proof. By (27.46), there exists a positive constant C such that 1 2 (r (x) − C) 4 for x ∈ M − BO˜ (C). Hence f is a proper function; i.e., f −1 ((−∞, c]) is compact for any c ∈ R. On the other hand, by (27.6a), (27.55), and (27.54), we have (27.55)

f (x) ≥

1 (r (x) − C)2 − α (r (x) + C)2 . 4 Thus we have |∇f |2 (x) > 0 provided r (x) is sufficiently large. Therefore, using the properness of f , we conclude that there exists k ∈ R such that the compact set K  f −1 ((−∞, k]) contains all of the critical points of f . By the “deformation lemma” in Morse theory, we may deform f in a neighborhood of K, so that all of the critical points of f are nondegenerate (and still lie in a compact set). The fact that M has finite topological type then follows from standard Morse theory (see Milnor [233]).  |∇f |2 = f − R ≥

2. ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS

13

Since the assumption (27.54) is on the edge of holding true in the sense that it is true for α = 14 (see Lemma 27.15), we ask the following. Problem 27.17 (Conjecture 1.3 in [105]). Can one remove the condition (27.54) with α < 14 in Corollary 27.16? A solution to Problem 27.17 would follow from obtaining a good lower estimate for |∇f |2 , i.e., one which would show that |∇f |2 is positive outside a compact set. More ambitiously, one may pose the following: Optimistic Conjecture 27.18. For any complete noncompact shrinking GRS, the scalar/Ricci/sectional curvature is necessarily bounded from above. One of the best results in this direction is due to O. Munteanu and J. Wang; they proved the following.  Theorem 27.19. Let M4 , g, f, −1 be a 4-dimensional complete noncompact shrinking GRS. If the scalar curvature R is bounded, then there exists a constant C < ∞ such that |Rm| ≤ CR on M. In particular, |Rm| is bounded. Some elementary evidence for Optimistic Conjecture 27.18 is given by inequality (27.79) below, which says that the average scalar curvatures on the sublevel sets of the potential function are bounded above by n2 . Next we improve the scalar curvature upper bound for a net of points. We say that a countable collection of points {xi } in a Riemannian manifold (Mn , g) is a δ-net if for every y ∈ M there exists i such that d (y, xi ) ≤ δ. The following result supports Problem 27.17. Lemma 27.20. Let (Mn , g, f, −1) be a complete normalized noncompact shrinking GRS and let O ∈ M be a minimum point of f . Then for any δ > 0 there exists a constant C(n, δ) < ∞ such that for any x ∈ M − BO (C(n, δ)), there exists y ∈ Bx (δ) such that (27.56)

R (y) ≤ C(n, δ) (d (y, O) + 1) .

In other words, the scalar curvature has at most linear growth on an δ-net, where the rate of growth depends on δ. Proof. Define

⎧ ⎪ ⎨ ζδ (s) =

⎪ ⎩

s δ

1 r(x)−s δ

if s ∈ [0, δ] , if s ∈ (δ, r (x) − δ], if s ∈ (r (x) − δ, r (x)].

As in (27.50), we obtain from (27.49) that  δ  r(x) r(x) 2δ s r (x) − s 2(n − 1)  ≥ − +2 (f ◦ γ) ds − 2 (f ◦ γ) ds. 2 2 δ 2 3 δ δ 0 r(x)−δ √  Using | (f ◦ γ) (s)| ≤ f (O) + 2s for s ∈ [0, δ], the above formula implies that    r(x)  n−1 δ r (x) − s r(x)  − 2 + (f ◦ γ) ds ≥ f (O) . 2 − δ2 2 δ 2 r(x)−δ

14

27. NONCOMPACT GRADIENT RICCI SOLITONS

Since 2 satisfies

 r(x) r(x)−δ

r(x)−s δ 2 ds

= 1, there exists sˆ ∈ [r (x) − δ, r (x)] such that y  γ(ˆ s)

   n−1 δ n r(y) −2 + |∇f | (y) ≥ − 2 δ 2 2 n √ since r(x) ≥ r(y) and f (O) ≤ 2 . Thus 2

|∇f | (y) ≥

1 2 ((d (y, O) − 2C(n, δ))+ ) , 4

n δ where C(n, δ)  2( n−1 δ + 2) + 2. On the other hand, by (27.44) we have f (y) ≤

√ 1 (d (y, O) + 2n)2 . 4

Therefore, if d (x, O) ≥ 2C(n, δ) + δ, then d (y, O) ≥ 2C(n, δ) and hence 2

R(y) = f (y) − |∇f | (y) √ 1 1 ≤ (d (y, O) + 2n)2 − (d (y, O) − 2C(n, δ))2 4 4   n n + C(n, δ) + − C(n, δ)2 . = d (y, O) 2 2



This motivates us to consider Conjecture 27.21 (Elliptic Harnack estimate for the scalar curvature). Let (Mn , g, f, −1) be a complete noncompact shrinking GRS. There exists const < ∞ such that for any x, y ∈ M with d (x, y) ≤ 1 we have (27.57)

R (x) ≤ const R (y) .

Exercise 27.22 (An elliptic Harnack estimate would imply finite topological type). Show that the truth of (27.57) would affirm Problem 27.17. Returning to the lower bound for |∇f | of a noncompact shrinker, note that the ˜ − C on M for some most optimistic conjecture would be that |∇f | (x) ≥ 12 d(x, O) constant C. By (27.6a) and (27.46), we have   2 1  ˜ −C d(x, O) (27.58) |∇f |2 ≥ − R, 4 + so that such an estimate would follow from a uniform upper bound for R. Regarding the scalar curvature of a noncompact steady or expanding GRS, if we further assume a positive Ricci pinching condition, then R attains its maximum and has quadratic exponential decay (see D. Chen and L. Ma [215] or Theorem 9.56 in [77]). In particular, we have Proposition 27.23. Let G = (Mn , g, f, 1) be an expanding GRS with Rc ≥ ηRg, where η > 0 and R > 0. Then, for any η¯ < η, there exists C < ∞ such that (27.59)

R(x) ≤ Ce−¯ηd

2

˜ (x,O)

.

3. LOWER BOUNDS FOR R OF NONFLAT NONEXPANDERS

15

We have the following results of Y. Deng and X. Zhu. Theorem 27.24. Let G = (M, g, f, 1) be an expanding K¨ ahler GRS with dimC M ˜ ∈ M. If R ∈ o(d−2 (x, O)), ˜ = n and let O then (M, g) is isometric to Cn . Corollary 27.25. There does not exist an expanding K¨ ahler GRS with dimC M ≥ 2 and Rc ≥ ηRg, where η > 0 and R > 0. 3. Lower bounds for the scalar curvature of nonflat nonexpanding gradient Ricci solitons In this section, in the case of nonflat nonexpanding (i.e., shrinking and steady) GRS and with the aid of the estimates for the potential function of the previous section, we sharpen the lower bounds for the scalar curvature given in §1. 3.1. Lower estimate of R for shrinkers. In this subsection we discuss a lower bound for the scalar curvatures of noncompact nonflat shrinkers. We have the following result due to the combined works of B. Wilking, B. Yang, and three of the authors. Theorem 27.26 (Scalar curvature of nonflat shrinkers decay at most quadratically). Let (Mn , g, f, −1) be a complete normalized noncompact nonflat shrinking ˜ ∈ M there exists a constant C0 > 0 such that GRS. Then for any point O ˜ R(x)  C0−1 d−2 (x, O) ˜  C0 . Consequently, the asymptotic scalar curvature ratio wherever d(x, O) ASCR (g) > 0 (see (19.8) in Part III for its definition) and the asymptotic cone, if it exists, is not flat. Proof. We modify the quantity R appearing in (27.10) by adding powers of f . For any p ∈ R, using (27.9a) we compute that

  2 1 −p |∇f | n −p −p−1 f − (p + 1) . (27.60) Δf =f −f p 2 f In particular, we shall use the formulas (p = 1, 2)

2  −1 |∇f | n −1 −2 Δf f (27.61) −2 , =f −f 2 f

2  −2 |∇f | −2 −3 Δf f = 2f − f (27.62) . n−6 f Using (27.10) and (27.61), we compute for any c > 0 that

2  |∇f | n −1 −1 −2 (27.63) Δf R − cf −2 .  R − cf + cf 2 f Keeping in mind that we wish to modify (27.63) so that more negative terms appear on the rhs, we define φ  R − cf −1 − cnf −2 . By (27.62), we obtain   f 2 −3 − n − cf −4 (2f + 6n) |∇f | . (27.64) Δf φ  φ − cnf 2

16

27. NONCOMPACT GRADIENT RICCI SOLITONS

By (27.42) and (27.46), we have good estimates on the potential function:   2 2  1  1 ˜ ˜ + 2 f (O) ˜ d(x, O) − C1 d(x, O) (27.65) ≤ f (x) ≤ 4 4 + for some constant C1 . Choosing c > 0 sufficiently small, we have φ > 0 inside BO˜ (C1 + 3n). If inf M−BO˜ (C1 +3n) φ  −δ < 0, then by (27.65) there exists ρ > C1 +3n such that φ > − 2δ in M−BO˜ (ρ). Thus a negative minimum of φ is attained at some point x0 outside of BO˜ (C1 + 3n). By the maximum principle, evaluating 2 (27.64) at x0 yields f (x2 0 ) − n ≤ 0. However, (27.65) implies that f (x0 )  9n4 , a contradiction. We conclude that R ≥ cf −1 + cnf −2

on M. 

The theorem now follows from (27.65).

Remark 27.27. M. Feldman, T. Ilmanen, and one of the authors [111] constructed complete noncompact K¨ ahler shrinkers on the total spaces of k-th powers of tautological line bundles over the complex projective space CPn−1 for 0 < k < n. These examples, which have Euclidean volume growth and quadratic scalar curvature decay, show that Theorem 27.26 is sharp. As a variation on the proof of Theorem 27.26, define on {f > n2 } the function ψ (f ) = f −c n , where c > 0. In general, we compute that 2  n  ψ (f ) − ψ  (f ) |∇f |2 . Δf (R − ψ (f )) = −2 |Rc|2 + R + f − 2 Since (f − n2 )ψ  (f ) = −ψ (f ) and ψ  ≥ 0, we obtain 2

Δf (R − ψ (f )) ≤ −2 |Rc| + R − ψ (f ) .

(27.66)

Choose c sufficiently small so that R ≥ c on {f = n2 + 1}, which implies that R − ψ (f ) ≥ 0 on {f = n2 + 1}. By applying the maximum principle to (27.66), since lim inf x→∞ (R − ψ (f ))(x) ≥ 0, we obtain a contradiction if R − ψ (f ) < 0 somewhere in {f ≥ n2 + 1}. Hence R − ψ (f ) ≥ 0 in {f ≥ n2 + 1}. 3.2. Lower estimate of R for steadies. By a similar argument to the previous subsection we may prove the following result, due to B. Yang and two of the authors, regarding steady GRS assuming a condition on the potential function. Theorem 27.28 (Scalar curvature of nonflat steadies decay at most exponentially). Let (Mn , g, f, 0) be a complete normalized steady GRS. If limx→∞ f (x) = −∞ and f ≤ 0, then R ≥ √ n1 ef . Since |∇f | ≤ 1, this implies that for any 2

+2

˜ ∈ M we have O (27.67) where c = (

R(x) ≥ ce−d(x,O) ˜

n 2

for all x ∈ M,

+ 2)−1 ef (O) . ˜

Proof. The idea is to find a lower barrier function (an expression of f ) for the scalar curvature R. Using (27.9b), we compute that Δf (ef ) = ef Δf f + ef |∇f |2 = −R ef < 0.

4. VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS

17

By this and (27.10), we obtain for c ∈ R,  2 nc2 2f Δf R − cef ≤ − R2 + cR ef ≤ e . n 8 Using Δf (e2f ) = 2e2f (1 − 2R), we compute for any constant b ∈ R that   2  nc f 2f (27.68) Δf R − ce − be − 2b + 4bR e2f . ≤ 8 Given b, c > 0 to be chosen below, suppose that R−cef −be2f is negative somewhere. Then, since R ≥ 0 by Theorem 27.2(1) and since limx→∞ ef (x) = 0 by hypothesis, a negative minimum of R − cef − be2f is attained at some point. By (27.68) and the maximum principle, at such a point we have 0≤

nc2 nc2 − 2b + 4bR < − 2b + 4b (c + b) 8 8 2

since f ≤ 0. Given c ∈ (0, 12 ], the minimizing choice b = 1−2c yields (1−2c) < 4 4 1 √ With this choice of b, we obtain a contradiction by then choosing c = n . 2 2

+2

nc2 8 .



2

dx +dy f Remark 27.29. For the cigar soliton (R2 , 1+x 2 +y 2 ) we have R = e , where  f (x, y) = − ln 1 + x2 + y 2 .

4. Volume growth of shrinking gradient Ricci solitons In this section we discuss the asymptotic volume ratio of noncompact shrinkers, including a Euclidean upper bound for their volume growth. We consider two approaches: sublevel sets of the potential function and the Riccati equation along geodesics. 4.1. A differential identity for the volume ratio of sublevel sets. Given a complete noncompact Riemannian manifold (N n , h) and a basepoint ˜ ∈ N , the asymptotic volume ratio (AVR) is defined by O AVR(h)  lim

(27.69)

r→∞

Vol BO˜ (r) , ωn r n

provided the limit exists, where ωn is the volume of the unit Euclidean n-ball. If Rc ≥ 0, then AVR(h) ∈ [0, 1] exists by the Bishop volume comparison theorem. In ˜ general, whenever the AVR exists, it is independent of O. In the sublevel set approach, we shall need the co-area formula (see Schoen and Yau [355, p. 89] or Lemma 5.4 of [77] for example), which says the following. Proposition 27.30 (Co-area formula). Let (Mn , g) be a compact manifold with or without boundary. If f is a Lipschitz function and if h is an L1 function or a nonnegative measurable function, then   ∞  (27.70) h |∇f | dμ = dc h dσ, M

−∞

{f =c}

where dμ is the Riemannian measure on M and where dσ is the induced measure on {f = c}.

18

27. NONCOMPACT GRADIENT RICCI SOLITONS

Henceforth, let (Mn , g, f, −1) be a complete normalized noncompact shrinking GRS structure. Extend this structure to a complete solution g (t), t ∈ (−∞, 1), to the Ricci flow with g (0) = g. By Remark 13.32 in Part II and by the local nature of real analyticity, we may extend Bando’s Theorem 13.21 in Part II to the noncompact case. Namely, using Shi’s local derivative estimates, one can show that M has a real analytic structure and that, given t, the metric components gij (t) are real analytic functions in any normal coordinate system. In particular, g is real analytic. We shall use this fact in the sublevel set approach. By (27.46) we have that f attains its minimum at some point O ∈ M. Thus, from (27.2), we have n n (27.71) inf R(x) ≤ R(O) = − Δf (O) ≤ . x∈M 2 2 We claim that, since M is noncompact, we have inf x∈M R(x) < n2 . If not, then on M we have Δf = n2 − R ≤ 0. Since f is superharmonic and attains its minimum, we conclude that f is a constant function by the strong maximum principle, which contradicts M being a noncompact shrinker. Define the functions V : R → (0, ∞) , R : R → (0, ∞) by

3

(27.72a)

 V (c) 

{f 0. This is due to B. Yang and two of the authors, based on earlier works of others. Proposition 27.35 (Necessary and sufficient condition for AVR > 0). Let (Mn , g, f, −1) be a complete normalized noncompact shrinking GRS. Then  ∞ R(c) (27.89) AVR(g) > 0 if and only if dc < ∞, c n+2 V(c) or, equivalently,



∞

 BO ˜ (r)

Rdμ dr

< ∞. Vol BO˜ (r) r ∞ That is, AVR (g) = 0 if and only if n+2 cR(c) V(c) dc = ∞, or, equivalently,  ∞ Rdμ dr BO ˜ (r) = ∞. Vol BO˜ (r) r 1 (27.90)

1

Proof. Integrating (27.83) yields (27.91)

P(c) = P(n + 2) e−

c n+2

N(c) (1− n+2 2c ) dc 1−N(c)

for c ≥ n + 2. Regarding the integral on the rhs, from (27.79) it is easy to see that for any c ∈ [n + 2, ∞) we have   c  c   N (c) 1 c n+2 dc ≤ 2 (27.92) N (c) dc ≤ N (c) dc. 1− 2 n+2 2c 1 − N (c) n+2 n+2 ∞ If n+2 N (c) dc = ∞, then by (27.91) we have AVR(g) = 2n1ωn limc→∞ P(c) = 0. ∞ On the other hand, if n+2 N (c) dc < ∞, then it follows from (27.91) and (27.92) that ∞ P(c) ≥ P(n + 2) e−2 n+2 N(c)dc > 0 for c ≥ n + 2; hence AVR(g) > 0. We have shown that AVR (g) = 0 if and only if  ∞ R(c) dc = ∞. n+2 c V(c) Finally, we observe that by inequalities (27.42) and (27.46) we have that   ∞ R(c)  ∞ BO˜ (r) Rdμ dr dc = ∞ if and only if 1 Vol B ˜ (r) r = ∞.  n+2 c V(c) O

Using Proposition 27.35, one may obtain the following qualitatively sharp result due to S.-J. Zhang.

22

27. NONCOMPACT GRADIENT RICCI SOLITONS

Corollary 27.36 (Volume growth of shrinkers with R ≥ δ ≥ 0). If G = (Mn , g, f, −1) is a complete noncompact normalized shrinking GRS with R ≥ δ ≥ 0 ˜ ∈ M, then and O (27.93)

˜ (1 + r)n−2δ . Vol BO˜ (r) ≤ const(G, O)

Proof. By hypothesis, N (c) = with (27.91), for c ≥ n + 2 we have P(c) ≤ P(n + 2) e−

c

From (27.85), we have P(c) ≥

n+2

R(c) c V(c)

≥ δc . Hence, combining this inequality

δ δ δ (1− n+2 2c ) c dc ≤ (n + 2) e 2 P(n + 2)c−δ .

V(c) 2cn/2

for c ≥ n + 2 and we conclude that

V (c) ≤ 2 (n + 2)δ e 2 P(n + 2)c 2 −δ . √ By (27.42) we obtain BO˜ (2 c − C) ⊂ {f < c}. Hence, using (27.94), we derive that  √ δ n Vol BO˜ 2 c − C ≤ 2 (n + 2)δ e 2 P(n + 2)c 2 −δ δ

(27.94)

holds for c ≥ n + 2 almost everywhere, so that δ

δ

Vol BO˜ (r) ≤ 2 (n + 2) e 2 P(n + 2)

n



r+C 2

n−2δ

√ for all r ≥ 2 n + 2 − C. This completes the proof of Corollary 27.36.



Remark 27.37. For 2 ≤ k ≤ n, we have the cylinder shrinkers (Mn , g) = (N , h) × Rn−k , where Rch ≡ 12 h and hence Rg ≡ k2 . Since N must be compact, n−k . we have Vol BO˜ (r) ≈ const (1 + r) k

We may ask the following: Question 27.38. Can one show that if a simply-connected noncompact shrinker (Mn , g, f, −1) satisfies R ≥ δ > 0, then it must have a compact factor (N k , h) with k ≥ min{m ∈ Z : m ≥ 2δ}? Question 27.39. Can one show that if a complete noncompact shrinking GRS (Mn , g, f, −1) satisfies AVR (g) > 0, then ASCR(g) < ∞? Question 27.40. Can one show that if a simply-connected noncompact shrinker (Mn , g, f, −1) does not have an R factor, then g has Euclidean volume growth? 4.4. Bakry–Emery volume comparison. The analytic essence of the Bishop volume comparison theorem is the Bochner formula 1 (27.95) Δ|∇u|2 = |∇2 u|2 + ∇u, ∇Δu + Rc(∇u, ∇u) 2 applied to the distance function. We now consider the Bakry–Emery volume comparison theorem, which was used to derive (27.88) above, from the same point of view. Let (Mn , g) be a complete Riemannian manifold and let f : M → R be a smooth function. Adding (27.95) and 1 − ∇f, ∇|∇u|2  = −∇u, ∇∇f, ∇u + ∇2 f (∇u, ∇u) 2

4. VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS

23

together, we obtain the f -Bochner formula (27.96)

1 Δf |∇u|2 = |∇2 u|2 + ∇u, ∇Δf u + Rcf (∇u, ∇u). 2

˜ ∈ M and let r(x) = d(x, O). ˜ Let S(O, ˜ r)  {x ∈ M : d(x, O) ˜ = r} be Let O ˜ r), wherever it is a the distance sphere. Let H denote the mean curvature of S(O, smooth hypersurface. The f -mean curvature is (27.97)

Hf = H − ∇f, ∇r = Δf r

since Δr = H. Recall that since |∇r|2 ≡ 1 and ∇2 r = II, (27.96) with u = r implies the f -Riccati equation     ∂ ∂ ∂ ∂ ∂Hf H2 2 = − Rcf , , . (27.98) − |II| ≤ − Rcf − ∂r ∂r ∂r ∂r ∂r n−1 Let J denote the Jacobian of the exponential map and let Jf  e−f J denote the ∂ ∂ f -Jacobian. We have ∂r ln J = H and ∂r ln Jf = Hf . Thus (27.99)

r 2 H (r)dr Jf (r2 ) = e r1 f . Jf (r1 )

If Rcf ≥ − 2ε g for some ε ∈ R, then ∂  2 r2 H 2 − r 2 Rc r H ≤ 2rH − ∂r n−1



∂ ∂ , ∂r ∂r



 ≤ n − 1 + r2

∂2f ε + ∂r 2 2

 .

Hence (27.100)

ε ∂f ∂  2 + r2 . r Hf ≤ n − 1 − 2r ∂r ∂r 2

Integrating this while using limr0 r 2 Hf (r) = 0, we obtain (27.101)

2 ∂f n−1 ε (r) = Hf (r) ≤ + r− 2 H (r) − ∂r r 6 r



r

r¯ 0

∂f (¯ r )d¯ r. ∂r

Now consider the case where ε = 0. Wherever we have the bound |∇f | ≤ A, (27.102)

Hf (r) ≤

n−1 + A. r

By substituting this into (27.99), we have  r n−1 2 Jf (r2 ) +A)dr = eA(r2 −r1 ) ≤ e r1 ( r Jf (r1 )

Taking r1 → 0 and calling r2 = r¯, we obtain (27.103)

r ) ≤ e−f (O)+A¯r r¯n−1 Jf (¯ ˜



r2 r1

n−1 .

24

27. NONCOMPACT GRADIENT RICCI SOLITONS

since limr→0 r n−1 Jf (r) = e−f (O) . Integrating this yields the following (one deals ˜ r) in the same way as the Bishop volume with the possible nonsmoothness of S(O, comparison theorem): ˜

Theorem 27.41 (Bakry–Emery volume comparison). Let (Mn , g) be a com˜ ∈M plete Riemannian manifold and let f : M → R be a smooth function. If O and r > 0 are such that Rcf ≥ 0 and |∇f | ≤ A in BO˜ (r) for some A > 0, then the f -volume (27.104)   Volf BO˜ (r) =

r

e−f dμ ≤ nωn

BO ˜ (r)

e−f (O)+A¯r r¯n−1 d¯ r ≤ ωn e−f (O)+Ar r n , ˜

˜

0

where ωn is the volume of the unit Euclidean n-ball. In particular, if f ≤ C in BO˜ (r), then Vol BO˜ (r) ≤ ωn e−f (O)+Ar+C r n . ˜

(27.105)

From the form of the estimates we see that Bakry–Emery volume comparison is more effective at bounded distances. 4.5. Euclidean volume growth via the Riccati equation. There is the following essentially equivalent version of the volume growth bound in Theorem 27.33. The proof we give is due to Munteanu and J. Wang. Theorem 27.42 (Shrinkers have at most Euclidean volume growth). Let (Mn , g, f, −1) be a complete normalized noncompact shrinking GRS. Then for any ˜ ∈ M we have O ˜

Vol BO˜ (r) ≤ ωn ef (O) r n

(27.106)

for all r > 0,

where ωn is the volume of the unit Euclidean n-ball. In particular, if O is a minin mum point of f , then by (27.43) we have that Vol BO (r) ≤ ωn e 2 r n for r > 0. Proof. On a shrinker we have Rcf = 12 g. By (27.101), we have   J(r) ∂ n−1 (27.107) ln = H (r) − n−1 ∂r r r  r ∂f 2 r ∂f ≤− + (r) − 2 r )d˜ r. r˜ (˜ 6 ∂r r 0 ∂r Since limr→0

J(r) r n−1

= 1, we have     r¯   J(¯ r) ∂f 2 r ∂f r (27.108) + (r) − (˜ r )d˜ r dr ln r ˜ ≤ − r¯n−1 6 ∂r r 2 0 ∂r 0  r¯2 2 r¯ ∂f = − − f (¯ r ) + f (0) + r (r)dr, 12 r¯ 0 ∂r r r )d˜ r = 0 to obtain the last where we integrated by parts and used limr→0 r1 0 r˜ ∂f ∂r (˜ ˜ line. Note that f (O) = f (0). Another version of this formula is    J(¯ r) 2 r¯ r¯2 + f (0) + f (¯ r ) − (27.109) ln f (r)dr. ≤ − r¯n−1 12 r¯ 0

4. VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS

25

By combining (27.107) and (27.108), we obtain   ∂ J(¯ r) J(¯ r) ∂ J(¯ r) r¯ ln n−1 = ln n−1 + r¯ ln n−1 (27.110) ∂ r¯ r¯ r¯ ∂ r¯ r¯ ∂f r¯2 r ) + f (0) + r¯ (¯ r) ≤ − − f (¯ 4 ∂r   2

2 ∂f ∂f r¯ ≤ f (0) − (¯ r) − − f (¯ r) − (¯ r) . ∂r 2 ∂r Let ∇f T be the tangential component to ∂BO˜ (r) of ∇f . Then f = |∇f |2 + R = 2 T 2 ¯ = 0 to r¯ = r, we obtain ( ∂f ∂r ) + |∇f | + R. By integrating (27.110) from r

 2  ∂f J(r) r¯ 1 r T 2 ln n−1 ≤ f (0) − (¯ r) − (27.111) + |∇f | (¯ r ) + R(¯ r ) d¯ r r r 0 ∂r 2 ≤ f (0). Therefore J(r) ≤ ef (0) r n−1 .

(27.112)

Since J is the volume density of g in spherical coordinates, by integrating (27.112), we obtain (27.106).  Exercise 27.43. Show that



r¯ →

J(¯ r) ef (0) r¯n−1

r¯

is nonincreasing and that the limit as r¯ → 0 is equal to 1. ˜ and that Remark 27.44. Suppose that the minimum of R is attained at O 1 r ˜ ˜ ˜ |∇f | (O) = 0. Then we have R(O) = f (O) = f (0). This implies that r 0 R(¯ r )d¯ r≥ ˜ = f (0). Therefore, by (27.111), R(O)  1 r J(r) R(¯ r )d¯ r ≤ 0. (27.113) ln n−1 ≤ f (0) − r r 0 This implies that Vol BO˜ (r) ≤ ωn r n . On the other hand, by Corollary 27.36, we ˜ ˜ n−2R(O) . know that Vol BO˜ (r) ≤ C(G, O)r Exercise 27.45. Show that (27.114)

r¯ →

e

r ¯2 r )+ r2¯ 12 −f (¯

 r¯ 0

f (r)dr

r¯n−1

J(¯ r)

=

e

1 r ¯

 r¯  0

2

( r2 −f  (r))

2

+f (r)−(f  (r))

 dr

J(¯ r)

r¯n−1

is nonincreasing and that the limit as r¯ → 0 is equal to ef (0) . ˜ ∈ M, x ∈ ˜ and r(x) = d(x, O), ˜ where Cut(O) ˜ is the cut locus of Let O / Cut(O), ˜ O. Define  r(x) 2 r 2 (x) − f (x) + h(x) = f (γ(s)) ds, 12 r(x) 0 ˜ to x. where γ : [0, r(x)] → M is the unique minimal unit speed geodesic joining O The measure corresponding to the numerator in (27.114) is dm (x) = eh(x) dμ (x)

˜ on M − Cut(O).

26

27. NONCOMPACT GRADIENT RICCI SOLITONS

In general, for shrinkers, although limr→∞

Vol BO ˜ (r) rn

˜ ∈ M and exists for each O Vol B (r)

˜ O is bounded above only depending on n, we do not know if is bounded rn ˜ above independent of O ∈ M and r ≥ 1. In particular, does there exist C < ∞ ˜ ∈ M? such that Vol BO˜ (1) ≤ C for all O

5. Logarithmic Sobolev inequality In this section we discuss the logarithmic Sobolev inequality on shrinkers due to Carrillo and one of the authors, which is based on the works of Bakry and Emery, Villani, and others. Let G = (Mn , g, f, −1) be a complete noncompact shrinking GRS structure. Taking τ = 1 in Perelman’s entropy functional, we define the (scale 1) entropy as  2 (R + |∇ϕ| + ϕ − n)(4π)−n/2 e−ϕ dμ. (27.115) W(g, ϕ)  W(g, ϕ, 1) = M

Define the μ-invariant (or logarithmic Sobolev constant) of g by μ(g)  inf W(g, ϕ), ϕ

where the infimum is taken over all ϕ : M → R ∪ {∞} such that w  e−ϕ/2 ∈  n/2 Cc∞ (M) satisfies the constraint M w2 dμ = (4π) .  Now assume that f satisfies the normalization M e−f dμ = (4π)n/2 . Then for some constant C1 (G) we have R + |∇f |2 − f ≡ C1 (G).

(27.116)

Define the entropy of G to be μ(G)  W(g, f ). In view of the exponentially decaying e−f factor and the volume bound (27.106), one can prove using the es2 timates for f , |∇f | , and Δf in §2 of this chapter that the integral defining μ(G) converges and that we have the equality   2 −f |∇f | e dμ = Δf e−f dμ. M

Hence (27.117)

 μ(G) = M

M

(R + 2Δf − |∇f |2 + f − n)(4π)−n/2 e−f dμ.

By (27.2) and (27.116), we have (27.118)

2

R + 2Δf − |∇f |2 + f − n = f − R − |∇f | = −C1 (G).

Therefore, by (27.117) we have that μ(G) = −C1 (G). Carrillo and one of the authors proved that μ(g) = μ(G), that is, Theorem 27.46 (Sharp logarithmic Sobolev inequality for shrinkers). If G = (Mn , g, f, −1) is a complete noncompact shrinking GRS, then inf W(g, ϕ) = μ(G) = −C1 (G), ϕ

where the infimum is taken over all ϕ : M → R ∪ {∞} such that w  e−ϕ/2 ∈  Cc∞ (M) satisfies the constraint M w2 dμ = (4π)n/2 .

5. LOGARITHMIC SOBOLEV INEQUALITY

27

Remark 27.47. In comparison, Corollary 6.38 in Part I is equivalent to the following statement. Given any closed Riemannian manifold (Mn , g) and  constant b > 0, there exists a constant C (b, g) such that if a function ϕ satisfies M e−ϕ dμ = (4π)n/2 , then  M

(b |∇ϕ|2 + ϕ − n)e−ϕ dμ ≥ −C (b, g) .

Here we give a heuristic proof of the theorem following the Bakry–Emery method, which can be made rigorous in our noncompact setting; detailed proofs ∂ are given in [53], [16], and [425]. Let f  ∂t − Δf denote the f -heat operator. Let ξ(x, t) be a solution to (27.119)

f ξ = |∇ξ|2 ,

equivalently,

f eξ = 0.

Henceforth we shall assume suitable growth conditions on ξ so that we can differentiate under the integral sign and so that we can integrate by parts to justify some of the equalities below. We first observe that      ∂ ξ −f d e e dμ = eξ−f dμ = (Δf eξ )e−f dμ = 0. (27.120) dt M ∂t M M  n/2 Now assume that ξ satisfies the normalization (4π) = M eξ−f dμ. Define the relative Fisher information functional  (27.121) I (ξ)  |∇ξ|2 eξ−f dμ. M

By (27.96), for any function u(x, t) we have the parabolic Bochner formula  2 1 2 f |∇u| = − ∇2 u + ∇u, ∇f u − Rcf (∇u, ∇u) . (27.122) 2 Hence, for ξ satisfying (27.119) on a shrinking GRS, we have  2 2 2 2 (27.123) f |∇ξ| = −2 ∇2 ξ  + 2∇ξ, ∇ |∇ξ|  − |∇ξ| . From this we compute that (27.124)

 2 2 2 f (|∇ξ| eξ ) = −eξ (2 ∇2 ξ  + |∇ξ| ).

Therefore (27.125)



∂ 2 (|∇ξ| eξ )e−f dμ ∂t M   2 eξ (2 ∇2 ξ  + |∇ξ|2 )e−f dμ =−

d I (ξ (t)) = dt

M

≤ − I (ξ (t)) . Now define the Boltzmann relative entropy (or Nash entropy) functional  ξeξ−f dμ. (27.126) H (ξ)  M

We compute using (27.119) that4 (27.127)     ∂ ξ d −f H(ξ (t)) = e (ξ + 1) e dμ = Δf (eξ ) (ξ + 1) e−f dμ = − I (ξ(t)). dt ∂t M M   |∇u|2 f = 0, then ξ  ln u gives H (ξ) = M u ln u dμ and I (ξ) = M u dμ. As a special case ∂ of (27.127), we have that ∂t H (ln u) = − I (ln u) provided u > 0 satisfies the heat equation. 4 If

28

27. NONCOMPACT GRADIENT RICCI SOLITONS

Therefore, by (27.125) we have d d H (ξ (t)) ≥ I (ξ (t)) . dt dt Under suitable growth conditions on the function ξ, it can be shown that limt→∞ ξ (t) = 0 and thus limt→∞ H (ξ (t)) = 0. We conclude that if ξ(t) satisfies (27.119), then (27.128)   H (ξ(t)) ≤ I (ξ (t)) ; 

Given ϕ with implies that 

M

e−ϕ dμ = (4π)

n/2

M

(f − ϕ) e−ϕ dμ ≤

M



and

2

that is,



−ϕ

2

(|∇ϕ| + ϕ)e M

|∇ξ(t)| eξ(t)−f dμ ≥

, take ξ(0) = f − ϕ. Then (27.128) at t = 0

(|∇f | + |∇ϕ| − 2 ∇f, ∇ϕ)e−ϕ dμ 2

M

ξ(t)eξ(t)−f dμ. M

 dμ ≥

M

2

  2Δf − |∇f |2 + f e−ϕ dμ.

We conclude from this and (27.118) that (27.129) 

−ϕ

2

M

(R + |∇ϕ| + ϕ − n)e

 dμ ≥

M

  R + 2Δf − |∇f |2 + f − n e−ϕ dμ n/2

= − (4π)

C1 (G).

This finishes our sketch of the proof of Theorem 27.46.  In terms of w = e−ϕ/2 , the logarithmic Sobolev inequality says that if M w2 dμ = (4π)n/2 , then   2 2 2 2 (Rw + 4 |∇w| − w ln(w ))dμ ≥ (n − C1 (G)) w2 dμ. M

M

If we do not impose any constraint on w, then this is equivalent to     2 (27.130) w2 ln(w2 )dμ − ln w2 dμ w2 dμ ≤ (4 |∇w| + Rw2 )dμ M M M M  + C2 (G) w2 dμ, M

n/2

where C2 (G) = C1 (G) − n − ln((4π) (27.131)   w2 ln(w2 )dμ − ln M

). Since R =



− Δf , we obtain



w2 dμ

M

n 2

M

w2 dμ ≤

  (4 |∇w|2 + ∇f, ∇(w2 ) )dμ M  + C3 (G) w2 dμ, M

where C3 (G) = C1 (G) −

n 2

n/2

− ln((4π)

).

Remark 27.48. Compare the derivation of inequality (27.125) with Bakry and Emery’s proof of their logarithmic Sobolev inequality [16] (for an exposition, see Proposition 5.40 in Part I).

6. GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE

29

By Perelman’s proof of his no local collapsing theorem, one can demonstrate the following. Corollary 27.49. If G = (Mn , g, f, −1) is a complete noncompact shrinking GRS, then there exists a constant κ > 0 depending only on C1 (G) satisfying the following property. If x0 ∈ M is a point and r0 > 0 are such that R ≤ r0−2 in Bx0 (r0 ), then Vol Bx0 (r0 ) ≥ κr0n . The logarithmic Sobolev inequality plays a crucial role in the following result of Munteanu and J. Wang. Theorem 27.50 (Shrinkers must have at least linear volume growth). If G = (Mn , g, f, −1) is a complete noncompact shrinking GRS and p ∈ M, then there  exists a constant c = c(n, f (p), e−f dμ) > 0 such that 1 . 2

Area S(p, r) ≥ 2c

for r ≥

Vol Bp (r) ≥ cr

for r ≥ 1.

Hence  The constant c is nondecreasing in f (p) and nonincreasing in e−f dμ. Hence, if G is also a singularity model that is κ-noncollapsed below all scales, then by taking p to be a minimum point of f , we have that c depends only on n and κ. Finally, we state a pair of important results on the geometry at infinity of shrinkers. First, we have the following uniqueness result of Kotschwar and L. Wang. Theorem 27.51. Any two complete noncompact shrinking GRS which have a pair of ends which are asymptotic to the same Euclidean cone over a smooth (n − 1)dimensional closed Riemannian manifold must have isometric universal covers. Second, we have the following result of Munteanu and J. Wang. Theorem 27.52. Let (Mn , g, f, −1) be a complete noncompact shrinking GRS. If |Rc| (x) → 0 as x → ∞, then (M, g) is asymptotic to a Euclidean cone over a smooth (n − 1)-dimensional closed Riemannian manifold. 6. Gradient shrinkers with nonnegative Ricci curvature If we consider shrinkers which are non-Ricci flat with nonnegative Ricci curvature, then we have the following improvement for the lower bound of the scalar curvature due to one of the authors. Theorem 27.53 (Rc ≥ 0 and Rc ≡ 0 shrinker ⇒ R ≥ δ > 0). Let (Mn , g, f, −1) be a complete noncompact non-Ricci flat shrinking GRS with nonnegative Ricci curvature. Then there exists δ > 0 such that (27.132)

R≥δ

on M.

˜ ∈ M and let x0 ∈ M− Proof of the theorem (modulo a claim). Let O BO˜ (8r1 ), where the constant r1 < ∞ is chosen to satisfy (27.147) below. Let σ be an integral curve of ∇f with σ (0) = x0 . Since ∇f is a complete vector field (by Corollary 27.7), σ (u) is defined for all u ∈ R. Using ∇R = 2 Rc (∇f ), we have (27.133)

d R (σ (u)) = ∇R, ∇f  = 2 Rc (∇f, ∇f ) ≥ 0 du

30

27. NONCOMPACT GRADIENT RICCI SOLITONS

and hence R (x0 ) ≥ R (σ (u))

(27.134)

for all u ∈ (−∞, 0].

Claim. For any x0 ∈ M − BO˜ (8r1 ), we must have either Case (1): R (σ (u1 )) ≥ 1 for some u1 ∈ (−∞, 0]

(27.135) or Case (2): (27.136)

¯ ˜ (8r1 ) σ (u2 ) ∈ B O

for some u2 ∈ (−∞, 0).

Now assume the claim. If Case (1) holds, then (27.134) implies R (x0 ) ≥ R (σ (u1 )) ≥ 1. On the other hand, if Case (2) holds, then (27.134) implies R (x0 ) ≥ R (σ (u2 )) ≥

(27.137)

min

¯ ˜ (8r1 ) x∈B O

R (x) .

Since g is not Ricci flat, by Proposition 27.8(3) the rhs is positive. Therefore, in either Case (1) or Case (2) we have   R (x0 ) ≥ min 1, min R (x)  δ > 0. ¯ ˜ (8r1 ) x∈B O

Since x0 ∈ M − BO˜ (8r1 ) is arbitrary, the theorem is proved, modulo the claim.



It remains for us to present the following: Proof of the claim. Suppose that the claim is false. Then there exists x0 ∈ M − BO˜ (8r1 ) such that R (σ (u)) < 1 ¯ ˜ (8r1 ) σ (u) ∈ /B

(27.138)

O

for all u ∈ (−∞, 0], for all u ∈ (−∞, 0).

˜ The whole aim of the proof is to show that Let r ( · ) = d( · , O). d− 1 r (σ (u)) ≥ r (σ (u)) du 8

(27.139)

− for all u ∈ (−∞, 0), where ddu denotes the lim inf of backward difference quotients. ¯ ˜ (8r1 ) for all u ∈ (−∞, 0). This will prove the claim since it contradicts σ (u) ∈ /B O Step 1. For x ∈ M − BO˜ (4 (n − 1)) with R (x) ≤ 1, we have

1 ˜ r (x) − C1 − |∇f | (O), 4 ˜ to x and where where γ : [0, r (x)] → M is a minimal unit speed geodesic joining O C1 is given by (27.146) below. Proof of (27.140). By integrating the shrinker equation Rc(γ  , γ  )+(f ◦γ) = 1 2 , we have  r(x) 1 Rc(γ  , γ  ) ds + ∇f, γ  (0). (27.141) ∇f, γ  (r (x)) = r (x) − 2 0

(27.140)

∇f, γ  (r (x)) ≥

Regarding the second term on the rhs, the second variation formula implies (see (27.49))  r(x)  r(x) 2 ζ 2 Rc (γ  , γ  ) ds ≤ (n − 1) (ζ  ) ds 0

0

6. GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE

31

for any continuous piecewise C ∞ function ζ : [0, r (x)] → R satisfying ζ (0) = ζ (r (x)) = 0. Now define ζ to be ⎧ s if 0 ≤ s ≤ 1, ⎪ ⎨ 1 if 1 < s ≤ r (x) − r0 , ζ (s) = ⎪ ⎩ r(x)−s if r (x) − r < s ≤ r (x) , 0 r0 where r0 is to be chosen below. We have  r(x)  ζ 2 Rc (γ  , γ  ) ds ≤ (n − 1) r0−1 + 1 . 0

From this we obtain (27.142)  r(x)   Rc (γ  , γ  ) ds ≤ (n − 1) r0−1 + 1 + 0





1

1 − ζ 2 Rc (γ  , γ  ) ds

0

 1 − ζ 2 Rc (γ  , γ  ) ds

r(x)

+ r(x)−r0

2  max Rc (V, V ) ≤ (n − 1) r0−1 + 1 + 3 V ∈Ty M, |V |=1, y∈B¯O˜ (1)  r(x) + R (γ(s)) ds r(x)−r0

since Rc ≥ 0. We now estimate have for all y ∈ M

 r(x) r(x)−r0

R (γ(s)) ds. Since ∇R = 2 Rc (∇f ) and Rc ≥ 0, we 

|∇R| (y) ≤ 2 |Rc| (y) |∇f | (y) ≤ 2R (y)

  1 ˜ r (y) + f (O) , 2

using (27.30). Thus, if y ∈ M satisfies r (y) ≥ 1, then  ˜ (y) . (27.143) |∇ ln R| (y) ≤ (1 + 2 f (O))r ≤ 1. Suppose that s¯ ∈ [r (x) − r0 , r (x)]. Using Now choose r0 = 4(n−1) r(x) r (γ(¯ s)) ≥ 1, the assumption that R (x) ≤ 1, and (27.143), we compute that ln R (γ(¯ s)) ≤ ln 

R (γ(¯ s)) R (x) r(x)

|∇ ln R| (γ(s)) ds

≤ s¯

  ˜ ≤ (1 + 2 f (O))

r(x)

r (γ(s)) ds

r(x)−r0

 ˜ 0 r (x) ≤ (1 + 2 f (O))r  ˜ ≤ 4 (n − 1) (1 + 2 f (O)). That is, for s¯ ∈ [r (x) − r0 , r (x)], √ ˜ f (O))

R (γ(¯ s)) ≤ e4(n−1)(1+2

.

32

27. NONCOMPACT GRADIENT RICCI SOLITONS

Therefore 

√ ˜ f (O))

r(x)

R (γ(s)) ds ≤ e4(n−1)(1+2

(27.144)

r0 .

r(x)−r0

Combining this with (27.142), we have 

r(x)

(27.145)

Rc (γ  , γ  ) ds ≤ (n − 1) r0−1 + C1 ,

0

where (27.146)

C1 = n − 1 +

√ ˜ 2 max Rc (V, V ) + e4(n−1)(1+2 f (O)) . ¯ 3 V ∈Ty M, |V |=1, y∈BO˜ (1)

Applying (27.145) to (27.141) while using r0−1 =

r(x) 4(n−1)

≥ 1, we obtain (27.140).

Step 2. Applying the inequality (27.140) to prove the claim. Now define   n−1 ˜ . , C1 + |∇f | (O) (27.147) r1 = max 2 By (27.138) and (27.140) with x = σ (u), we have that ∇f, γ  (r (σ (u))) ≥

1 ˜ ≥ 1 r (σ (u)) r (σ (u)) − C1 − |∇f | (O) 4 8

for any u ∈ (−∞, 0]. Here, γ : [0, r (σ (u))] → M is a minimal unit speed geodesic ˜ to σ (u). Since d− r (σ (u)) ≥ ∇f, γ  (r (σ (u))), we obtain (27.139) for joining O du all u ∈ (−∞, 0).  As a consequence, we have the following result, originally due to Carrillo and one of the authors. Theorem 27.54 (Ric ≥ 0 and Ric ≡ 0 shrinker ⇒ AVR = 0). If (Mn , g, f, −1) is a complete noncompact non-Ricci flat shrinking GRS with nonnegative Ricci curvature, then AVR (g) = 0. Proof. This follows directly from a combination of Theorem 27.53 and Corollary 27.36.  Remark 27.55 (Ricci flat manifolds with AVR > 0). There are many examples of 4-dimensional complete noncompact Ricci flat manifolds (hyper-K¨ ahler and asymptotically locally Euclidean) with AVR (g) > 0. Note that Feldman, Ilmanen, and one of the authors [111] have described examples of complete noncompact K¨ ahler shrinkers, which have AVR > 0 and have Ricci curvature which is negative somewhere. Related to Remark 27.37, we ask the following: Problem 27.56. Is there an example of a noncompact shrinker with Rc ≥ 0 which does not split as the product of a compact shrinker and a Euclidean space?

7. NOTES AND COMMENTARY

33

7. Notes and commentary There are many works on GRS which we have not discussed in this chapter. We have not discussed some important advances in the contruction of K¨ahler and nonK¨ ahler Ricci solitons. Some works we would like to mention are Xu-Jia Wang and Xiaohua Zhu [431], Fabio Podest` a and Andrea Spiro [329], and Andrew Dancer and McKenzie Wang [90]. The results in this chapter are due to various authors. We give some citations below. §1. For Exercise 27.1, see §1.3 of [312]. The proof of Theorem 27.2(1), using the elliptic maximum principle, follows Theorem 1.3(ii) in [455]. The proof of Theorem 27.2(2) follows Theorem 0.5 in [451]. For an extension to ancient solutions (discussed in the next chapter), see [61]. Theorem 27.4 follows from the formula (27.6) for GRS due to Hamilton combined with Theorem 27.2. For the shrinking case, see (2.3) in [48] (see also [45]). Corollary 27.7 is Theorem 1.3(i) in [455]. For Proposition 27.8, see [326] or [451]. For the claim stated in its proof, see also Proposition 2 in [319]. §2. Regarding shrinkers, (27.42) is (2.2) in [48]. For Theorem 27.11, see Theorem 1.1 in [48] and the refinement in [146]. If the Ricci curvature of (Mn , g) is bounded, then this estimate is in [312]. For Corollary 27.16, see [105]. Also related is the work in [105], where the properness of f is proven and from which a nonsharp form of the quadratic growth of f may be derived. Theorem 27.19 is in [268]. Theorem 27.24 and Corollary 27.25 are in [95]. §3. For Theorem 27.26, see [291] and [78]. See [441] and [264] for an estimate for the potential functions of steady GRS. See [32], [46], [132], [138], and [264] for further works on the qualitative aspects of steady GRS. §4. For (27.81), see (3.7) of [48]. The result that shrinkers have at most Euclidean volume growth, in Theorem 27.33 and Theorem 27.42, is primarily due to [48], with a technical hypothesis they first assumed removed in [263]. That the lim sup of the volume ratios, as the radius tends to infinity, is bounded above by a constant depending only on dimension is in [146]. The part of Theorem 27.33 that the AVR of a shrinker exists is in [79]. Their work built upon the eariler works in [53], [45], [451], [64], and [290]. Some work related to the above is in [105] and is by Hamilton (see Proposition 9.46 in [77]). For Proposition 27.35, see [79]. Corollary 27.36 is in [451]. It extends the earlier results in [48] and [263]. The alternate proof we give is due to Bo Yang. For Theorem 27.41, see Theorem 1.2(a) in [434] for example. See also [442]. For some earlier related work, see [202] and [16]. The proof of Theorem 27.42 that we present, using the Riccati equation, is in [267]. For further work on GRS, see [453].

34

27. NONCOMPACT GRADIENT RICCI SOLITONS

§5. Theorem 27.46 is in [53]. Their proof is based on the earlier work in [16] and [425], among others. Theorem 27.50 is in [265] (see Theorem 1.4(2) in [267] for a generalization). Theorem 27.51 is in [170]. §6. Theorem 27.53 is Proposition 1.1 in [280]. Its proof is a modification of an idea of Perelman (see Proposition 9.81 in [77]). Theorem 27.54 was proved in [53].

CHAPTER 28

Special Ancient Solutions When you’re ridin’ sixteen hours and there’s nothin’ much to do And you don’t feel much like ridin’, you just wish the trip was through. – From “Turn the Page” by Bob Seger

The ancient solutions arising as singularity models (associated to finite-time solutions of the Ricci flow on closed manifolds) are necessarily κ-noncollapsed. In Part III our discussion of ancient solutions was primarily focused on Perelman’s theory of 3-dimensional ancient κ-solutions. This theory is partly based on the reduced volume monotonicity formula and on compactness arguments by subtle point picking. On the other hand, in the physics literature, ancient solutions without the κ-noncollapsed condition are also considered. This partially motivates us to consider general ancient solutions in this chapter. In §1, we discuss a local estimate for the scalar curvature. A consequence of this estimate is that any complete ancient solution must have nonnegative scalar curvature. In §2, after mentioning some properties and conjectures about singularity models which are reminiscent of those for gradient Ricci solitons (GRS), we prove some results about 3-dimensional singularity models which were not discussed in the earlier parts of this volume. In §3 we discuss the classification of noncompact 2-dimensional ancient solutions. In §4 we discuss some positive curvature conditions on ancient solutions which imply that they are shrinking spherical space forms. 1. Local estimate for the scalar curvature under Ricci flow A priori, noncompact singularity models in dimensions at least 4 may not have bounded curvature. The following result of B.-L. Chen implies that singularity models always have nonnegative scalar curvature. The idea of its proof is to localize the estimate that if Rmin (0) < 0 on a closed manifold, then R≥

1 Rmin (0)−1 − n2 t

.

Theorem 28.1 (Sharp lower bound for R on complete solutions). If (Mn , g (t)), t ∈ (α, ω), is a complete solution to the Ricci flow, then n on M × (α, ω). (28.1) R≥− 2 (t − α) Proof. Without loss of generality, we may assume that the solution is defined on the time interval [α, ω). 35

36

28. SPECIAL ANCIENT SOLUTIONS

Step 1. The cutoff function. First we improve the standard cutoff function slightly. Given p ∈ (2, ∞], let η : R → [0, 1] be a nonincreasing C 2 function such that |η  (u)| and |η  (u)| are bounded and ⎧ ⎪ if u ∈ (−∞, 1], ⎨ 1 p (2 − u) if u ∈ [ 32 , 2), (28.2) η (u) = ⎪ ⎩ 0 if u ∈ [2, ∞).   Note that η (u) ∈ 21p , 1 for u ∈ [0, 32 ). We have η  (u) = −p (2 − u)p−1 , η  (u) = p (p − 1) (2 − u)p−2 for u ∈ [ 32 , 2), from which we may easily deduce that there exists const < ∞ such that |η  (u)| ≤ const · η (u)

(28.3a)



(28.3b)

p−2 p

,

2

p−2 (η (u)) ≤ const · η (u) p η (u)

for all u ∈ [0, 2). Step 2. Defining the localization S of R. Now let (O, t0 ) ∈ M × (α, ω) be any point with R (O, t0 ) < 0. Choose r0 ∈ (0, ∞) so that (28.4)

Rcg(t) ≤ (n − 1)r0−2

¯g(t) (O, r0 ) for t ∈ [α, t0 ] in B

(clearly such an r0 exists). Given any ρ > 0, define S : M × [α, t0 ] → R by   r˜ (x, t) (28.5) S (x, t) = η R (x, t) , ρ where (28.6)

5 r˜ (x, t)  dg(t) (x, O) + (n − 1)r0−1 (t − t0 ) . 3

Note that S has compact support. By (28.4) and Theorem 18.7(1) in Part III, we have that for all x ∈ M − Bg(t) (O, r0 ) and t ∈ [α, t0 ],   ∂ − Δ r˜ (x, t) ≥ 0 (28.7) ∂t in the barrier sense. From now on we shall assume that 5 (28.8) ρ ≥ r0 + (n − 1)r0−1 (t0 − α) . 3 Step 3. The heat operator  ΔS = η ◦ (28.9) +

acting on S. We calculate    r˜ r˜ 2 ΔR + η ◦ ∇˜ r, ∇R ρ ρ ρ

η  ◦ ρr˜ η  ◦ ρr˜ 2 |∇˜ r | R, Δ˜ r+ ρ ρ2

1. LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER RICCI FLOW

so that



(28.10)

  ∂ −Δ S = η◦ ∂t  = η◦





37

η ◦ ρ ∂ r˜ ∂R +R − ΔS ∂t ρ ∂t     η  ◦ ρr˜ ∂ r˜ r˜ ∂R − ΔR + R − Δ˜ r ρ ∂t ρ ∂t   r˜  η ◦ρ 2 r˜ − R η ◦ ∇˜ r, ∇R − ρ ρ ρ2 r˜ ρ

r˜

2

since |∇˜ r | = 1. Step 4. An ordinary differential inequality (odi) for Smin . Let Smin (t) = miny∈M S (y, t). Note that (28.11)

Smin (t0 ) ≤ R (O, t0 ) < 0.

For any t ∈ [α, t0 ] where Smin (t) < 0, let xt be such that S (xt , t) = Smin (t). Note that R (xt , t) ≤ 0. We consider two cases while calculating at the point (xt , t).   (1) If dg(t) (xt , O) ≤ r0 , then r˜ (xt , t) ≤ ρ. This implies η ◦ ρr˜ (xt , t) = 1 and     η  ◦ ρr˜ (xt , t) = η  ◦ ρr˜ (xt , t) = 0. Thus (28.10) says that at (xt , t) we have   ∂ ∂R 2 −Δ S = − ΔR ≥ R2 . (28.12) ∂t ∂t n 2 2 (2) If dg(t) (xt , O) ≥ r0 , then we may apply (28.7), η  ≤ 0, and ∂R ∂t ≥ ΔR+ n R to obtain that at (xt , t)     r˜  ∂ r˜ 2 2 2 η ◦ ρ (28.13) −Δ S ≥ η◦ R − ∇˜ r, ∇S ∂t ρ n ρ η

(η  ◦ ρr˜ )2 1 r˜ − η  ◦ + 2 2 R. ρ η ρ

Now take p = 4 in (28.2). Using (28.12) and applying (28.3) to (28.13), we have at (xt , t) in either Case (1) or Case (2) that      1 ∂ r˜ 2 2 const r˜ 2 −Δ S ≥ η◦ R + R, (28.14) η◦ ∂t ρ n ρ2 ρ where const > 0 is a universal constantand where we used ∇S (xt , t) = 0. By (28.14) and (28.8), for any ε ∈ 0, n2 we have     r˜ (xt , t) d− 2 const 1 r˜ (xt , t) 2 Smin (t) ≥ η (28.15) η R (xt , t)2 + R (xt , t) dt n ρ ρ2 ρ     2 r˜ (xt , t) const2 2 ≥ −ε η R (xt , t) − n ρ 4ερ4   2 2 const − ε Smin (t)2 − ≥ ; n 4ερ4 1 2 here we have used ab ≥ −εa2 − 4ε b and 0 ≤ η ≤ 1, where of backward difference quotients.

d− dt

denotes the lim inf

38

28. SPECIAL ANCIENT SOLUTIONS

Step 5. odi comparison gives a lower bound for Smin . Recall that in general if we have a solution to the ode  ds = A s2 − B 2 (28.16) dt on [α, t0 ] with s (t0 ) = s0 , where A, B > 0 and s0 < 0, then s (t) = −B

(28.17) where C = Take

s0 −B s0 +B

Ce−2BA(t0 −t) + 1 , Ce−2BA(t0 −t) − 1

provided B = −s0 . If B = −s0 , then s (t) ≡ s0 . 2 − ε, n

1 const B = √ √ 2, A 2 ερ and s0 = Smin (t0 ) < 0. It follows from (28.15) and the odi comparison theorem that we have Smin (t) ≤ s (t) and s (t) > −∞ for t ∈ [α, t0 ] (the latter inequality is since Smin (t) > −∞). Case 1. If s0 ≥ −B, then we have the estimate

(28.18)

(28.19)

A=

R (O, t0 ) ≥ Smin (t0 ) ≥ −B.

Case 2. If s0 < −B, then C > 1 and since s (t) > −∞ for t ∈ [α, t0 ], we have that the denominator of S (α) satisfies s0 − B −2BA(t0 −α) e − 1 > 0, s0 + B so that e2BA(t0 −α) + 1 . e2BA(t0 −α) − 1 Since (28.19) is a stronger estimate than (28.20), we conclude that (28.20) holds in either case for any ρ ≥ r0 + 53 (n − 1)r0−1 (t0 − α). Note that, from (28.18), A depends only on n and ε, whereas B depends only on n, ε, and ρ. Further note that although r0 is used to define r˜ and hence S, in the end, we shall obtain a lower estimate for R independent of r0 .  Step 6. Sharp lower bound for R. Now fix ε ∈ 0, n2 and let ρ → ∞. We then have B → 0. Since by l’Hˆopital’s rule (28.20)

R (O, t0 ) ≥ s0 > −B

lim B

B→0

e2BA(t0 −α) + 1 1 = A (t0 − α) e2BA(t0 −α) − 1

and since (28.20) holds for all ρ, we conclude that 1 R (O, t0 ) ≥ −  2 . − ε (t0 − α) n Finally, taking ε → 0, we obtain R (O, t0 ) ≥ −

n . 2 (t0 − α)

Since our only assumption on (O, t0 ) ∈ M × (α, ω) was R (O, t0 ) < 0, this estimate holds everywhere.  By taking α → −∞ in Theorem 28.1, we obtain the following, which does not assume a curvature bound.

1. LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER RICCI FLOW

39

Corollary 28.2 (Ancient solutions have nonnegative scalar curvature). If (Mn , g (t)), t ∈ (−∞, 0], is a complete ancient solution to the Ricci flow, then (28.21)

R≥0

on M × (−∞, 0].

As a further corollary, we obtain Theorem 27.2 since we may use Theorem 4.1 in [77] to put a complete GRS in canonical form. For example, both shrinking and steady GRS are ancient solutions and hence have R ≥ 0. Likewise, for an expander metric g with ε > 0, the corresponding solutionng (t) with g (0) = g and in particular exists on the time interval − 1ε , ∞ , so that R (g (t)) ≥ − 2 t+ ( 1ε ) nε R (g) = R (g (0)) ≥ − 2 . We may attempt to use the same localization method as in the proof of Theorem 28.1 to study the following.  Problem 28.3. Let M2 , g (t) , t ∈ (0, T ), be a complete solution to the Ricci flow with positive scalar curvature on a noncompact surface. Does the estimate Q

1 ∂ 2 ln R − |∇ ln R| ≥ − ∂t t

hold? The issue is that we do not assume that the scalar curvature is bounded. Note that ∂Q ≥ ΔQ + 2∇ ln R · ∇Q + Q2 . ∂t In attempting to localize this formula, we note that the gradient term 2∇ ln R · ∇Q poses a problem. Regarding incomplete ancient solutions, Peter Topping has pointed out to us the following examples of incomplete ancient solutions with scalar curvature negative somewhere.  Example 28.4. Consider the 2-dimensional hyperbolic disk D2 , gH and modify the metric gH only in the ball of radius 1 centered at the origin O (measured with respect to gH ) so that the resulting metric g0 on D2 (1) is rotationally symmetric, (2) has nonpositive curvature everywhere, (3) is flat in the ball of radius 1/10 (measured with respect to the flat metric) centered at O. Then,  2 by W.-X. Shi’s short-time existence theorem, there exists a complete solution D , g (t) to the Ricci flow satisfying the following properties: (i) g (t) is defined for t ∈ [0, ε) for some ε > 0, (ii) g (0) = g0 , (iii) g (t) has bounded curvature for all t ∈ [0, ε). 2 Since ∂R ∂t = ΔR + R (we are here in dimension 2), it can be proved by the weak and strong maximum principles that −2 < R (g (t)) < 0 for all t ∈ (0, ε). If we let B  Bg(0) (O, 1/10), then (B, g (t)|B ), t ∈ [0, ε), is an incomplete solution to the Ricci flow with −2 < R ( g (t)|B ) < 0 for t ∈ (0, ε) and R ( g (0)|B ) = 0. Defining the metric on B at all times t < 0 to be identically equal to g (0)|B , we obtain an incomplete ancient solution with scalar curvature equal to 0 for all t ≤ 0 and scalar curvature negative for 0 < t < ε.

40

28. SPECIAL ANCIENT SOLUTIONS

Regarding the sectional curvatures of ancient solutions, B.-L. Chen has proved the following. Theorem 28.5 (Ancient 3-dimensional solutions have sect ≥ 0). Any complete 3-dimensional ancient solution must have nonnegative sectional curvature. We omit the proof. We just remark that it may be of interest to find sharper forms of the local estimate that B.-L. Chen uses to prove this theorem. 2. Properties of singularity models In this section we recall how the study of singular solutions on closed manifolds leads us to consider both κ-solutions and GRS. We show that singularity models with bounded scalar curvature have at least linear volume growth. In dimension 3, we prove (1) the existence of singularity models at all curvature scales, (2) that singularity models must have bounded curvature, (3) that the asymptotic cone of a κ-solution is either a half-line or a line. 2.1. Singularity models are noncollapsed below all scales. We now turn to singularity models. Using his entropy monotonicity formula, in [312] Perelman proved that all finite-time solutions on closed manifolds have the following fundamental property (see also Theorem 6.74 in Part I). Theorem 28.6 (No local collapsing using the scalar curvature). Let (Mn , g (t)), 0 ≤ t < T < ∞, be a solution to the Ricci flow on a closed manifold and let ρ ∈ (0, ∞). There exists κ = κ (n, g (0) , T, ρ) > 0 such that if x0 ∈ M, t0 ∈ [0, T ), and r0 ∈ (0, ρ) are such that the scalar curvature R satisfies the condition R ≤ r0−2 in the open geodesic ball Bg(t0 ) (x0 , r0 ), then Vol g(t0 ) Bg(t0 ) (x0 , r) ≥ κ for all 0 < r ≤ r0 . rn By definition, we say that g (t), t ∈ [0, T ), is κ-noncollapsed below the scale ρ. A finite-time solution (Mn , g (t)), t ∈ [0, T ), to the Ricci flow on a closed manifold is called singular if supM×[0,T ) |Rm| = ∞, where Rm denotes the Riemann curvature tensor. We are most interested in such solutions. By the work of Sesum [365], any finite-time singular solution on a closed manifold must have supM×[0,T ) |Rc| = ∞. To understand singular solutions, we rescale and take limits. Definition 28.7 (Finite-time singularity model). Given a finite-time singular solution g (t) on a closed manifold Mn , an associated singularity model is a complete nonflat ancient solution to the  Ricci flow which is the pointed Cheeger– Gromov limit of rescalings gi (t)  Ki g ti + Ki−1 t , where ti → T and the rescaling factors satisfy Ki → ∞. Perelman’s no local collapsing theorem and Hamilton’s Cheeger–Gromov-type compactness theorem for solutions to Ricci flow imply the following (a slightly different statement is given in Theorem 19.4 in Part III). Theorem 28.8 (Existence of singularity models with bounded curvatures). Let (Mn , g (t)), for t ∈ [0, T ), with T < ∞, be a singular solution to the Ricci flow on a closed manifold. Then there exists (xi , ti ) with ti → T and Ki  |Rm| (xi , ti ) →

2. PROPERTIES OF SINGULARITY MODELS

41

 ∞ such that if we set gi (t)  Ki g ti + Ki−1 t , then (Mn , gi (t) , (xi , 0)) converges in the C ∞ pointed Cheeger–Gromov sense to a nonflat complete solution (Mn∞ , g∞ (t) , (x∞ , 0)), t ∈ (−∞, 0], with uniformly bounded sectional curvature. A consequence of Theorem 28.6 is that any singularity model is κ-noncollapsed below all scales for some κ > 0. This is true because being κ-noncollapsed below a fixed scale is preserved under Cheeger–Gromov limits and “below all scales” follows from the rescaling factors tending to infinity. Theorem 28.9 (Singularity models are κ-noncollapsed below all scales). Let (Mn , g(t)), t ∈ [0, T ), be a finite-time singular solution on a closed manifold. Then there exists κ = κ (n, g(0), T ) > 0 with the following property. For any associated singularity model (Mn∞ , g∞ (t)), t ∈ (−∞, 0], and for any x0 ∈ M∞ , t0 ∈ (−∞, 0], and r0 > 0 such that the scalar curvature satisfies Rg∞ (t0 ) ≤ r0−2 in Bg∞ (t0 ) (x0 , r0 ), we have Volg∞ (t0 ) Bg∞ (t0 ) (x0 , r) ≥κ rn

for all 0 < r ≤ r0 .

Note that we may choose the constant κ (n, g(0), T, ρ) in Theorem 28.6 so that it is nonincreasing in ρ. We may then choose κ (n, g(0), T ) in Theorem 28.9 to be equal to limρ→0 κ (n, g(0), T, ρ). Remark 28.10. The above discussion extends, with modifications, to complete solutions of the Ricci flow on noncompact manifolds (using Theorem 8.26 in Part I). When the underlying manifold M∞ of a singularity model is compact, it is diffeomorphic to the underlying manifold M of the finite-time singular solution from which it arises. Remarkably, Z.-L. Zhang has classified compact singularity models in the following sense. Theorem 28.11 (Compact singularity models). Any compact singularity model is a shrinking GRS. Exercise 28.12. Show that a compact singularity model cannot be Ricci flat. We have the following consequence of no local collapsing regarding the volume growth of noncompact singularity models. Corollary 28.13 (Linear volume growth if R ≤ C). If (Mn∞ , g∞ (t)) is a noncompact singularity model and if t is such that the scalar curvature Rg∞ (t) is bounded from above, then for any O ∈ M∞ the volume of Bg∞ (t) (O, r) grows at least linearly in r. Proof. By Theorem 28.9, if Rg∞ (t) ≤ C on M∞ , then for any x0 ∈ M∞ ,   Volg∞ (t) Bg∞ (t) x0 , C −1/2 ≥ κC −n/2 for some κ > 0. Since g∞ (t) is complete and M∞ is noncompact, there exists a unit speed ray γ emanating from O. The corollary follows from summing this

42

28. SPECIAL ANCIENT SOLUTIONS

 r estimate for x0 = γ 2kC −1/2 , where 0 ≤ k ≤  2C −1/2 − 12   .1 That is, Volg∞ (t) Bg∞ (t) (O, r) ≥

 

    Volg∞ (t) Bg∞ (t) γ 2kC −1/2 , C −1/2

k=0

≥ ( + 1) κC −n/2   r 1 ≥ − κC −n/2 . 2 2C −1/2



Partially motivated by Corollary 28.13, we have the following. Stronger motivation comes from work of Perelman, which implies that the conjecture is true in dimension n = 3; see Theorem 28.20 below. Optimistic Conjecture 28.14 (R and Vol of noncompact singularity models). Any noncompact singularity model defined on a time interval (−∞, ω) must have scalar curvature bounded from above on each interval (−∞, ω0 ], where ω0 < ω. Consequently, the volume growth of any noncompact singularity model must be at least linear. To better characterize when a singularity forms, it is natural to consider the following. Optimistic Conjecture 28.15. For any finite-time singular solution g (t), t ∈ [0, T ), on a closed manifold Mn , its scalar curvature must satisfy sup M×[0,T )

R = ∞.

By the Hamilton–Ivey estimate (see Theorem 9.4 in Volume One), this is true when dim M = 3. Zhou Zhang [454] has proved this conjecture for solutions to the K¨ ahler–Ricci flow on closed manifolds. We also have the following criterion for a singular solution to have unbounded scalar curvature. Lemma 28.16 (Criterion for long-time existence under bounded scalar curvature). Given a finite-time singular solution (Mn , g (t)), t ∈ [0, T ), on a closed manifold, if there are no Ricci flat singularity models associated to it, then supM×[0,T ) R = ∞. Proof. By Theorem 28.8, there exists an associated singularity model (Mn∞ , g∞ (t)), which is nonflat by definition. Suppose that supM×[0,T ) R < ∞. Then, by the standard lower estimate for R, we have supM×[0,T ) |R| < ∞. This im∂R

plies that the scalar curvature of g∞ (t) is Rg∞ (t) ≡ 0. By g∂t∞ (t) = Δg∞ (t) Rg∞ (t) +   Rcg (t) 2 , this implies Rcg (t) ≡ 0, contradicting the hypothesis.  ∞ ∞ A primary goal in studying singularity models is to understand their geometry as well as their topology. For example, one anticipates the need to understand the geometry at infinity of noncompact singularity models. The following may be a small step in this direction. Problem 28.17. What can one say about the asymptotic cones of noncompact singularity models? 1 Here,  ·  : R → Z denotes the floor function; i.e., x is the greatest integer less than or equal to x. In particular, x + 1 > x.

2. PROPERTIES OF SINGULARITY MODELS

43

Optimistic Conjecture 28.18. The asymptotic cone of a noncompact singularity model at any fixed time is well defined (i.e., unique). This conjecture is true in dimension 3 (see Theorem 28.30 below). 2.2. Singularity models and κ-solutions in dimension 3. Singularity models in dimension 3 are particularly nice. For our discussion of these objects, we recall the following class of ancient solutions introduced by Perelman. Definition 28.19 (κ-solution). Given a positive constant κ, a complete ancient solution (Mn , g˜(t)), t ∈ (−∞, 0], of the Ricci flow is called a κ-solution if it satisfies the following: (i) For each t ∈ (−∞, 0] the metric g˜(t) is nonflat with nonnegative curvature operator and is κ-noncollapsed at all scales. (ii) There is a constant C < ∞ such that the scalar curvature Rg˜ (x, t) ≤ C for all (x, t) ∈ M × (−∞, 0]. The work of Perelman implies the following, which is equivalent to saying that Optimistic Conjecture 28.14 is true when n = 3 (see [76]). Theorem 28.20 (3-dimensional singularity models must have bounded curvature). Any 3-dimensional singularity model must be a κ-solution. To prove this theorem, we shall use the following canonical neighborhood theorem of Perelman. Theorem 28.21. For any  ε > 0, ρ > 0, and κ > 0, there exists r0 ∈ (0, 1] with the following property. Let M3 , g (t) , t ∈ [0, T ) with T ∈ (1, ∞), be a solution to the Ricci flow on a closed 3-manifold which is κ-noncollapsed below the scale ρ and suppose that (x0 , t0 ) is such that t0 ∈ [1, T ) and Q  R (x0 , t0 ) ≥ r0−2 . Then the solution  on Bgˆ(0) (x0 , ε−1/2 ) × [−ε−1 , 0] gˆ (t)  Qg t0 + Q−1 t −1 is ε-close in the C ε +1 -topology2 to the corresponding subset of some κ-solution.3 We first prove, in dimension 3, the following strengthening of Theorem 28.8. Theorem of 3-dimensional singularity models at all curvature  28.22 (Existence scales). Let M3 , g (t) , t ∈ [0, T ), be a finite-time singular solution to the Ricci flow on a closed 3-manifold. Given any sequence of points (xi , ti ) ∈ M × [0, T ) with Ri  R (xi , ti ) → ∞, define  g˜i (t)  Ri g ti + Ri−1 t . Then there a subsequence of (M, g˜i (t) , (xi , 0)) which converges to some κ  exists 3 , h∞ (t) , t ∈ (−∞, 0]. solution N∞  Proof. By Theorem 28.6, there exists κ > 0 such that M3 , g (t) , t ∈ [0, T ), is κ-noncollapsed below the scale 1. Theorem 28.21 then says that for any j ∈ N and for such a κ, there exists rj ∈ (0, 1] such that if ij is chosen large enough so that Rij ≥ rj−2 (this is possible since Ri → ∞ as i → ∞), then the solution g˜ij (t) on  ε−1 denotes the least integer greater than or equal to ε−1 . −1/2 3 Of course, B −1/2 ) = B ). g ˆ(0) (x0 , ε g(t0 ) (x0 , (εQ) 2 Here



44

28. SPECIAL ANCIENT SOLUTIONS

Bg˜ij (0) (xij , j 1/2 ) × [−j, 0] is 1j -close to the corresponding subset of some κ-solution (Nj3 , hj (t)) centered at some yj ∈ Nj . Now, from Perelman’s compactness theorem for κ-solutions in dimension 3 (see Theorem 11.7 in Perelman [312] or the expository  Corollary 20.10 in Part III) and since limj→∞ Rhj (yj , 0) = limj→∞ Rg˜ij xij , 0 = 1, we have that (Nj , hj (t) , ∞ (yj ,30)) subconverges in the C pointed Cheeger–Gromov sense to some κ-solution N∞ , h∞ (t) , (y∞ , 0) , t ∈ (−∞, 0]. Since j → ∞, we conclude that (M, g˜ij (t),  (xij , 0)) subconverges to (N∞ , h∞ (t) , (y∞ , 0)), t ∈ (−∞, 0]. With this result, we may give the  Proof of Theorem 28.20. Let M3∞ , g∞ (t) , t ∈ (−∞, 0], be a singularity model of a finite-time singular solution M3 , g (t) , t ∈ [0, T ), on a closed manifold. Then, by definition there exists (xi , ti ) and Ki → ∞ such that (M, gi (t) , (xi , 0)),  where gi(t)  Ki g ti + Ki−1 t , converges in the C ∞ pointed Cheeger–Gromov sense to M3∞ , g∞ (t) , (x∞ , 0) , where x∞ ∈ M∞ and g∞ (t) is nonflat. Let Ri  Rg (xi , ti ). Since the limit lim Ki−1 Ri = Rg∞ (x∞ , 0) ∈ (0, ∞)

i→∞

exists (here Rg∞ (x∞ , 0) > 0 follows from the strong maximum principle since g∞ (t) is nonflat), we also have that  g˜i (t)  Ri g ti + Ri−1 t sense to (M∞ , g˜∞ (t),(x∞ , 0)), t ∈ converges in the C ∞ pointed Cheeger–Gromov  (−∞, 0], where g˜∞ (t) = cg∞ c−1 t and c  Rg∞ (x∞ , 0). On  the other hand, } such that M, g˜ij (t) , (xij , 0) by Theorem 28.22, there exists a subsequence {i j  3 , h∞ (t) , (y∞ , 0) , t ∈ (−∞, 0]. It then follows converges to a κ-solution N∞ from the definition of Cheeger–Gromov convergence that (M∞ , g˜∞ (t)) is isometric to (N∞ , h∞ (t)) as solutions on the time interval (−∞, 0]. Therefore g∞ (t) =  c−1 g˜∞ (ct), t ∈ (−∞, 0], is a κ-solution. 2.3. Relation between singularity models and gradient solitons. As evidenced below, the GRS are typical singularity models. However, although there is a rich theory of singularity analysis in Ricci flow by Hamilton and Perelman, there are not many isometry classes of noncompact singularity models which are known to exist. Besides κ-solutions, an important class of potential singularity models consists of the GRS. First we state the following important classification result for nonflat 3-dimensional steadies due to Brendle. Theorem 28.23. Any κ-noncollapsed nonflat complete noncompact 3-dimensional steady GRS must be the Bryant soliton. Recall from Theorem 28.20 that any 3-dimensional singularity model must be a κ-solution. The following result concretely relates singularity models and gradient solitons. It says that backward blow-down limits of κ-solutions, based at points where the reduced distances are bounded, are nonflat shrinkers (see Chapter 7 of Part I for the definition of the reduced distance ).

2. PROPERTIES OF SINGULARITY MODELS

45

Theorem 28.24 (Existence of asymptotic shrinker). Let (Mn , g (τ )), τ ∈ [0, ∞), be a κ-solution to the backward Ricci flow and let p ∈ M. For any sequences τi → ∞ and qi ∈ M such that the reduced distance based at (p, 0) satisfies the estimate (p,0) (qi , τi ) ≤ C for some constant C independent of i (note that for any τi such qi exist with C = n2 ),  there exists a subsequence such that Mn , τi−1 g (τi τ ) , (qi , 1) converges to a complete nonflat κ-noncollapsed shrinking gradient Ricci soliton (Mn∞ , g∞ (τ ) , (q∞ , 1)) with nonnegative curvature operator. Moreover, if n = 3, then the curvature of g∞ (τ ) is bounded. Regarding the geometric properties of singularity models, some in relation to GRS, we have the following questions. Recall that an ancient solution (Mn , g (t)), C on t ∈ (−∞, 0), is called Type I if there exists C < ∞ such that |Rm| (x, t) ≤ |t| M × (−∞, −1]. Otherwise the ancient solution is called Type II. Problem 28.25 (Singularity models which are not solitons). Does there exist a noncompact Type I singularity model which is not a shrinker? Does there exist a noncompact Type II singularity model which is not a steady? Problem 28.26 (Can a singularity model be Ricci flat?). Can a noncompact singularity model be Ricci flat? Hopefully the answer is no (by Exercise 28.12 and Lemma 28.16, this would imply Optimistic Conjecture 28.15). Regarding the volume growth of singularity models, note the following. Example 28.27 (Volume growth of the Bryant soliton). The n-dimensional Bryant soliton (n ≥ 3) satisfies lim

r→∞

Vol B (p, r) ∈ (0, ∞) . r (n+1)/2

On the other hand, for the cylinder S n−1 ×R, where S n−1 is the unit (n − 1)-sphere, the volumes of balls grow linearly. Because of this example, qualitatively speaking, the following is the best one can hope for. Optimistic Conjecture 28.28 (Lower bound for volume growth of nonsplitting singularity models). The volume growth of any n-dimensional noncompact singularity model, provided its universal cover is not isometric to the product of R with an (n − 1)-dimensional solution, is at least of the order r (n+1)/2 . The best possible upper bound for the volume growth of singularity models is less clear. One may start with this (compare with Theorem 27.42): Optimistic Conjecture 28.29 (Upper bound for volume growth of singularity models). The volume growth of any n-dimensional noncompact singularity model is at most of the order rn .

46

28. SPECIAL ANCIENT SOLUTIONS

2.4. Classification of asymptotic cones of 3-dimensional κ-solutions. Recall that 3-dimensional singularity models are κ-solutions. The following result gives a classification of the asymptotic cones of noncompact 3-dimensional κ-solutions. In particular, it gives an affirmative answer to Optimistic Conjecture 28.18 in dimension 3. Theorem 28.30 (Asymptotic cones of 3-dimensional κ-solutions). The asymptotic cone of a noncompact orientable 3-dimensional κ-solution must be either a line or a half-line. Proof. Step 1. Reduction of the problem. Let (M3 , g (t)), t ∈ (−∞, 0], be a noncompact orientable 3-dimensional κ-solution. Since g (t) has nonnegative curvature operator, by the strong maximum principle for Rm (see Theorem 12.53 in Part II) and by the classification of 2-dimensional κ-solutions (see Corollary 9.19 in [77]), (M3 , g (t)) is isometric to either (1) S 2 × R or (S 2 × R)/Z2 , where S 2 is the shrinking round 2-sphere and Z2 is generated by the isometry (x, y) → (−x, −y) or (2) a noncompact κ-solution with positive sectional curvature, where M3 is diffeomorphic to R3 . In case (1), the asymptotic cone of (M3 , g (t)) is either a line or a half-line (exercise). A well-known fact is that for any complete noncompact Riemannian manifold with nonnegative sectional curvature, the asymptotic cone exists (see Theorem I.26 in Part III for example). The rest of the proof is devoted to showing that in case (2) the asymptotic cone is a half-line. We first recall some basic facts. M (O) denote the space of (unit speed) rays emanating from O in  3Let Ray M , g (0) and let d0  dg(0) . Recall that a pseudo-metric d˜∞ on RayM (O) is defined by (I.7) in Part III; i.e., ˜ 1 (s) O γ2 (t) (28.22) d˜∞ (γ1 , γ2 )  lim γ s,t→∞

˜ is defined by for γ1 , γ2 ∈ RayM (O), where the Euclidean comparison angle  (28.23)

d0 (x, y)2 + d0 (y, z)2 − d0 (x, z)2 ˜ ∈ [0, π] xyz  cos−1 2d0 (x, y) d0 (y, z)

for x, y, z ∈ M. Recall also that the asymptotic cone of (M, g(0)) is isometric to the Euclidean metric cone Cone (M (∞) , d∞ ), where (M (∞) , d∞ ) is the quotient metric space induced by (RayM (O) , d˜∞ ) (see Theorem I.26 in Part III). Thus, the conclusion that the asymptotic cone of (M, g(0)) is a half-line (and hence the theorem) shall follow from showing that (28.24) d˜∞ (γ1 , γ2 ) = 0 for all γ1 , γ2 ∈ RayM (O). Now, by (I.8) in Part III, we have that  1/2 d0 (γ1 (at) , γ2 (bt))  2 = a + b2 − 2ab cos d˜∞ (γ1 , γ2 ) lim . t→∞ t Hence the desired equality d˜∞ (γ1 , γ2 ) = 0 is true if and only if (28.25)

lim

t→∞

d0 (γ1 (t) , γ2 (t)) = 0. t

2. PROPERTIES OF SINGULARITY MODELS

47

Step 2. Existence of ε-necks. Fix a time t, which without loss of generality we may assume is equal to 0, and fix a point O ∈ M. Recall that the notion of an ε-neck in a Riemannian manifold is given in Definition 18.26 of Part III. Claim 1. For any ε > 0 there exists an ε-neck Nε contained in (M3 , g (0)). Proof of Claim 1. Recall from Theorem 20.1 in Part III that the asymptotic scalar curvature ratio ASCR(g(t)) = ∞ for all t ∈ (−∞, 0]. Hence, by Corollary 18.21 in Part III on dimension reduction for noncompact κ-solutions with ASCR =  ∞, there exists a sequence {xi }i∈N in M such that the sequence gi (t) = Ri g Ri−1 t , where Ri  Rg (xi , 0), on M × (−∞, 0] and based at the point (xi , 0), converges in the C ∞ pointed Cheeger–Gromov sense to the product of R with a 2-dimensional κ-solution (which must be the shrinking round 2-sphere by a result of Hamilton (see Corollary 9.19 in [77])). The existence of ε-necks in (M, g (t)), for any ε > 0 and any t ∈ (−∞, 0], now follows from the definition of Cheeger–Gromov convergence. This finishes the proof of Claim 1. Since Nε is diffeomorphic to S 2 × R and since ∂Nε is smoothly embedded in ∼ onflies theorem we have that M − Nε has exactly M = R3 ,4 by the smooth Sch¨ two components, a compact component Bε diffeomorphic to a closed 3-ball and a noncompact component Cε diffeomorphic to S 2 × [0, 1). Furthermore, since Nε is an ε-neck, there exists an embedding (see p. 63 in Part III)   (28.26) ψε : S 2 × −ε−1 + 4, ε−1 − 4 → Nε −1 and a radius rε ∈ (0, ∞) such that rε−2 ψε∗ g(0) is ε-close in the C ε +1 -topology to the standard unit cylinder metric gS 2 + du2 . Without loss of generality, we may assume that ψε (S 2 × {−ε−1 + 4}) is closer to ∂Bε and ψε (S 2 × {ε−1 − 4}) is closer to ∂Cε . Note that since g (0) has bounded curvature, the radius of Nε satisfies

rε ≥ c,

(28.27)

where c > 0 is independent of ε sufficiently small.5 Step 3. Rays and necks. First observe the following. Claim 2. For ε > 0 sufficiently small, O ∈ Bε . Proof of Claim 2. Since g (0) has positive sectional curvature (using this only at O) and since each point of Nε has a small sectional curvature,6 for ε > 0 sufficiently small we have that O ∈ / Nε . Hence, if the claim is false, then there exists a sequence εi  0 such that O ∈ Cεi for all i ∈ N. We may pass to a subsequence {ki }i∈N so that Nεki ⊂ Cεkj

(28.28)

for j < i

(in particular, the Nεki are pairwise disjoint). Indeed, suppose that we have chosen  (M − Cεkj ) is compact, the sectional curvatures 1  k1 < · · · < ki−1 . Since Ki  j ki−1 such that Nεki ∩ Ki = ∅, which in turn implies (28.28). Now, by again using the smooth Sch¨ onflies theorem, for each i and j with j < i we have Cεkj − Nεki  Ki,j ∪ Li,j , where Ki,j ∼ = S 2 × [0, 1) has compact closure in M and where Li,j ∼ = S 2 × [0, 1) 3 ∼ 3 satisfies Li,j = Li,j . Since M = R , we conclude that B ε kj ⊂ Bε ki

(28.29)

and

Cεkj ⊃ Cεki

for all j < i. Claim 2 follows from  Cεki = ∅. Subclaim. i∈N

Proof of the subclaim. Fix p ∈ Bεk1. By (28.29) we then have p ∈ Bεki for all i ∈ N. Suppose the subclaim is false; then there exists x ∈ Cεki for all i. Let γ be a minimal geodesic from p to x with respect to g (0). Then γ must pass from one end of the εki -neck Nεki to the other end. Hence we have for i sufficiently large, 1 1 c diamg(0) (Nεki ) ≥ ε−1 rε ≥ ε−1 , 2 2 k i ki 2 ki where c > 0 is independent of i. The subclaim follows easily from the fact that ε−1 ki → ∞. dg(0) (p, x) = L g(0) (γ) ≥

Since O ∈ Bε for ε > 0 sufficiently small, we have for such ε that any γ ∈ RayM (O) passes from one end of Nε to the other end. Recall that ψε is the embedding in (28.26). Claim 3. For any ε > 0 sufficiently small, we have that any γ ∈ RayM (O) intersects ψε (S 2 × {ε−1 − 4}) at exactly one point, which we define to be γ(tγ,ε ). Proof of Claim 3. This follows from the facts that rays are minimizing and that the geometry of any ε-neck is, after rescaling, ε-close to that of the standard unit cylinder (of length 2ε−1 ); we leave a detailed proof of Claim 3 as an exercise. Step 4. Proof of the theorem. By the reduction to (28.25) in Step 1, it is now easy to see that (28.24) and the theorem are consequences of the following. Claim 4. For any γ1 , γ2 ∈ RayM (O), (28.30)

d0 (γ1 (tγ1 ,ε ) , γ2 (tγ2 ,ε )) = 0. ε→0 min{tγ1 ,ε , tγ2 ,ε } lim

Proof of Claim 4. Since Nε is an ε-neck, we have that for ε sufficiently small d0 (γ1 (tγ1 ,ε ) , γ2 (tγ2 ,ε )) ≤ 2πrε , where rε is the radius of Nε , since the intrinsic diameter of a round 2-sphere of radius r is πr. Since γ1 and γ2 are rays emanating from the same point, this implies by the triangle inequality that (28.31)

|tγ1 ,ε − tγ2 ,ε | ≤ 2πrε .

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

49

On the other hand, since the geodesic segments γ1 |[0,tγ ,ε ] and γ2 |[0,tγ ,ε ] both pass 1 2 through most of Nε (in particular, they both intersect ψε (S 2 × {−ε−1 + 4}) and ψε (S 2 × {ε−1 − 4})), we have (28.32)

min{tγ1 ,ε , tγ2 ,ε } ≥ ε−1 rε .

Now (28.30) follows immediately from (28.31) and (28.32).



Remark 28.31. Theorem 28.30 may also be proved using Corollary 9.88 in Morgan and Tian [251]. In contrast to Theorem 28.30, in dimension 4 it is expected that the asymptotic cones of singularity models can be top-dimensional. For example, under the K¨ ahler– Ricci flow of complex surfaces, consistent with Mori’s minimal model program, it is expected that one of the shrinkers of Feldman, Ilmanen, and one of the authors [111] occurs as a singularity model. 3. Noncompact 2-dimensional ancient solutions with finite width Singularity models, whether they be compact or noncompact, which correspond to the Ricci flow on closed manifolds are necessarily κ-noncollapsed at all scales. Moreover, those singularity models which dimension reduce to dimension 2 have been classified by the works of Hamilton and Perelman as having the round shrinking 2-sphere (or its Z2 quotient) as their nonflat factor. This uses the fact that for each κ > 0 the cigar is κ-collapsed at all large enough scales. However, it is still of substantial interest to understand ancient solutions which are not κ-noncollapsed for any κ > 0. In this section, all the results shall pertain to 2-dimensional complete ancient solutions with bounded curvature. A result of Hamilton (see Theorem 9.14 in [77]) states that, in the Type I case, the universal cover of any such solution must be either the Euclidean plane or the shrinking round 2-sphere. Hence we consider the Type II case. By the work of one of the authors (Sun-Chin Chu) in [83], any such noncompact Type II ancient solution must have finite width (see Definition 28.39 below). Furthermore, Panagiota Daskalopoulos and Natasa Sesum proved that any noncompact Type II ancient solution with finite width is isometric to a multiple of the cigar; this is the result we presently discuss (see Theorem 28.41 below). In the remaining case of a compact Type II ancient solution, Daskalopoulos, Hamilton, and Sesum proved that it must be isometric to a multiple of the King– Rosenau solution; we discuss this result in the next chapter. The above works combine to give us a complete classification of 2-dimensional complete ancient solutions to the Ricci flow with bounded curvature. 3.1. Relative isoperimetric inequalities. Let (Mn , g) be a Riemannian manifold and let Vol denote the Riemannian measure for measurable subsets. Given an (n − 1)-dimensional C ∞ submanifold N n−1 , let Area(N ) denote its (n − 1)-dimensional volume with respect to the induced metric. Definition 28.32. We say that a sequence of measurable sets {Ei } in (Mn , g) volume converges to a measurable set E if limi→∞ Vol(Ei  E) = 0, where A  B = (A − B) ∪ (B − A) denotes the symmetric difference of two subsets A and B.

50

28. SPECIAL ANCIENT SOLUTIONS

Consider the following notion of the area of the boundary of a set. Definition 28.33. The geometric perimeter of a measurable set E in (Mn , g) is (28.33)

P GEO (E) = inf lim inf Area(∂Ei ), {Ei } i→∞

where the infimum is taken over all sequences {Ei } of open subsets with C ∞ boundary which volume converges to E. If Ω is an open set, define the relative geometric perimeter of E with respect to Ω to be (28.34)

P GEO (E, Ω) = inf lim inf Area(∂Ei ∩ Ω). {Ei } i→∞

The functions E → P GEO (E) and E → P GEO (E, Ω) are lower semicontinuous with respect to volume convergence. Given x ∈ M, we say that a unit vector ν ∈ Tx M is normal to a measurable subset E ⊂ M if Vol(E ∩ {expx V : V, ν < 0, |V | < ε}) ωn lim = n ε→0 ε 2 and Vol(E ∩ {expx V : V, ν ≥ 0, |V | < ε}) lim = 0, ε→0 εn where ωn is the volume of the unit Euclidean n-ball; note that ν is unique if it exists. The reduced boundary of E is the set ∂ ∗ E of all points x in M such that there exists a normal unit vector in Tx M. Clearly, ∂E ∗ ⊂ ∂E. Let Hk denote the k-dimensional Hausdorff measure (see, e.g., p. 100 of [39]). We have the following result (see p. 112 of [39]). Lemma 28.34. If E is a measurable set, then PGEO (E) = Hn−1 (∂ ∗ E) ≤ Hn−1 (∂E). Similarly, if Ω is an open set, then PGEO (E, Ω) = Hn−1 (∂ ∗ E ∩ Ω) ≤ Hn−1 (∂E ∩ Ω). Remark 28.35. There exists open sets E with compact closure, with PGEO (E) < ∞, and with Hn (∂E) > 0, so that Hn−1 (∂E) = ∞. Let B(r) denote a ball of radius r in R2 . The following is the n = 2 and K = 0 special case of Theorem 18.1.3 in Burago and Zalgaller’s book [39]. Since n = 2, we use Area = H2 to denote Vol = Hn . Theorem 28.36 (Relative isoperimetric inequality for the 2-ball). If E is a measurable subset of B(r) ⊂ R2 and Area(E) ≤ 12 πr 2 , then 8 Area(E). π By Lemma 28.34, we have the following level set version of the above result.

(28.35)

PGEO (E, B(r))2 ≥

Corollary 28.37 (Level set version of the relative isoperimetric inequality for the 2-ball). Suppose that f : B(r) → [0, ∞) is a smooth function satisfying H1 (f −1 (c)) < ∞ for all c ∈ [0, ∞). Then for each c ∈ [0, ∞) we have that 8 (28.36) (H1 (f −1 (c)))2 ≥ min{H2 (f −1 [0, c) ∩ B(r)), H2 (f −1 [c, ∞) ∩ B(r))}. π

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

51

Choosing c0 ∈ (0, ∞) so that H2 (f −1 [0, c0 )∩B(r)) = H2 (f −1 [c0 , ∞)∩B(r)) = 12 πr 2 , we have that H1 (f −1 (c0 )) ≥ 2r. Remark 28.38. See General Isoperimetric Theorem 4.4.2 in [110] for a general result that encompasses relative isoperimetric inequalities, but without the sharp constants. 3.2. Width. Let (M2 , g) be an orientable complete noncompact Riemannian surface. Given a C ∞ proper function F : M → [0, ∞), its width w (F, g) with respect to g is defined to be the supremum of the lengths of the level sets of F ; i.e.,  (28.37) w (F, g)  sup L g F −1 (c) , c∈[0,∞)

where L g (S) = H1 (S) ∈ [0, ∞] denotes the 1-dimensional Hausdorff measure of a subset S ⊂ M. Note that S ⊂ M is the image of a rectifiable curve if and only if S is compact, connected, and has L g (S) < ∞. Definition 28.39. The width of the metric g is the infimum of the widths of all C ∞ nonnegative proper functions F with respect to g; i.e., (28.38)

w (g) = inf w (F, g) ∈ [0, ∞]. F

The Euclidean plane (R2 , dx2 +dy 2 ) has infinite width. To see this, note that for any smooth proper function F , the set F −1 (c) bounds the compact set F −1 ([0, c]). By Sard’s theorem, almost every c is a regular value of F , in which case F −1 (c) is a (possibly disconnected) smooth curve. By the Euclidean isoperimetric inequality, we have (28.39)

L2 (F −1 (c)) ≥ 4π Area(F −1 ([0, c]))

for such c; the rhs approaches infinity as such c → ∞. In fact, by the more general isoperimetric inequality for currents (see Theorem 4.5.9(31) in Federer’s book [110]), we have that (28.39) holds for all c ∈ [0, ∞). We have the following property of width under Cheeger–Gromov convergence. Lemma 28.40 (The limit of surfaces with uniformly bounded widths cannot be Euclidean). Let (M2i , gi , pi ) be a sequence of complete Riemannian surfaces which converge in the pointed C ∞ Cheeger–Gromov sense to (M2∞ , g∞ , p∞ ). If the widths of (Mi , gi , pi ) are bounded independent of i, then (M∞ , g∞ ) cannot be isometric to the Euclidean plane. Proof. Suppose that (M∞ , g∞ ) is isometric to theEuclidean plane. Then, for any k ∈ N, there exists i(k) ∈ N such that Bk  Bgi(k) pi(k) , k ⊂ Mi(k) is k−1 of radius k. Let Fk : Mi(k) → [0, ∞) close in the C k -norm to the Euclidean 2-ball  be any C ∞ proper function satisfying w Fk , gi(k) < ∞. The function Gk : [0, ∞) → [0, Areagi(k) (Bk )] defined by Gk (c)  Areagi(k) (Fk−1 ([0, c]) ∩ Bk )

52

28. SPECIAL ANCIENT SOLUTIONS

is continuous, nondecreasing, and surjective.  To see the continuity of Gk we argue as follows. Given any c0 ∈ [0, ∞), since Fk−1 ([0, c]) ∩ Bk = Fk−1 ([0, c0 )) ∩ Bk c 0 and M is noncompact, M is diffeomorphic to R2 and Area(g(t)) = ∞ for each t (in fact, the areas of geodesic balls grow linearly in terms of their radii). Furthermore, a consequence of the uniformization theorem is that (M, g (t)) is conformal to the plane for each t < 0 (see Theorem 15 in Huber [151]). Since Ricci flow preserves the pointwise conformal class for 2-dimensional complete solutions with bounded curvature, there exist global conformal coordinates (x, y) : M → R2 such that g (x, y, t) = e−f (x,y,t) (dx2 + dy 2 ) for some C ∞ function f : M × (−∞, 0) → R. Hence Rg = ef Δeuc f . By the Ricci ∂ flow equation ∂t g = −Rg, we then obtain (28.42)

∂f = R = Δg f. ∂t

∂ ∂ Thus, the conformal factor u  e−f (x,y,t) satisfies ∂t ln u = Δg ln u; i.e., ∂t u = Δeuc ln u. The following inequality is a geometric basis for the proof of Theorem 28.41.

Lemma 28.42 (Bochner-type inequality). Let (M2 , g(t)), t ∈ (−∞, 0), be a complete noncompact ancient solution with positive curvature. Suppose that f : 7 In the proof of the claim inside the proof of Lemma 28.43 below, we consider a similar situation in more detail.

54

28. SPECIAL ANCIENT SOLUTIONS

M × (−∞, 0) → R satisfies Δf = R. Then, for any smooth bounded domain Ω ⊂ M and any time t < 0, we have  2

  2  2  1 |∇R + R∇f |   + 2 ∇ f − Δf g  dμ ≤ ν(R + |∇f |2 )dσ, (28.43) R 2 Ω ∂Ω where ν is the unit outward normal to ∂Ω; all norms, covariant derivatives, and Laplacians are with respect to g(t). Proof. Fix t < 0. Recall the everywhere present Bochner formula (28.44)

 2 2 Δ |∇f | = 2 ∇(Δf ), ∇f  + 2 Rc (∇f, ∇f ) + 2 ∇2 f   2 2 = 2 ∇R, ∇f  + R |∇f | + 2 ∇2 f  .

From this we obtain (28.45) Δ(R + |∇f |2 ) = ΔR + R2 − (Δf )2 −

|∇R|2 R

 2 |∇R|2 2 + 2 ∇R, ∇f  + R |∇f | + 2 ∇2 f  R  2   |∇R + R∇f |2 1 |∇R|2 + 2 ∇2 f − Δf g  + ΔR + R2 − . = R 2 R +

Hamilton’s trace Harnack estimate (see Proposition 15.7 in Part II) says (28.46)

∂R |∇R|2 = ΔR + R2 ≥ ≥ 0. ∂t R

Thus, by applying the divergence theorem to the integral of (28.45) over Ω, we obtain (28.43).  Modulo some estimates to be subsequently proved, we can now give the Proof of Theorem 28.41. Fix any time t < 0. Choose Ω = Ωi in (28.43), where Ωi is as in Lemma 28.45. By taking i → ∞ we obtain  2

   2 |∇R + R∇f |2 1  2  + 2 ∇ f − Rg  dμ ≤ lim ν(R + |∇f | )dσ. (28.47) i→∞ ∂Ω R 2 M i Now, (28.51) and (28.70) below imply that the rhs of this tends to zero. We conclude that ∇2 f = 12 Rg = Rc on M × (−∞, 0); i.e., g (t) is a steady GRS flowing along −∇f . By Proposition 1.25 in Part I, (M, g(t)) must be a constant multiple of the cigar soliton.    We shall call ∂Ωi ν(R)dσ and ∂Ωi ν(|∇f |2 )dσ on the rhs of (28.47) to be the first and second boundary terms, respectively. In the following subsections, we prove that both of these boundary terms tend to zero.

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

55

3.4. Vanishing of the first boundary term in the limit.  First, we show that M2 , g(t) is asymptotically cylindrical at spatial infinity in the following sequential sense. Lemma 28.43 (Asymptotically cylindrical). Assume the hypothesis of Theorem 28.41. Given any sequence of basepoints pi → ∞, the sequence of pointed solutions (M, g (t) , pi ), t ∈ (−∞, 0), subconverges in the C ∞ Cheeger–Gromov sense to a static flat cylinder. Proof. By (28.46) we have that R is uniformly bounded and positive on M × (−∞, ω] for each ω < 0. Hence, by Shi’s local first derivative estimate (see Theorem 14.10 in Part II) there exists a constant C < ∞ depending on ω such that |∇R| ≤ C

(28.48)

on M × (−∞, ω].

Claim. For each t < 0, (28.49)

lim R(p, t) = 0.

p→∞

Proof of the claim. By a result of Cheeger and Gromoll (see Theorem B.65 in Volume One), we have ρ  inj (g(t)) > 0. By this and the volume comparison for R ≤ C, there exists c > 0 such that Area(Bq (ρ, t)) ≥ c for each q ∈ M, where Bq (ρ, t) denotes the geodesic ball of radius ρ centered at q with respect to g(t). Now suppose that (28.49) is false. Then, by using (28.48), we see that there exist c > 0 and a sequence of points qi → ∞ such that the balls Bqi (ρ, t) are  pairwise disjoint and Bq (ρ,t) Rdμ(t) ≥ c for each i. This contradicts the Cohni  Vossen inequality M Rdμ(t) ≤ 4π and so the claim is proved. Now, by Hamilton’s Cheeger–Gromov compactness theorem, for any sequence of basepoints pi → ∞, the sequence of pointed solutions (M, g (t) , pi ), t ∈ (−∞, 0), subconverges in C ∞ to a complete noncompact pointed solution M2∞ , g∞ , p∞ , t ∈ (−∞, 0), with Rg∞ ≥ 0. From (28.49), we have Rg∞ (p∞ , t) = 0. By the strong maximum principle, we conclude that g∞ (t) ≡ g∞ is flat and static. By Lemma 28.40 we have that (M∞ , g∞ ) cannot be isometric to the Euclidean plane. Since (M∞ , g∞ ) is orientable, flat, complete, and noncompact, by the classification of such Riemannian surfaces we then have that (M∞ , g∞ ) is isometric to a flat cylinder.  Next, infer from limp→∞ R(p, t) = 0 that at each time the higher covariant derivatives of R decay to zero at spatial infinity. Lemma 28.44 (Higher derivatives of curvature decay). For each t ∈ (−∞, 0) and j ≥ 1 we have   (28.50) lim ∇j R (p, t) = 0. p→∞

Proof. Given t < 0 and O ∈ M, by (28.49) we have that for any ε > 0 there exists Cε < ∞ such that 0 < R(p, t) ≤ ε

for p ∈ M − BO (Cε , t).

Thus the trace Harnack estimate (28.46) implies that 0 < R(p, t¯) ≤ ε

for p ∈ M − BO (Cε , t)

and

t¯ ∈ (−∞, t].

56

28. SPECIAL ANCIENT SOLUTIONS

Since R > 0, distances are decreasing under the Ricci flow. Hence, if p ∈ M − BO (Cε + ε−1/2 , t), then Bp (ε−1/2 , t¯) ⊂ Bp (ε−1/2 , t) ⊂ M − BO (Cε , t)

for all t¯ ∈ (−∞, t].

In particular, for such p we have 0 < R(x, t¯) ≤ ε

for x ∈ Bp (ε−1/2 , t − ε−1 )

and

t¯ ∈ [t − ε−1 , t].

Now, from scaling by ε Shi’s local higher derivative estimates (see Theorem 14.14 in Part II), we obtain that for each j ≥ 1 there exists Cj < ∞ such that  j  ∇ R (p, t) ≤ Cj ε1+ 2j for p ∈ M − BO (Cε + ε−1/2 , t).  Finally, we consider the first boundary term in (28.47). By the definition ∞ of Cheeger–Gromov convergence, there exists an exhaustion {Ui }i=1 of R × S 1 ∗ and embeddings ϕi : (Ui , p∞ ) → (M,  pi ) such that (Ui , ϕi g (t)) subconverges in C ∞ uniformly on compact sets to R × S 1 , g∞ . Let (s∞ ,θ∞ ) be the cylindrical coordinates of p∞ ∈ R × S 1 and define the curve γi  ϕi {s∞ } × S 1 to be the push forward of the distance circle passing through p∞ . Note that pi ∈ γi . For i sufficiently large, {s∞ }×S 1 ⊂ Ui and hence γi is an embedded loop in M bounding a disk Ωi . Lemma 28.45 (Vanishing of the first boundary term in the limit). We have      → 0 as i → ∞,  ν (R) dσ (28.51)   ∂Ωi

where ν is the unit outward normal to ∂Ωi . Proof. Fix O ∈ M. We have that limi→∞ dg (pi , O) = ∞ and that Lg (∂Ωi ) ≤ C since it is approximately equal to the length of {s∞ } × S 1 with respect to g∞ . Therefore ∂Ωi → ∞ and       ≤ L(∂Ωi ) sup |∇R| → 0 as i → ∞ ν (R) dσ   ∂Ω ∂Ωi

i



by (28.50) for j = 1.

3.5. Vanishing of the second boundary term in the limit. In this subsection we shall control the second boundary term in (28.47). To do this, we shall show that the spatial infinity of g(t) is pointwise asymptotically cylindrical. In particular, we shall use a crucial estimate for the conformal factor of g(t) relative to the cylinder metric (see (28.62) below). Let (x, y) : M → R2 be global conformal coordinates as before and from now on set M = R2 . On R2 − {0} let g  g(t)  u ˆ(t)gcyl , 2

2

u−1 Δcyl ln u ˆ. Thus where gcyl = dxx2 +dy +y 2 . Since Rcyl = 0, we have R = −ˆ ∂ ˆ = Δcyl ln u ˆ and hence −Rg implies ∂t u

∂ ∂t g

=

∂ ln u ˆ = Δg ln u ˆ = −R < 0. ∂t First, for each time t we shall obtain a uniform lower bound for u ˆ (see (28.56) below). We shall begin with a weaker estimate obtained from bounding g(t) from below by a hyperbolic cusp. Since the noncompactness of R2 makes the application (28.52)

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

57

of a comparison principle for metrics more technical, we instead use complex geometry. Recall the following complex 1-dimensional special case of a result of Yau (see Theorem 2 in [444]). Lemma 28.46 (Generalization of the Schwarz–Ahlfors–Pick lemma). Let (M21 , g1 ) and (M22 , g2 ) be Riemannian surfaces such that g1 is a complete metric, Rg1 ≥ −c1 , and Rg2 ≤ −c2 , where c1 ≥ 0 and c2 > 0. If φ : (M1 , g1 ) → (M2 , g2 ) is a conformal map, then c1 (28.53) φ∗ (g2 ) ≤ g1 . c2 In particular, if g2 and g1 = vg2 are Riemannian metrics on a surface M2 , where g1 is complete with Rg1 ≥ −c1 and Rg2 ≤ −c2 for constants c1 , c2 > 0, then v ≥ cc21 . By the Schwarz lemma and the method of Giesen and Topping [120], one obtains the following consequence of an estimate of Ana Rodriguez, Juan Luis Vazquez, and Juan R. Esteban. Lemma 28.47 (The hyperbolic cusp is a lower barrier). For each t < 0 there exists c(t) > 0 such that u ˆ(s, θ, t) ≥ c(t)s−2

(28.54)

for all s = ln r sufficiently large and θ ∈ S 1 . The constants c(t) are bounded from below by a positive constant on each compact subinterval of (−∞, 0). Since u = ruˆ2 , this is equivalent to c(t) . r 2 (ln r)2

u(r, θ, t) ≥

(28.55)

Proof. Let (R2 , g(t)), t ∈ (−∞, 0), be our solution. On R2 − BO (1) consider the metric ghyp = r2 (ln1 r)2 gE . This is the complete hyperbolic cusp as can be seen from the change of variables σ = ln (ln r), which yields ghyp = dσ 2 + e−2σ dθ 2

for − ∞ < σ < ∞, θ ∈ S 1 .

Let η : R2 → [0, 1] be a smooth radial cutoff function with η = 1 in BO (2) and η = 0 in R2 − BO (3). On R2 − BO (1) define the “blended” complete metric −2η

gble (t) = (r ln r)

u(t)1−η gE .

We have Rgble (r, θ, t) ≡ −2 for r ∈ (1, 2] and Rgble (r, θ, t) = Rg(t) > 0 for r ∈ [3, ∞). Since the blended region is compact, there exists a constant C(t) < ∞ depending only on η and g(t) such that Rgble (r, θ, t) ≥ −C(t)

in (BO (3) − BO (2)) × (−∞, 0).

Hence Rgble ≥ −C(t) in (R − BO (1)) × (−∞, 0), where C(t) is uniformly bounded on compact subintervals of (−∞, 0). Now, by applying Lemma 28.46 to g1 = gble (t) and g2 = ghyp with φ = id, we obtain 2 in R2 − BO (1). (r ln r)2(1−η) u(t)1−η ≥ C(t) In particular, we have 2 2 in R2 − BO (3).  (r ln r) u(t) ≥ C(t) 2

58

28. SPECIAL ANCIENT SOLUTIONS 2s

By (28.41), we have for the cigar soliton that u ˆ(t) = e4te+e2s . In particular, 1 ˆ(t) ≤ 1 on [0, ∞) × S 1 . With the help of the above lemma, we shall now e4t +1 ≤ u prove the following. Lemma 28.48 (Lower bound for conformal factor of g(t) relative to gcyl ). There exists c(t) > 0, which is positively bounded from below on finite closed intervals, such that (28.56)

u ˆ(s, θ, t) ≥ c(t)

in [1, ∞) × S 1 .

Proof. Let (ln u ˆ)−  max{− ln u ˆ, 0} be the negative part of ln u ˆ. Then (ln u ˆ)− ≥ 0 is subharmonic with respect to gcyl = ds2 + dθ 2 since by (28.52) it is the maximum of two subharmonic functions. Define the circular averages (times 2π)  (28.57) W− (s, t)  (ln u ˆ)− (s, θ, t)dθ ≥ 0. S1

Since (ln u ˆ)− is subharmonic, s → W− (s, t) is weakly convex. Thus s → (W− )s (s, t) is nondecreasing and hence the limit λ(t)  lim (W− )s (s, t) ∈ [0, ∞] s→∞

exists (if λ(t) < 0, then W− (s, t) < 0 for s sufficiently large, a contradiction). Claim. For each t we have λ(t) = 0. Proof of the claim. Suppose that for some t we have λ(t) ∈ (0, ∞]. Then there exists s0 < ∞ such that   λ(t) , 1 ∈ (0, ∞) for s ≥ s0 . (W− )s (s, t) ≥ κ(t)  min 2 Since W− (s, t) ≥ 0, this implies that W− (s, t) ≥ exponential decay of the circular minimums: (28.58)

ˆ(s, θ, t) ≤ e− min u

κ(t) 4π s

θ∈[0,2π]

κ(t) 2 s

for s ≥ 2s0 . Hence we have

for each s ≥ 2s0 .

This contradicts the estimate in Lemma 28.47 and hence proves the claim. Now λ(t) = 0 implies that s → W− (s, t) is nonincreasing. In particular, (28.59)

W− (s, t) ≤ W− (0, t)

for s ≥ 0.

Now let (¯ s, θ¯ ) ∈ [1, ∞) × S 1 . We have that B(¯s,θ¯ ) (1) is embedded in [0, ∞) × S 1 and hence is isometric to a unit Euclidean 2-ball. Since (ln u ˆ)− is subharmonic, we can apply the mean value inequality to obtain that for any (¯ s, θ¯ ) ∈ [1, ∞) × S 1 ,  ¯ t) ≤ 1 (ln u ˆ)− (¯ s, θ, (ln u ˆ)− (s, θ, t)dsdθ π B(¯s,θ¯ ) (1)   s¯+1 1 (ln u ˆ)− (s, θ, t)dsdθ ≤ π S 1 s¯−1  1 s¯+1 = W− (s, t)ds π s¯−1 2 ≤ W− (0, t). π

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

59

We conclude that we have the lower bound u ˆ(s, θ, t) ≥ e− π W− (0,t) 2

on [1, ∞) × S 1 × (−∞, 0).



Next, we turn to the upper bound for u ˆ. First, we have the following. Lemma 28.49 (A bound for the circular averages of (ln u ˆ)+ ). For each t ∈ (−∞, 0) there exists C(t) < ∞ such that the circular averages of the positive part of ln u ˆ satisfy  (ln u ˆ)+ (s, θ, t)dθ ≤ C(t) for all s ∈ [0, ∞). (28.60) S1

Proof. Fix t ∈ (−∞, 0). Define  ln u ˆ(s, θ, t)dθ, W (s, t)  1 S W+ (s, t)  (ln u ˆ)+ (s, θ, t)dθ, S1

so that W = W+ − W− , where W− is defined by (28.57). By (28.59) we have that W (s, t) ≥ −C for s ∈ [0, ∞). Since Δ ln u ˆ < 0 and smooth, we have that Wss (s, t) < 0; i.e., s → Ws (s, t) is strictly decreasing. Hence μ(t)  lim Ws (s, t) ∈ [0, ∞) s→∞

exists (μ cannot be negative by the lower bound for W ). Claim. μ(t) = 0 for each t < 0. Proof of the claim. Suppose that μ(t) > 0 for some t. Since Ws (s, t) > μ, we then have that lims→∞ W (s, t) = ∞. Hence, given any k ∈ N, there exists sk < ∞ such that W (s, t) ≥ k for all s ≥ sk . We shall now derive a contradiction. Denote π : [1, ∞) × R → [1, ∞) × S 1 to be the universal cover with the lifted Euclidean metric on the total space and define u ˜ to be the lift of u ˆ; i.e., u ˜=u ˆ ◦ π. ˜ = (¯ s, θ) s, θ¯ ). By Let (¯ s, θ¯ ) ∈ [sk + π, ∞) × S 1 and choose θ˜ ∈ R such that π(¯ Δ ln u ˜ < 0 and the mean value inequality, we have  ¯ t) = ln u ˜ t) ≥ 1 ln u ˜(s, θ, t)dsdθ. ln u ˆ(¯ s, θ, ˜(¯ s, θ, √ 2π 3 B(¯s,θ) ˜ ( 2π) √ ˜ is bounded from Observe that (¯ s − π, s¯ + π) × S 1 ⊂ π(B(¯s,θ) ˜ ( 2π)). Since ln u below by (28.56), there exists C < ∞ independent of k such that   s¯+π  ln u ˜(s, θ, t)dsdθ ≥ ln u ˆ(s, θ, t)dsdθ − C √ S1

B(¯ ˜ ( 2π) s,θ)



s¯−π

s¯+π

= s¯−π

W (s, t)ds − C

≥ 2πk − C. ¯ t) ≥ k2 − C3 for any (¯ s, θ¯ ) ∈ [sk + π, ∞) × S 1 . Finally, because Therefore ln u ˆ(¯ s, θ, π 2π k is arbitrary, this contradicts the finite-width condition in the same way as in the proof of Lemma 28.43. 

60

28. SPECIAL ANCIENT SOLUTIONS

We shall use the following Trudinger–Moser-type inequality of Brezis and Merle [36]. ¯ → Lemma 28.50. Suppose that B(r) ⊂ R2 is a ball of radius r and that w : B(r) R is a C 2,α function satisfying w = 0 on ∂B(r). Then for each δ ∈ (0, 4π) we have the estimate    δ |w(x)| 16π 2 r 2 . (28.61) exp dμE (x) ≤ ||Δw||L1 (B(r)) 4π − δ B(r) Proof. Define w ¯ : R2 → R by    2r 1 w(x) ¯ = ln |Δw(y)|dμE (y). 2π B(r) |x − y| 1 ln |x − y| is the fundamental solution to the Laplace equation on R2 , we Since − 2π have (see Theorem 4.3 of [122] for example)

Δw ¯ = −|Δw|

in B(r).

≥ 1 for y ∈ B(r). Since Δw ¯ ≤ ±Δw in B(r), If x ∈ B(r), then w(x) ¯ ≥ 0 since w ¯ ≥ 0 on ∂B(r), and ±w = 0 on ∂B(r), by the maximum principle we conclude that w ¯ ≥ ±w; that is, w ¯ ≥ |w| in B(r). Recall that by Jensen’s inequality we have for functions ϕ and ψ ≥ 0, with  ψdμ = 1, that E B(r)

  2r |x−y|

ϕψdμE

exp

≤

B(r)

Taking ϕ(y) = e

δ 2π

B(r)

2r ln ( |x−y| ) and ψ(y) =

δ w(x) ¯ ||Δw|| 1 L (B(r))

eϕ ψdμE .



|Δw(y)| ||Δw||L1 (B(r)) ,

we obtain

  2r |Δw(y)| δ ln = exp dμE (y) |x − y| ||Δw||L1 (B(r)) B(r) 2π δ   2π  2r |Δw(y)| dμE (y). ≤ |x − y| ||Δw||L1 (B(r)) B(r)

Using w ¯ ≥ |w| and integrating the above inequality yields  e

δ|w(x)| ||Δw|| 1 L (B(r))

dμE (x)

B(r)



 ≤ B(r)

B(r)



2r |x − y|

δ  2π

dμE (x)

|Δw(y)| dμE (y). ||Δw||L1 (B(r))

On the other hand, since x, y ∈ B(r), δ δ    2π  2π   2r 2r dμE (x) ≤ dμE (x) |x − y| |x| B(r) B(r) δ  r   2π δ 2r 4π 2 2 2π 2 = 2π r . r¯d¯ r= r¯ 4π − δ 0

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH

61

δ We conclude that since 2π < 2,     δ δ |w(x)| |Δw(y)| 4π 2 2 2π 2 r exp dμE (y) dμE (x) ≤ ||Δw||L1 (B(r)) 4π − δ B(r) B(r) ||Δw||L1 (B(r))


0 and t < 0 there exists sε (t) ∈ (0, ∞) such that   

R2

∞

Rˆ udθds < ε. sε (t)

S1

cyl (π), so that (Bπ , gcyl ) is isometric Let s ∈ [sε (t)+π, ∞) and θ ∈ S 1 . Let Bπ = B(s,θ) to a Euclidean 2-ball. Let w ˆ be the unique solution to

−Δcyl w ˆ = Rˆ u

in Bπ ,

w ˆ = 0 on ∂Bπ .  By Lemma 28.50 with δ = 1 and since Bπ |Rˆ u|dμcyl < ε, we have  −1 16π 4 ˆ < 135. eε |w| dμcyl ≤ (28.63) 4π − 1 Bπ   −1 ˆ −3 By Jensen’s inequality we have Bπ ε−1 |w|π ˆ −3 dμcyl ≤ ln( Bπ eε |w| π dμcyl ), so that   16π (28.64) ||w|| ˆ L1 (Bπ ) ≤ επ 3 ln < 46ε. 4π − 1 z , 0} is Now let zˆ = ln u ˆ − w. ˆ Then Δcyl zˆ = 0 in Bπ . Since zˆ+ = max{ˆ subharmonic on Bπ , by the mean value inequality we have that ||ˆ z+ ||L∞ (Bπ/2 ) ≤

1 4 z+ ||L1 (Bπ ) , ||ˆ z+ ||L1 (Bπ ) = 3 ||ˆ Area(Bπ/2 ) π

cyl where Bπ/2 = B(s,θ) (π/2). On the other hand, since zˆ+ ≤ (ln u ˆ)+ + |w|, ˆ we have

||ˆ z+ ||L1 (Bπ ) ≤ ||(ln u ˆ)+ ||L1 (Bπ ) + ||w|| ˆ L1 (Bπ ) ≤ C by (28.60) and (28.64). Hence ||ˆ z+ ||L∞ (Bπ/2 ) ≤ (28.65)

4C π3 .

This implies that

0 < −Δcyl ln u ˆ = Rˆ u ≤ Rezˆ+ +wˆ ≤ C1 ewˆ ,

62

28. SPECIAL ANCIENT SOLUTIONS 3

where C1 = e4C/π supR2 R. By (28.63) with ε = 12 we have that ewˆ ∈ L2 (Bπ ). Now we may apply a standard interior elliptic estimate to (28.65) to obtain (see Theorem 4.1 in [144]) ||(ln u ˆ)+ ||L∞ (Bπ/2 ) ≤ C||(ln u ˆ)+ ||L1 (Bπ ) + C||Δcyl ln u ˆ||L2 (Bπ ) ≤ C.



Let fˆ  − ln u ˆ. We now estimate the higher derivatives of fˆ. Lemma 28.53 (Higher derivative bounds for fˆ). For any compact time interval [t1 , t2 ] ⊂ (−∞, 0) and k ∈ N there exists a constant Ck < ∞ such that |∇kcyl fˆ|2cyl ≤ Ck

on [2, ∞) × S 1 × [t1 , t2 ].

Proof. By Lemmas 28.48 and 28.52, we have that for each compact time interval [t1 , t2 ] ⊂ (−∞, 0) there exists a constant C < ∞ such that (28.66)

|fˆ| ≤ C

in [1, ∞) × S 1 × [t1 − 1, t2 ].

∂ ˆ ∂ By (28.52), we have that ∂t g = −Rg. Since R and its covariant f = Δg fˆ, where ∂t derivatives are uniformly bounded on ([1, ∞) × S 1 ) × [t1 − 1, t2 ], one can prove uniform higher derivative bounds for fˆ in the interior by the Bernstein technique (compare with Section 3 of Chapter 14 in Part II). Namely, there exist constants Ck < ∞ such that

(28.67)

|∇kg fˆ|g ≤ C

in [2, ∞) × S 1 × [t1 , t2 ].

For example, using (28.44), we compute that (the quantities are with respect to g(t)) ∂ |∇fˆ|2 = R|∇fˆ|2 + 2∇(Δfˆ), ∇fˆ = Δ|∇fˆ|2 − 2|∇2 fˆ|2 . ∂t Using |fˆ| ≤ C and defining P = (4C 2 + fˆ2 )|∇fˆ|2 , we compute that (28.68)

∂P = ΔP − 2(4C 2 + fˆ2 )|∇2 fˆ|2 − 2|∇fˆ|4 − 2∇(fˆ2 ) · ∇|∇fˆ|2 ∂t 2 P 2. ≤ ΔP − 125C 4 We can localize (28.69) to obtain a uniform estimate for |∇fˆ| on [2, ∞)×S 1 ×[t1 , t2 ]. Similarly, one can prove uniform bounds for the higher covariant derivatives of fˆ(t) with respect to g(t) on [2, ∞) × S 1 × [t1 , t2 ]. Now (28.66) and (28.67) imply that (28.69)

|∇kcyl fˆ|cyl ≤ C

in [2, ∞) × S 1 × [t1 , t2 ].

For example, for k = 1 we have that |∇cyl fˆ|cyl = u ˆ1/2 |∇g fˆ|g is bounded. For higher k k derivatives, we compare ∇cyl with ∇g .  By the derivative estimates for fˆ = − ln u ˆ of Lemma 28.53 and by the Arzela– Ascoli theorem, we can improve the Cheeger–Gromov convergence to pointwise convergence; i.e., we may choose the diffeomorphisms to be translations in the s direction of the cylinder. Recall that the cylindrical coordinates of the basepoints pi are (si , θi ), where si → ∞.

4. ANCIENT SOLUTIONS WITH POSITIVE CURVATURE

63

Lemma 28.54 (Pointwise subconvergence to a flat cylinder). The sequence of ˆ (s + si , θ, t) gcyl ), t ∈ (−∞, 0), subconverges in each C k solutions ([2, ∞) × S 1 , u  norm on compact sets to a complete static flat solution g∞ =u ˆ∞ gcyl on R × S 1 , where ˆ (s + si , θ, t) . u ˆ∞ (s, θ, t) = lim u i→∞

 is flat. Hence By the fact that limp→∞ R(p, t) = 0 for each t, we have that g∞  there exists an isometry ϕ between g∞ and cgcyl , for some constant c > 0. Since ϕ is an orientation-preserving conformal diffeomorphism of (R2 − {0} , gcyl ), it must be given by multiplication by a nonzero complex number; this implies that u ˆ∞ is a constant. Using Lemma 28.54, we now prove

Lemma 28.55 (Vanishing of the second boundary term in the limit). At each time t ∈ (−∞, 0) we have  ν(|∇f |2 )dσ = 0, (28.70) lim i→∞

∂Ωi

where ∂Ωi is as in Lemma 28.45. ˆ + 2s. Since u ˆ (s + si , θ, t) Proof. From g = e−f +2s gcyl we have f = − ln u converges as i → ∞ in each C k on compact sets to a constant a, on ∂Ωi we have  2  ∇g f  ≤ 2 |∇2g f |g = 2 |∇2g ln u ˆ|gcyl → 0 cyl cyl cyl g a a ˆ|gcyl → 0 as i → ∞, we have as i → ∞. Because max∂Ωi |∇gcyl ln u      ∂  −1/2  −1/2   ˆ+2  ≤ 6a−1/2 on ∂Ωi ∇gcyl f g = 2a |∇g f |g ≤ 2a −∇gcyl ln u cyl ∂s gcyl

for i large enough. Hence   lim ν(|∇f |2 )dσ = 2 lim i→∞

∂Ωi

i→∞



 ∇2 f, ∇f ⊗ ν dσ = 0.



∂Ωi

4. Ancient solutions with positive curvature In Chapter 19 of Part III we discussed Perelman’s nonround ancient κ-solution on S n for n ≥ 3. His solution has positive curvature operator and is Type II. In this section we discuss two results which classify positively curved ancient solutions as shrinking spherical space forms under certain hypotheses. 4.1. Type I κ-noncollapsed ancient solutions with positive curvature operator (PCO). To obtain more detailed information of Type I κ-noncollapsed ancient solutions, we make some extra assumptions. In this subsection we classify such solutions with positive curvature operator. We have the following result proven by one of the authors [285]. The rough idea is to use a diameter bound to apply the equality case of Perelman’s ν-invariant. Theorem 28.56 (Compact Type I κ-solutions with PCO). If (Mn , g (t)), t ∈ (−∞, 0), is a Type I κ-noncollapsed (on all scales) ancient solution to the Ricci flow with positive curvature operator and is defined on a compact manifold, then (M, g (t)) is isometric to a shrinking spherical space form.

64

28. SPECIAL ANCIENT SOLUTIONS

Proof. Step 1. Diameter bound. Let g˜ (τ ) = g (−τ ). By the Type I assumption, we have Rc g˜(τ ) ≤ (n − 1) Cτ1 on M × [1, ∞) for some C1 . Suppose that q1 , q2 ∈ M and τ¯ ∈ [1, ∞) are such that  τ¯ . dg˜(¯τ ) (q1 , q2 ) > 2 C1 Let (τ∗ , τ¯], where τ∗ ∈ [1, τ¯], be the maximal interval on which we have dg˜(τ ) (q1 , q2 ) !   > 2 Cτ1 for τ ∈ (τ∗ , τ¯]. Let γ : 0, dg˜(τ ) (q1 , q2 ) → M be a unit speed minimal geodesic joining q1 to q2 with respect to g˜ (τ ). By Proposition 18.8 in Part III, we have  d const (n, C1 ) √ dg˜(τ ) (q1 , q2 ) ≤ (28.71) Rcg˜(τ ) (γ  (s) , γ  (s)) ds ≤ dτ τ γ for τ ∈ (τ∗ , τ¯]. This implies



   τ¯ τ∗ const (n, C1 ) √ dτ dg˜(¯τ ) (q1 , q2 ) ≤ max dg˜(1) (q1 , q2 ) , 2 + C1 τ τ∗ √ ≤ diam (˜ g (1)) + const (n, C1 ) τ¯.

Hence (28.72)

   √ τ¯ , diam (˜ g (1)) + const (n, C1 ) τ¯ diam (˜ g (¯ τ )) ≤ max 2 C1

for τ¯ ∈ [1, ∞). Step 2. A backwards limit is a shrinker. By Theorem 30.31 in Chapter 30, there exists a backwards limit Mn−∞ , g−∞ (t) , t ∈ (−∞, 0), which is a gradient shrinker with nonnegative curvature operator. From (28.72) and scaling,  diam (g−∞ (t)) ≤ const (n, C1 ) |t|, so that M−∞ is compact and hence diffeomorphic to M. Step 3. The solution has constant sectional curvature. We have the following: (1) Since g (t) has positive curvature operator (PCO), M is a topological spherical space form (see B¨ohm and Wilking’s Theorem 11.2 in Part II). (2) Any shrinking Ricci soliton gˆ (t) on M with nonnegative curvature operator must have constant sectional curvature; in particular, g−∞ (t) has constant sectional curvature. More specifically, Theorem 11.2 in Part II says that this is true under the stronger assumption that gˆ (t) has PCO. On the other hand, if gˆ (t) does not have PCO, then the combination of the strong maximum principle for systems (see Theorem 12.53 in Part II), Berger’s holonomy classification theorem (see Theorem 7.35 in [77]), and (1) lead to a contradiction. (3) Let ν (g) be the ν-invariant as defined in (6.50) of Part I. By ν-invariant monotonicity (see Lemma 6.35 in Part I) and by Theorem 11.2 in Part II, we have that g) , sup ν (g) = ν (¯ g

where g¯ has constant sectional curvature and the supremum is taken over all metrics with PCO on M. Thus, by ν-monotonicity again and (2), we have (28.73)

g) . ν (¯ g ) ≥ ν (g (t)) ≥ ν (g−∞ (t)) ≡ ν (¯

4. ANCIENT SOLUTIONS WITH POSITIVE CURVATURE

65

By Lemma 6.35(2) in Part I (note that λ (g (t)) > 0 since R (g (t)) > 0), we conclude that g (t) is a gradient shrinker and, by (2), that g (t) must have constant sectional curvature.  Problem 28.57. Can one remove the κ-noncollapsed condition in Theorem 28.56? 4.2. Ancient 3-dimensional solutions with pinched Ricci curvatures. In the presence of positive curvature, instead of assuming the Type I condition as in the previous subsection, we may assume a curvature pinching condition. The following result is due to Brendle, Huisken, and Sinestrari [34] (they also prove a higher-dimensional analogue). This result is related to Hamilton’s “necklike points theorem” (see Theorem 9.19 in Volume One). Theorem 28.58 (Compact 3-dimensional ancient solutions with pinched Ricci  curvatures). If M3 , g (t) , t ∈ (−∞, 0], is an ancient solution of the Ricci flow on a closed 3-manifold with Rc ≥ cRg, where c > 0 is a constant and the scalar curvature R is positive, then g (t) has constant positive sectional curvature. Proof. Let λ ≥ μ ≥ ν denote the eigenvalues of Rm (equal to twice the ◦

principal sectional curvatures) and let Rm denote the trace-free part of Rm. Given ε > 0, define ◦

2

2

2

(μ − ν) + (λ − ν) + (λ − μ) |Rm|2 ≥ 0. (28.74) G  2−ε = R 3R2−ε By the formula on p. 273 of Volume One with γ = 0, we obtain 2 (1 − ε) ∂ G ≤ ΔG + ∇G, ∇R + 2J, (28.75) ∂t R where   ◦ 1 2 2 (28.76) J  3−ε ε|Rm| |Rm| − P R and (28.77)

P  λ2 (μ − ν)2 + μ2 (λ − ν)2 + ν 2 (λ − μ)2 ≥ 0.

On the other hand, the hypothesis Rc ≥ cRg yields μ+ν ≥ cR ≥ c |Rm| . μ≥ 2 Thus ◦

P ≥ μ2 (λ − ν)2 ≥ c2 |Rm|2 |Rm|2 ,

(28.78) ◦

using (λ − ν)2 ≥ |Rm|2 . Now take ε =

c2 2 .

Then, by (28.78), we have

◦

(28.79) ◦

where we used fore (28.80)

|Rm|2 R2−ε

1 |Rm|2 |Rm|2 ≤ −εG1+ ε , J ≤ −ε 2−ε R R   ε ε ◦ 2 |Rm| |Rm| ≤ |Rm| (which is equivalent to ≤ ). ThereR R R

1 2 (1 − ε) ∂ G ≤ ΔG + ∇G, ∇R − 2εG1+ ε . ∂t R

66

28. SPECIAL ANCIENT SOLUTIONS

By what amounts to the maximum principle, the function Gmax (t)  max G (x, t) x∈M

satisfies the odi (28.81)

1+ 1 d Gmax (t) ≤ −2εGmaxε (t) dt

for all t, in the sense of the lim sup difference quotients. We conclude  of forward that Gmax (t) ≡ 0 and hence that M3 , g(t) is isometric to a spherical space form S 3 /Γ. Indeed, to see this, we note that the solution to the comparison ode 1+ 1 dΓα (t) = −2εΓα ε (t) , dt lim Γα (t) = ∞

tα

is Γα (t) = (2 (t − α))−ε . By the ode comparison principle, we then have Gmax (t) ≤ −ε (2 (t − α)) for all t > α and α > −∞. Hence for all t ∈ (−∞, 0] we conclude that −ε

Gmax (t) ≤ lim (2 (t − α)) α→−∞

=0 

as desired. 5. Notes and commentary

§1. Theorem 28.1 is due to Bing-Long Chen (see Corollary 2.3(i) in [61] and also Proposition 5.5 in Huai-Dong Cao, B.-L. Chen, and Xi-Ping Zhu [45]); our presentation of the proof follows more closely Takumi Yokota [450]. Corollary 28.2 is Corollary 2.5 in [61]. Theorem 28.5 is Corollary 2.4 in [61]. See B.-L. Chen, Guoyi Xu, and Zhuhong Zhang [62] for a localization of the Hamilton–Ivey estimate. §2. For Theorem 28.11, see the original Zhenlei Zhang [452] or the expository Theorem 17.13 in Part III. Besides Natasa Sesum [365] and Zhou Zhang [454], results which support Optimistic Conjecture 28.15 have appeared in a number of recent works, including Bing Wang [429], Nam Le and Sesum [184], Joerg Enders, Reto M¨ uller, and Peter Topping [104] (the latter two references are discussed in the next chapter), and one of the authors (see Theorem 6.40 of [77]). For Theorem 28.21, see the original §12.1 of Perelman [312] or the expository Theorem 52.7 in Bruce Kleiner and John Lott [161]. Theorem 28.24 is Proposition 11.2 in Perelman [312]; see also the expository Theorem 19.53 in Part III. Theorem 28.23 is due to Simon Brendle [32]. For Theorem 28.30 see [76]. §3. The complete classification of 2-dimensional ancient solutions to the Ricci flow with bounded curvature is due to the combined works of Daskalopoulos and Sesum [94] (compact case), Daskalopoulos, Hamilton, and Sesum [93] (noncompact

5. NOTES AND COMMENTARY

67

finite-width case), and one of the authors [83] (noncompact infinite-width case). The last result is in part based on earlier work of W.-X. Shi [374] and Tam and one of the authors [289]. One may ask if the isoperimetric profile approach of Andrews and Bryan [8] can be applied to study ancient solutions on surfaces.

CHAPTER 29

Compact 2-Dimensional Ancient Solutions Picture paragraphs unloaded, wise words being quoted. – From “Hail Mary” by Tupac Shakur

Recall that, besides the round shrinking metric, there is another ancient solution on the 2-sphere S 2 , called the King–Rosenau solution or sausage model. We discussed this solution in Subsection 3.3 of Chapter 2 in Volume One. In this chapter we shall show that the King–Rosenau solution is the only nonround ancient solution on S 2 . Thus, by the results on noncompact ancient solutions stated in the previous chapter, we have a complete classification of 2-dimensional complete ancient solutions. One of the main ideas of the proof is the study of a scalar invariant, which we call Q(x, t). We show that Q is a subsolution to the heat equation and moreover that maxx Q(x, t) → 0 as t → −∞. This implies that Q ≡ 0. A classification result then yields that g(t) is the King–Rosenau solution. Geometric methods, such as the application of isoperimetric monotonicity, are also important to the proof. 1. Statement of the classification result and outline of its proof In this section we state the main theorem and we give an outline of the proof of the theorem. 1.1. Statement of the main theorem. The main theorem is the following result of P. Daskalopoulos, R. Hamilton, and N. Sesum. Their proof involves an eclectic combination of monotonicity formulas, a priori estimates, and geometric arguments, which we shall discuss. Theorem 29.1 (Classification of ancient solutions on the 2-sphere). Any compact simply-connected ancient solution (M2 , g (t)) to the Ricci flow must be either a round shrinking 2-sphere or the rotationally symmetric King–Rosenau solution. Note that any compact simply-connected surface is diffeomorphic to the 2sphere. 1.2. Outline of the proof of the main theorem. The proof of the main theorem involves a combination of fine and coarse estimates, where the words “fine” and “course” are used in the sense of Chapter 14 in Part II. Roughly, we have the following utilities. Coarse estimates are used to prove convergence to a limit, in our case, backward limits as time tends to minus infinity. Fine estimates, usually called monotonicity formulas, are used to classify these limits. Fine estimates are also used to obtain stronger control of geometric quantities for the solution g(t). 69

70

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

In §2 we review the various standard coordinates in which we express the Ricci flow equation on S 2 . Based on Cheeger–Gromov limits for times ti → −∞, we develop intuition to guide the proof. In §3 we describe the King–Rosenau solution in natural coordinate systems on the sphere, the plane, and the cylinder. We consider various backward limits of the King–Rosenau solution, which help our study of nonround ancient solutions on S 2 . In §4 we prove a priori estimates for the pressure function v (reciprocal of the conformal factor relative to the round metric) of an ancient solution. In §5, we give two proofs that the backward limit of the scalar curvature is equal to zero a.e. The first proof uses the evolution of the area form and the second proof uses an energy monotonicity formula (a fine estimate). In §6, by applying the a priori estimates in the previous sections, we show that the backward limit of the pressure function corresponds to a flat (possibly incomplete) metric in a weak sense. In §7, we discuss some basic properties of the isoperimetric constant of metrics on S 2 . In §8, using isoperimetric monotonicity, we prove that if one has infinite expansion of the conformal factor backward in time, then the solution must be round. In §9, by improving the result of a previous section, we show either that the backward limit of the pressure function is zero (i.e., the metric has infinite expansion backward in time) or that the backward limit corresponds to a complete flat cylinder metric. Some key tools are a concentration-compactness-type result, a monotonicity formula for circular averages, and the classification of harmonic functions on R2 with at most linear growth. In §10 we prove that, for nonround solutions, we also obtain cigar backward limits at the poles, where the locations of the poles are determined by the cylinder limits. In §11 we consider the key quantity Q, which depends on the third covariant derivative of the pressure function. In §12 we prove that Q is a subsolution of the heat equation. In §13 we show that if Q is identically zero, then the ancient solution is the King–Rosenau solution. ¯ on the plane analogous to Q. In §14 we consider the quantity Q In §15 we prove that Q is identically zero. In §16 we show that the quantity Q on S 2 is the pull-back by stereographic ¯ on R2 . projection of the quantity Q 2. The Ricci flow equation on S 2 and some intuition In this section we review the spherical, Euclidean, cylindrical, and polar coordinates on S 2 minus 0, 1, or 2 points that we shall use to analyze ancient solutions to the Ricci flow on S 2 . We introduce the backward limit of the pressure function and we also discuss some intution which will guide our analysis. 2.1. The pressure function and its equation in various standard coordinates. Let S 2 denote the unit 2-sphere centered at the origin in R3 , let N = (0, 0, 1) be its north pole, and let S = (0, 0, −1) be its south pole. Let S 1 (r) denote the

2. THE RICCI FLOW EQUATION ON S 2 AND SOME INTUITION

71

circle of length 2πr and let S 1 = S 1 (1). We recall the following standard coordinate systems. (i) Spherical coordinates. Let ψ ∈ [− π2 , π2 ] be the latitude and let θ ∈ [0, 2π) be the longitude on S 2 , so that ψ(N ) = π2 and ψ(S) = − π2 . On S 2 − {N, S} ∼ = (− π2 , π2 ) × S 1 the standard metric gS 2 , which has constant curvature 1, is given by gS 2 (ψ, θ) = dψ 2 + cos2 ψ dθ 2 with associated Laplacian ΔS 2 . (ii) Cylindrical coordinates. Let (s, θ) be the standard coordinates on R × S 1 . The standard cylinder metric is gcyl = ds2 +dθ 2 with associated Laplacian Δcyl . (iii) Euclidean and polar coordinates. Let (x, y) be Euclidean coordinates on R2 . Let (r, θ) be polar coordinates on R2 − {0}. The Euclidean metric is geuc = dx2 + dy 2 = dr 2 + r 2 dθ 2 with associated Laplacian Δeuc . The above metrics are related as follows. (i ) Define the plane to cylinder diffeomorphism τ : R2 − {0} → R × S 1 by τ (r, θ) = (ln r, θ)  (s(r), θ), so that τ

−1

(s, θ) = (e , θ)  (r(s), θ). We have s

dr 2 + r 2 dθ 2 = e2s (ds2 + dθ 2 ). In other words, (τ −1 )∗ geuc = e2s gcyl and τ ∗ gcyl = r −2 geuc . (ii ) Let σ : S 2 − {S} → R2 denote stereographic projection: (29.1)

σ(p) =

p − p, S S 1 − p, S

for p ∈ S 2 − {S},

which is a conformal diffeomorphism with respect to gS 2 and geuc . Using p, S = − sin ψ(p), we see that r(p)  |σ(p)| =

(29.2)

cos ψ(p) −1 = (sec ψ(p) + tan ψ(p)) . 1 + sin ψ(p)

With respect to spherical and polar coordinates, we have that −1

σ (ψ, θ) = ((sec ψ + tan ψ) Since

dψ dr

=

2 − 1+r 2

and cos ψ =

2r 1+r 2 ,

we have that

dψ 2 + cos2 ψ dθ 2 = That is, (σ −1 )∗ gS 2 =

, θ)  (r(ψ), θ).

 2 4 dr + r 2 dθ 2 . 2 2 (1 + r )

4 (1+r 2 )2 geuc .

(iii ) Mercator projection is the diffeomorphism m : S 2 − {N, S} → R × S 1 defined by m = τ ◦ σ|S 2 −{N,S} ; that is, m (ψ, θ) = (− ln (sec ψ + tan ψ) , θ)  (s(ψ), θ). From this we obtain the relations cosh s = sec ψ One computes that

dψ ds

sinh s = − tan ψ.

and

= − sech s, so that

 dψ 2 + cos2 ψ dθ 2 = sech2 s ds2 + dθ 2 .

Equivalently, (29.3)

(m−1 )∗ gS 2 = sech2 s gcyl

and

m∗ gcyl = sec2 ψ gS 2 .

72

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We shall find it useful to consider solutions (including the King–Rosenau solution) in these various coordinate systems. Let g(t), t ∈ (−∞, 0), be a maximal solution to the Ricci flow on S 2 , so that the singularity time is 0. By applying the weak and strong maximum principles to the equation for ∂R ∂t , any compact ancient solution must have scalar curvature R either zero everywhere or positive everywhere. Since we are on S 2 , we must have R > 0. By Hamilton [137], g (t) must shrink to a round point as t → 0. We may write (29.4)

g(t) =

1 1 1 gS 2 = gcyl = geuc , v (t) vˆ (t) v¯ (t)

where v (t) is defined on S 2 , vˆ (t) is defined on S 1 × R, and v¯ (t) is defined on R2 . 1 1 gcyl and g¯(t)  v¯(t) geuc we have suppressed our need in (29.4) to For gˆ(t)  vˆ(t) pull-back by the Mercator projection and stereographic projection, respectively. The conformal factors u, u ˆ, and u ¯ are defined by g(t) = u (t) gS 2 = u ˆ (t) gcyl = u ¯ (t) geuc . The Ricci flow is equivalent to (the limiting case of) the porous medium equation ∂u ¯ = Δeuc ln u ¯. ∂t In the study of the porous medium equation, the function v¯ = u1¯ is called the pressure function. In the analysis of ancient solutions on S 2 , it seems most natural to consider v and its covariant derivatives with respect to gS 2 as well. We shall also call v and vˆ pressure functions. From the above discussion it is not difficult to see that we have the following relations relating either the pressure functions or the conformal factors in various coordinates: (29.5)

vˆ(s, θ, t) = cosh2 s v(ψ(s), θ, t),

i.e., vˆ(t) = cosh2 s v(t) ◦ m−1 ,

and (29.6)

u ¯=

4u ◦ σ −1 , (1 + r 2 )2

equivalently, v¯ =

1 (1 + r 2 )2 v ◦ σ −1 . 4

The scalar curvature of g (t) and the evolution of v (t) may be expressed as (29.7)

0 < Rg =

v v 1 ∂v 1 ∂ˆ 1 ∂¯ = = v ∂t vˆ ∂t v¯ ∂t

= ΔS 2 v −

2 |∇ˆ v |2gcyl |∇¯ v |2geuc |∇v|S 2 + 2v = Δcyl vˆ − = Δeuc v¯ − . v vˆ v¯

A significant part of this chapter is devoted to understanding the consequences of this equation in the three different coordinate systems.  d Area(g (t)) = − S 2 R dμ, by the Gauss– A fact that we shall use is that since dt Bonnet formula and the extinction being at time 0 we have that the area of g (t) is 8π |t|.

3. THE KING–ROSENAU SOLUTION IN THE VARIOUS COORDINATES

73

2.2. Intuition based on earlier results. To gain intuition, we note the following earlier results of Hamilton on backward Cheeger–Gromov limits. Theorem 29.2 (2-dimensional Type I and II ancient solutions). (1) If K  supS 2 ×(−∞,−1] |t|R < ∞ (i.e., g (t) is Type I ), then g (t) must be a round shrinking 2-sphere. In this case, a backward pointed limit is the flat plane. (2) If K = ∞ (i.e., g (t) is Type II ), then there exists a backward Cheeger– Gromov limit which is the cigar soliton. By taking a second limit of points tending to infinity in the cigar, we obtain a flat cylinder. Since the limit of a limit is a limit, we have that the flat cylinder is a backward Cheeger– Gromov limit of g(t). Part (1) can be proved by using Hamilton’s entropy monotonicity and part (2) can be proved by using Hamilton’s trace Harnack estimate. For expositions, see Theorem 9.14 and Proposition 9.18, both in [77]. A priori, it may require rescaling to obtain the limit in part (2). However, Proposition 29.38 below obtains cigar soliton limits without needing to rescale. Note that the main theorem, Theorem 29.1, says that if g (t) is Type II, then g (t) must actually be the King–Rosenau solution. One should make the distinction between Cheeger–Gromov limits, which use the pull-backs by diffeomorphisms, and pointwise limits. We shall use both types of limits in the proof of the main theorem. Furthermore, we expect to have (and in fact do have) the following correspondences: (1) If a backward Cheeger–Gromov limit is the flat plane, then g(t) is a round shrinking 2-sphere. (2) If a backward Cheeger–Gromov limit is a flat cylinder or a cigar, then g(t) is the King–Rosenau solution. 3. The King–Rosenau solution in the various coordinates In this section we characterize the King–Rosenau solution as the only solution, up to a constant scale factor, having a specific rotationally symmetric form given below. This characterization is used in the proof of the main theorem. We also make some observations about the behavior of certain quantities on the King–Rosenau solution. 3.1. Characterization of the King–Rosenau solution. We shall use the superscript or subscript KR for quantities on the King– Rosenau solution. A general method for finding explicit solutions of pde, including those which arise in geometry, is to find a suitable ansatz. The ansatz for the King–Rosenau solution on S 2 is gKR (t) = vKR1 (t) gS 2 , where (29.8)

v KR (ψ, θ, t) = 2α (t) + β (t) cos2 ψ

and where α (t) and β (t) are positive functions of time. In terms of the induced coordinates on the unit 2-sphere as a subset of R3 , we have  (29.9) v KR (x, y, z, t) = 2α (t) + β (t) 1 − z 2 , which is evidently smooth on S 2 .

74

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Let gKR  (ˆ v KR )−1 gcyl  (¯ v KR )−1 geuc . The ansatz (29.8) is equivalent to (29.10)

vˆKR (s, θ, t) = 2α (t) cosh2 s + β (t)

in cylindrical coordinates on S 1 × R and is also equivalent to (29.11)

v¯KR (r, θ, t) =

α (t) α (t) 4 + (α (t) + β (t)) r 2 + r 2 2

in polar coordinates on R2 . Substituting (29.8) and the scalar curvature formula (29.12) into

∂v KR ∂t

(29.13)

RKR = 4α + 4β

2α sin2 ψ 2α + β cos2 ψ

= RKR v KR , we see that v KR is a solution to the Ricci flow if and only if α = 4α (α + β)

and

β  = −4αβ,

where  denotes a time derivative. Let α (t) a (t)  , b (t)  β (t) + α (t) , 2 so that (29.14)

v¯KR (r, θ, t) = a (t) + b (t) r 2 + a (t) r 4 .

Then (29.13) is equivalent to a = 4ab

(29.15)

and

b = 16a2 .

Since α and β are positive, we have that a > 0 and b > 2a. Hence a > 8a2 , which implies that there exists a time t at which both a(t) and b(t) tend to infinity; by translating time we may assume that this time is 0.  We compute that b2 − 4a2 = 0. Define μ > 0 by b(t)2 − 4a(t)2 ≡ μ2 . 1+Ce8μt Hence b = 4(b2 − μ2 ). The general solution to this ode is b(t) = μ 1−Ce 8μt . Since limt 0 b(t) = ∞, we have C = 1 and hence μ (29.16) b (t) = −μ coth(4μt), a (t) = − csch(4μt), where μ > 0; 2 that is, (29.17)

α (t) = −μ csch(4μt),

β (t) = −μ tanh(2μt).

In conclusion, we have Lemma 29.3 (Characterization of the King–Rosenau solution). Let (S 2 , gKR (t)), t ∈ (−∞, 0), be an ancient solution to the Ricci flow of the form (29.8), where α, β > 0 and limt 0 α(t) = ∞. Then gKR (t) is the King–Rosenau solution, where for some μ > 0 we have that −1 1 gS 2 gKR (t) = − 2 csch(4μt) + tanh(2μt) cos2 ψ μ −1  1 1 1 csch(4μt) + coth(4μt)r 2 + csch(4μt)r 4 =− geuc μ 2 2 1 −1 = − (csch(4μt) cosh(2s) + coth(4μt)) gcyl . μ

3. THE KING–ROSENAU SOLUTION IN THE VARIOUS COORDINATES

75

3.2. A quantity that is constant on the King–Rosenau solution. Now we discuss some properties of the King–Rosenau solution. Given the coor dinates x1 , x2 = (ψ, θ) on S 2 −{N, S}, we have that the only nonzero components of the Christoffel symbols Γkij of gS 2 are sin 2ψ . 2 Hence, for any radial function f = f (ψ) on S 2 we have  2  ∂ f ij k ∂ 2 ΔS f = g − Γij k = fψψ − tan ψ fψ . ∂xi ∂xj ∂x Γ212 = Γ221 = − tan ψ,

(29.18)

Γ122 =

Thus the pressure function v KR of the King–Rosenau solution satisfies the equation F  ΔS 2 v KR + 6v KR = 4β + 12α.

(29.19)

This is significant since quantities which are constant in space on the desired special solutions often are good quantities to which to apply the maximum principle for general solutions. 3.3. Backward limits of the King–Rosenau solution. We describe how the cigar soliton is a backward Cheeger–Gromov limit of the King–Rosenau solution. For simplicity, let μ = 1 and let N = (0, 0) in Euclidean coordinates be our basepoint. Then by Lemma 29.3 we have that −1  1 r4 csch(4t) + r 2 coth(4t) + csch(4t) . u ¯KR (r, θ, t) = − 2 2 Let ti → −∞ and let1 g˜iKR (t)  u ˜KR i (t)geuc , 2 KR ¯ (Ki x, Ki y, t + ti ) and Ki2  (¯ uKR )−1 (0, 0, ti ). where u ˜KR i (x, y, t)  Ki u 1 2 Then Ki = − 2 csch(4ti ) = a (ti ) and  −1 −1 a (t + ti ) + r 2 b (t + ti ) + r 4 a (ti ) a (t + ti ) , u ˜KR i (r, θ, t) = a (ti )

where b (t) = − coth(4t). Thus (29.20)

 4t 2 −1 ˜KR , u ˜KR ∞ (r, θ, t)  lim u i (r, θ, t) = e + r i→∞

or, equivalently, (29.21)

KR −1 v˜∞ (r, θ, t)  (˜ uKR (r, θ, t) = e4t + r 2 . ∞ )

KR This says that g˜∞ u ˜KR ∞ geuc = gcig . We have proven the following:

Lemma 29.4 (Cheeger–Gromov convergence to a cigar at the poles). Let (S 2 , gKR ) be the King–Rosenau solution. For any sequence ti → −∞, let Φi : S 2 → S 2 be the conformal diffeomorphisms of S 2 characterized by Φi (N ) = N , Φi (S) = S, and Φ∗i gKR (N, ti ) = gS 2 (N ) for each i. Then (S 2 , Φ∗i gKR (t + ti ), N ) converges in the C ∞ pointed Cheeger–Gromov sense to a cigar soliton solution. The same result holds true for N replaced by S. 1 We use a “tilde” instead of a “bar” in our notation for g and u to remind us that the i i sequence involves a conformal diffeomorphism change.

76

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Now we consider some other Cheeger–Gromov limits. Let ti → −∞ and define viKR )−1 gcyl . Note that, geometrically, v´iKR (s, t) = vˆKR (s + si , t + ti ) and g´iKR = (´ KR ∗ 1 g´i (t) = Ψi gˆKR (t + ti ), where Ψi : R × S → R × S 1 is the translation defined by Ψi (s, θ) = (s + si , θ). Then (with μ = 1) v´iKR (s, t) = − csch(4(t + ti )) cosh(2(s + si )) − coth(4(t + ti )). KR KR KR −1 Let v´∞ (s, t)  limi→∞ v´iKR (s, t) and g´∞  (´ v∞ ) gcyl . Assume for simplicity that si ≤ 0 for all i. Let

(29.22) Then (29.23)

σ∞  lim (2ti − si ). i→∞

⎧ ⎪ ⎨ 1 1 KR v´∞ (s, t) = 1 + e4(t+ 2 σ∞ )−2s ⎪ ⎩ ∞

if σ∞ = −∞, if σ∞ ∈ R, if σ∞ = ∞.

To summarize, we have the following. Lemma 29.5. Corresponding to the limiting value of 2ti − si , we have three KR (t) based at ((si , θi ), ti ): different limits g´∞ (1) If σ∞ = −∞, then the Cheeger–Gromov limit is a flat cylinder. (2) If σ∞ ∈ R, then the Cheeger–Gromov limit is the cigar soliton. (3) If σ∞ = ∞, then the limit vanishes. 4. A priori estimates for the pressure function Let g(t), t ∈ (−∞, 0), be a maximal ancient solution to the Ricci flow on S 2 . In this section we derive estimates for the pressure function v(t) defined in (29.4) and its first few derivatives. Since various geometric quantities may not converge even in C 0 as t → −∞, in order to obtain uniform estimates on the time interval (−∞, −1] we need to consider appropriate quantities. With the aid of these uniform estimates we shall understand the backward limits of the solution g(t) as t → −∞. The choices of quantities which we now consider are affected by the form of the nonlinear parabolic equation (29.7). In what follows, C denotes a finite constant which may depend on the solution v. When C also depends on other parameters, such as p ∈ [1, ∞) and α ∈ (0, 1), we write C(p) and C(α), respectively. By the trace Harnack estimate (28.46), ∂R ∂t ≥ 0. Hence the scalar curvature is uniformly bounded backward in time: (29.24)

sup S 2 ×(−∞,−1]

Since

∂v ∂t

(29.25)

R < ∞.

= Rv > 0, we obtain on S 2 × (−∞, −1] the uniform bound 0 < v ≤ C.

We shall derive a number of elliptic inequalities and identities for quantities involving at most three derivatives of v. The main estimates derive from " we shall " these are (29.28) for v (t)W 2,p (S 2 ) ≤ C(p) and (29.38) for "v∇2 v "C α (S 2 ) ≤ C(α), both given below. In the following, all of the Sobolev and H¨ older spaces are defined using gS 2 .

4. A PRIORI ESTIMATES FOR THE PRESSURE FUNCTION

77

We start with a second-order quantity in v. Let ϕ : S 2 → R be any C 3 function. Recall the ubiquitous Bochner formula  2 2 2 (29.26) Δg |∇ϕ|g = 2g (∇(Δg ϕ), ∇ϕ) + R |∇ϕ|g + 2 ∇2 ϕg , where we used Rc = 12 Rg for dimension 2. From this we obtain (29.27)

Δg (R + |∇ϕ|2g ) = A + 2g (∇(Δg ϕ − R), ∇ϕ) + (Δg ϕ)2 − R2 ,

where the first term on the rhs, defined by 2

A  Δg R + R2 −

|∇R|g R

2

+

|∇R + R∇ϕ|g R

 2   1 + 2 ∇2 ϕ − Δg ϕg  , 2 g

is nonnegative by the trace Harnack estimate (28.46). Note that (29.27) generalizes (28.45) in the previous chapter. The a priori estimates we prove below are uniform on S 2 × (−∞, −1]. Lemma 29.6. For any p ∈ [1, ∞) there exists C(p) < ∞ such that for each t ∈ (−∞, −1], v (t)W 2,p ≤ C(p).

(29.28)

For any α ∈ (0, 1) there exists C(α) < ∞ such that for t ∈ (−∞, −1], v(t)C 1,α ≤ C(α).

(29.29)

Proof. Taking ϕ = ln v in (29.27) yields 2

2

Δg (R + |∇ ln v|g ) ≥ 2g (∇(Δg ln v − R), ∇ ln v) + (Δg ln v) − R2 . Note that |∇ψ|2g = v |∇ψ|2S 2 for any function ψ. Since 2

R + |∇ ln v|g = ΔS 2 v + 2v by (29.7), we obtain that 2

vΔS 2 (ΔS 2 v + 2v) ≥ −4 |∇v|S 2 − 4vR + 4v 2 = −4vΔS 2 v − 4v 2 . Define the quantity (29.30)

F  ΔS 2 v + 6v,

which is constant on the King–Rosenau solution. On S 2 × (−∞, −1] we have (29.31)

ΔS 2 F ≥ −4v ≥ −C.

By (29.7), there exists a constant C < ∞such that ΔS 2 v > −2v ≥ −C on S 2 × (−∞, −1]. In particular, F > 0. Since  S 2 F dμS 2  ≤ C, by applying the mean value inequality to (29.31) we conclude that 0 < F ≤ C (see Theorem 4.1 in [144]). Hence the Laplacian of v is bounded: (29.32)

|ΔS 2 v| ≤ C.

By applying standard elliptic estimates to (29.25) and (29.32), we see that (29.28) holds for any p ∈ [1, ∞) and t ∈ (−∞, −1]. Finally, the estimate (29.29) follows from the Sobolev inequality.  Remark 29.7. Compare Δg ln v = R − 2v with Δf = R in (28.42). In the compact case we had g = e−f geuc , whereas in the present noncompact case we have g = e− ln v gS 2 . So ln v in the noncompact case plays the role of f in the compact case.

78

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Lemma 29.8. (29.33) " " " 2 " "|∇v|S 2 " 2,p ≤ C(p) W

" " " 2 " "|∇v|S 2 "

for p ≥ 1,

C 1,α

≤ C(α)

for α ∈ (0, 1) .

2

Proof. By the Bochner formula for ΔS 2 |∇v|S 2 (see (29.26)) and by applying |∇v|2S 2

− 2v to the term 2 ∇(ΔS 2 v), ∇vS 2 , we compute that  2 ΔS 2 |∇v|2S 2 = 2 ∇2 v S 2 + 2 ∇(ΔS 2 v), ∇vS 2 + 2 |∇v|2S 2 $  2 |∇v|4S 2 2# 2 ∇ |∇v|S 2 , ∇v 2 − 2 = 2 ∇2 v S 2 + v v2 S 2 − 2 |∇v|S 2 + 2 ∇R, ∇vS 2  B.

ΔS 2 v = R +

v

By (29.32), (29.25), and (29.7), we have 2

2

|∇ ln v|g =

(29.34)

|∇v|S 2 ≤C v |∇v|

on S 2 × (−∞, −1]. From this, (29.28), and |∇R, ∇vS 2 | ≤ |∇R|g v1/2S 2 ≤ C (see the remark below), we obtain BLp ≤ C(p). Thus we conclude (29.33) for t ∈ (−∞, −1].  Remark 29.9. Since 0 < R ≤ C on (−∞, −1], the imply   derivative estimates 2  that there exist constants Cm < ∞ such that ∇m g R g ≤ Cm on S × (−∞, −1]. However, this does not imply bounds for |∇m S 2 R|S 2 . Lemma 29.10. " " " 3/2 " "v " 2,α ≤ C(α),

"√ 2 " " v∇ v "

C

Cα

≤ C(α),

and

" 2" "v "

C 3,α

    |∇v|3  |∇v|2  Proof. We have that ∇ v1/2S 2  ≤ 12 v3/2S 2 + 2 ∇2 v S 2 Lp by (29.34) and (29.28) and we have that

|∇v|S 2 v 1/2

≤ C(α). is bounded in

2 2 3 |∇v|S 2 ΔS 2 (v 3/2 ) = v 1/2 R − 2v 3/2 + , 3 2 v 1/2

"  " so that "ΔS 2 v 3/2 "W 1,p ≤ C. Hence, on the time interval (−∞, −1], we have that " " " " " 3/2 " " " (29.35) "v " 3,p ≤ C(p), so that "v 3/2 " 2,α ≤ C(α). W

C

3 1/2 2 ∇ v 2v 1,p

3 −1/2 ∇v 4v

+ ⊗ ∇v = ∇2 (v 3/2 ) ∈ W 1,p as well as Since we have v −1/2 ∇v ⊗ ∇v ∈ W , it follows from (29.35) that on (−∞, −1], "√ 2 " "√ " " v∇ v " 1,p ≤ C(p), so " v∇2 v " α ≤ C(α). (29.36) W C Since ΔS 2 (v 2 ) = 2Rv − 4v 2 + 4 |∇v|2S 2  D, where DW 2,p ≤ C, we obtain " 2" " " "v " 4,p ≤ C(p), "v 2 " 3,α ≤ C(α). (29.37)  W C Lemma 29.11. " 2 " "v∇ v " (29.38)

W 1,p

≤ C(p),

" 2 " "v∇ v " α ≤ C(α), C

vΔS 2 vC α ≤ C(α).

5. THE ALMOST EVERYWHERE VANISHING OF R∞

79

Proof. We compute that  2 v 3 ∇ = ∇2 (v∇v) = v∇3 v + 2∇v ⊗ ∇2 v + ∇2 v ⊗ ∇v, 2 so that, by (29.34) and Lemma 29.10, we have on (−∞, −1], " " " 3 " " "√ 2 " " "  " " " v∇ v " 0 ≤ C. "v∇ v " 0 ≤ 1 "∇3 v 2 " 0 + 3 " ∇v √ (29.39) " C C C 2 v "C 0  2 Hence, from ∇ v∇ v = v∇3 v + ∇v ⊗ ∇2 v ∈ Lp for p ≥ 1, we have (29.38) on (−∞, −1].  The above estimates shall be combined in the following sections with various monotonicity formulas to strengthen our understanding of the ancient solution g(t) and especially its backward limits. 5. The almost everywhere vanishing of R∞ Let (S , g (t)), t ∈ (−∞, 0), be an ancient solution to the Ricci flow on a maximal time interval. By the trace Harnack estimate ∂R ∂t ≥ 0, the backward limit 2

R∞ (x)  lim R (x, t)

(29.40)

t→−∞

exists and is a bounded nonnegative upper semicontinuous measurable function. However, R∞ might not be continuous, which can be seen by the King–Rosenau example. Adopting the notation of §3 of this chapter, we have that KR (t)  max RKR (x, t) = 4 (α + β) (t) Rmax x

KR (attained at the north and south poles) and Rmin (t)  minx RKR (x, t) = 4α(t) (attained on the equator, which is invariant under the Z2 reflectional symmetry). Let KR (x)  lim RKR (x, t) . R∞ KR that R∞ 2

t→−∞ KR R∞ (S)

KR (N ) = = 4μ = limt→−∞ Rmax (t), whereas We also observe KR R∞ (x) = 0 for x ∈ S − {N, S}. In this section we give two proofs that R∞ = 0 a.e. on S 2 . The first proof uses the evolution of the area form, whereas the second proof relies on an energy monotonicity formula.

5.1. Proof that R∞ = 0 a.e. using area evolution. Define Ec  {x ∈ S 2 : R∞ (x) > c}, which is a measurable set for each c ≥ 0. Let mg denote the Riemannian measure with respect to g. For any i ∈ N and t ∈ (−∞, −1] we have  d 1 − mg(t) (E1/i ) = Rg(t) dAg(t) ≥ mg(t) (E1/i ), dt i E1/i where we used R (x, t) ≥ R∞ (x). This implies that mg(t) (E1/i ) ≥ e|t+1|/i mg(−1) (E1/i ) for t ∈ (−∞, −1]. On the other hand, mg(t) (E1/i ) ≤ Area(g(t)) = 8π|t|.

80

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Therefore mg(−1) (E1/i ) = 0 for each i ∈ N. Since E0 =



E1/i , we obtain

i∈N

mg(−1) (E0 ) = 0. We conclude that mgS 2 (E0 ) = 0; that is, (29.41)

R∞ = 0

a.e. on S 2 .

We remark that by the strong maximum principle and Klingenberg’s theorem 2 (see (29.69)  2below), we see that for any x0 ∈ S ∞− E0 and any ti → −∞, the solutions S , g (t + ti ) , x0 subconverge in the C Cheeger–Gromov sense to a complete flat solution, which must be R2 or a cylinder. This is consistent with the conclusions of Theorem 29.2. Later, we shall show that R∞ = 0 on S 2 except at a pair of points; see Proposition 29.36 below. 5.2. Proof using an energy monotonicity formula. To understand the backward limits of g (t) better (i.e., working towards their classification), it is useful to consider monotonicity formulas for quantities associated to g(t). In the Type I case, entropy monotonicity has previously been applied. In this subsection we consider a 1-parameter family of monotone functionals which generalize Polyakov’s energy functional for the Ricci flow on surfaces. The monotonicity of one of these functionals gives another proof that R∞ = 0 a.e. on S 2 . Define a family Jβ of Dirichlet-type energy functionals on the space of Riemannian metrics g˜ = v1˜ gS 2 on S 2 by

 2 |∇˜ v |S 2 (29.42) Jβ (˜ g)  + Fβ (˜ v ) dμS 2 , v˜β S2 4 where Fβ (˜ v ) = − 2−β v˜2−β for β = 2 and where F2 (˜ v ) = −4 ln v˜. We compute for the ancient solution g(t) that

   2 |∇v|S 2 ∇v d ∂v 1−β Jβ (g (t)) = dμS 2 (29.43) −β β+1 − 2 div − 4v β dt v v ∂t S2  ∂v   ∂v ∂t − (2 − β) |∇v|2S 2 = dμS 2 , −2 ∂t v β+1 S2

which is nonpositive whenever β ≤ 2. We note two important special cases. First, the functional  (29.44) J2 (g)  (|∇ ln v|2S 2 − 4 ln v)dμS 2 S2

is equivalent to Polyakov’s energy, i.e., the logarithm of the determinant of the Laplacian of g. Indeed, a standard formula (see (A.17) in [77] for example) yields 1 J2 (g) + ln Area(g) − ln(4π). 48π  In this case the rhs of (29.43) can be rewritten as −2 S 2 R2 dμg (see (B.24) in [77] for example). In the space Met of metrics in a conformal class on S 2 , the Ricci flow is the negative gradient flow of J2 with respect to the L2 -metric on Met. Second, for the present purposes, Daskalopoulos, Hamilton, and Sesum consider the functional     1 2 |∇v|S 2 − 4v dμS 2 = − (R + 2v)dμS 2 . (29.45) J1 (g (t)) = v S2 S2 log det Δg − log det ΔS 2 = −

6. FIRST PROPERTIES OF THE BACKWARD LIMIT v∞

81

Since J1 (g (t)) ∈ [−C, 0) for t ∈ (−∞, −1] and since

  2 |∇v| d 2 2 S J1 (g (t)) = R dμS 2 ≤ −2 R2 dμS 2 , −2R − dt v S2 S2  −1  we obtain −∞ S 2 R2 dμS 2 dt ≤ C. Since at each point x ∈ S 2 we have that R (x, t)  2 dμS 2 = 0, monotonically decreases to R∞ (x) as t → −∞, we conclude that S 2 R∞ 2 which yields R∞ = 0 a.e. on S . 6. First properties of the backward limit v∞ Let (S , g (t)), t ∈ (−∞, 0), be an ancient solution to the Ricci flow on a maximal time interval. By (29.25), the backward limit of the pressure function 2

(29.46)

v∞  lim v t→−∞

exists and is a bounded nonnegative upper semicontinuous function on S 2 . The understanding of v∞ is a crucial part of the proof of the main theorem. 1 , In the case of a round 2-sphere shrinking to a point at t = 0, we have v (t) = 2|t| t ∈ (−∞, 0), and we have v∞ ≡ 0. We shall show in Proposition 29.16 below that this property characterizes the round shrinking 2-sphere. On the other hand, for the King–Rosenau solution, the C ∞ backward limit of KR v (ψ, θ, t) is (29.47)

KR v∞ (ψ, θ)  lim v KR (ψ, θ) = μ cos2 ψ. t→−∞

This corresponds to a backward Cheeger–Gromov limit being a flat cylinder, where the diffeomorphisms are identity maps. By (29.11), the corresponding limit on R2 is (29.48)

KR (r, θ) = μr 2 . v¯∞

KR KR (s, θ)  limt→−∞ vˆ∞ (s, θ, t) = μ from (29.10). Equivalently, vˆ∞ In this section, using the estimates proved in the previous two sections, we show that v1∞ gS 2 is a flat metric in a weak sense. In §9 we shall classify v∞ . Unless otherwise indicated, all of the norms and inner products in this section are with respect to gS 2 . By the estimates in §4 and by the Arzela–Ascoli theorem, we have the following.

Lemma 29.12 (Convergence of v(t) to v∞ ). The limit v∞ is contained in C 1,α for all α ∈ (0, 1) and has the properties that as t → −∞, (1) v(t) converges to v∞ in C 1,α , 2 (2) v 2 (t) converges to v∞ in C 3,α ; hence v (t) converges uniformly to v∞ in 3,α on compact subsets of the open subset Ω = {x : v∞ (x) > 0}, C (3) |∇v(t)|2 converges to |∇v∞ |2 in C 1,α , (4) v (t) converges to v∞ weakly in W 2,2 . Since |∇(vRg )| ≤ Rg |∇v| + v |∇Rg | ≤ Rg |∇v| + v 1/2 |∇Rg |g ≤ C, we have that (vRg ) (t) converges in C α as t → −∞. Since v is bounded and R∞ = 0 a.e., we conclude that (vRg ) (t) converges to 0 in C α . By (29.38), vΔS 2 v converges in C α to a function w∞ ∈ C α as t → −∞.

82

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Recall that from (29.7) we have vRg = vΔS 2 v − |∇v|2 + 2v 2 .

(29.49) Hence we have that

2

2 w∞ − |∇v∞ | + 2v∞ = 0.

(29.50)

Definition 29.13. We say that a function v˜ ∈ W 1,2 is a weak solution of the equation v˜ΔS 2 v˜ − |∇˜ v |2 + 2˜ v2 = 0

(29.51)

if for any ϕ ∈ C ∞ (S 2 ) we have that    2 v 2 ϕ dμS 2 = 0. −˜ v ∇ϕ, ∇˜ v  − 2 |∇˜ v | ϕ + 2˜ S2

By (29.50), showing that v∞ is a weak solution of (29.51) is equivalent to showing that     2 −v∞ ∇ϕ, ∇v∞  − |∇v∞ | ϕ dμS 2 = w∞ ϕdμS 2 S2

∞

S2

for any ϕ ∈ C (S ). Now, since vΔS 2 v → w∞ in C , we have     −v(t) ∇ϕ, ∇v(t) − |∇v(t)|2 ϕ dμS 2 = (vΔS 2 v)(t)ϕdμS 2 S2 S2  → w∞ ϕdμS 2 . 2

α

S2

But since we have v(t) → v∞ in C , we also have    2 −v(t) ∇ϕ, ∇v(t) − |∇v(t)| ϕ dμS 2 S2    −v∞ ∇ϕ, ∇v∞  − |∇v∞ |2 ϕ dμS 2 → 1,α

S2

as t → −∞. To summarize, Lemma 29.14. The limit v∞ ∈ C 1,α is a weak solution to 2 v∞ ΔS 2 v∞ − |∇v∞ |2 + 2v∞ = 0.

By (29.7), this may be interpreted as saying that in a weak sense.

1 2 v∞ gS

has zero scalar curvature

In fact, we have that v∞ is a strong solution in Ω = {x : v∞ (x) > 0} of S 2 . 2 One way to see this is that v∞ ∈ C 3,α is a strong solution to   2 2  2 3 ∇(v∞ ) 2 (29.52) ΔS 2 v∞ − + 2v∞ =0 2 4 v∞ 2 in Ω. From a standard interior regularity theory, we obtain v∞ ∈ C ∞ in Ω. There∞ fore v∞ ∈ C in {x : v∞ (x) > 0}.

Problem 29.15. Can one directly show that (Ω, v1∞ gS 2 |Ω ) is a complete metric? Later we shall see that if Ω is nonempty, then (Ω, v1∞ gS 2 |Ω ) is a flat cylinder.

7. ISOPERIMETRIC CONSTANT OF METRICS ON S 2

83

7. Isoperimetric constant of metrics on S 2 An important result for the Ricci flow on S 2 is Hamilton’s isoperimetric monotonicity formula, which we discussed in §14 of Chapter 5 in Volume One. This monotonicity was used in [141] to give another proof of the result in [67] that for any initial metric on S 2 the solution to the Ricci flow eventually has positive curvature and by [137] converges to a round point. In this section we discuss the basic properties of isoperimetric monotonicity. Using this, we carry out the proof of the following proposition in the next section. Let (S 2 , g(t)) be an ancient solution defined on a maximal time interval (−∞, 0), where g(t)  v(t)−1 gS 2 . Recall that if g(t) is a round shrinking metric, then v∞  limt→−∞ v(t) ≡ 0. The converse is also true: Proposition 29.16 (Infinite expansion implies being the round 2-sphere). If v∞ ≡ 0, then g(t) is a round shrinking 2-sphere. Let g˜ be a C ∞ Riemannian metric on S 2 . Consider a smooth embedded closed 2 (γ) and curve γ (with a finite number of components) and two open domains S+ 2 2 2 2 S− (γ) with the properties that S = S+ (γ) ∪ S− (γ) ∪ γ is a disjoint union and 2 that ∂S± (γ) = γ. Let L (γ) denote the length of γ and let A + (γ) = A + (γ; g˜) and 2 2 (γ) and S− (γ), respectively. Define the A − (γ) = A − (γ; g˜) denote the areas of S+ isoperimetric ratio of γ with respect to g˜ by (29.53)

−1 I(γ; g˜)  L2 (γ) (A −1 + (γ) + A − (γ))

= L2 (γ)

A(˜ g) , A + (γ) A − (γ)

where A(˜ g ) = A+ (γ) + A− (γ). Define the isoperimetric constant of g˜ by (29.54)

I(˜ g )  inf I(γ; g˜), γ

where the infimum is taken over all smooth embedded closed curves γ. Example 29.17 (Isoperimetric constant of the unit 2-sphere). For the unit sphere (S 2 , gS 2 ), let (ψ, θ) ∈ [− π2 , π2 ] × [0, 2π) be the latitude and longitude. Given ψ0 ∈ (− π2 , π2 ), let γψ0 be the circle of points with ψ = ψ0 , dividing S 2 into the disks Sψ2 0 ,+ where ψ > ψ0 and Sψ2 0 ,− where ψ < ψ0 . Then L(γψ0 ) = 2π cos ψ0 , A(Sψ2 0 ,+ ) = 2π (1 − sin ψ0 ), and A(Sψ2 0 ,− ) = 2π (1 + sin ψ0 ). Therefore I (γψ0 ; gS 2 ) = 4π. It is a classical result that I (γ; gS 2 ) ≥ 4π with equality if and only if γ is a round circle (see (4.2) in Osserman [299]). In particular, I (gS 2 ) = 4π. We have the following basic results. Lemma 29.18 (Isoperimetric constant of metrics on S 2 ). Let g˜ be any C ∞ Riemannian metric on S 2 . (1) We have I(˜ g ) ≤ 4π. (2) The functional g˜ → I(˜ g ) is continuous with respect to the C 0 -topology on the space of smooth metrics on S 2 . (3) Any minimizer γ of γ → I(γ; g˜) in the space of smooth embedded closed curves must be a loop, i.e., connected. (4) If I(˜ g ) < 4π, then there exists a minimizer γ of the functional γ → I(γ; g˜) in the space of smooth embedded closed curves. Any minimizer must have constant geodesic curvature.

84

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Proof. (1) This follows easily from considering small distance circles; see Lemma 5.84 in Volume One for example. (2) This follows directly from how the length of any curve and the area of any domain depend on the metric. We leave the details to the reader. (3) Let Lk denote the space of smooth embedded closed curves with k components. Define Ik  inf γ∈Lk I(γ; g˜). It suffices to show that if Ij ≥ I1 for some j ≥ 1, then for any γ ∈ Lj+1 we have I(γ; g˜) > I1 (see Yau [445] for a similar proof). % 2 2 Let γ ∈ Lk+1 , dividing S 2 into domains S+ (γ) and S− (γ). Then γ = k+1 a=1 γa , where the γa are disjoint smooth embedded loops. Since k ≥ 1, without loss of 2 2 (γ) is disconnected and S+ (γ) = A ∪ B, where generality we may assume that S+ %k B is a disk with ∂B = γk+1 . Note that ∂A = a=1 γa . Since Ik ≥ I1 , we have   2 %k 2 L γ a a=1 L (γ) L2 (γk+1 ) ≥ + I1 Ik I1 1 ≥  2 (γ)) −1 Area(A)−1 + Area(B) + Area(S− 1 +  2 (γ)) −1 −1 Area(B) + Area(A) + Area(S− 1 > , 2 (γ))−1 (Area(A) + Area(B))−1 + Area(S− 2 2 (γ))−1 +Area(S− (γ))−1 ) > I1 , as desired. which implies that I (γ) = L2 (γ) (Area(S+ The last inequality follows from the fact that, for any x, y, z > 0,

(29.55)

1 x−1

+ (y + z)

−1

+

1 y −1

+ (x + z)

−1

−

1 z −1

+ (x + y)

−1

> 0.

(4) When L(γ) is small, each of the components of γ lies in a small almost Euclidean ball. On the other hand, for the Euclidean metric, I(geuc ) = 4π. Hence, g )) > 0 that there exists δ = δ(ε) > 0 such by part (2), we have for ε  12 (4π − I(˜ that if L(γ) ≤ δ, then I(γ; g˜) > 4π − ε. Let {γi } be a minimizing sequence of I( · ; g˜) in the space of smooth embedded closed curves. By applying the proof of part (3), we may assume that each γi is connected. For i large enough, we have that I(γi ; g˜) < 4π − ε. Thus, by (29.53), we have 1 2 δ2 L (γi ) > . 4π 4π In order to take a limit, we shall obtain a new minimizing sequence with better regularity by evolving the γi by the curve shortening flow. Recall that if βt is a C 1 1-parameter family of curves on a Riemannian surface ∂ βt = f N , where N is a continuous choice of unit normal to βt and where f with ∂t is any C 1 function along βt , then   ∂  (29.57a) L(β ) = f κ ds, t ∂t t=0 β0   ∂  (29.57b) A ± (βt ) = ± f ds. ∂t t=0 β0

(29.56)

A(˜ g ) ≥ A ± (γi ) >

7. ISOPERIMETRIC CONSTANT OF METRICS ON S 2

85

The curve shortening flow is the case where f N = −κN is the curvature vector of βt . Let γi (t) be the solution to the curve shortening flow with γi (0) = γi . Let Li (t) = L(γi (t)) and A±,i (t) = A± (γi (t)). By (29.57), we have  d Li (29.58a) κ2 ds ≤ 0, =− dt γi (t)   d A±,i 1 =∓ (29.58b) κds = Rg˜ dμg˜ − 2π ∈ [−C, C], dt 2 S±2 (γi (t)) γi (t) 2 (γi (t)) is an where the last equality is by the Gauss–Bonnet formula and since S± embedded disk. Here, we have chosen the unit normal N to yield the given signs dA for dt±,i . δ2 By differentiating (29.53), we have that as long as A±,i (t) ≥ 4π > 0 holds,

1 d A+,i 1 d A−,i d 2 d Li ln I(γi (t); g˜) = − − dt Li dt A+,i dt A−,i dt  2 ≤− κ2 ds + C C γi (t) since Li (t) ≤ Li (0) ≤ C. Let C0 = C 2 /2. For each i, suppose that the solution γi (t) to the curve shortening flow exists on a maximal time interval [0, Ti ). Let Ti δ2 be the maximum time for which A±,i (t) ≥ 4π holds for t ∈ [0, Ti ). Claim. There exist ti ∈ [0, Ti ) such that γi (ti ) is a minimizing sequence with L(γi (ti )) > δ and  (29.59) κ2 ds ≤ C0 . 

γi (ti )

Suppose that γi (t) κ ds > C0 for all t ∈ [0, Ti ). We then have the following inequalities for all t ∈ [0, Ti ). The isoperimetric constant is nonincreasing: d ˜) ≤ 0, so that I(γi (t); g˜) < 4π − ε. Thus Li (t) > δ, which implies that dt I(γi (t); g δ2 . Hence Ti = Ti . Now, by Grayson’s theorem for the curve shortening A±,i (t) > 4π flow of embedded curves on surfaces (see [123] and [112]), since γi (t) cannot shrink to a round point, we conclude that  γi (t) converges to a closed embedded geodesic as t → ∞. In particular, limt→∞ γi (t) κ2 ds = 0, which is a contradiction. Therefore  there exists first time ti ∈ [0, Ti ) such that γi (ti ) κ2 ds ≤ C0 . Then L(γi (ti )) > δ and I(γi (ti ); g˜) ≤ I(γi ; g˜), so that {γi (ti )} is a minimizing sequence. This proves the claim. Let γi  γi (ti ), let {xα } be local coordinates on S 2 , and let γiα = xα ◦γi . Let κi and Ni denote the geodesic curvature and unit normal of γi , respectively. Assume that the metric components and Christoffel symbols satisfy C −1 δαβ ≤ g˜αβ ≤ Cδαβ  α 2 dγi ˜ α | ≤ C, so that 2 and |Γ ≤ C. We have βδ

2

α=1

 −κi Niα

=

ds

∇ dγi ds

dγi ds

α =

d2 γiα ˜ α dγiβ dγiδ , + Γβδ ds2 ds ds

so that κ2i = g˜αβ (κi Niα )(κi Niβ ) ≥ c

2  2 α 2  d γ i

α=1

ds2

− C.

86

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We thus obtain from (29.59) and Li (ti ) ≤ C that   2  2 α 2 d γi ds ≤ C. ds2 γi α=1 From this we can conclude that the sequence {γi } is bounded in W 2,2 and hence bounded in C 1,α . By the Arzela–Ascoli theorem and by passing to a subsequence, we have that the γi converge to a curve γ∞ in C 1,α , where α ∈ (0, 1). Since the γi are embedded, γ∞ has at most self-tangencies and no transverse intersections. Then Lemma A.1.2 in [140] implies that γ∞ is an embedded C 1,α minimizer of I( · ; g˜). The fact that γ∞ is C ∞ follows from its having constant geodesic curvature (see the following claim) and from regularity theory for the system of ode satisfied by γ∞ . Claim. The geodesic curvature of any C 1 minimizer γ0 of I( · ; g˜) is constant and equal to Lg˜ (γ0 )  −1 A + (γ0 ) − A −1 (29.60) κ(γ0 ) ≡ − (γ0 ) . 2 Proof of the claim. Extend γ0 to a C 1 1-parameter family of curves γσ ∂  in S 2 with ∂σ γ = f N , where N is continuous and f is C 1 . Since γ0 is a σ=0 σ minimizer, using (29.57), we compute that  ∂  ln I(γσ ; g˜) 0= ∂σ σ=0   −2 2 γ0 f κ ds − A −2 + (γ0 ) + A − (γ0 ) = + f ds −1 L (γ0 ) A −1 γ0 + (γ0 ) + A − (γ0 )     2κ −1 = + − A −1 (γ ) + A (γ ) f ds. 0 0 + − L (γ0 ) γ0 The claim follows.



Lemma 29.19. Let g˜ be any C ∞ Riemannian metric on S 2 . (1) If I(˜ g ) = 4π, then g˜ is a round metric. g (t0 )) = 4π (2) If g˜ (t), t ∈ (α, ω), is a solution of the Ricci flow on S 2 with I(˜ for some t0 ∈ (α, ω), then g˜ (t) is a round shrinking metric. Proof. (1) Since the equality I(˜ g ) = 4π is scale invariant, we may assume that Area(˜ g ) = 4π. Claim. R ≤ 2 on S 2 .  Then, from Area(˜ g ) = 4π and the Gauss–Bonnet formula, i.e., S 2 Rdμ = 8π, we conclude that R ≡ 2 on S 2 . Proof of the claim. Let p ∈ S 2 . For ε ∈ (0, injg˜ (p)), let γ be the distance circle of radius ε centered at p. In normal coordinates centered at p, we have for ¯p (ε) that x∈B   1 g˜ij (x) = δij − R(p) r 2 δij − xi xj + O r 3 , 6 !  1 det(˜ gij (x)) = 1 − R(p)r 2 + O r 3 , 12

8. CHARACTERIZING ROUND SOLUTIONS

87

where r = d(x, p). Using this, we compute that π L(γ) = 2πε − R(p)ε3 + O(ε4 ), 6 π Area(Bp (ε)) = πε2 − R(p)ε4 + O(ε5 ). 24 From the above we have −1

I(γ; g˜) = L2 (γ) (Area(Bp (ε))−1 + (4π − Area(Bp (ε)))     1 1 2 3 = 4π 1 + − R(p) ε + O(ε ) . 4 8

)

From I(γ; g˜) ≥ 4π we conclude that R(p) ≤ 2 for each p ∈ S 2 . (2) By part (1), we have that g˜ (t), t ∈ [t0 , ω), is a round shrinking metric. It follows from Kotschwar’s backwards uniqueness theorem that g˜ (t) is a round shrinking metric for all t ∈ (α, ω) (see [169]).  8. Characterizing round solutions  2 Let S , g (t) , t ∈ (−∞, 0), be a nonround ancient solution of the Ricci flow on a maximal time interval. In this section, relying on the isoperimetric monotonicity developed in the last section, we give a proof of Proposition 29.16. We adopt the notation of §7 of this chapter. 8.1. Estimating minimizing loops. Recall that we have the isoperimetric constant I (t)  I (g(t))

for t ∈ (−∞, 0) .

Let t0 ∈ (−∞, 0) be any given time and let γ0 be any smooth embedded loop. For ρ small enough, let γρ be the nearby equidistant curve of signed distance ρ from γ0 2 2 (with respect to g(t0 )), where γρ ⊂ S+ (γ0 ) for ρ > 0 and γρ ⊂ S− (γ0 ) for ρ < 0. For t near t0 , let (29.61)

I (ρ, t)  I (γρ ; g(t))

and let (29.62)

2 ), A ± (ρ, t) = Areag(t) (S±,ρ

2 2 2 where γρ separates S 2 into S+,ρ and S−,ρ and where S±,ρ continuously starts from 2 S± (γ0 ) at ρ = 0. Recall that Lemma 5.92 in Volume One says the following.

Lemma 29.20 (Heat-type equation for isoperimetric ratios of parallel closed curves). Let t0 ∈ (−∞, 0) and let γρ be a family of equidistant curves defined as above. Then the isoperimetric ratios I (ρ, t) of γρ satisfy 

 κ ds ∂  ∂2 ∂ γρ ρ − (29.63) − ln I (ρ, t)  ∂t ∂ρ2 Lg(t) (γρ ) ∂ρ  t=t0   4π − I (ρ, t0 ) A + (ρ, t0 ) A − (ρ, t0 ) + = 8π |t0 | A − (ρ, t0 ) A + (ρ, t0 ) 4π − I (ρ, t0 ) , ≥ 4π |t0 | where κρ is the geodesic curvature of γρ .

88

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Since the function t → I (t) may not be differentiable, given a time t¯, we define t¯ )  lim inf t t¯ I(t)−I( to be the lower converse Dini derivative. By t−t¯ applying the first and second derivative test to Lemma 29.20, we have that the isoperimetric constant worsens at least at a certain rate going backward in time. d− I ¯ dt (t )

Proposition 29.21 (Decay of the isoperimetric constant). (1) The isoperimetric constant I (t) of a nonround solution g (t) satisfies (29.64)

4π − I (t) d− I (t) ≥ I(t) dt 4π |t|

for t ∈ (−∞, 0) .

In particular, the function t → I (t) is strictly increasing. (2) For any t1 ∈ (−∞, −1] and all t ∈ (−∞, t1 ] we have that (29.65)

I (t) ≤

4π , 1 + c |t|

where c 

4π − I (t1 ) . I (t1 ) |t1 |

Proof. (1) By Lemma 29.18(4) and since g (t) is not a round shrinking 2sphere, we have that I (t) < 4π for all t ∈ (−∞, 0). Then by Lemma 29.18(2), given any time t0 ∈ (−∞, 0), there exists a smooth embedded  closed loop γ0 such ∂  ∂2  that I (γ0 ; g(t0 )) = I(t0 ). Since ∂ρ ln I(ρ, t0 ) = 0 and ∂ρ ln I(ρ, t0 ) ≥ 0, by  2 ρ=0

Lemma 29.20 we have

ρ=0

 1 ∂  4π − I(0, t0 ) . I (0, t) ≥ I(0, t0 ) ∂t t=t0 4π |t0 |

Since I (t) ≤ I (0, t) for t ≤ t0 and I(t0 ) = I (0, t0 ), we obtain (29.64). (2) Let J (t)  I (t) −

4π 1+c|t| ,

so that J (t1 ) = 0. From part (1) we have that 2

d− J J (t) 1 c |t| − 1 (t) ≥ − + J (t) . dt 4π |t| |t| c |t| + 1 Inequality (29.65) now follows from Corollary 10.26 in Part II.



Let γt be any C ∞ embedded loop minimizing I ( · ; g(t)); we estimate its length. Proposition 29.22 (γt is uniformly bounded from above and below). There exists a constant C ∈ [1, ∞) such that for t ∈ (−∞, −1], (29.66)

C −1 ≤ Lg(t) (γt ) ≤ C.

Proof. 1. Upper bound. By Proposition 29.21(2), for t ∈ (−∞, −1] we have that (29.67)

 4π 1 −1 L2g(t) (γt ) ≤ L2g(t) (γt ) A −1 , + (γt ) + A − (γt ) ≤ 4π |t| 1 + c |t|

where the areas A± (γt ) are measured with respect to g(t). 2. Lower bound. The proof is by contradiction. Suppose that there exists a sequence of times {ti } with ti → −∞ such that Li  Lg(ti ) (γti ) → 0, where

8. CHARACTERIZING ROUND SOLUTIONS

89

each γti is a minimizer for I ( · ; g(ti )). Consider the pointed sequence of solutions (S 2 , gi (t) , pi ), t ∈ (−∞, − L−2 i ti ), where  2 gi (t)  L−2 i g ti + L i t and pi ∈ γti . Since 0 < Rg (x, t) ≤ C on S 2 × (−∞, −1], we have (29.68)

Rgi (x, t) ≤ C L2i

on S 2 × (−∞, − L−2 i (ti + 1)]

(note that − L−2 i (ti + 1) → +∞). Furthermore, by Klingenberg’s theorem, we have (29.69)

π inj(S 2 , g(t)) ≥ √ C

for t ∈ (−∞, −1].

Hence π inj(S 2 , gi (0)) ≥ √ L−1 → ∞. i C

(29.70)

By Hamilton’s compactness theorem for the Ricci flow, there exists a subsequence {gi (t)} which converges in the C ∞ Cheeger–Gromov sense to a solution (M2∞ , g∞ (t) , p∞ ), t ∈ (−∞, ∞). By (29.68) and (29.70), we have that g∞ (t) is flat and inj (g∞ (0)) = ∞. This implies that (M∞ , g∞ (t)) is the static Euclidean plane. Now the isoperimetric constant at time ti is  −1 I(ti ) = L2g(ti ) (γti ) A −1 (29.71) + (γti ; g(ti )) + A − (γti ; g(ti )) −1 2 2 = Area−1 gi (0) (S+ (ti )) + Areagi (0) (S− (ti )), 2 2 (ti ) and S− (ti ). where γti divides S 2 into S+ On the other hand, since Lgi (0) (γti ) = 1, since γti is a C ∞ embedded loop with constant geodesic curvature with respect to gi (0), and since pi ∈ γti , we have that under the Cheeger–Gromov convergence of gi (0) to g∞ (0) the sequence of loops γti subconverges to some smooth embedded loop γ∞ , where Lg∞ (0) (γ∞ ) = 1.2 Note that by (29.60) we have that κgi (0) (γti ) is uniformly bounded. The loop γ∞ divides M∞ into two regions, one of which is bounded, which we call M∞,+ , and one of which is unbounded, which we call M∞,− . Without loss of generality, we may assume that 2 2 (ti )) ≤ Areagi (0) (S− (ti )). Areagi (0) (S+

Then, by this and the convergence of γti to γ∞ , we have that 2 lim Areagi (0) (S+ (ti )) = Areag∞ (0) (M∞,+ ) > 0

i→∞

and 2 (ti )) = ∞. lim Areagi (0) (S−

i→∞

Therefore, by applying this to (29.71), we conclude that I(ti ) ≥ c for some constant c > 0. (Actually, limi→∞ I(ti ) = 4π.) This contradiction to (29.65) completes the proof of the lower bound.  2 As in the proof of Lemma 29.18(4), one first observes that γ ∞ is an immersed closed curve with at most self-tangencies and then shows that γ∞ is embedded.

90

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

A consequence of the above proposition is the following. Lemma 29.23 (Areas of the two isoperimetric regions are comparable). There exists a constant c > 0 such that for any minimizer γt of I ( · ; g(t)) the areas of the isoperimetric regions are bounded from below by (29.72)

c|t| ≤ A ± (γt ) ≤ 8π|t|

for t ∈ (−∞, −1].

Proof. By Proposition 29.22, we have that −1 −1 A −1 ± (γt ) ≤ A + (γt ) + A − (γt ) ≤

4π L−2 g(t) (γt ) 1 + c |t|

≤

4πC 2 1 + c |t|

for t ∈ (−∞, −1]. The second inequality in (29.72) follows from A ± (γt ) ≤ Area (g (t)).  8.2. Backward cylinder limit. In this subsection we prove the following. Lemma 29.24 (Existence of an asymptotic cylinder). For any ti → −∞, any minimizer γti of I ( · ; g(ti )), and any pi ∈ γti , the sequence of solutions (S 2 , g(t + ti ), pi ) subconverges in the Cheeger–Gromov sense to a static flat cylinder (R × S 1 (r∞ ), g∞ (t), p∞ ), t ∈ (−∞, ∞), where r∞ is the radius of the circle. Moreover, under this convergence, γti converges to an embedded geodesic loop γ∞ , which we may assume to be {0} × S 1 (r∞ ). Hence, for any ε > 0 there exists Tε < 0 such that for each t ≤ Tε there exists an ε-neck in (S 2 , g(t)) centered at any pt ∈ γt in the sense of Definition 18.26 of Part III. Let ti → −∞, let the loop γti be a minimizer of I( · ; g(ti )), and choose a point pi ∈ γti . Recall from (29.69) that inj(S 2 , g(t)) ≥ √πC for t ∈ (−∞, −1]. From this and the boundedness and positivity of the curvature, we may apply Hamilton’s compactness theorem to conclude that the sequence of solutions (S 2 , g(t+ti ), pi ) subcon2 , g∞ (t), p∞ ) verges in the Cheeger–Gromov sense to a complete eternal solution (N∞ with bounded nonnegative curvature. That is, after passing to a subsequence, there exists an exhaustion Ui of N∞ by open sets and diffeomorphisms (29.73)

ϕ i : Ui → S 2

such that ϕ∗i g(t + ti ) converges to g∞ (t) in each C k on compact subsets of N∞ × (−∞, ∞). Now, by (29.60), (29.66), and (29.72), we have that the geodesic curvatures satisfy (29.74)

lim κ(γti ) = 0.

i→∞

From this and from (29.66) again, we have that γti converges to a geodesic loop γ∞ ; ∞ that is, ϕ−1 i (γti ) converges to γ∞ in C . Since γ∞ is the limit of embedded loops, γ∞ has at most self-tangencies and no transverse intersections. Then Lemma A.1.2 in [140] implies that γ∞ is an embedded geodesic loop. 2 2 (t) and S− (t) By (29.72) we have that γt divides S 2 into differentiable disks S+ + with areas satisfying A± (γt ) ≥ c|t|, where c > 0. Thus there exist points qi ∈ 2 2 S+ (ti ) and qi− ∈ S− (ti ) such that limi→∞ dg(ti ) (qi+ , pi ) = ∞ and limi→∞ dg(ti ) (qi− , pi ) = ∞. Let i = dg(ti ) (qi+ , qi− ) and let βi : [0, i ] → S 2 be a unit speed minimal geodesic from qi+ to qi− . Since βi ∩ γti = ∅, we may choose a point oi ∈ βi ∩ γti . Then there exists si ∈ (0, i ) such that oi = βi (si ). By passing to a subsequence, we may assume under the Cheeger–Gromov convergence of (S 2 , g(ti ), pi )

8. CHARACTERIZING ROUND SOLUTIONS

91

to (N∞ , g∞ (0), p∞ ) that oi converges to a point o∞ ∈ γ∞ ⊂ N∞ and that βi (si ) converges to a tangent vector V∞ ∈ To∞ N∞ which is unit with respect to g∞ (0). This implies that αi (s)  β(si + s) converges to a path α∞ : (−∞, ∞) → N∞ , with  (0) = V∞ , which is a geodesic line with respect to g∞ (0). α∞ Since g∞ (0) has nonnegative curvature, by the Aleksandrov splitting theorem we obtain that (N∞ , g∞ (0)) splits off a line. Since we are in dimension 2 and since N∞ contains the embedded geodesic loop γ∞ with respect to g∞ (0), we conclude that (N∞ , g∞ (0)) is a flat cylinder R × S 1 (r∞ ) with a product metric. Hence, by translating the s coordinate, we may assume that γ∞ = {0} × S 1 (r∞ ). This completes the proof of Lemma 29.24. 8.3. Bounding the areas of Bx± (ρ± t ) by time. g(t) t

In this subsection we further explore how any minimizer γt of I ( · ; g(t)) divides S 2 . For this purpose, we first prove results about ε-necks. To garner some intuition, we first consider the flat cylinder (R × S 1 , gcyl ). In this case, for any points (s1 , θ1 ) and (s2 , θ2 ), we have ! 2 |s2 − s1 | ≤ dgcyl ((s1 , θ1 ) , (s2 , θ2 )) ≤ (s2 − s1 ) + π 2 . Hence (29.75)

sup dgcyl ((s1 , θ1 ) , (s2 , θ)) − inf1 dgcyl ((s1 , θ1 ) , (s2 , θ))

θ∈S 1

θ∈S

!

= ≤

2

(s2 − s1 ) + π 2 − |s2 − s1 | π2 . 2 |s2 − s1 |

From this we see that if σ  |s2 − s1 | ≥ N , then   π2 g g (σ) for each θ1 , θ1 ∈ S 1 . (29.76) {s2 } × S 1 ⊂ B scyl,θ σ + − B(scyl 1 ,θ1 ) ( 1 1) 2N Similarly, for s2 = s1 + σ and where σ ≥ N , if we define γ to be the component of g (σ) close to {s2 } × S 1 , then ∂B(scyl 1 ,θ1 ) (29.77)

  π2 , s2 × S 1 γ ⊂ s2 − N

provided N ≥ π.

Recall that a normalized ε-neck Nε in a Riemannian2-sphere (S 2 , g), cen tered at a point p ∈ S 2 , is given by an embedding ϕ : Bp ε−1 → R × S 1 with ϕ(p) ∈ {0} × S 1 and such that |ϕ∗ (ds2 + dθ 2 ) − g|C ε−1 +1 (g) < ε.  Let (s, θ) = (s(ϕ(x)), θ(ϕ(x))) be the coordinates on Bp ε−1 induced by the ε-neck structure. We shall also denote γs  ϕ−1 ({s} × S 1 ) for s ∈ (−ε−1 + 4, 2 2 ε−1 − 4). Each embedded loop γs divides S 2 into two open disks Ss,+ and Ss,− ,  −1 −1 1 2 (s, ±(ε − 4)) × S ⊂ Ss,± . where ϕ

92

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We first prove the following result. Lemma 29.25 (Connectedness of distance circles). There exists ε > 0 with the following property. Let g be a Riemannian metric on S 2 with scalar curvature R > 0, let p ∈ S 2 , and let Nε be a normalized ε-neck centered at p. Then, for 2 2 x ∈ S0,+ − Bp ε−1 with d (x, p) ≥ 23 maxz∈S0,+ d (z, p) and for y ∈ Bp (100), we have that ∂Bx (ry ) is connected, where ry  d (y, x).

Bp(ε−1) (S 2, g) y

x

p

γ0 S 20,+ − Bp(ε−1) ∂Bx(ry) is connected

Bp(100)

Figure 29.1.

Proof. Assume that ε > 0 is sufficiently small. Define s0 so that γs0 satisfies minz∈γs0 d (x, z) = ry . Note that |s0 | ≤ 101. Let D  Ss20 ,+ , which is an open disk containing x. We claim that Bx (ry ) ⊂ D. If not, then there exists w ∈ ¯ Let δ : [0, d(x, w)] → S 2 be a minimal geodesic from x to w. Since Bx (ry ) − D.  2 ¯ the function t → s(ϕ(δ(t))) is decreasing for t such / D, x ∈ S0,+ − Bp ε−1 and w ∈  −1  2 that δ(t) ∈ Cε  ϕ(S × −ε + 4, ε−1 − 4 ). Since s(ϕ(w)) < s0 , there exists a unique t0 ∈ (0, d(x, w)) such that δ(t0 ) ∈ γs0 . This implies that d(x, w) > t0 = d(x, δ(t0 )) ≥ ry , which contradicts w ∈ Bx (ry ). D

γs0

Cε (S 2, g)

y x

δ

δ(t0) w

Bx(ry)

∂Bx(ry)

Figure 29.2. Now let α denote the component of ∂Bx (ry ) containing y. Suppose that there is another component β of ∂Bx (ry ). Since α is approximately at the center of an

8. CHARACTERIZING ROUND SOLUTIONS

93

ε-neck, we have β ∩ Cε = ∅. Let u ∈ β. Since β ⊂ Ss20 ,+ , we have d (u, α) ≤ d (u, p) + d (p, y) ≤

sup d (z, p) + 100 z∈Ss2

0 ,+

≤ sup d (z, p) + 202, 2 z∈S0,+

where we used |s0 | ≤ 101 and provided ε is small enough. By our hypothesis we then obtain 3 d (u, α) ≤ d (x, p) + 202 ≤ 1.6ry . 2 It follows that there exist points a ∈ α and b ∈ β such that d (a, b) = d (α, β)  σ ≤ 1.6ry . Consider a geodesic triangle in (S 2 , g) with vertices x, b, a and minimal geodesic ˜, ˜b, a ˜ be a sides xb, xa, ba having corresponding side lengths ry , ry , σ. Let x x˜b) = ry , L(˜ xa ˜) = ry , L(˜b˜ a) = σ. Euclidean triangle with the same side lengths L(˜ Then the angle at a ˜ satisfies ˜ xa ˜˜b ≥ cos−1 (0.8) since σ ≤ 1.6ry . By the triangle version of the Toponogov comparison theorem, we conclude that xab ≥ ˜ xa ˜˜b ≥ cos−1 (0.8). However, since a is near the center of an ε-neck and since x and b are outside the ε-neck, the angle xab must be small and we obtain a contradiction.  In view of Lemma 29.24, the following result is useful. Lemma 29.26 (Distance circles in the vicinity of the center of an ε-neck). Let g be a Riemannian metric on S 2 with scalar curvature R > 0 and let p ∈ S 2 . For any δ > 0 there exists ε ∈ (0,0.01) such that if Nε is a normalized ε-neck centered at p, then for any x ∈ / Bp ε−1 , y ∈ Bp (100), and for almost every s ∈ (d(y, x) − ε, d(y, x) + ε), we have the following. Let αs be the connected component of ∂Bx (s) with d (αs , y) < ε. Then the loop αs is smooth except at finitely many points and αs is δ-close to γsy  ϕ−1 ({sy } × S 1 ) in C 0 , where (sy , θy ) = ϕ(y). Furthermore, the values L(αs ), L(γsy ), and 2π are all δ-close to each other. Proof. Step 1. C 0 closeness. By [376], for any ε > 0 and for a.e. s ∈ (d(y, x) − ε, d(y, x) + ε) we have that ∂Bx (s) is a disjoint union of piecewise smooth embedded loops. By choosing ε sufficiently small, we guarantee that there exists a unique connected component αs of ∂Bx (s) with d (αs , y) < ε. −1 Let g˜  (ϕ−1 )∗ (g). Then g˜ is ε-close in the C ε +1 -topology to gcyl on (−ε−1 + 4, ε−1 − 4) × S 1 ; compare with (28.26). Since (29.77) holds approximately with gcyl replaced by g˜, we have that for s ∈ (d(y, x) − ε, d(y, x) + ε) that ϕ(αs ) is 10ε-close to {sy } × S 1 in C 0 with respect to g˜. That is, αs is 10ε-close to γsy = ϕ−1 ({sy } × S 1 ) in C 0 with respect to g. So, first of all, we require that ε < δ/10. Step 2. Closeness of the lengths. Parametrize αs by arc length u and choose any point z  αs (uz ) at which αs is smooth. Join z to x by a minimal unit speed geodesic βz : [0, s] → S 2 , so that βz (0) = z and βz (s) = x. Let [0, L1 ) be the largest subinterval for which βz ([0, L1 )) ⊂ Bp ε−1 . Without loss of generality, we may

94

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

assume that s(ϕ(βz (t))) ≥ s for all t ∈ [1, L1 ). Define L2 to be the first number t for which s(ϕ(βz (t))) = ε−1 − 4. ϕ −1((−ε −1 + 4, ε −1− 4)×S 1) Bp(100) βz(L2)

(S 2, g) y

x

βz βz(L1)

p

z γs

y

αs Bp(ε −1)

Figure 29.3. Note that αs (uz ) is perpendicular to βz (0) with respect to g. By choosing ε ∈ (0, δ/40) small enough, from the geometry of a normalized ε-neck and since βz |[0,L2 ] is a minimal geodesic with L2 large, we have that       βz (0) − (ϕ−1 )∗ ∂  ≤ δ .  ∂s g 40 By considering vectors perpendicular to those on the lhs, we obtain that       αs (uz ) − (ϕ−1 )∗ ∂  ≤ δ . (29.78)  ∂θ  20 g

By (29.78), the curve αs can be parametrized in the neck coordinates by (s (θ) , θ), θ ∈ S 1 . Moreover, at any smooth point of αs , the tangent vector δ . Hence Lgcyl (αs ) = S 1 |αs (θ)| dθ is 2δ -close αs (θ) = (s (θ) , 1) satisfies |s (θ)| ≤ 15 to 2π. On the other hand, since we are in an ε-neck, Lg (γsy ) is also 2δ -close to 2π.  Now we consider an ancient solution (S 2 , g(t)). We first prove that areas of balls are bounded from below linearly in their radii. Lemma 29.27 (Area lower bound for balls). There exists a constant c0 > 0 such that (29.79)

Areag(t) (Bpg(t) (r)) ≥ c0 r

for any (p, t) ∈ S 2 × (−∞, −1] and 3 ≤ r ≤ diam(g(t)). Proof. Recall that by Rg(t) ≥ 0 and the relative Bishop–Gromov volume comparison theorem, we have for x ∈ S 2 that g(t)

(29.80)

g(t)

Areag(t) (Bx (r4 ) − Bx (r3 )) g(t)

g(t)

Areag(t) (Bx (r2 ) − Bx (r1 ))

≤

r42 − r32 r22 − r12

8. CHARACTERIZING ROUND SOLUTIONS

95

for 0 ≤ r1 ≤ r2 ≤ r3 ≤ r4 . Hence, for r ≥ 2,  (r − 1)2  Areag(t) (Bxg(t) (r − 1)) ≥ Areag(t) (Bxg(t) (r + 1)) − Areag(t) (Bxg(t) (r − 1)) . 4r Let p ∈ S 2 . It suffices to prove the lemma for r ≤ 12 diam(g(t)). Choose x ∈ g(t) g(t) g(t) S 2 with dg(t) (x, p) = r ≥ 2. Then Bp (1) ⊂ Bx (r + 1) − Bx (r − 1) and g(t) g(t) Bx (r − 1) ⊂ Bp (2r − 1), so we obtain Yau’s “at least linear volume growth” estimate: r Areag(t) (Bpg(t) (1)). Areag(t) (Bpg(t) (2r − 1)) ≥ 16 Since 0 < R ≤ C on S 2 × (−∞, −1] and by the injectivity radius estimate (29.69), g(t) this implies that Areag(t) (Bp (1)) ≥ c for some positive constant c. We conclude g(t) that Areag(t) (Bp (r)) ≥ cr for all p ∈ S 2 , r ≥ 3, and t ∈ (−∞, −1].  Now we are ready to prove that the lower bound for the areas of balls in Lemma 29.27 can be expressed in terms of time instead of radius. By a parabolic rescaling of the solution g(t), we may assume without loss of generality that the S 1 (r∞ ) in Lemma 29.24 and throughout this discussion has radius 1. 2 Recall that γt denotes a minimizer of I ( · ; g(t)). Choose points x± t ∈ S± (γt ) so that (29.81)

ρ± t 

max dg(t) (x, γt ) = dg(t) (x± t , γt ).

2 (γ ) x∈S± t

Since (S 2 , g(t), pt ) sequentially converges as t → −∞ to cylinders (R×S 1 , g∞ (0), p∞ ), we have limt→−∞ ρ± t = ∞. (In the case of the King–Rosenau solution, γt is the equator and x± t are the poles.) Lemma 29.28 (Area bounds in terms of time). There exist T0 < 0 and c0 > 0 such that (29.82)

c0 |t| ≤ Areag(t) (Bx± (ρ± t )) ≤ 8π |t| g(t) t

for all t ≤ T0 . ± Proof. Since ρ± t = dg(t) (xt , γt ) by definition and since Lg(t) (γt ) is uniformly bounded by (29.66), there exists a constant C < ∞ such that for all t ∈ (−∞, −1] we have that

(29.83)

± 2 Bx± (ρ± t ) ⊂ S± (γt ) ⊂ Bx± (ρt + C). g(t)

g(t)

t

t

On the other hand, since the distance circle ∂Bx± (ρ± t ) is connected (and near g(t) t

γt ) by Lemma 29.25, we have that ± Area(Bx± (ρ± t + C) − Bx± (ρt )) ≤ C g(t)

g(t)

t

t

independent of t. Hence, by (29.83), we have (29.84)

± A ± (γt ) ≤ Area(Bx± (ρ± t + C)) ≤ Area(Bx± (ρt )) + C. g(t)

g(t)

t

t

The lemma now follows from Lemma 29.23. We may also bound the distance ρ± t by a constant times |t|.



96

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Lemma 29.29. There exists a constant c1 > 0 such that for all t ∈ (−∞, −1], ± c−1 1 |t| ≤ ρt ≤ c1 |t| .

(29.85) Hence diam(g(t)) ≤ C|t|.

Proof. Step 1. There exists a constant C < ∞ such that for each t, − + C −1 ρ+ t ≤ ρt ≤ Cρt .

(29.86)

Since Lg(t) (γt ) ≤ C by (29.66), we have − diam(g(t)) ≤ ρ+ t + ρt + C.

(29.87)

2 Since Bx± (ρ± t ) ⊂ S± (t) and by Lemma 29.23, there exists a constant c > 0 t such that for all t, g(t)

c≤

A ± (γt ) A ∓ (γt ) − ∓ Areag(t) (Bx∓ (ρ+ t + ρt + C) − Bx∓ (ρt )) g(t)

≤

g(t)

t

t

Areag(t) (Bx∓ (ρ∓ t )) g(t) t

∓ 2 2 (ρ+ + ρ− t + C) − (ρt ) ≤ t ∓ 2 (ρt )

by the relative Bishop–Gromov volume comparison theorem. The above inequality says that √ + − c + 1ρ∓ t ≤ ρt + ρt + C. ± Hence ρ∓ t ≤ Cρt for some constant C < ∞. ± 8π Step 2. Bounds for ρ± t . By (29.79) and (29.82), we have ρt ≤ c0 |t|. By (29.87) and the relative Bishop–Gromov volume comparison theorem, we have A ± (γt ) ∓ Areag(t) (Bx∓ (ρ∓ t ) − Bx∓ (ρt − 1)) g(t)

g(t)

t

t

− ∓ Areag(t) (Bx∓ (ρ+ t + ρt + C) − Bx∓ (ρt )) g(t)

≤

g(t)

t

t

∓ Areag(t) (Bx∓ (ρ∓ t ) − Bx∓ (ρt − 1)) g(t)

g(t)

t

t

2 (ρ+ + ρ− + C)2 − (ρ∓ t ) ≤ t ∓ t2 2 (ρt ) − (ρ∓ t − 1)

≤ Cρ± t by (29.86). From the geometry of the neck we obtain (29.88)

± Areag(t) (Bx± (ρ± t ) − Bx± (ρt − 1)) ≤ C. g(t)

g(t)

t

t

Hence (29.89)

Cρ± t ≥ A ± (γt ) ≥ c|t|,

where the last inequality follows from (29.72). This completes the proof of the lemma. 

8. CHARACTERIZING ROUND SOLUTIONS

97

8.4. Proof of the main proposition on characterizing round solutions. First, we observe a basic fact about lengths of distance circles which follows from the proof of the Bishop–Gromov relative volume comparison theorem. Let (M2 , g˜) be a complete Riemannian surface with scalar curvature R ≥ 0. ∂ log J(β(s)) ≤ 0 Along minimal geodesics β(s) emanating from x we have that ∂s s (see, e.g., Lemma 1.126 in [77]), where J is the Jacobian of the exponential map; is nonincreasing. Let thus s → J(β(s)) s Dx  {V ∈ Tx S 2 : d(expx (V ) , x) = |V |}. We have that expx : int (Dx ) → S 2 − Cut (x) is a diffeomorphism, where Cut (x) denotes the cut locus of x. Let Ux S 2 denote the unit tangent space of S 2 at x and let Dx (s) = {V ∈ Ux S 2 : sV ∈ int (Dx )}. Note that if s1 < s2 , then Dx (s2 ) ⊂ Dx (s1 ). Then  L(∂Bx (s)) =

J(expx (sV ))dV. Dx (s)

We may now conclude from the monotonicity of s1 < s2 of d(·, x),

J(β(s)) s

and Dx (s) that for any

L(∂Bx (s2 )) L(∂Bx (s1 )) ≤ ≤ 2π. s2 s1

(29.90)

We are now in a position to prove that if v∞ ≡ 0, then g(t) is a round shrinking 2-sphere. Proof of Proposition 29.16. Suppose that v∞ ≡ 0. By Lemma 29.19(2) we may assume that I(g(t)) < 4π for all t. g(t) ± Let −1 ≥ t → −∞. For r > 0 define the distance circles St,r = ∂Bx± (r), t

± where x± t is as in (29.81). It is well known that the St,r are piecewise smooth for g(t)

a.e. r (see [376]). Since AreagS 2 (Bx± (r)) is a continuous nondecreasing function t

of r, there exists rt± > 0 so that

AreagS 2 (Bx± (rt± )) = 2(1 + εt )π, g(t) t

± where εt ∈ [0, 0.01) is chosen so that St,r ± is piecewise smooth. Note that if we take t

the infimum in (29.54) over piecewise smooth embedded closed curves, we obtain the same infimum. Since I (gS 2 ) = 4π, we have for all t ≤ −1 that ± L gS 2 (St,r ±) ≥ ! t

2π 1 2(1+εt )

+

1 2(1−εt )

≥ 1.999π.

Claim. There exists a constant C < ∞ such that for each t ≤ −1, + L g(t) (St,r +) ≤ C

(29.91)

t

or

− L g(t) (St,r − ) ≤ C. t

The proposition follows from the claim for the following reasons. We may assume without loss of generality that there exists a sequence ti → −∞ such that L g(ti ) (St−,r− ) ≤ C. Since v (t) converges uniformly to v∞ ≡ 0 as t → −∞, we have i

ti

98

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

that for any ε > 0 there exists Tε ≤ −1 such that g(t) ≥ ε−1 gS 2 for t ≤ Tε . Hence C ≥ L g(ti ) (St−,r− ) ≥ ε−1/2 L gS 2 (St−,r− ) ≥ 1.999πε−1/2 , i

i

ti

ti

which is a contradiction.  Proof of the claim. Since Rg(t) ≥ 0, from (29.90) we obtain that the function L

(S ∓ )

r → g(t)r t,r is nonincreasing. Now choose the basepoint pt in the statement of ± ± ± ± Lemma 29.24 to be p± t ∈ γt , so that dg(t) (pt , xt ) = ρt , where ρt is defined by ± (29.81) and where ρt ≥ c|t| holds by (29.85). Note that, by Lemma 29.26, the minimizer satisfies ± γt ⊂ Bx± (ρ± t + 1) − Bx± (ρt )

(29.92)

g(t)

g(t)

t

t

± for all t ≤ −1. Without loss of generality, we may assume that the ∂Bx± (ρ± t )  αt g(t) t

are piecewise smooth in addition to being connected; otherwise we replace ρ± t by a suitably close value (the reader may check that the argument still goes through without change). By Lemma 29.26 we have Lg(t) (αt± ) ≤ C. To complete the proof of the proposition, we consider three cases. (1) If rt− ≥ ρ− t , then (29.90) tells us that − Lg(t) (St,r −) ≤

(29.93)

t

ρ− t

rt−

− 10 < (2) If − Lg(t) (St,r − ) is nearly 2π.


0 and where σ : S 2 − {p} → R2 denotes stereographic projection. 1 Proof. Let g¯(t) = (σ −1 )∗ g(t) = v¯(t) geuc . We prove the proposition by contra−1 diction. Suppose that v¯ (t) converges to a positive constant A as t → −∞. Then, for each k ∈ N (which we shall assume is sufficiently large), there exists tk ≤ −1 such that

(29.94)

¯ g (t) − Ageuc C k (B g¯(t) (k),¯g(t)) ≤ k−1 O

for t ∈ (−∞, tk ],

where O denotes the origin of R2 . Let γt be an isoperimetric curve. We may assume, without loss of generality, that the antipodal point q  σ −1 (O) of p satisfies 2 (γ ). q ∈ S+ t Define tk  min{tk , − 2c πk2 }, where c is given by (29.72). We claim that if t ∈ (−∞, tk ], then (29.95)

γt ⊂ Bqg(t) (k).

8. CHARACTERIZING ROUND SOLUTIONS g(t)

Indeed, suppose that γt ⊂ Bq

99

(k) for some t ∈ (−∞, tk ]. By (29.94) we have g ¯(t)

Areag(t) (Bqg(t) (k)) = Areag¯(t) (BO (k)) < 2πk2 . Thus A+ (γt ) < 2πk2 , which contradicts (29.72) since t ≤ tk . Recall from (29.66) that Lg(t) (γt ) ≤ C for t ∈ (−∞, tk ]. By choosing k large enough, we have that (29.95) implies γt ∩ Bqg(t) (3k/4) = ∅.

(29.96)

g(t)

2 2 (γ ) (γt ) contains the large almost Euclidean ball Bq (3k/4). Let xt ∈ S− Thus S+ t be such that (we drop the minus sign superscripts from our notation for xt and ρt )

ρt  dg(t) (xt , γt ) =

sup

dg(t) (z, γt ).

2 (γ ) z∈S− t

Let wt ∈ γt be such that ρt = dg(t) (xt , wt ). Let (ρt ). αt = {z ∈ S 2 : dg(t) (z, xt ) = ρt } = Sxg(t) t 2 (γ ) and Note that wt ∈ αt ⊂ S− t

αt ∩ Bqg(t) (3k/4) = ∅.

(29.97)

S 2−(γt)

Bg(t) q (3k/4)

γt L = rt xt

L = ρt

q

wt αt

S xg(t)(rt) t

S 2+(γt)

Figure 29.4.

From Lemmas 29.26 and 29.29, we know that Lg(t) (αt ) ≤ C and c |t| ≤ ρt ≤ g(t) C |t|. Let rt = dg(t) (q, xt ). Then rt > ρt and q ∈ Sxt (rt ). By (29.90) and by rt ≤ diam(g(t)) ≤ C|t| from Lemma 29.29, we have that Lg(t) (Sxg(t) (rt )) ≤ t

rt Lg(t) (αt ) ≤ C. ρt

g(t) g(t) ¯qg(t) (C) ⊂ Bqg(t) (k/2) for k large Since q ∈ Sxt (rt ), this implies that Sxt (rt ) ⊂ B enough.

100

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Bqg(t)(k/2) γt q

xt S g(t) xt (ρt)=αt

S g(t) xt (st)

S g(t) xt (rt)

Figure 29.5. g(t) ¯qg(t) (k/2) and By (29.97), there exists st ∈ (ρt , rt ) such that Sxt (st ) ⊂ B

Sxg(t) (st ) ∩ ∂Bqg(t) (k/2) = ∅. t

(29.98)

¯qg(t) (k/2), g(t)) is a nearly Euclidean closed disk, the sets Sxg(t) Since (B t (rt ) and g(t) g(t) ¯ Sxt (st ) are loops bounding disks D1 and D2 in Bq (k/2), respectively. We also g(t) g(t) have that Sxt (rt ) ∪ Sxt (st ) bounds an embedded annulus. This and (29.98) ¯ 2 ∩ ∂Bqg(t) (k/2) = ∅. Since imply that D2 contains D1 . Hence q ∈ D2 as well as D g(t) g(t) Bq (k/2) is almost Euclidean, we conclude that Lg(t) (Sxt (st )) ≥ k2 . We now obtain a contradiction by choosing k sufficiently large since Lg(t) (Sxg(t) (st )) ≤ t

st Lg(t) (αt ) ≤ C ρt

holds by (29.90).



Using the proof of Proposition 29.30, we can obtain the following. Lemma 29.31 (Nonround solutions cannot have plane limits). Assume that (S 2 , g (t)) is not a round shrinking 2-sphere. Suppose that ti → −∞ and qi ∈ S 2 are such that (S 2 , g(ti + t), qi ) converges in the pointed C ∞ Cheeger–Gromov sense to (M2∞ , g∞ (t), q∞ ). Then (M2∞ , g∞ (t)) cannot be the flat plane. Proof. Suppose that (M2∞ , g∞ (t)) is the flat plane. The key point is that the pointed Cheeger–Gromov convergence does not use rescaling of the sequence of solutions g(t + ti ). By the proof of Proposition 29.30 we may assume that, for 2 (γt ) contains a large almost Euclidean ball. We may now |t| sufficiently large, S+ repeat the rest of the proof to obtain the lemma. 

9. Classifying the backward pointwise limit Let g(t) = v(t)−1 gS 2 , t ∈ (−∞, 0), be a maximal solution of the Ricci flow on S . The main goal of this section is to prove Proposition 29.36 below. This result classifies the specific form of the backward limit of v(t) either as vanishing or as corresponding to a flat cylinder. The proof hinges on the monotonicity of certain 2

9. CLASSIFYING THE BACKWARD POINTWISE LIMIT

101

circular averages as well as on a concentration-compactness-type result which implies that v∞  limt→−∞ v(t) has at most two zeros if it does not vanish identically. The classification then uses the fact that an entire harmonic function on the plane with at most linear growth must be a linear function. 9.1. An estimate from monotonicity of circular averages. Let p ∈ S 2 . We shall find it convenient to consider the conformal factors relative to the Euclidean and cylinder metrics; see (29.4), defined via stereographic ¯  − ln v¯ = ln u ¯ on R2 and w ¯±  max{±w, ¯ 0}. projection σ : S 2 − {p} → R2 . Let w ¯(r, θ, t). We have the following Define the backward limit u ¯∞ (r, θ) = limt→−∞ u result, which is similar to Lemma 28.49. ¯∞ (O) < Lemma 29.32 (A bound for integrals of w ¯+ over balls). Suppose that u ∞, where O ∈ R2 is the origin. Then for any r0 > 0 and t ∈ (−∞, −1]   r0 (29.99) w ¯+ (r, θ, t)rdrdθ ≤ πr02 ln u ¯∞ (O) + C(r0 , umin (−1)), S1

0

where umin (t)  minx∈S 2 u(x, t) > 0. Proof. Define the circular averages  w(r, ¯ θ, t)dθ W (r, t)  S1

for r > 0. By (29.7), we have that w ¯ is superharmonic:   ¯ 1 ∂ ∂w ¯ 1 ∂2w 0 > −Rg u ¯ = Δeuc w ¯= r + 2 2 r ∂r ∂r r ∂θ ∂ on R2 − {0}. Integrating this inequality over circles yields ∂r (r ∂W ∂r (r, t)) < 0. ∂W ∂W Since limr0 (r ∂r (r, t)) = 0, we obtain the monotonicity: ∂r (r, t) < 0 for r > 0. Therefore  w(r, ¯ θ, t)dθ < lim W (¯ r, t) = 2π w(O, ¯ t) ≤ 2π ln u ¯∞ (O) < ∞, r¯0

S1

where the last inequality is true because ∂∂tu¯ < 0. Integrating this inequality with respect to r yields   r0 (29.100) w(r, ¯ θ, t)rdrdθ ≤ πr02 ln u ¯∞ (O). S1

Since

∂u ¯ ∂t

0

< 0 and by (29.6), we have that u ¯(r, θ, t) ≥

(29.101)

4umin (−1) (1 + r 2 )2

Hence, for t ∈ (−∞, −1],    r0 (29.102) w ¯− (r, θ, t)rdrdθ ≤ S1

0

S1

for t ≤ −1.  0

r0

  4umin (−1) rdrdθ ln (1 + r 2 )2 −

≤ C(r0 , umin (−1)). Since w ¯+ = w ¯+w ¯− , the lemma follows from (29.100) and (29.102).



102

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

9.2. A concentration-compactness result. The following is a concentration-compactness-type or bubbling-type result. In view of the geometric perspective, note that the area form of the pull-back g¯(t) ¯(t)dμeuc and also that the Cohn-Vossen inequality says that for of g(t) to R2 is u a complete noncompact Riemannian surface (M2 , g) with positive curvature, we  have M Rg dμg ≤ 4π. Throughout this subsection, all balls are with respect to the Euclidean metric on R2 . Lemma 29.33 (¯ u unbounded implies a concentration of integral of curvature). Suppose that u ¯∞ (O) < ∞ and let z0 ∈ R2 , r > 0, and ε > 0. Then there exists a ¯∞ (O), umin (−1)) such that if constant C = C(|z0 |, r, ε, u  Rg¯(t) u ¯(t)dμeuc ≤ 4π − ε Bz0 (r)

at some t ∈ (−∞, −1], then at this time t we have u ¯(t) ≤ eC

in Bz0 (r/2).

Remark 29.34. In the case of a round 2-sphere shrinking to a point at time 0, we have by (29.6) that 8 |t| . u ¯(r, θ, t) = (1 + r 2 )2 In this case, u ¯∞ (r, θ) = ∞. Moreover,  r  16π 8πr 2 Rg¯(t) u ¯(t)dμeuc = rdr = . 2 2 1 + r2 BO (r) 0 (1 + r ) So we cannot simply remove the condition u ¯∞ (O) < ∞ from the hypothesis of the lemma. Proof of Lemma 29.33. Let w ˜ be the unique solution to ˜ = Rg¯ u ¯ −Δeuc w w ˜=0

in Bz0 (r),

on ∂Bz0 (r).

By Lemma 28.50 we have that for each δ ∈ (0, 4π),

 δ |w(x)| ˜ 16π 2 r 2 (29.103) . exp dμeuc ≤ ||Rg¯ u ¯||L1 (Bz0 (r)) 4π − δ Bz0 (r) Since ||Rg¯ u ¯||L1 (Bz0 (r)) ≤ 4π − ε from the hypothesis, by choosing δ = 4π − 2ε , we have  32π 2 r 2 ˜ (29.104) eq|w(x)| dμeuc ≤ ε Bz0 (r) for some q ≥ (29.105)

8π−ε 8π−2ε

> 1. By Jensen’s inequality we have

  1 πr 2 q|w| ˜ ln |w|dμ ˜ euc ≤ e dμeuc q πr 2 Bz0 (r) Bz0 (r)   32π 2 8π − 2ε ln ≤ πr  C(r, ε). 8π − ε ε

9. CLASSIFYING THE BACKWARD POINTWISE LIMIT

103

Now let z˜ = w ¯ − w. ˜ Then Δeuc z˜ = 0 in Bz0 (r). Since z˜+ is subharmonic on Bz0 (r), by the mean value inequality we have that ||˜ z+ ||L∞ (Bz0 (3r/4)) ≤

1 16 ||˜ z+ ||L1 (Bz0 (r)) = 2 ||˜ z+ ||L1 (Bz0 (r)) . Area(Bz0 (r/4)) πr

¯+ + |w|, ˜ we have On the other hand, since z˜+ ≤ w ¯+ ||L1 (BO (|z0 |+r)) + ||w|| ˜ L1 (Bz0 (r)) ||˜ z+ ||L1 (Bz0 (r)) ≤ ||w ≤ π ln u ¯∞ (O)(|z0 | + r)2 + C(|z0 | + r, umin (−1)) + C(r, ε)  C by (29.99) and (29.105). Hence ||˜ z+ ||L∞ (Bz0 (3r/4)) ≤ This implies that

16  πr 2 C .

¯ = Rg¯ ew¯ ≤ Rg¯ ez˜+ +w˜ ≤ C1 ew˜ 0 < −Δeuc w

(29.106) 16

in Bz0 (3r/4) ,



˜ ∈ Lq (Bz0 (r)). where C1 = e πr2 C supS 2 ×(−∞,−1] R. By (29.104), we have that e|w| Now we may apply a standard interior elliptic estimate to (29.106) to obtain (see Theorem 4.1 in [144])

||w ¯+ ||L∞ (Bz0 (r/2)) ≤ C||w ¯+ ||L1 (Bz0 (3r/4)) + C||Δeuc w|| ¯ Lq (Bz0 (3r/4)) ≤ C. That is, u ¯ ≤ eC in Bz0 (r/2).



9.3. A nontrivial v∞ must vanish at no more than two zeros. With the above lemma we can prove the following result. Note that by (29.47) we have that v∞ has exactly two zeros for the King–Rosenau solution. Lemma 29.35 (Nontrivial v∞ have at most two zeros). For any ancient solution on S 2 , either v∞ ≡ 0 or v∞ has at most two zeros. Proof. Suppose that v∞ is zero at distinct points N1 , N2 , and N3 . We shall then show that v∞ ≡ 0. By applying a conformal diffeomorphism of S 2 , we may choose the north and south poles to satisfy v∞ (N ) > 0 and v∞ (S) > 0. Note that the points N , S, N1 , N2 , N3 are distinct. Let σ : S 2 − {N } → R2 be stereographic ¯∞ (O) < ∞. Let zk = σ(Nk ) for projection, so that σ(S) = O ∈ R2 satisfies u k = 1, 2, 3. Let r > 0 be such that the balls Bzk (r) are disjoint. Let ε ∈ (0, 4π/3). Consider any sequence of times ti → −∞. Since limi→∞ v¯(zk , ti ) = 0, by Lemma 29.33 we have that for i sufficiently large  8π Rg¯(ti ) u ¯(ti )dμeuc > 4π − ε > 3 Bzk (r) for each k = 1, 2, 3. Since the Bzk (r) are disjoint, we conclude that  S2

Rg(ti ) dμg(ti ) ≥

3   k=1

Rg¯(ti ) u ¯(ti )dμeuc > 8π,

Bzk (r)

which contradicts the Gauss–Bonnet formula.



104

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

9.4. Proof of the classification of v∞ . We now classify the backward pointwise limit v∞ . Let ψ ∈ [− π2 , π2 ] denote the latitude on S 2 . Proposition 29.36 (Infinite expansion or cylinder limit). Assume that the ancient solution (S 2 , g(t)) is not round. Then, after pulling back by a suitable conformal diffeomorphism Φ of S 2 , we have the following. The v(t) converge in C 1,α on S 2 and in C ∞ on compact subsets of S 2 − {N, S} to (29.107)

v∞ (ψ, θ) = μ cos2 ψ

for some μ > 0. In particular, v∞ ∈ C ∞ (S 2 ) and R∞ ≡ 0 on S 2 − {N, S}. Proof. By Proposition 29.16, Lemma 29.35, and applying a conformal diffeomorphism, we may assume that v∞ is nonzero on Ω  S 2 − {N, S}. With this, we have that vˆ∞  limt→−∞ vˆ(t) > 0 on all of R × S 1 , where vˆ is defined in (29.4). Recall from §6 of this chapter that v∞ ∈ C ∞ (Ω). From Lemma 29.12(2), we have that v (t) converges to v∞ in C 3,α on compact subsets of Ω. By this and (29.52), we conclude that R∞ ≡ 0 in Ω. Hence Δcyl ln vˆ∞ = 0 on R × S 1 . By (29.5) and v∞ ≤ C, we have that (29.108)

ln vˆ∞ (s, θ) = ln(cosh2 s v∞ (ψ, θ)) ≤ C1 |s| + C2 ,

where C1 and C2 are nonnegative constants. Now pull back ln vˆ∞ by the covering map π : R × R → R × S 1 to a harmonic function w ˜∞ on R2 . Then (29.109)

˜ ≤ C1 |s| + C2 w ˜∞ (s, θ)

for s, θ˜ ∈ R.

Claim 1. There exist nonnegative constants C3 and C4 such that (29.110)

|w ˜∞ |(z) ≤ C3 |z| + C4

˜ ∈ R2 . for z = (s, θ)

Proof of Claim 1. By (29.109), we only need to prove a lower bound for w ˜∞ . By ˜∞ − 2(C1 |s| + C2 ) applying the mean value property and then using − |w ˜∞ | ≥ w from (29.109), we have that for any r > 0 and z1 ∈ Br (O),   (29.111) πr 2 (w ˜∞ (z1 ) − w ˜∞ (O)) = w ˜∞ (z)dμeuc − w ˜∞ (z)dμeuc Bz1 (r)

BO (r)



≥− 

|w ˜∞ (z)| dμeuc Bz1 (r)BO (r)

≥

(w ˜∞ (z) − 2 (C1 |s| + C2 ))dμeuc . Bz1 (r)BO (r)

Note that the symmetric difference satisfies Bz1 (r)BO (r) ⊂ BO (r + |z1 |) − BO (r − |z1 |). Now (29.111) yields  ˜∞ (z1 ) − w ˜∞ (0)) ≥ w ˜∞ (z)dμeuc πr 2 (w BO (r+|z1 |)−BO (r−|z1 |)  −2 (C1 |s| + C2 ) dμeuc . BO (r+|z1 |)−BO (r−|z1 |)

Applying the mean value theorem to the first term on the rhs, we obtain ˜∞ (z1 ) − w ˜∞ (O)) ≥ 4πr|z1 |(w ˜∞ (O) − 2C1 (r + |z1 |) − 2C2 ). πr 2 (w

9. CLASSIFYING THE BACKWARD POINTWISE LIMIT

105

Thus w ˜∞ (z1 ) − w ˜∞ (O) ≥ lim 4r −1 |z1 |(w ˜∞ (O) − 2C1 (r + |z1 |) − 2C2 ) = −8C1 |z1 |. r→∞

This completes the proof of Claim 1. w ˜∞ Now, by Corollary 6.3 in [194], (29.110) implies that ∂ ∂s and ∂ ∂w˜θ˜∞ are bounded 2 harmonic functions on R and hence are constant. Thus ln vˆ∞ (s, θ) = C5 s + C6 . We conclude that

vˆ∞ = μeBs ,

(29.112) where μ > 0 and B are constants.

In view of (29.108), the proposition follows from Claim 2. B = 0. Proof of Claim 2. By (29.112) and (29.108) we have that v∞ (ψ, θ) = μeBs sech2 s. Since we have C 3,α convergence of v(t) to v∞ on compact subsets of S 2 − {N, S}, the gradient estimate (29.34) holds for v∞ , so that 2

|∇v∞ |S 2 cosh2 s C≥ = v∞ v∞



∂v∞ ∂s

2 2

= μeBs (B − 2 tanh s)

for s ∈ R.

This implies that B = 0 or B = ±2. Suppose that B = ±2. Then vˆ∞ = μe±2s 1 implies that as t → −∞ we have that v(t) gS 2 limits on S 2 − {N } to −1 gcyl = μe∓2s (ds2 + dθ 2 ) = μgeuc . vˆ∞

This contradiction to Proposition 29.30 yields B = 0.



As a consequence of the proposition, we can interpret the gradient estimate (29.34) as follows. Corollary 29.37. Suppose that (S 2 , g(t)), t ∈ (−∞, 0), is a nonround solution. Then there exists C < ∞ such that 2  ∂ ln v ≤ C on S 2 × (−∞, −1]. (29.113) ∂θ Proof. Since gS 2 (ψ, θ) = dψ 2 + cos2 ψ dθ 2 , (29.34) says that 

∂v ∂θ

2 2

≤ |∇v|S 2 cos2 ψ ≤ Cv cos2 ψ

on S 2 × (−∞, −1].

On the other hand, since g(t) is nonround, by Proposition 29.36, we have that μ cos2 ψ = v∞ (ψ, θ) ≤ v(ψ, θ, t) for some μ > 0.



106

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

10. An unrescaled cigar backward Cheeger–Gromov limit Let (S 2 , g(t)), t ∈ (−∞, 0), be a nonround maximal solution to the Ricci flow. Recall from Proposition 29.36 that we may assume that v(ψ, θ, t) converges on S 2 − {N, S} to v∞ (ψ, θ) = μ cos2 ψ, where μ > 0. Geometrically, this says that the metrics g(t) converge pointwise on S 2 −{N, S} to a flat cylinder metric as t → −∞. In this section we show that g(t) pointed at N or S subconverges, after being pulled back by conformal diffeomorphisms fixing N and S, to the cigar soliton as t → −∞; see Proposition 29.38 below. Motivations for this result come from both Theorem 29.2(2) and Lemma 29.4. We consider g(t) pointed at N (at S is essentially the same). Recall that σ : S 2 − {S} → R2 denotes stereographic projection and g¯(t) = (σ −1 )∗ (g(t)) = u ¯(t)geuc = v¯−1 (t)geuc . Let ti → −∞ be any sequence of times. Define Ki  u ¯−1/2 (O, ti ). Note that by (29.6) and v∞ (N ) = 0, we have Ki → 0 as i → ∞. Consider the sequence of pulled-back solutions (29.114)

¯i (t)geuc  v¯i−1 (t)geuc , g¯i (t)  u

where (29.115)

¯i (x, y, t)  Ki2 u u ¯(Ki x, Ki y, t + ti ).

¯i (0, 0, 0) = 1 for each i. By (29.7), we have that Note that u (29.116)

v¯i (x, y, t) = Ki−2 v¯(Ki x, Ki y, t + ti )

is a solution to ∂ v¯i = v¯i2 Δeuc ln v¯i = Rg¯i v¯i > 0. ∂t We have the following.

(29.117)

Proposition 29.38 (Backward convergence based at the poles to cigar solitons). For any nonround ancient solution there exists a subsequence such that the functions v¯i (x, y, t) in (29.116) converge in C ∞ on compact subsets of R2 ×(−∞, ∞) to  (29.118) v¯∞ (x, y, t) = A BeCt + (x − x0 )2 + (y − y0 )2 , where A, B, C are positive constants and (x0 , y0 ) ∈ R2 . Hence (S 2 , g(t+ti ), N ) subconverges in the Cheeger–Gromov sense to a cigar soliton solution (R2 , g¯∞ (t), O), where g¯∞ (t)  v¯∞1(t) geuc . As in Lemma 29.4, we may interpret the sequence of solutions in (29.116) in the context of Cheeger–Gromov convergence. Define the homotheties Φi : R2 → R2 by Φi (x, y) = (Ki x, Ki y), which via stereographic projection correspond to conformal diffeomorphisms of S 2 fixing N and S. Then (29.119)

g¯i (t) = v¯i−1 (t)geuc = Φ∗i (¯ g (t + ti )).

So the sequence of solutions gi (t)  σ ∗ g¯i (t) on S 2 (they extend smoothly over S) are diffeomorphism equivalent to g(t + ti ) via Ψi  σ −1 ◦ Φi ◦ σ. Note that this sequence has a time translation but no rescaling of the size of the metric. Hence the convergence of v¯i to v¯∞ in (29.118) says that the solutions g(t+ti ) on S 2 , based

10. AN UNRESCALED CIGAR BACKWARD CHEEGER–GROMOV LIMIT

107

at the point N and using the diffeomorphisms Ψi  σ −1 ◦ Φi ◦ σ, converge in the −1 (t)geuc . pointed C ∞ Cheeger–Gromov sense to a cigar soliton g¯∞ (t)  v¯∞ The proof of this proposition occupies the rest of the section. By (29.24), there exists a constant C < ∞ such that 0 < Rg¯i (t) ≤ C

(29.120)

for t ∈ (−∞, −1 − ti ].

We have the following rough, but uniform, estimate. Lemma 29.39. There exists C < ∞ such that on R2 ,  (29.121) −C ≤ ln v¯i (r, θ, 0) ≤ C 1 + r 2 in polar coordinates. Proof. Similarly to as in the proof of Lemma 29.32, we define  wi (r)  ln v¯i (r, θ, 0) dθ. S1

By (29.117), we have (29.122)

 Δeuc ln v¯i (r, θ, 0) dθ    1 ∂ ∂ ln v¯i = r (r, θ, 0) dθ ∂r S 1 r ∂r   dwi 1 d (r) . r = r dr dr

0
0 for r > 0, which implies that r → wi (r) is a strictly increasing function. By this and v¯i (O, 0) = 1, we obtain  (29.123) ln v¯i (r, θ, 0) dθ > 0. S1

Rewriting (29.113) using v¯ = 14 (1 + r 2 )2 v ◦ σ −1 and (29.116), we obtain  2 ∂ ¯ ln vi ≤ C on R2 × (−∞, −1 − ti ]. (29.124) ∂θ This and (29.123) imply that ln v¯i (r, θ, 0) ≥ −C. By (29.122), (29.117), and (29.120), we then have    Rg¯i 1 d dwi (r) = (r, θ, 0) dθ ≤ C r ¯ r dr dr S 1 vi i for some constant C < ∞. Since limr→0 r dw dr (r) = 0, integrating this yields dwi C C 2 ¯ dr (r) ≤ 2 r. Since wi (0) = 2π ln vi (O, 0) = 0, we obtain that wi (r) ≤ 4 r . We conclude by (29.124) the upper bound in (29.121), which proves the lemma. 

Now, by (29.117), (29.121), and (29.120), we obtain that there exists a constant C < ∞ such that for any T ∈ (1, ∞) and any i sufficiently large so that ti +T ≤ −1, (29.125)

−CT ≤ ln v¯i (r, θ, t) ≤ C(T + r 2 )

on R2 × [−T, T ].

By standard parabolic estimates similar to those in the proof of Lemma 28.53, we obtain uniform estimates for the higher derivatives of v¯i on compact subsets of R2 × (−∞, ∞). Thus, by the Arzela–Ascoli theorem, there exists a subsequence

108

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

such that v¯i converges uniformly to a C ∞ function v¯∞ : R2 × (−∞, ∞) → R in each C k on compact subsets. Equation (29.117) and estimate (29.125) imply that ∂ v¯∞ 2 = v¯∞ Δ ln v¯∞ , ∂t where e−CT ≤ v¯∞ (r, θ, t) ≤ eC(T +r

(29.126)

2

)

on R2 × [−T, T ].

Since g¯i (t) = (σ −1 ◦ Φi )∗ g(t + ti ), we have that their limit g¯∞ (t) = v¯∞1(t) geuc is complete and satisfies 0 ≤ Rg¯∞ (t) ≤ C for t ∈ (−∞, ∞). By Theorem 28.41 together with the work of one of the authors in [83], since we have an eternal solution with nonnegative bounded curvature, we conclude that g¯∞ (t) is either a cigar soliton or the flat plane. Proposition 29.38 now follows from Lemma 29.31. 11. Irreducible components of ∇3 v  2 Let S , g (t) , t ∈ (−∞, 0), be an ancient solution of the Ricci flow on a maximal time interval. The trace-free symmetric part of the third covariant derivative of the pressure function v has a remarkable property. Namely, the quantity Q, equaling v times the norm squared of this tensor, is a subsolution to the heat equation. We shall prove this by long but straightforward computations in §12 of this chapter. Moreover, in §13 we show that a nonround g(t) is the King–Rosenau solution if and only if Q vanishes. The main theorem of this chapter then follows from proving that Q = 0. This will be done toward the end of the chapter. 11.1. The quantity Q.  Let x1 , x2 be local coordinates on S 2 . For a covariant 3-tensor α=

2 

αijk dxi ⊗ dxj ⊗ dxk

i,j,k=1

which is symmetric in j and k, its symmetrization is given by 1 1 (29.127) S(α)ijk  α[ijk] = (αijk + αjki + αkij ), 3 3 where [ijk] here and below represents cyclically permuting i, j, and k and summing. Let v(t) be the pressure function of g(t). The third covariant derivative ∇3ijk v = ∂ ∂ ∂ ∇3 v( ∂x i , ∂xj , ∂xk ) is symmetric in its last two components, where ∇ denotes the Riemannian covariant derivative of gS 2 . Define the symmetric 3-tensor B  S(∇3 v).

(29.128)

By the commutator formulas for covariant derivatives, we have (lowering indices and summing over repeated indices) 1 S2 S2 ∇ v + Rkij ∇ v). Bijk = ∇3ijk v − (Rjik 3 S Hence, by Rijk = (gS 2 )i (gS 2 )jk − (gS 2 )ik (gS 2 )j , we have that 2

(29.129a) (29.129b)

Bijk = ∇3ijk v + Aijk , 1 Aijk  − (∇j v(gS 2 )ik + ∇k v(gS 2 )ij − 2∇i v(gS 2 )jk ) . 3

11. IRREDUCIBLE COMPONENTS OF ∇3 v

Since |A|2 =

2 3

109

|∇v|2 and

 3  2 2 ∇ v, A = − (∇3ijk v − ∇3jik v)∇j v(gS 2 )ik = − |∇v|2 , 3 3 we conclude that  2  2 2   (29.130) |B|2 = ∇3 v  + 2 ∇3 v, A + |A|2 = ∇3 v  − |∇v|2 . 3 Let TF (β) stand for the totally trace-free part of a covariant 3-tensor β. We have the orthogonal irreducible decomposition (29.131a) (29.131b)

Bijk = TF(B)ijk + Zi (gS 2 )jk + Zj (gS 2 )ik + Zk (gS 2 )ij ,   1 2,3 1 2 Z  tr (B) = ∇ΔS 2 v + dv , 4 4 3

where trp,q denotes the trace with respect to gS 2 acting on a tensor by metrically contracting its p-th and q-th components. Note that from TF(B), Z ⊗ gS 2  = 0 we see that 2

2

2

|B| = |TF(B)| + 12 |Z| .

(29.132)

Substituting (29.130) into this, we obtain Lemma 29.40. The trace-free part of B = S(∇3 v) satisfies  2 3 2 2 2 (29.133) |TF(B)| = ∇3 v  − |∇ΔS 2 v| − |∇v| − ∇ΔS 2 v, ∇v . 4 Define the key quantity 2

Q  v |TF(B)| .

(29.134)

Note that Q = 0 on a round 2-sphere. In the next section we shall compute the evolution of Q under the Ricci flow on S 2 . Exercise 29.41. Let pij  ∇2ij v + v(gS 2 )ij and P  g ij pij . Show that 1 (∇i P (gS 2 )jk + ∇j P (gS 2 )ik + ∇k P (gS 2 )ij ) . 4 is totally symmetric. Hence   3 TF(B) = S ∇(p − P g) . 4

TF(B)ijk = ∇i pjk − Show also that ∇i pjk

Remark 29.42. Alternatively, we can obtain TF(B) as follows. The irreducible decomposition for ∇3 v on S 2 is given by (29.135)

Hijk = ∇3ijk v + Xi (gS 2 )jk + Yj (gS 2 )ik + Yk (gS 2 )ij ,

where the 1-forms X and Y are defined so that 0 = tr2,3 (H) = ∇ΔS 2 v + 2X + 2Y

and

0 = tr1,2 (H) = ΔS 2 ∇v + X + 3Y.

This implies that (29.136)

Y =−

∇ΔS 2 v dv − 4 2

and

X=−

∇ΔS 2 v dv + . 4 2

110

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

The 3-tensor H is fully symmetric and trace-free and is easily seen to be H = TF(B). By (29.135), (29.136), and H, X ⊗ gS 2  = H, Y ⊗ gS 2  = 0, we see that (29.133) also follows from  2 (29.137) |TF(B)|2 = |H|2 = ∇3 v  − 2 |X|2 − 6 |Y |2 − 4 X, Y  .

11.2. The vanishing of Q on the King–Rosenau solution. Consider the King–Rosenau ancient solution (S 2 , gKR (t)). For simplicity, we take μ = 1 in (29.17). Differentiating (29.8) then yields dv KR = −β sin 2ψdψ, where ψ is the latitude. In this discussion we shall treat β as a positive constant, although we may assume that it satisfies (29.17). Using formula (29.18) for the Christoffel symbols, we compute that the Hessian and Laplacian of v are given by 1 2 KR 1 ∇ v = −2 cos 2ψdψ ⊗ dψ + sin2 2ψdθ ⊗ dθ, β 2 1 ΔS 2 v KR = −2 cos2 ψ + 4 sin2 ψ, (29.138b) β  where θ is the longitude. Let x1 , x2 = (ψ, θ). Since ∇2ij ψ = −Γ1ij and ∇2ij θ = −Γ2ij , the Hessian of the latitude and the longitude are (29.138a)

∇2 ψ = −

sin 2ψ dθ ⊗ dθ, 2

∇2 θ = tan ψ (dψ ⊗ dθ + dθ ⊗ dψ) .

Hence, covariantly differentiating (29.138a), we obtain (29.139)

1 3 KR ∇ v = 4 sin 2ψdψ ⊗ dψ ⊗ dψ + 2 sin 2ψ cos2 ψdψ ⊗ dθ ⊗ dθ β + sin 2ψ cos2 ψ (dθ ⊗ dθ ⊗ dψ + dθ ⊗ dψ ⊗ dθ) .

Equivalently, we may compute directly that the components of ∇3 v KR are given by ∇1 ∇1 ∇1 v KR = 4β sin 2ψ, ∇1 ∇2 ∇2 v KR = 2β sin 2ψ cos2 ψ = 2∇2 ∇1 ∇2 v, ∇1 ∇1 ∇2 v KR = ∇2 ∇1 ∇1 v = ∇2 ∇2 ∇2 v = 0. From this we can derive the following identities for the King–Rosenau solution: 2 1  3 KR 2 1  ∇ v ∇ΔS 2 v KR  = 22 36  KR 2   1 = ∇v  = − ∇ΔS 2 v KR , ∇v KR . 6

β 2 sin2 2ψ =

Substituting these formulas into (29.133) yields that 2  QKR  v KR TF(B KR ) = 0. Alternatively, from β1 dv KR = − sin 2ψdψ and β1 ∇ΔS 2 v KR = 6 sin 2ψdψ, we find this vanishing of QKR directly from (29.139), (29.135), and H = TF(B). Let TFS  TF ◦ S be the trace-free part of the symmetrization.

12. THE HEAT-TYPE EQUATION SATISFIED BY Q

111

Exercise 29.43. (1) Show that

 S ∇3 v KR = 4β sin 2ψ S (dψ ⊗ gS 2 ) .  From this we again see that TF(B KR ) = TFS ∇3 v KR = 0. (2) Using (29.9) and that ∇2 z + zgS 2 = 0 on S 2 , where {x, y, z} are the Euclidean coordinates on R3 , show that 1 1 KR ∇3 v KR = −dv KR ⊗ gS 2 − gS 2 ⊗ dv KR − gS 2 (13) ⊗ dv(2) , 2 2 where (gS 2 (13) ⊗ dv(2) ) (X1 , X2 , X3 )  gS 2 (X1 , X3 ) dv (X2 ). 12. The heat-type equation satisfied by Q

Let g(t), t ∈ (−∞, 0), be a maximal ancient solution to the Ricci flow on S 2 and let v(t) be its pressure function. Throughout this section, the covariant derivative ∇, Laplacian ΔS 2 , and norms of tensors are all defined with respect to gS 2 . 2 We have the following nice evolution equation for Q = v |TF(B)| , as defined in (29.134). Proposition 29.44 (Q is a subsolution of the heat equation). We have (29.140)

∂ 2 Q = vΔS 2 Q − 4RQ − 2 |TF (v∇ TF(B) + 2dv ⊗ TF(B))| ∂t  2 − 2 tr1,2 (v∇ TF(B) − dv ⊗ TF(B)) .

In particular, since R > 0, we have that

∂ ∂t Q

≤ Δg Q.

The rest of this section is devoted to the proof of this proposition. We begin by calculating the evolution of B = S(∇3 v). Using (29.7), we compute that   ∂ (∇3ijk v) = ∇3ijk vΔS 2 v − |∇v|2 + 2v 2 ∂t   2

= v∇3ijk ΔS 2 v + ΔS 2 v∇3ijk v + ∇3ijk − |∇v| + 2v 2

+ ∇i (∇k v∇j ΔS 2 v) + ∇j (∇i v∇k ΔS 2 v) + ∇k (∇j v∇i ΔS 2 v) . By symmetrizing both sides of this equation, we obtain ∂ S(∇3 v) = v S(∇3 ΔS 2 v) + ΔS 2 v S(∇3 v) + 3 S(∇ (dv ⊗ ∇ΔS 2 v)) (29.141) ∂t + S(∇3 (− |∇v|2 + 2v 2 )). On the other hand, for a function f on a Riemannian manifold (Mn , g), we have (see (2.34) in [77] for example) ∇2ij Δg f = Δg ∇2ij f + (−∇j Ri + ∇ Rij − ∇i Rj )∇ f − 2Rikjp ∇2pk f − Ri ∇2j f − Rj ∇2i f. Hence, for v on (S 2 , gS 2 ), we have (29.142)

∇2jk ΔS 2 v = ΔS 2 ∇2jk v + 2ΔS 2 v(gS 2 )jk − 4∇2jk v.

Taking another covariant derivative, we have (29.143)

∇3ijk ΔS 2 v = ∇i ΔS 2 ∇2jk v + 2∇i ΔS 2 v(gS 2 )jk − 4∇3ijk v.

112

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

The first term on the rhs is equal to ∇i ΔS 2 ∇2jk v = ∇5ijk v S S S = ∇5ijk v − Rim ∇3mjk v − Rijm ∇3mk v − Rikm ∇3jm v 2

2

2

S S S = ΔS 2 ∇3ijk v − Rim ∇3mjk v − 2Rijm ∇3mk v − 2Rikm ∇3jm v 2

2

2

= ΔS 2 ∇3ijk v − ∇3ijk v + 2(−∇3jik v − ∇3kij v) + 2 ((gS 2 )ij (∇k ΔS 2 v + ∇k v) + (gS 2 )ik (∇j ΔS 2 v + ∇j v)) . Symmetrizing this formula, we obtain S(∇ΔS 2 ∇2 v) = ΔS 2 S(∇3 v) − 5 S(∇3 v) + 4 S((∇ΔS 2 v + dv) ⊗ gS 2 ) since S and ΔS 2 commute. Therefore, by (29.143), we have (29.144)

S(∇3 ΔS 2 v) = ΔS 2 S(∇3 v) − 9 S(∇3 v) + 24 S(Z ⊗ gS 2 ),

where Z = 14 (∇ΔS 2 v + 23 dv) as in (29.131b). Now substituting (29.144) into (29.141) yields Lemma 29.45. The 3-tensor B = S(∇3 v) satisfies the following heat-type equation (29.145)

∂ B = vΔS 2 B + (ΔS 2 v − 9v) B + 24v S(Z ⊗ gS 2 ) ∂t 2 + 3 S(∇ (dv ⊗ ∇ΔS 2 v)) + S(∇3 (− |∇v| + 2v 2 )).

We now proceed to prove the proposition by carrying out a long derivation of the formula for Q = v |TF(B)|2 . We are guided by collecting terms to form complete squares. Step 1. Taking the trace-free part of (29.145) yields (29.146) ∂ TF(B) = vΔS 2 TF(B) + (ΔS 2 v − 9v) TF(B) + 24v TFS(Z ⊗ gS 2 ) ∂t + TFS(3∇2 v ⊗ ∇ΔS 2 v + 3dv ⊗ ∇2 ΔS 2 v − ∇3 |∇v|2 + 2∇3 (v 2 )), where TFS = TF ◦ S. On the other hand,     ∂ ∂ − vΔS 2 Q = 2v TF(B) · − vΔS 2 (TF(B)) − 2v 2 |∇ TF(B)|2 ∂t ∂t   ∂ 2 2 − vΔS 2 v. − 2v∇v · ∇ |TF(B)| + |TF(B)| ∂t By combining the above two displays, while applying (29.7) and the fact that TFS(Z ⊗ gS 2 ) = 0, we obtain   ∂ 2 − vΔS 2 Q = −2v 2 |∇ TF(B)| − 4v(dv ⊗ TF(B)) · ∇ TF(B) (29.147) ∂t + (2vΔS 2 v − |∇v|2 − 16v 2 ) |TF(B)|2 + 4v TF(B) · TFS(∇3 (v 2 )) + 6v TF(B) · TFS(∇2 v ⊗ ∇ΔS 2 v) − 2v TF(B) · TFS(∇3 |∇v|2 ) + 6v TF(B) · TFS(dv ⊗ ∇2 ΔS 2 v).

12. THE HEAT-TYPE EQUATION SATISFIED BY Q

113

Toward the goal of obtaining nonpositive terms on the rhs of (29.147), we first rewrite the last four lines therein.  Third line. Since S(∇3 (v 2 )) = 2vB + 6 S dv ⊗ ∇2 v by a direct calculation, we have  (29.148) TF(B) · TFS(∇3 (v 2 )) = 2v |TF(B)|2 + 6 TF(B) · TFS dv ⊗ ∇2 v . Fourth line. Since TF(B) is totally trace-free and symmetric, we have  (29.149) TF(B) · TFS(∇2 v ⊗ ∇ΔS 2 v) = TF(B) · ∇ΔS 2 v ⊗ ∇2 v . Fifth line. We compute that (29.150)

 1 2 TFS(∇3 |∇v| ) = TFS(dv 1,4 ∇4 v) + 3 TFS ∇3 v 3,2 ∇2 v , 2

where U p,q V denotes a single tensor contraction of the p-th component of U with the q-th component of V . A computation yields the commutator formula (we use ∇ RmS 2 = 0 in the first line) (29.151)

S ∇2im v ∇4ijk v = ∇4ijk v − Rjkm 2

S S S = ∇4ijk v − Rijm ∇2mk v − Rikm ∇2jm v − Rjkm ∇2im v 2

2

2

= ∇4ijk v − 2(gS 2 )k ∇2ij v − (gS 2 )j ∇2ik v + (gS 2 )ij ∇2k v + (gS 2 )ik ∇2j v + (gS 2 )jk ∇2i v, and consequently, ∇ v∇4ijk v = ∇ v∇4ijk v − 2∇k v∇2ij v − ∇j v∇2ik v + (gS 2 )ij ∇ v∇2k v + (gS 2 )ik ∇ v∇2j v + (gS 2 )jk ∇ v∇2i v. Thus, by taking trace-free symmetric parts of this, we have  TFS(dv 1,4 ∇4 v) = dv 1,1 ∇ TF(B) − 3 TFS dv ⊗ ∇2 v . Substituting this into (29.150) and taking the inner product with −4v TF(B) yields (29.152)

2

−2v TF(B) · TFS(∇3 |∇v| ) = −4v(dv ⊗ TF(B)) · ∇ TF(B)  + 12v TF(B) · TFS dv ⊗ ∇2 v  − 12v TF(B) · TFS ∇3 v 3,2 ∇2 v .

Sixth line. Since TF(B) is totally trace-free and symmetric, we have (29.153)

   1 2 2 TF(B) · TFS(dv ⊗ ∇ ΔS 2 v) = TF(B) · dv ⊗ ∇ ΔS 2 v − ΔS 2 vgS 2 2   1 2 1,2 2 = tr (dv ⊗ TF(B)) · ∇ ΔS 2 v − ΔS 2 vgS 2 . 2 2

114

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Applying (29.148), (29.149), (29.152), (29.153), and Rv = (vΔS 2 v−|∇v|2 +2v 2 ) to (29.147) and completing a square, we obtain   ∂ 2 2 − vΔS 2 Q = −2 |v∇ TF(B) + 2dv ⊗ TF(B)| − 4Rv |TF(B)| (29.154) ∂t + (6vΔS 2 v + 3 |∇v|2 ) |TF(B)|2  + 36v TF(B) · TFS dv ⊗ ∇2 v  + 6v TF(B) · ∇ΔS 2 v ⊗ ∇2 v  − 12v TF(B) · TFS ∇3 v 3,2 ∇2 v   1 2 1,2 2 + 6v tr (dv ⊗ TF(B)) · ∇ ΔS 2 v − ΔS 2 vgS 2 . 2 Step 2. To evaluate the norm of (29.155)

α  v∇ TF(B) + 2dv ⊗ TF(B),

which occurs in the first term on the rhs of (29.154), we need the following. Given  a 4-tensor c = 2i,j,k,=1 cijk dx ⊗ dxi ⊗ dxj ⊗ dxk which is both symmetric and trace-free in the last three slots, an orthogonal decomposition for c is given by   1 (29.156) cijk = TF(c)ijk + δ[i ejk] − δ[jk ei] , 2 where 1 i 1 δ cijk = tr1,2 (c)jk . 3 3 The tensors TF(c) and e are totally trace-free. We compute that ejk 

(29.157)

2

2

|c| − | TF(c)|2 = 9 |e| = | tr1,2 (c)|2 .

In particular, (29.158)

 2 |α|2 = |TF(α)|2 + tr1,2 (α) .

First, we compute tr1,2 (α). We have   1,2 tr (∇B) jk = ∇ S(∇3 v) iijk 1 4 (∇ v + ∇4ijik v + ∇4ikij v) 3 iijk 1 S2 2 1 S2 2 = ∇4iijk v − Rjik ∇i v − Rkij ∇i v 3 3 2 2 = ΔS 2 ∇2jk v − ∇2jk v + ΔS 2 v(gS 2 )jk . 3 3 =

Now applying (29.142) to this yields tr1,2 (∇B) = ∇2 ΔS 2 v +

10 2 4 ∇ v − ΔS 2 vgS 2 . 3 3

Since by (29.131a) we have that B = TF(B) + 3 S(Z ⊗ gS 2 ), it follows that tr1,2 (∇ TF(B)) = tr1,2 (∇B) − 3 tr1,2 (∇ S(Z ⊗ gS 2 )),

12. THE HEAT-TYPE EQUATION SATISFIED BY Q

115

where Z = 14 ∇ΔS 2 v + 16 dv. Now 3 tr1,2 (∇ S(Z ⊗ gS 2 ))jk = ∇i (Zi (gS 2 )jk + Zj (gS 2 )ik + Zk (gS 2 )ij ) = div(Z)(gS 2 )jk + ∇k Zj + ∇j Zk . Therefore (29.159) 10 4 tr1,2 (∇ TF(B)) = ∇2 ΔS 2 v + ∇2 v − ΔS 2 vgS 2 − div(Z)gS 2 − 2 S(∇Z) 3 3     1 2 1 1 2 2 = ∇ ΔS 2 v − ΔS 2 vgS 2 + 3 ∇ v − ΔS 2 vgS 2 . 2 2 2 We conclude that α in (29.155) satisfies     v 1 1 tr1,2 (α) = (29.160) ∇2 ΔS 2 v − Δ2S 2 vgS 2 + 3v ∇2 v − ΔS 2 vgS 2 2 2 2 + 2 tr1,2 (dv ⊗ TF(B)). Second, for the second line of (29.154), we shall derive  2 |∇v|2 |TF(B)|2 = 2 tr1,2 (dv ⊗ TF(B)) .

(29.161)

Given any point p ∈ S 2 , we shall compute in local coordinates where gij = δij at p. We have 1 3 ∇ v− 4 111 1 = ∇3222 v − 4

(29.162a) TF(B)111 = − TF(B)122 = − TF(B)212 = (29.162b) TF(B)222 = − TF(B)211 = − TF(B)121

3 3 ∇ v− 4 122 3 3 ∇ v− 4 211

1 ∇1 v, 2 1 ∇2 v. 2

This implies that (29.163)

2 

|TF(B)|2 =

(TF(B)ijk )2 = 4((TF(B)111 )2 + (TF(B)222 )2 ).

i,j,k=1

Using (29.162) and (29.163), we calculate   1,2 tr (dv ⊗ TF(B))2 = (∇1 v TF(B)111 )2 + (∇1 v TF(B)122 )2 2

2

+ 2 (∇1 v TF(B)112 ) + (∇2 v TF(B)211 ) 2

2

+ (∇2 v TF(B)222 ) + 2 (∇2 v TF(B)212 ) 2

2

2

= 2 (∇1 v) (TF(B)111 ) + 2 (∇1 v) (TF(B)222 ) 2

2

2

2 2

+ 2 (∇2 v) (TF(B)222 ) + 2 (∇2 v) (TF(B)111 ) 1 = |∇v|2 |TF(B)|2 . 2 This establishes (29.161).

116

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Third, regarding the third line of equation (29.154), observe that by using that TF(B) is totally trace-free and symmetric, we have (29.164)

    1 2 2 TF(B) · TFS dv ⊗ ∇ v = TF(B) · dv ⊗ ∇ v − ΔS 2 vgS 2 2   1 1,2 2 2 2 = tr (dv ⊗ TF(B)) · ∇ v − ΔS vgS . 2

Now, by applying to (29.154) the formulas (29.160), (29.161), and (29.164), we obtain (29.165)   ∂ − vΔS 2 + 4R Q + 2 |TF (v∇ TF(B) + 2dv ⊗ TF(B))|2 ∂t   2    v  Δ2S 2 vgS 2 ΔS 2 vgS 2 2 2 1,2  = −2  ∇ ΔS 2 v − + 3v ∇ v − + 2 tr (dv ⊗ TF(B)) 2 2 2  1,2 2 2   + 6vΔS 2 v |TF(B)| + 6 tr (dv ⊗ TF(B))    ΔS 2 vgS 2 1,2 2 + 36v tr (dv ⊗ TF(B)) · ∇ v − + 6v TF(B) · ∇ΔS 2 v ⊗ ∇2 v 2  − 12v TF(B) · TFS ∇3 v 3,2 ∇2 v   1 2 1,2 2 + 6v tr (dv ⊗ TF(B)) · ∇ ΔS 2 v − ΔS 2 vgS 2 . 2 Step 3. We can improve this by completing the square on the second line in a slightly different way, which reduces the number of terms, as follows: (29.166)   ∂ 2 − vΔS 2 + 4R Q + 2 |TF (v∇ TF(B) + 2dv ⊗ TF(B))| ∂t 2       v Δ2 2 vgS 2 Δ 2 vg 2 = −2  ∇2 ΔS 2 v − S + 3v ∇2 v − S S − tr1,2 (dv ⊗ TF(B)) 2 2 2  2 2 + 6vΔS 2 v |TF(B)| + 6v TF(B) · ∇ΔS 2 v ⊗ ∇ v  − 12v TF(B) · TFS ∇3 v 3,2 ∇2 v . We claim that (29.167)   ΔS 2 v |TF(B)|2 + TF(B) · ∇ΔS 2 v ⊗ ∇2 v − 2 TF(B) · TFS ∇3 v 3,2 ∇2 v = 0. By (29.159), this completes the proof of Proposition 29.44. To see the claim, we compute in local coordinates that (29.168)

TF(B) · (∇ΔS 2 v ⊗ ∇2 v) = TF(B)111 ((∇211 v − ∇222 v)∇1 ΔS 2 v − 2∇212 v∇2 ΔS 2 v) + TF(B)222 ((∇222 v − ∇211 v)∇2 ΔS 2 v − 2∇212 v∇1 ΔS 2 v).

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING–ROSENAU SOLUTION

117

We also compute  TF(B) · TFS ∇3 v 3,2 ∇2 v = TF(B) · ∇3 v 3,2 ∇2 v  = TF(B)111 ∇311 v∇21 v − ∇312 v∇22 v − ∇321 v∇22 v − ∇322 v∇21 v  + TF(B)222 −∇311 v∇22 v − ∇312 v∇21 v − ∇321 v∇21 v + ∇322 v∇22 v  = TF(B)111 ∇3111 v − ∇3221 v ∇211 v  + TF(B)111 ∇3112 v − ∇3121 v − ∇3211 v − ∇3222 v ∇212 v   + TF(B)111 −∇3122 v − ∇3212 v ∇222 v + TF(B)222 ∇3222 v − ∇3112 v ∇222 v  + TF(B)222 ∇3221 v − ∇3212 v − ∇3122 v − ∇3111 v ∇212 v  + TF(B)222 −∇3211 v − ∇3121 v ∇211 v. Hence, by this and (29.168), while using ∇i ΔS 2 v = ∇3i11 v + ∇3i22 v and ΔS 2 v = ∇211 v + ∇222 v, we obtain   TF(B) · ∇ΔS 2 v ⊗ ∇2 v − 2 TF(B) · TFS ∇3 v 3,2 ∇2 v  = − TF(B)111 ∇3111 v − ∇3122 v − 2∇3221 v ΔS 2 v  − TF(B)222 ∇3222 v − ∇3211 v − 2∇3112 v ΔS 2 v. Now, commuting covariant derivatives and using (29.162), we have (29.169a)

∇3111 v − ∇3122 v − 2∇3221 v = ∇3111 v − 3∇3122 v − 2∇1 v = 4 TF(B)111 ,

(29.169b)

∇3222 v − ∇3211 v − 2∇3112 v = ∇3222 v − 3∇3211 v − 2∇2 v = 4 TF(B)222 .

From the above formulas and (29.163), we obtain (29.167). This finishes the proof of Proposition 29.44.  13. That Q = 0 implies the solution is the King–Rosenau solution Let Q be the quantity defined by (29.134) for a maximal ancient solution (S 2 , g(t)), t ∈ (−∞, 0). In this section we prove the following characterization of the King–Rosenau solution. Proposition 29.46 (Q ≡ 0 implies being the King–Rosenau solution). We have that Q ≡ 0 if and only if g (t) is either round or the King–Rosenau solution. Throughout this section, let x = x1 and y = x2 denote the standard Euclidean coordinates, let ∇ = ∂, and let | · | denote the Euclidean norm. 13.1. Plane version of Q. To prove the main proposition of this section, we shall find it convenient to consider the plane version of Q. Computations in the plane, as compared to computations on the round 2-sphere, are slightly simplified since partial derivatives commute. Recall by (29.7) that the pressure function v¯ satisfies (29.170)

∂¯ v 2 = v¯Δeuc v¯ − |∇¯ v| . ∂t

118

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

¯ of Q, we first consider the irreducible orthogonal To define the plane version Q 2 ¯ijk dxi ⊗ dxj ⊗ dxk on R2 . The decomposition for the 3-tensor ∂ 3 v¯  i,j,k=1 v decomposition is given via defining 1 (29.171) a ¯ijk  v¯ijk − (Δeuc v¯i δjk +Δeuc v¯j δik +Δeuc v¯k δij ) . 4 2 ¯ijk dxi ⊗ dxj ⊗ dxk is totally trace-free and totally The 3-tensor a ¯ = i,j,k=1 a 2 symmetric. Define on R the quantity ¯  v¯ |¯ (29.172) Q a |2 . ¯ is scale-invariant in the following sense. Given K > 0 and Observe that Q t0 < 0, define ¯(Kx, Ky, t0 + t), u ˜(x, y, t) = K 2 u

v˜(x, y, t) = K −2 v¯(Kx, Ky, t0 + t).

Then (29.173)

2 ¯ ˜ y, t)  v˜ |˜ Ky, t0 + t), Q(x, a| (x, y, t) = Q(Kx,

where a ˜ijk  v˜ijk −

1 (Δeuc v˜i δjk +Δeuc v˜j δik +Δeuc v˜k δij ) . 4

Example 29.47. (1) On the King–Rosenau solution we compute, using (29.11), that v¯111 = ¯ = 0. v112 , so that Q 12αx = 3¯ v221 and that v¯222 = 12αy = 3¯ 2 −1 (2) The cigar soliton metric on R is gcig = v¯ geuc , where v¯ = 1 + r 2 . Thus, ¯ = 0. for the cigar soliton we have that v¯ijk = 0 and hence Q ¯ (3) Q = 0 for a round 2-sphere. Next, we observe formulas for a ¯ and |¯ a|2 . We compute that 1 1 (29.174) a ¯111 = (¯ v111 − 3¯ v222 − 3¯ v122 ) = −¯ a122 and a ¯222 = (¯ v112 ) = −¯ a112 4 4 are all of the components of a ¯ up to symmetry. Since the decomposition (29.171) is orthogonal, we find that |∂ 3 v¯|2 =

2 

(¯ vijk )2

and

i,j,k=1

|¯ a|2 =

2 

(¯ aijk )2

i,j,k=1

satisfy

 3 1 (¯ v111 − 3¯ |∂ 3 v¯|2 − |∂Δeuc v¯|2 = |¯ a|2 = v221 )2 + (¯ v222 − 3¯ v112 )2 . 4 4 13.2. Proof of Proposition 29.46. ¯ ≡ 0. By Lemma 29.50 below, our hypothesis is equivalent to Q Step 1. For each t ∈ (−∞, 0) there exist (x0 (t), y0 (t)) ∈ R2 , A(t) > 0, and a quadratic polynomial q(x, y, t) such that

(29.175)

v¯(x, y, t) = A(t)((x − x0 (t))2 + (y − y0 (t))2 )2 + q(x, y, t). ¯ ≡ 0, the components of a Fix a time t. Since Q ¯ in (29.174) vanish and we have

(29.176) (29.177)

v122 = 0 v¯111 − 3¯

and

v¯222 − 3¯ v112 = 0.

Differentiating this implies that (29.178)

v1221 = 3¯ v1122 = v¯2222 . v¯1111 = 3¯

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING–ROSENAU SOLUTION

119

We also have that v¯1112 = 3¯ v1222

and

v¯2221 = 3¯ v1121 ,

which imply that v¯1112 = v¯2221 = 0. Hence we have that v¯111 is a function of x only and that v¯222 is a function of y only. Thus, the v¯1111 = v¯2222 in (29.178) is a constant. We obtain that v¯111 = 2(C1 x + C2 )

and

v¯222 = 2(C1 y + C3 )

for some constants C1 (t), C2 (t), and C3 (t). Integrating this yields v¯11 = C1 x2 + 2C2 x + f1 , v¯22 = C1 y 2 + 2C3 y + f2 for some functions f1 (y, t) and f2 (x, t). Then (29.177) implies 2C1 x + 2C2 = 3∂x f2

and

2C1 y + 2C3 = 3∂y f1 .

We conclude that 1 v¯11 = C1 x2 + 2C2 x + (C1 y 2 + 2C3 y) + C4 , 3 1 2 v¯22 = C1 y + 2C3 y + (C1 x2 + 2C2 x) + C5 , 3 where C4 (t) and C5 (t) are constants. Taking the derivative of this, we have that 2 (C1 y + C3 ) 3 Integrating this, we obtain v¯112 =

v¯12 =

and

v¯221 =

2 (C1 x + C2 ). 3

2 (C1 xy + C3 x + C2 y) + C6 , 3

where C6 (t) is a constant. Summarizing, we have with (29.179)

x ˜x+

C2 C1

and

y˜  y +

C3 C1

that 1 v¯11 = C1 x ˜2 + C1 y˜2 + C7 , 3 1 ˜2 + C1 y˜2 + C8 , v¯22 = C1 x 3 2 v¯12 = C1 x ˜y˜ + C9 , 3 for some constants C7 (t), C8 (t), and C9 (t). From this we conclude that q  v¯ −

2 C1  2 x ˜ + y˜2 12

satisfies q11 = C7 ,

q22 = C8 ,

q12 = C9 .

Therefore q is a quadratic polynomial in x and y. This completes Step 1.

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29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

By Proposition 29.36 and by assuming without loss of generality that μ = 1, we have that (29.180)

lim v¯(x, y, t) = x2 + y 2

t→−∞

uniformly in C ∞ on compact subsets of R2 . Step 2. There exist smooth functions A(t) > 0, B(t), and C(t) > 0 such that  2  (29.181) v¯(x, y, t) = A(t) x2 + y 2 + B(t) x2 + y 2 + C(t). By (29.176), we have that  2 2 2 v¯(x, y, t) = A(t) (x − x0 (t)) + (y − y0 (t)) (29.182) 2

2

+ B(t) (x − x1 (t)) + E(t) (y − y1 (t)) + F (t)xy + C(t) for some smooth functions A(t), B(t), C(t), E(t), F (t), x0 (t), y0 (t), x1 (t), and y1 (t). We will use equation (29.170) to determine these functions. First, by substituting (29.182) into the terms on the rhs of (29.170), we have that   2 2 2 2 v¯Δeuc v¯ = A((x − x0 ) + (y − y0 ) )2 + B(x − x1 ) + E(y − y1 ) + F xy + C   × 16A((x − x0 )2 + (y − y0 )2 ) + 2B + 2E and

 2 |∇¯ v |2 = 4A((x − x0 )2 + (y − y0 )2 ) (x − x0 ) + 2B (x − x1 ) + F y  2 + 4A((x − x0 )2 + (y − y0 )2 ) (y − y0 ) + 2E (y − y1 ) + F x .

Hence, by canceling terms and expanding in powers of x − x0 and y − y0 (except for the xy term), we obtain (29.183) v¯Δeuc v¯ − |∇¯ v |2 = 2A (B + E) ((x − x0 )2 + (y − y0 )2 )2 + 8A (F y0 + 2B (x0 − x1 )) (x − x0 ) ((x − x0 )2 + (y − y0 )2 ) + 8A (F x0 + 2E (y0 − y1 )) (y − y0 ) ((x − x0 )2 + (y − y0 )2 )  

2 2 16A B (x0 − x1 ) + E (y0 − y1 ) + (x − x0 )2 +2B (E − B) − F 2 + 16A (C + F x0 y0 )  

16A B (x0 − x1 )2 + E (y0 − y1 )2 2 + (y − y0 ) 2 +2E (B − E) − F + 16A (C + F x0 y0 ) − 2F (B + E) xy  + 4B (E − B) (x0 − x1 ) − 2F 2 x0 + 4EF y1 (x − x0 )  + 4E (B − E) (y0 − y1 ) − 2F 2 y0 + 4BF x1 (y − y0 ) + 2B (E − B) (x0 − x1 )2 + 2E (B − E) (y0 − y1 )2  + 2C (B + E) − F 2 x20 + y02 + 4BF x1 y0 + 4EF x0 y1 .

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING–ROSENAU SOLUTION

121

Second, by substituting (29.182) into the lhs of (29.170), we have that (where  denotes a time derivative) (29.184) ∂¯ v 2 2 2 2 = A ((x − x0 ) + (y − y0 ) )2 + B  (x − x1 ) + E  (y − y1 ) + F  xy + C  ∂t − 4Ax0 (x − x0 ) ((x − x0 )2 + (y − y0 )2 ) − 4Ay0 (y − y0 ) ((x − x0 )2 + (y − y0 )2 ) − 2B (x − x1 ) x1 − 2E (y − y1 ) y1 . We rewrite this in a form analogous to (29.183): (29.185) ∂¯ v 2 2 = A ((x − x0 ) + (y − y0 ) )2 ∂t 2 2 − 4Ax0 (x − x0 ) ((x − x0 ) + (y − y0 ) ) − 4Ay0 (y − y0 ) ((x − x0 )2 + (y − y0 )2 ) + B  (x − x0 )2 + E  (y − y0 )2 + F  xy + 2 (B  (x0 − x1 ) − Bx1 ) (x − x0 ) + 2 (E  (y0 − y1 ) − Ey1 ) (y − y0 ) + B  (x0 − x1 ) + E  (y0 − y1 ) − 2Bx1 (x0 − x1 ) − 2Ey1 (y0 − y1 ) + C  . 2

2

By (29.170) and by equating the coefficients in (29.183) and (29.185), we obtain (29.186a)

A = 2A (B + E) ,

(29.186b)

x0 = −2F y0 + 4B (x1 − x0 ) ,

(29.186c) (29.186d)

y0 = −2F x0 + 4E (y1 − y0 ) ,   2 2 B  = 16A B (x0 − x1 ) + E (y0 − y1 ) + F x0 y0 + C

(29.186e)

+ 2B (E − B) − F 2 ,   E  = 16A B (x0 − x1 )2 + E (y0 − y1 )2 + F x0 y0 + C + 2E (B − E) − F 2 ,

(29.186f)

F  = −2F (B + E) ,

(29.186g)

x1 =

(29.186h)

B 2EF y1 − F 2 x0 (x0 − x1 ) − 2 (E − B) (x0 − x1 ) − , B B  2 E 2BF x1 − F y0 y1 = (y0 − y1 ) − 2 (B − E) (y0 − y1 ) − , E E

and (29.187) C  = −B  (x0 − x1 )2 − E  (y0 − y1 )2 + 2Bx1 (x0 − x1 ) + 2Ey1 (y0 − y1 ) + 2 (E − B) B (x0 − x1 )2 + 2 (B − E) E (y0 − y1 )2 + 2 (B + E) C  − F 2 x20 + y02 + 4BF x1 y0 + 4EF x0 y1 .

122

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Observe that by taking the difference of (29.186d) and (29.186e), we obtain the simple equation (29.188)

(B − E) = −2 (B + E) (B − E) .

Now, expanding (29.182) yields  2    v¯ = A x2 + y 2 − 4A (x0 x + y0 y) x2 + y 2 + 2A x20 + y02 x2 + y 2  + 4A x20 x2 + y02 y 2 + Bx2 + Ey 2 + 8Ax0 y0 xy + F xy  − 2Bx1 x − 2Ey1 y − 4A x20 + y02 (x0 x + y0 y)  2 + A x20 + y02 + Bx21 + Ey12 + C. Reconciling this with just the C 0 convergence in (29.180), we obtain the following limits as t → −∞: (29.189a)

A → 0,

(29.189b)

Ax0 → 0,

(29.189c)

Ay0 → 0,  2 B + 2A 3x0 + y02 → 1,  E + 2A x20 + 3y02 → 1,

(29.189d) (29.189e) (29.189f) (29.189g) (29.189h) (29.189i)

8Ax0 y0 + F  2 −2Bx1 − 4A x0 + y02 x0  −2Ey1 − 4A x20 + y02 y0 2  A x20 + y02 + Bx21 + Ey12 + C

→ 0, → 0, → 0, → 0.



By (29.186a) and (29.186f), we have (AF ) = 0, so that AF is constant. By (29.189a), (29.189b), (29.189c), and (29.189f), we obtain that  0 = lim 8A2 x0 y0 + AF = lim AF. t→−∞

t→−∞

Hence AF ≡ 0. Claim. A(t) > 0, which implies F ≡ 0. To see the claim, first note that v¯ > 0 implies that A(t) ≥ 0. Now suppose that A(t0 ) = 0 for some t0 . Then (29.186a) implies that A(t) ≡ 0. Hence (29.189d) and (29.189e) imply that B(t) → 1 and E(t) → 1 as t → −∞. Hence, by (29.188), we have B (t) = E (t). We also have that (29.186f) and (29.189f) imply that F (t) ≡ 0. So (29.186d) implies that B(t) ≡ 1. Therefore v¯(x, y, t) = (x − x1 (t))2 + (y − y1 (t))2 + C(t), which yields a noncompact solution. This contradiction proves the claim. Now (29.189f) implies that Ax0 y0 → 0.

(29.190)

By (29.189a), (29.189b), (29.189c), (29.189d), and (29.189e), we have   (29.191) 0 = lim A = lim AB + 2A2 3x20 + y02 = lim AB t→−∞

t→−∞

and (29.192)

0 = lim A = lim t→−∞

t→−∞

t→−∞

  AE + 2A2 x20 + 3y02 = lim AE. t→−∞

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING–ROSENAU SOLUTION

123

By (29.186a) and (29.188), we find that A (B − E) is constant. This and (29.191) and (29.192) imply that B = E. By (29.189d), (29.189e), and B = E, we have A(x20 − y02 ) → 0. On the other hand, (29.190) implies that A2 x20 y02 → 0. Therefore 0 = lim A2 (x20 − y02 )2 = lim A2 (x20 + y02 )2 , t→−∞

t→−∞

so that A(x20 + y02 ) → 0.

(29.193)

From (29.189d), we conclude that B = E → 1. This, (29.189g), (29.189h), and (29.193) imply that (29.194a)

A1/2 x1 → 0,

(29.194b)

A1/2 y1 → 0.

Thus, by (29.189i), we have AC → 0. Let X (t)  x0 (t) − x1 (t) and Y (t)  y0 (t) − y1 (t). Since F = 0 and B = E, by (29.186) we have (29.195a)

x0 = −4BX,

(29.195b)

y0 = −4BY,   C 2 2 = 16AX X + Y + , B   C . y1 = 16AY X 2 + Y 2 + B

x1

Thus

  B C 2 2 , X = −16AX X + Y + + B 4A   B C . Y  = −16AY X 2 + Y 2 + + B 4A 

Define Z  X 2 + Y 2 . Then

  AC B Z  = −32Z AZ + + . B 4

Combining this with A = 4AB, we obtain   B AC  (AZ) = −32AZ AZ + + . B 8 Since limt→−∞ AZ = 0 by (29.193) and (29.194), we obtain AZ ≡ 0. Because Z(t) = 0, by (29.195) we have that x0 (t) = x1 (t) and y0 (t) = y1 (t) are independent of t. Then (29.180) implies that x0 (t) ≡ y0 (t) ≡ 0 and hence x1 (t) ≡ y1 (t) ≡ 0. We conclude that v¯(t) is of the form (29.181). Since v¯ > 0, we have C > 0. This completes Step 2. Step 3. If g (t) is not round, then it is the King–Rosenau solution. Regarding the coefficients in (29.181), we have by (29.186a) and (29.186d) that A = 4BA,

B  = 16AC,

C  = 4BC.

124

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS 



A(t) Since AA = CC , we have that C(t) is independent of t. On the other hand, by pulling back by a conformal diffeomorphism, we can rescale the metric and coordinates by defining for any λ > 0

v¯λ (x, y, t)  λ−2 v¯(λx, λy, t) 2   = λ2 A x2 + y 2 + B x2 + y 2 + λ−2 C. So, without loss of generality, we may assume that A (t) = C (t) for all t. We then obtain 2   v¯(x, y, t) = A x2 + y 2 + B x2 + y 2 + A, A = 4BA,

B  = 16A2 .

This is (29.14) and (29.15); so we have obtained the King–Rosenau solution. ¯ 14. The evolution equation for Q ¯ which is the plane In this section we consider the evolution equation for Q, version of equation (29.140) for Q. We calculate this directly for the sake of comparison; some aspects of the calculation are simpler because the background space is flat. The reader may skip this section if he or she likes since the discussion is not necessary for the proof of the main theorem of this chapter. Let a ¯ be the 3-tensor defined by (29.171) and let TF (β) denote the trace-free part of a tensor β of degree 3 or 4 as defined in (29.131) or (29.156), respectively. ¯ = v¯ |¯ Lemma 29.48. The quantity Q a|2 on R2 defined in (29.172) evolves by   v¯ 2  ¯ ∂Q   2 ¯ ¯ v ∇¯ a + 2d¯ v⊗a ¯)| +  z − d¯ = v¯Δeuc Q − 4RQ − 2 | TF(¯ v 1,1 a ¯ (29.196) ∂t 2 ¯ ≤ v¯Δeuc Q, where z  ∇2 Δeuc v¯ − 12 Δ2euc v¯gR2 and (¯ v 1,1 a ¯)ij =

2

¯ a ¯ij . =1 v

We now sketch a proof of the lemma. Taking three partial derivatives of (29.170), we obtain   ∂ 2 (29.197) v| v¯ijk = v¯Δeuc v¯ − |∇¯ ∂t ijk = v¯Δeuc v¯ijk + v¯ijk Δeuc v¯ + v¯[i Δeuc v¯jk] + v¯[jk Δeuc v¯i] − 2¯ vijk v¯ − 2¯ v[jk v¯i] , where A[ijk]  Aijk + Ajki + Akij for a 3-tensor A. Thus (29.198) ∂ |¯ vijk |2 = v¯Δeuc |¯ vijk |2 − 2¯ v |¯ vijk |2 + 2Δeuc v¯ |¯ vijk |2 ∂t vi Δeuc v¯jk + v¯jk Δeuc v¯i ) − 4¯ vijk v¯ijk v¯ − 12¯ vijk v¯jk v¯i . + 6¯ vijk (¯ Here and throughout, we include  2the indices in our notation for the norm of a tensor; for example, |¯ vijk |2 = ∂ 3 v¯ . Tracing (29.197) yields ∂ (Δeuc v¯i ) = v¯Δeuc (Δeuc v¯i ) + 2Δeuc v¯i Δeuc v¯ + v¯i Δ2euc v¯ − 4¯ vji v¯j ∂t

15. THE QUANTITY Q MUST BE IDENTICALLY ZERO

125

and (29.199)

∂ 2 2 2 2 |Δeuc v¯i | = v¯Δeuc |Δeuc v¯i | − 2¯ v |Δeuc v¯i | + 4Δeuc v¯ |Δeuc v¯i | ∂t + 2¯ vi Δeuc v¯i Δ2euc v¯ − 8Δeuc v¯i v¯ji v¯j .

From (29.198), (29.199), and dropping the vanishing terms 2

2

Δeuc v¯(|¯ vijk | − |Δeuc v¯i | ) − 2¯ vi v¯ijk v¯jk + 2¯ vjk v¯ijk Δeuc v¯i = 0, we compute using (29.175) that ∂ 2 2 2 aijk | − 4R |¯ aijk | |¯ aijk | = v¯Δeuc |¯ ∂t 2 va ¯ijk )|2 + |∇¯ v |2 |¯ aijk |2 − 3¯ va ¯jk v¯ Δeuc v¯jk ). − (|∂ (¯ v¯ To analyze the rhs of (29.200), let zjk = Δeuc v¯jk − 12 Δ2euc v¯δjk . Since

(29.200)

1 2 2 |∇¯ v | |¯ aijk | 2 (which is analogous to (29.161)) and since δ i ∂ (¯ va ¯ijk ) = v¯ a ¯jk + v2¯ zjk , by (29.157) we have 2

¯jk | = |¯ v a

2

va ¯ijk )| − | TF(∂ (¯ va ¯))ijk |2 = |∂i (¯ va ¯ijk )| |∂ (¯ =

2

1 v¯2 |∇¯ v |2 |¯ aijk |2 + v¯i a ¯ijk v¯zjk + |zjk |2 . 2 4

Applying 2 |∇¯ v |2 |¯ aijk |2 + v¯v¯ |¯ aijk |2 = 2¯ v a ¯ijk TF(∂(¯ va ¯))ijk + 2 |¯ v a ¯jk |2 + v¯ a ¯jk v¯zjk to (29.200), we calculate that ¯ ∂Q ¯ − 4RQ ¯ = v¯Δeuc Q ∂t   3 2 2 2 2 v | |¯ − 2 |∂ (¯ va ¯ijk )| + |∇¯ aijk | − 3¯ v a ¯jk v¯Δeuc v¯jk + v¯v¯ |¯ aijk | 2 ¯ − 4RQ ¯ − 2(| TF(∂(¯ va ¯))ijk + v¯ a ¯ijk |2 − |¯ v a ¯jk |2 ) = v¯Δeuc Q  v¯ 2   − 2  zjk − v¯ a ¯jk  . 2 Again applying (29.157) yields (29.196). This completes the proof of Lemma 29.48. 15. The quantity Q must be identically zero Let Q = v |TF(B)|2 , as defined in (29.134). In this section we shall prove the following. Since we have shown in Proposition 29.46 that Q being zero implies that g(t) is either a shrinking round 2-sphere or the King–Rosenau solution, this will complete the proof of the main Theorem 29.1. Proposition 29.49. If μ > 0 in Proposition 29.36, then Qmax (t)  maxx Q(x, t) is nonincreasing and satisfies (29.201)

lim Qmax (t) = 0.

t→−∞

Hence Q(x, t) ≡ 0 on S 2 × (−∞, 0).

126

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Proof. Since Q is a subsolution of the heat equation by (29.140), the maximum principle implies that the function t → Qmax (t) is nonincreasing. Recall that by (29.107) we have that v∞ (ψ, θ) = μ cos2 ψ, where ψ is the latitude. Since Q vanishes on a flat cylinder, we thus have 2  (29.202) Q(v∞ )  v∞ TFS(∇3 v∞ ) ≡ 0 on S 2 −{N, S}. Geometrically, this says that Q is zero on the flat cylinder backward limit. Suppose that the proposition is false. Then there exists ε > 0 and sequences of points qi ∈ S 2 and times ti → −∞ such that (29.203)

Q(qi , ti ) ≥ ε for all i.

By passing to a subsequence, we may assume that q∞  limi→∞ qi exists. Since v(t) converges to v∞ in C 3,α on compact subsets of S 2 − {N, S} and by (29.202), we have that q∞ is either N or S. Without loss of generality, we may assume ¯ i  σ(qi ) ∈ R2 , where σ : S 2 − {S} → R2 is stereographic that q∞ = N . Let O ¯ projection. Then Oi → σ(N ) = O is the origin. Now we recall the cigar limits at the poles. Let Ki  u ¯−1/2 (O, ti ) = v¯1/2 (O, ti ). 2 2 Since v¯∞ (x, y) = μ(x + y ), we have that Ki → 0. By Proposition 29.38 we have that the sequence v¯i (x, y, t)  Ki−2 v¯(Ki x, Ki y, t + ti ), as defined in (29.116), converges in C ∞ on compact subsets of R2 to a positive function v¯∞ (x, y, t) satisfying (29.118). Analogously, there is a cigar limit based at S. Oi | < ∞. Case 1. The Q ≥ ε points are in the cigar region: lim inf i→∞ |K i ¯ Oi ¯ By passing to a subsequence, we may assume that Oi  Ki converges to a point ¯ ∈ R2 as i → ∞. By (29.173), we have that O ∞ ¯

¯ (x, y, t)  v¯ |a(v¯ )|2 (x, y, t) = Q(K ¯ i x, Ki y, ti + t). Q i i i ¯ , 0) = Q( ¯ i (O ¯ O ¯ i , ti ). On the other hand, we have In particular, Q i ¯ (O ¯ , 0) = 0, ¯ i , 0) → Q ¯ ∞ (O ¯ ∞ , 0)  v¯∞ |a(v¯∞ )|2 (O Q i ∞ −1 ¯ O ¯ i , ti ) = (t)geuc is a cigar soliton. Since Q( where the last equality is true because v¯∞ Q(qi , ti ) by Lemma 29.50 below, we obtain a contradiction to (29.203). We conclude that Case 1 is impossible. Oi | = ∞. Case 2. The Q ≥ ε points are outside the cigar region: limi→∞ |K i The main idea is to show that rescalings of the solution about the points (qi , ti ) subconverge to a flat cylinder, which has Q = 0. ¯

Step 1. A general estimate for circular averages (used to study backward limits). Recall that g(t) = vˆ (t)−1 gcyl and define  ˆ (s, t)  (29.204) W ln vˆ(s, θ, t)dθ S1

for s ∈ R and t < 0. Recall that by Proposition 29.36 we have vˆ∞ (s, θ) ≡ μ. Since ∂v ˆ ˆ ∂t > 0, this implies that W (s, t) > 2π ln μ. Moreover, since   2 ∂ ∂2 + 2 ln vˆ = Δcyl ln vˆ = Rg vˆ−1 > 0 ∂s2 ∂θ

15. THE QUANTITY Q MUST BE IDENTICALLY ZERO

127

2 ˆ ˆ by (29.7), we have ∂∂sW 2 > 0; i.e., s → W (s, t) is convex. On the other hand, from (29.5), we have that

ln vˆ(s, θ, t) = 2 ln cosh s + ln v(ψ(s), θ, t), where ψ is the latitude on S 2 . Thus ∂ ln vˆ ∂ ln v (s, θ, t) = 2 tanh s + sech s (ψ, θ, t) ∂s ∂ψ ∂ ln v 1/2 since dψ ≤ C for each t ∈ ds = sech s. Since | ∂ψ |(t) ≤ |∇ ln v|gS 2 (t) ≤ Cv (−∞, −1] by (29.34), this implies that

ˆ ∂W (s, t) = −4π s→−∞ ∂s

(29.205) Therefore (29.206)

lim

ˆ ∂2W ∂s2

ˆ ∂W (s, t) = 4π. s→∞ ∂s

and

lim

> 0 implies that the derivative of the circular averages satisfies    ∂W   ˆ  (s, t) < 4π for t ∈ (−∞, −1].   ∂s 

In cylindrical coordinates, define vˆi (s, θ, t)  Ki−2 vˆ(s + ln Ki , θ, t + ti ), so that vˆi−1 (t)gcyl = v¯i−1 (t)geuc is the sequence of solutions in (29.119) and let vˆ∞ (t) = limi→∞ vˆi (t). Formula (29.118) implies that vˆ∞ (s, θ, 0) converges to A uniformly as s → ∞. Hence there exist s0 and i0 such that vˆi (s, θ, 0) ≤ 2A for all i ≥ i0 , s ≥ s0 , and θ. In particular, vˆ(s0 + ln Ki , θ, ti ) ≤ 2AKi2 , which tends to zero. (This is related to the fact that, backward in time and in the cigar region, the conformal factor tends to infinity.) So we have ˆ (s0 + ln Ki , ti ) ≤ 2π ln(2AKi2 ), W which tends to −∞. We shall compare the averages over the circle {s = s0 + ln Ki } in the cigar region and over the circle {s = si } passing through m(qi ), where m is Mercator projection. Let (29.207)

¯ i | = r(O ¯ i ), ri = |O

¯ i ), θi = θ(O

si = ln ri .

Since ri → 0, we have si → −∞. By the Case 2 hypothesis, si −ln Ki → ∞. Hence, si > s0 + ln Ki for i sufficiently large. Claim 1. There exists a constant C < ∞ such that (29.208)

ˆ (si , ti ) ≤ C 2π ln μ < W

for all i.

Proof of Claim 1. We have already observed the lower bound. To prove the ˆ (si , ti ) → ∞. upper bound by contradiction, we assume that W 2s ˆ (s, t) + 4πs) ∈ R for Since lims→−∞ e vˆ(s, θ, t) ∈ (0, ∞), we have lims→−∞ (W ˆ (s, t) = ∞. We also have each t. In particular, lims→−∞ W  ˆ (s, ti ) = (29.209) lim W ln vˆ∞ (s, θ)dθ = 2π ln μ i→∞

S1

128

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

for each s. We claim, for i sufficiently large, that ˆ ∂W (s, ti ) ≤ 0 for s ≤ si . ∂s

(29.210) ˆ

2

ˆ

ˆ

W ∂W Indeed, if ∂∂s (si , ti ) > 0 for some si ≤ si , then ∂∂sW 2 > 0 implies that ∂s (si , ti ) > 0. ˆ (s, ti ) ≥ W ˆ (si , ti ) → ∞, contradicting This in turn implies for s ≥ si that W (29.209) if i is sufficiently large. Now, by (29.210) we have that

ˆ (s0 + ln Ki , ti ) ≥ W ˆ (si , ti ) → ∞, 2π ln(2AKi2 ) ≥ W which contradicts Ki → 0. This completes the proof of Claim 1. Step 2. Pointed limits about (qi , ti ). Let vˆi (s, θ, t)  vˆ(s + si , θ + θi , t + ti ), where si and θi are as in (29.207) and ti → −∞ is as before. Note that gˆi  g (t + ti )), where Ψi is translation by (si , θi ) on the cylinder vˆi−1 (t)gcyl = e2si Ψ∗i (ˆ (on R2 this is multiplication by esi and rotation by θi ). Define  ˆ ˆ (s + si , t + ti ). Wi (s, t)  ln vˆi (s, θ, t)dθ = W S1

ˆ i (0, 0) ≤ C. By (29.206) we have From (29.208), we have that 2π ln μ < W    ∂W   ˆi  (s, t) < 4π for s ∈ R and t ∈ (−∞, −1 − ti ].   ∂s  = Rgˆ ∈ (0, C],  ˆi ∂W (s, t) = Rgˆ (s + si , θ + θi , t + ti )dθ ∈ (0, 2πC] ∂t S1

Moreover, since

1 ∂v ˆ v ˆ ∂t

ˆ i (s, t) are uniformly bounded on for s ∈ R and t ∈ (−∞, −1 − ti ]. Thus, the W 2 compact  ∂subsets  of R × (−∞, ∞) = R . On the other hand, from (29.113) we obtain  ∂θ ln vˆi  ≤ C. Hence, the ln vˆi are uniformly bounded on compact subsets of R × S 1 × (−∞, ∞). Since each vˆi is a solution of (29.7), we obtain uniform estimates for the higher derivatives of vˆi on compact subsets (see Lemma 28.53). Therefore, there exists a subsequence such that the vˆi converge in C ∞ on compact subsets to a smooth positive function vˆ∞ on R × S 1 × (−∞, ∞) satisfying (29.7). That is, gˆ∞  vˆ∞ (t)gcyl is a solution to the Ricci flow on R × S 1 × (−∞, ∞). Note that Rgˆ∞ (0, 0, 0) = limi→∞ Rgˆ (si , θi , ti ). Step 3. The pointed limit is a flat cylinder. Claim 2. Rgˆ∞ (0, 0, 0) = 0. Proof of Claim 2. Suppose that Rgˆ∞ (0, 0, 0) > 0. We shall show that the two cigar regions together with a neighborhood of (qi , ti ) contribute integral scalar curvature greater than 8π, yielding a contradiction. We then have Rgˆ∞ (0) > 0 on R × S 1 . Hence there exists δ > 0 such that Rgˆi (0) ≥ δ in [−1, 1] × S 1 for i sufficiently large. Moreover, μ < vˆi (s, θ, 0) ≤ C in [−1, 1] × S 1 . Therefore we have a given amount of total scalar curvature in a region containing qi which is away from the “cigar region” near N :  4πδ ˆ  δ. Rgˆi (0) vˆi−1 (0)dsdθ ≥ C 1 [−1,1]×S

¯ 16. THE EQUIVALENCE OF Q AND Q

129

The annulus ([−1, 1] × S 1 , gˆi (0)) is isometric to (BO (esi +1 ) − BO (esi −1 ), g¯(ti )). So we have  ˆ Rg¯(ti ) v¯−1 (ti )dμeuc ≥ δ. BO (esi +1 )−BO (esi −1 ) hand, since g¯i (t) = v¯i−1 (t)geuc

On the other converges to a cigar soliton (this is the cigar limit based at N ), for any η > 0 there exists rη < ∞ such that  Rg¯i (0) v¯i−1 (0)dμeuc > 4π − η. BO (rη )

The Riemannian ball (BO (rη ), g¯i (0)) is isometric to (BO (Ki rη ), g¯(ti )), where g¯(t)  1 v ¯(t) geuc . So we have  Rg¯(ti ) v¯−1 (ti )dμeuc > 4π − η. BO (Ki rη )

Similarly, the cigar region near S yields integral scalar curvature greater than 4π−η. More precisely, there exist r¯i → ∞ such that  Rg¯(ti ) v¯−1 (ti )dμeuc > 4π − η. R2 −BO (¯ ri )

si

Recall that Ki → 0, esi → 0, and eKi → ∞. Therefore the sets BO (Ki rη ), R2 − ri ), BO (esi +1 ) − BO (esi −1 ) are disjoint and we conclude that BO (¯  8π = Rg(ti ) dμg(ti ) > 8π − 2η + δˆ > 8π S2

ˆ by choosing η < δ/2, which is a contradiction. This completes the proof of Claim 2. By Claim 2 and the strong maximum principle, we must have Rgˆ∞ ≡ 0 on R×S 1 ×(−∞, ∞). From (29.7), we then obtain Δcyl ln vˆ∞ = 0. Since vˆi (s, θ, t) > μ, we have the global lower bound ln vˆ∞ (s, θ) ≥ ln μ. Lifting ln vˆ∞ from R × S 1 to R2 , we may apply the Liouville theorem for entire harmonic functions that are bounded −1 (t)gcyl is a flat cylinder. from below to conclude that ln vˆ∞ is constant. Thus vˆ∞ Step 4. Case 2 is impossible. Now let v¯i (r, θ, t)  r 2 vˆi (ln r, θ, t) = vˆ(ln r + si , θ + θi , t + ti ). −1 (0)geuc , where v¯∞  limi→∞ v¯i , is a flat cylinder, we have Since v¯∞ ¯ vi )(1, 0, 0) = Q(¯ ¯ v∞ )(1, 0, 0) = 0. lim Q(¯ i→∞

¯ vi )(1, 0, 0) = Q(v)(qi , ti ) ≥ ε; we obtain a contradiction. On the other hand, Q(¯ Since Case 1 and Case 2 are impossible, the proposition is proved.  ¯ 16. The equivalence of Q and Q Let σ : S 2 − {S} → R2 denote stereographic projection and let ϕ(x, y)  4 −1 gS 2  σ ∗ (¯ v −1 geuc ) be a Riemannian metric on S 2 . By (x2 +y 2 +1)2 . Let g = v ¯ be as defined in (29.134) and v . Let Q and Q (29.6), on R2 we have v ◦ σ −1 = ϕ¯ (29.172), respectively. We have the following. ¯ We have Q ◦ σ −1 = Q. ¯ Lemma 29.50 (Equivalence of Q and Q).

130

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

The proof of this lemma shall occupy the rest of this section. The totally symmetric and trace-free 3-tensor a ¯, defined by (29.171) in Euclidean coordinates, v ⊗geuc ), where ∇ denotes the Euclidean is given invariantly by a ¯ = ∇3 v¯− 34 S(Δeuc ∇¯ covariant derivative. Let (r, θ) be polar coordinates. By (29.175), now using the ∂ ∂ ∂ 1 ∂ 2 that orthonormal frame ∂x 1 = ∂r and ∂x2 = r ∂θ , we have on R v¯  2 ¯2 2 ¯ = v¯ |¯ Q a| = α ¯ +β , 4 where α ¯  ∇3rrr v¯ − 3r −2 ∇3θθr v¯, β¯  r −3 ∇3θθθ v¯ − 3r −1 ∇2rrθ v¯. Next, we express α ¯ and β¯ in terms of the partial derivatives of v¯ with respect to r and θ. We have Γθrθ = Γθθr = 1r and Γrθθ = −r and the rest of the Euclidean Christoffel symbols are zero. Therefore the components of the Hessian of v¯ are given by ∇2rr v¯ = v¯rr , vr , ∇2θθ v¯ = v¯θθ + r¯ 1 ∇2rθ v¯ = v¯θr − v¯θ . r Taking another covariant derivative, we have ∇3rrr v¯ = v¯rrr , vrθ − 2¯ vθ , ∇3θθθ v¯ = v¯θθθ + 3r¯ 2 ∇3θθr v¯ = v¯rθθ − v¯θθ + r¯ vrr − v¯r , r 2 2 ∇3rrθ v¯ = v¯θrr − v¯θr + 2 v¯θ . r r Therefore, α ¯ = v¯rrr − 3r −2 v¯rθθ + 6r −3 v¯θθ − 3r −1 v¯rr + 3r −2 v¯r , β¯ = r −3 v¯θθθ − 3r −1 v¯θrr + 9r −2 v¯rθ − 8r −3 v¯θ . To express Q ◦ σ −1 in terms of the partial derivatives of v¯ with respect to r and θ, we first express Q in terms of the partial derivatives of v with respect to ψ and ∂ ∂ ∂ ∂ θ. Since ∂x 1 = ∂ψ and ∂x2 = sec ψ ∂θ are orthonormal with respect to gS 2 , from (29.163) and (29.169), we have on S 2 that 2

Q = v |TF(B)| =

v 2 α + β2 , 4

where α  ∇3ψψψ v − 3 sec2 ψ∇3ψθθ v − 2∇ψ v, β  sec3 ψ∇3θθθ v − 3 sec ψ∇3θψψ v − 2 sec ψ∇θ v.

¯ 16. THE EQUIVALENCE OF Q AND Q

131

Using (29.18), we compute that the components of the second covariant derivative of v are ∇2ψψ v = vψψ , sin 2ψ vψ , 2 + tan ψ vθ .

∇2θθ v = vθθ − ∇2ψθ v = vθψ

For the third covariant derivative, we obtain ∇3ψψψ v = vψψψ , 3 sin 2ψ vψθ − 2 sin2 ψ vθ , 2 sin 2ψ vψψ + 2 tan ψ vθθ − vψ , ∇3ψθθ v = vθθψ − 2 ∇3θψψ v = vψψθ + 2 tan ψ vθψ + 2 tan2 ψ vθ . ∇3θθθ v = vθθθ −

Therefore

  sin 2ψ vψψ + 2 tan ψ vθθ − vψ α = vψψψ − 2vψ − 3 sec2 ψ vθθψ − 2

and

 cos ψ · β = sec2 ψvθθθ − 3vψψθ − 9 tan ψvψθ − 2 4 tan2 ψ + 1 vθ .

We now evaluate α◦σ −1 and β ◦σ −1 using v ◦σ −1 = ϕ¯ v . Since ϕ is independent of θ, we have that vθ , vθ ◦ σ −1 = ϕ¯ vθθ , vθθ ◦ σ −1 = ϕ¯ vθθθ . vθθθ ◦ σ −1 = ϕ¯ Recall that

dψ dr

= − r22+1 . We compute that vψ ◦ σ −1 = −

and hence vψθ ◦ σ −1 = −

r2

2 8r v¯rθ + ¯θ 2v 2 r2 + 1 (r + 1)

and vψθθ ◦ σ −1 = − We then compute that vψψ ◦ σ

(29.211)

−1

and hence vψψθ ◦ σ and vψψψ ◦ σ

−1

−1

2 8r v¯r + ¯, 2v 2 +1 (r + 1)

r2

2 8r v¯rθθ + ¯θθ . 2v 2 +1 (r + 1)

 4 3r 2 − 1 6r = v¯rr − 2 ¯ v¯r + 2 v r +1 (r 2 + 1)

 4 3r 2 − 1 6r v¯rθ + = v¯rrθ − 2 v¯θ r +1 (r 2 + 1)2

 4r 3r 2 − 5 r2 + 1 9r 2 − 5 v¯rrr + 3r¯ v¯r + =− vrr − 2 v¯. 2 r +1 (r 2 + 1)2

132

29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Therefore, in terms of v¯ we may rewrite α and β as (using also that cos ψ = sec ψ =

2

r +1 2r ,

sin ψ =

2

1−r 1+r 2 ,

and tan ψ =

1−r 2r

2

2r r 2 +1 ,

, where ψ = ψ ◦ σ −1 )

vrrr + 3r −1 v¯rθθ − 6r −2 v¯θθ + 3¯ vrr − 3r −1 v¯r = −r α ¯ (29.212) (cos ψ · α) ◦ σ −1 = −r¯ and (29.213)

¯ vrrθ + 9r −1 v¯rθ − 8r −2 v¯θ = r β. (cos ψ · β) ◦ σ −1 = r −2 v¯θθθ − 3¯ 2

v = cosr2 ψ v¯, we obtain that Finally, from (29.212), (29.213), and v ◦ σ −1 = ϕ¯ v 2 v¯  2 ¯2 ¯ α + β 2 ◦ σ −1 = α ¯ + β = Q. Q ◦ σ −1 = 4 4 17. Notes and commentary For the King–Rosenau solution or sausage model, see King [160], Rosenau [338], and Fateev, Onofri, and Zamolodchikov [109]. Theorem 29.1 is due to Daskalopoulos, Hamilton, and Sesum [93]. See Vazquez [424] for a comprehensive treatment of the porous medium equation. We would like to thank Frank Morgan for showing us the proof of Lemma 29.19.

CHAPTER 30

Type I Singularities and Ancient Solutions It’s a tad bit late. – From “Regulate” by Warren G featuring Nate Dogg

The classification of 3-dimensional singularity models is used to understand the high-curvature regions of singular solutions on closed 3-manifolds. In this chapter we mainly study various properties of Type I singular solutions and Type I ancient solutions in higher dimensions, where much less is known. In §1 we consider a condition for singular solutions that is weaker than Type I. We develop the tool of the reduced distance based at the singular time. In §2 we prove the monotonicity of the reduced volume based at the singular time for Type I singular solutions. In §3, by using the aforementioned monotonicity, we show that for any Type I singular solution, (1) there exists a nonflat shrinker singularity model and (2) the solution must have unbounded scalar curvature. In §4 we discuss some results about κ-noncollapsed Type I ancient solutions using the ideas developed in the previous sections. 1. Reduced distance of Type A solutions In the study of singular solutions it is helpful to impose restrictions on the growth rates of their curvatures. In this section, under the Type I assumption, and more generally under the so-called Type A assumption defined below, we shall estimate the reduced distance function. This enables us to define the reduced distance based at the singular time (as a limit) and to use it as a tool in singularity analysis. In the next section we consider the reduced volume based at the singular time. Up to this date, there are more results about Type I solutions than Type II solutions. Since Type II solutions lack curvature growth control, results about them are generally more difficult to prove; they are also expected to be nongeneric in some sense. 1.1. Derivative estimates for Type A singular solutions. Generalizing the notion of a Type I singular solution, we have the following condition, which is geared toward the study of the reduced distance function based at the singular time. Definition 30.1 (Type A singular solution). We say that a singular solution of the Ricci flow (Mn , g (t)), t ∈ (α, ω), where −∞ < α < ω < ∞, is Type A if 133

134

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

there exist M < ∞ and r ∈ [1, 32 ) such that (30.1)

|Rm| (x, t) ≤

M (ω − t)r

on M × (α, ω). Note that the time interval (α, ω) is finite and the Type A condition is essentially for the solution near time ω. We could impose the Type A condition for an M ancient solution by requiring |Rm| (x, t) ≤ |t| r on M × (−∞, 0). However, if t = 0 is the singular time, then the conditions near times 0 and −∞ are competing in the sense that increasing r makes the condition weaker near 0 but stronger near −∞. So, except for the r = 1 (i.e., Type I) case, one usually imposes the Type A condition either near the singular time or near time −∞. We observe that the Type A condition on the curvature leads to corresponding bounds for the derivatives of curvature. In general, suppose that (Mn , g (t)), t ∈ (α, ω), is a complete solution to the Ricci flow. Fix any η ∈ (0, ω−α 2 ] and let t¯ ∈ [α + η, ω). Recall that Shi’s local derivative estimates imply that if |Rm| ≤ K on M × (α, t¯], where K ≥ 1, then for any k,  ∈ N ∪ {0}, there exists Cn,k,,η < ∞ (independent of t¯ and K) such that  k  ∂   1+k+ 2   ¯ (30.2)  ∂tk ∇ Rm (x, t ) ≤ Cn,k,,η K for all x ∈ M (this follows from applying (20.38) in Part III to the solution with a time lag ηK −1 ≤ η; i.e., choose the initial time for Shi’s estimates to be t¯− ηK −1 ). Now suppose that the solution g (t) is Type A. Since (30.1) implies that |Rm| (x, t) ≤

M r (ω − t¯)

for (x, t) ∈ M × (α, t¯], we conclude from (30.2) with K = following derivatives of curvature estimates.

M (ω−t¯ )r

that we have the

Lemma 30.2 (Shi’s estimates for Type A solutions). If (Mn , g (t)), t ∈ (α, ω), is a complete Type A solution of the Ricci flow, then for any k,  ≥ 0 and η ∈ (0, ω−α 2 ] there exists Cn,k,,η,M < ∞ such that   k  ∂ Cn,k,,η,M    ¯ (30.3)  ∂tk ∇ Rm (x, t ) ≤ (1+k+ 2 )r (ω − t¯) on M × [α + η, ω). In particular, taking k = 0 and  = 1, we obtain (30.4)

|∇R| (x, t¯) ≤ Cn |∇ Rm| (x, t¯) ≤

Cn,η,M 3 r (ω − t¯) 2

on M × [α + η, ω). The reader may wish to compare this with the modified Shi’s derivative estimates in Chapter 14 of Part II.

1. REDUCED DISTANCE OF TYPE A SOLUTIONS

135

1.2. Motivation and set-up for the reduced distance based at the singular time. Let (Mn , g (t)), t ∈ (α, ω), where −∞ < α < ω < ∞, be a complete singular solution of the Ricci flow. Recall from Chapter 7 in Part I that for any p0 ∈ M and t0 ∈ (α, ω), the reduced distance based at (p0 , t0 ) p0 ,t0 : M × (α, t0 ) → R is defined by (30.5)

1 p0 ,t0 (q, t¯) = ! inf γ ¯ 2 t0 − t



t0 t¯

√

  t0 − t R (γ (t) , t) + |γ˙ (t)|2g(t) dt,

where the infimum is taken over γ : [t¯, t0 ] → M with γ(t¯) = q and γ (t0 ) = p0 . ω, we may ask if there exists Given a sequence (pi , ti ) ∈ M × (α, ω) with ti a subsequence such that the reduced distance functions pi ,ti converge to a limit function defined on M × (α, ω). Definition 30.3. If such a limit exists, we shall call it a reduced distance based at the singular time. We first consider a general setting which includes the case for which we can prove existence; see Theorem 30.10 below. Let (Mni , gi (t) , pi ), t ∈ [αi , ωi ], pi ∈ ω∞ Mi , be a sequence of complete pointed solutions to the Ricci flow with ωi and αi ≤ α∞  limi→∞ αi . Suppose that this sequence converges in the C ∞ pointed Cheeger–Gromov sense to a complete limit solution (Mn∞ , g∞ (t) , p∞ ), t ∈ [α∞ , ω∞ ). That is, suppose that there exist smooth embeddings (30.6)

Φi : (Ui , p∞ ) ⊂ (M∞ , p∞ ) → (Mi , pi ) ,

where {Ui }i∈N is an exhaustion of M∞ by open sets with compact closure, such that g˜i (t)  Φ∗i gi (t) → g∞ (t) in C k (K), for each k ∈ N and for each compact subset K of M∞ × [α∞ , ω∞ ). Furthermore, we shall assume that each solution gi (t) has bounded curvature on Mi × [αi , ωi ], where the bound may depend on i. Let the functions i  gpii ,ωi

and Li  Lgpii ,ωi

on Mi × [αi , ωi ) be the reduced distance and the L-distance for gi (t) based at (pi , ωi ), respectively (see Chapter 7 of Part I for the definition of the L-distance). ˜ i on Ui × [αi , ωi ) by Define the corresponding “transplanted” functions ˜i and L (30.7)

˜ i (q, t¯)  Li (Φi (q), t¯). ˜i (q, t¯)  i (Φi (q), t¯) and L

˜ i on Ui × [α∞ , ωi ) ⊂ M∞ × [α∞ , ω∞ ) and we shall prove We shall estimate ˜i and L their convergence under the Type A curvature assumption. 1.3. Pointwise bounds for the reduced distances. ˜ i under the By a rather direct method we first establish a pointwise bound for L Type A assumption, which is uniform (i.e., independent of i) on compact subsets of M∞ × [α∞ , ω∞ ). This, along with the derivative estimates in the next subsection, enables us to take limits of the reduced distance functions. In the next section we shall obtain sharper bounds in the Type I case (see Lemma 30.15 below). This

136

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

leads to the well-definedness and monotonicity of the reduced volume at the singular time. ˜ i on compact sets). Under the Cheeger– Lemma 30.4 (Uniform upper bound for L Gromov convergence set-up in the previous subsection, assume the Type A condition that there exists M < ∞ and r ∈ [1, 32 ) such that M (ωi − t)r

|Rmgi | (x, t) ≤

(30.8)

for all (x, t) ∈ Mi × [α∞ , ωi ) and i.1 Fix t0 ∈ (α∞ , ω∞ ). (1) If r > 1, then for all (q, t¯) ∈ M∞ × [t0 , ω∞ ) we have (30.9)

˜ i (q, t¯) ≤ 2 exp L +



2nM ω∞ −t¯ −(r−1) r−1 ( 2 )

 √ d2g (t ) (q, p∞ ) · ω∞ − α∞ ∞ 0 ω∞ − t¯

3 n2 M −r (ω∞ − t¯) 2 3 − r 2

for i sufficiently large. (2) If r = 1, then for all (q, t¯) ∈ M∞ × [t0 , ω∞ ) we have  2nM d2g∞ (t0 ) (q, p∞ ) √ ˜ i (q, t¯) ≤ 2 2 (ω∞ − t0 ) L (30.10) ω − α ∞ ∞ ω∞ − t¯ ω∞ − t¯ 1

+ 2n2 M (ω∞ − t¯) 2 . A simple consequence is Corollary 30.5. Under the assumptions of Lemma 30.4, for any compact set K ⊂ M∞ , ε ∈ (0, ω∞ − α∞ ], and t0 ∈ (α∞ , ω∞ − ε), there exists C < ∞ depending on K, ε, and t0 such that for any q ∈ K and any t¯ ∈ [t0 , ω∞ − ε], we have ˜ i (q, t¯) ≤ C L

(30.11) for all i sufficiently large.

˜ i by considering Proof of Lemma 30.4. We shall prove the upper bound for L elementary “test paths” for the L-length of gi (t). Let q ∈ M∞ and t¯ ∈ [t0 , ω∞ ) and define ε  ω∞ − t¯. Throughout this proof, we shall assume that i is sufficiently large and that ε ωi > t0 , ωi > ω∞ − > t¯ = ω∞ − ε, and q ∈ Ui . 2 Step 1. Bounds for the evolving metric. Let d0 denote the Riemannian distance with respect to g∞ (t0 ). By assumption (30.8), we have |Rcgi | (x, t) ≤

(30.12) 1 Observe

nM r (ωi − t)

that (30.8) implies that |Rmg∞ | (x, t) ≤

on M∞ × [α∞ , ω∞ ).

M (ω∞ − t)r

1. REDUCED DISTANCE OF TYPE A SOLUTIONS

137

on Mi × [α∞ , ωi ). Given any (p, t) ∈ M∞ × [t0 , ω∞ ) and any nonzero tangent vector V ∈ Tp M∞ , by the Ricci flow equation and (30.12) we have      g˜i (t) (V, V )   gi (t) ((Φi )∗ V, (Φi )∗ V )   = ln  ln (30.13)  g˜i (t0 ) (V, V )   gi (t0 ) ((Φi ) V, (Φi ) V )  ∗ ∗  t ≤2 |Rcgi | (Φi (p), s) ds t0 & −(r−1) 2nM if r > 1, r−1 (ωi − t) ≤ 0 2nM ln ωωii−t if r = 1. −t Since g˜i (t0 ) → g∞ (t0 ) and since ωi → ω∞ , we conclude from (30.13) that for any point (p, t) in any compact subset K of M∞ × [t0 , ω∞ ) we have 1 g∞ (p, t0 ) ≤ g˜i (p, t) ≤ K (t) g∞ (p, t0 ) , K (t)

(30.14) where

  ⎧ −(r−1) ⎨ exp 2nM if r > 1, r−1 (ω∞ − t)  2nM K (t)  ⎩ ω∞ −t0 if r = 1. ω∞ −t

(30.15)

Step 2. Test path for the L-length. Recall that (q, t¯) ∈ M × [t0 , ω∞ ). By the definition of the L-distance of gi (t), we have ˜ i (q, t¯) = Li (Φi (q), t¯) L

(30.16)

≤ Lgi (γ)  ωi   √ 2 = ωi − t Rgi (γ (t) , t) + |γ  (t)|gi (t) dt t¯

for any piecewise smooth path γ : [t¯, ωi ] → Mi with γ (ωi ) = pi and γ(t¯) = Φi (q). Now choose γ in (30.16) to be the “test path” uniquely defined so that γ|[ω∞ − ε ,ωi ] ≡ pi and so that γ|[t¯,ω∞ − ε ]  η is a constant speed minimal geodesic 2

2

∗ from Φi (q) to pi with respect to (Φ−1 i ) g∞ (t0 ). Using the definition of Cheeger– Gromov convergence, we see that such a minimal geodesic exists. We then have  ω∞ − ε2   √ 2 ˜ i (q, t¯) ≤ L ωi − t Rgi (η (t) , t) + K (t) |η  (t)|(Φ−1 )∗ g∞ (t0 ) dt i t¯  ωi √ ωi − t Rgi (pi , t) dt + ω∞ − ε2

by (30.16) and by the upper bound in (30.14). Since |η  (t)|(Φ−1 )∗ g∞ (t0 ) = i

=

d(Φ−1 )∗ g∞ (t0 ) (Φi (q), pi ) i

ε 2

2dg∞ (t0 ) (q, p∞ ) , ω∞ − t¯

138

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

we obtain (30.17)

 ˜ i (q, t¯) ≤ L

ω∞ − ε2

d2g (t ) (q, p∞ ) √ 4K (t) ωi − t ∞ 0 dt 2 (ω∞ − t¯) t¯  ωi √ + ωi − t sup Rgi (x, t) dt. t¯

By (30.12) and r <  ωi t¯

3 2,

√

x∈Mi

we have

ωi − t sup Rgi (x, t) dt ≤ x∈Mi

≤

3 n2 M −r (ωi − t¯) 2 3 − r 2 3 n2 M −r (ω∞ − t¯) 2 . 3 − r 2

¯ Applying this to (30.17) while using ω∞ − 2ε − t¯ = ω∞2−t and K (t) ≤ K(ω∞ − 2ε ) ε ¯ ¯ for t ∈ [t, ω∞ − 2 ], we find that for all q ∈ M∞ and t ∈ [t0 , ω∞ ) (30.18)  √ d2g (t ) (q, p∞ ) 3 n2 M −r ˜ i (q, t¯) ≤ 2K ω∞ − ε L + ω∞ − α∞ ∞ 0 (ω∞ − t¯) 2 . 3 ¯ 2 ω∞ − t − r 2

This implies (30.9) if r > 1 and (30.10) if r = 1. The proof of Lemma 30.4 is complete.  Exercise 30.6. Determine if one can qualitatively strengthen the estimate (30.9) by considering test paths γ with γ|[ω∞ −δ,ωi ] ≡ pi , where δ ∈ (0, ε) is chosen optimally. Note that in the proof of Lemma 30.4 we chose δ = 2ε . 1.4. First derivative bounds for the reduced distances. ˜ i , which Next we establish bounds for the gradients and time derivatives of the L are uniform (i.e., independent of i) on compact subsets of M∞ × (α∞ , ω∞ ). ˜ i on compact sets). Assume Lemma 30.7 (Uniform bounds for the gradient of L the same hypothesis as in Lemma 30.4. Then, for any compact set K ⊂ M∞ and ∞ ), there exists const < ∞ such that for any q ∈ K and any any ε ∈ (0, ω∞ −α 2 t¯ ∈ [α∞ + ε, ω∞ − ε], we have ˜ i |Φ∗ g (q, t¯) ≤ const |∇L i i for all i sufficiently large. Idea of the proof. The gradient of the L-distance is basically given by the tangent vector of a minimal L-geodesic. Hence we can estimate its norm by the L-geodesic equation with the aid of the Type A condition, Shi’s estimate for |∇R|, and the previous pointwise bound for the L-distance. ˜ i |Φ∗ g . Let q ∈ M∞ and let Proof. Step 1. The standard formula for |∇L i i ¯ t ∈ [α∞ + ε, ω∞ − ε]. From now on assume that i is sufficiently large and that ωi > ω∞ − 2ε and q ∈ Ui , where Ui is defined in (30.6). Since the (Mi , gi (t)), t ∈ [αi , ωi ], are complete solutions with bounded curvatures, for each i there exists a minimal L-geodesic with respect to gi (t) γi : [ t¯, ωi ] → Mi

1. REDUCED DISTANCE OF TYPE A SOLUTIONS

139

whose graph joins (Φi (q), t¯) to (pi , ωi ).2 By Lemma 7.32 in Part I, we have that the gradient of Li ( · , t¯) with respect to gi (t¯) is given by  (30.19) ∇Li (Φi (q), t¯) = −2 ωi − t¯γi (t¯). In order to estimate ˜ i |Φ∗ g (q, t¯) = |∇Li | (Φi (q), t¯) , |∇L gi i i

(30.20)

it suffices to estimate the speed of γi ; we now proceed to do this. Step 2. An odi for |∇Li |gi (γi (t) , t). Let Xi (t)  γi (t) and let √ 1 ωi − tXi = − ∇Li (γi (t) , t) . 2 Our goal is to estimate |Yi (t¯)|. Using the L-geodesic equation for the solution gi (t) Yi (t) 

(30.21)

1 1 Xi = 0 ∇gXii Xi − ∇Rgi − 2 Rc gi (Xi ) − 2 2 (ωi − t) (see (7.32) in Part I, with τ = ωi − t), we compute that  d d  2 2 |Yi |gi = (30.22) (ωi − t) |Xi |g i dt dt 2 = − |Xi |g + 2 (ωi − t) ∇Xi Xi , Xi g − 2 (ωi − t) Rc gi (Xi , Xi ) i i   + 2 (ωi − t) Rc gi (Xi , Xi ) . = (ωi − t) ∇Rgi , Xi gi

Hence (30.23)

   d Cn Cn 2  |Yi |2  ≤ |Yi |g (t) , 3 1 |Yi |g (t) + gi   dt r− i i (ωi − t)r (ωi − t) 2 2

where we used (30.8) and (30.4). 2

Step 3. Estimating |∇Li |gi at some time. To apply the odi (30.23) to estimate |γi (t¯)|, we just need to estimate |γi (t)| for some t ∈ [t¯, ωi ]. We have   ωi  √ 1 2 ˜ ¯ Li (q, t ) = Lgi (γi ) = ωi − t Rgi (γi (t) , t) + √ |Yi (t)|g (t) dt i ωi − t t¯ and (using (30.8) and r < 32 )   ωi √ ωi − t Rgi (γi (t) , t) dt ≤ − t¯

ωi

√

t¯

≤

ωi − t

Cn dt (ωi − t)r

3 2Cn −r (ω∞ − α∞ ) 2 . 3 − 2r

˜ i, Hence, by the pointwise bound (30.9) for L  ωi 1 2 √ (30.24) |Yi (t)|g (t) dt i ω − t i t¯  ωi √ ˜ i (q, t¯) − ωi − t Rgi (γi (t) , t) dt =L ≤ D, 2 Note

t¯

that γi ([ t¯, ωi ]) need not be contained in Φi (Ui ).

140

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

where 

 √ d2g (t ) (q, p∞ ) · ω∞ − α∞ ∞ 0 ω∞ − t¯ 2 3 3 2n M 2Cn (ω∞ − α∞ ) 2 −r + 3 (ω∞ − t¯) 2 −r + 3 − 2r 2 −r

D  2 exp

if r > 1 and

 D2

2nM ω∞ −t¯ −(r−1) r−1 ( 2 )

2 (ω∞ − t0 ) ω∞ − t¯

nM ·

d2g (t ) (q, p∞ ) √ ω∞ − α∞ ∞ 0 ω∞ − t¯

1

1

+ 4n2 M (ω∞ − t¯) 2 + 2Cn (ω∞ − α∞ ) 2 if r = 1. If we assume that q is contained in a compact set K ⊂ M∞ , then D depends on K and not on q. By the mean value theorem for integrals, there exists ti ∈ [t¯, ω∞ − 3ε 4 ] such that  ω∞ − 3ε 4 1 1 1 4D √ √ |Yi (t)|2g (t) dt ≤ |Yi (ti )|2gi (ti ) ≤ 3ε i ε ωi − ti ωi − t ω∞ − 4 − t¯ t¯ since t¯ ≤ ω∞ − ε. Thus (30.25)

|Yi (ti )|2gi (ti ) ≤

√ 4D ; ω∞ − α∞ ε

i.e., we have estimated |∇Li |2gi at some time ti ∈ [t¯, ω∞ −

3ε 4 ].

Step 4. Estimating |∇Li |gi using the odi. Let I be the maximum subinterval ¯ of [t, ti ] containing t¯ and such that |Yi (t)|gi (t) ≥ 1 for all t ∈ I. Then by (30.23), for all t ∈ I, we have that    d 2 2   (30.26)  dt |Yi (t)|gi (t)  ≤ E |Yi (t)|gi (t) , where E

Cn ε 23 r− 12 (4)

+

Cn < ∞. ( 4ε )r

We conclude from (30.26) on the interval I and from (30.25) that 1 |∇Li |2gi (Φi (q), t¯) = |Yi (t¯)|2gi (t¯) ≤ const, 4 √ E(ω∞ −α∞ ) where const  max ω∞ − α∞ 4D . ε ,1 e

(30.27)



We have a similar estimate for the time derivatives of the reduced distances. ˜ i on compact sets). Lemma 30.8 (Uniform bounds for the time derivative of L Under the assumptions of Lemma 30.4, for any compact set K ⊂ M∞ and any ∞ ), there exists const < ∞ such that for any q ∈ K and any t¯ ∈ ε ∈ (0, ω∞ −α 2 [α∞ + ε, ω∞ − ε], we have     ∂L  ˜i  ¯  ¯  (q, t ) ≤ const  ∂t  for all i sufficiently large.

1. REDUCED DISTANCE OF TYPE A SOLUTIONS

141

Proof. Let (q, t¯) ∈ K × [α∞ + ε, ω∞ − ε]. By Lemma 7.34 in Part I, the time derivative of Li is given by   ∂Li 2 (Φi (q), t¯) = − ωi − t¯Rgi (Φi (q), t¯) + ωi − t¯|γi |gi (t¯). (30.28) ¯ ∂t By the Type A assumption and by Lemma 30.7, we have     const Cn 2   − ωi − t¯Rgi (Φi (q), t¯) + ωi − t¯|γi |gi (t¯) ≤ 1 + 1 . r− (ωi − t¯) 2 (ωi − t¯) 2    ˜ i | and  ∂ L˜¯i , in terms of time Exercise 30.9. Derive explicit bounds for |∇L ∂t and the distance to a fixed point, as in Lemma 30.4.

The lemma easily follows.

1.5. The limit reduced distance and its properties. Now that we have estimated the L-distances for the sequence of solutions gi (t), we shall investigate some of the properties of the limit of the reduced distance functions. Theorem 30.10 (Reduced distance based at the singular time for a sequence of solutions). Suppose that complete solutions with bounded curvature (Mni , gi (t) , pi ), t ∈ [αi , ωi ], converge in the C ∞ pointed Cheeger–Gromov sense to (Mn∞ , g∞ (t) , p∞ ), ω∞ and αi ≤ α∞ . Assume the Type A condition that t ∈ [α∞ , ω∞ ), where ωi |Rmgi | ≤

M r (ωi − t)

on Mi × (α∞ , ωi )

for some M < ∞ and r ∈ [1, 32 ) independent of i. Then there exists a subsequence α such that ˜i (q, t¯) = gpii ,ωi (Φi (q), t¯), where the Φi are as in (30.6), converges in Cloc for α ∈ (0, 1) to a function ∞  p∞ ,ω∞ : M∞ × (α∞ , ω∞ ) → R,

(30.29)

which is called a reduced distance based at the singular time and which satisfies the following properties: (i) ∞ is locally Lipschitz. (ii) ∞ satisfies (30.30)

−

n ∂∞ 2 − Δg∞ ∞ + |∇∞ |g∞ − Rg∞ + ≥0 ∂t 2 (ω∞ − t¯)

and holds both in the support sense and in the weak (distributional ) sense, respectively. A direct consequence is the following. Corollary 30.11 (Reduced distance based at the singular time for a Type A solution). Let (Mn , g (t)), t ∈ [0, T ), be a complete Type A singular solution with T < ∞. For any p ∈ M and Ti T , there exists a subsequence such that the α for α ∈ (0, 1) to a reduced distance based at reduced distances p,Ti converge in Cloc the singular time, i.e., (30.31)

p,T : M × [0, T ) → R,

142

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

satisfying the following properties: (i) p,T is locally Lipschitz. (ii) The inequality −

(30.32)

n ∂p,T 2 − Δp,T + |∇p,T | − R + ≥0 ∂t 2 (T − t)

holds both in the support sense and in the weak sense. Proof of Theorem 30.10. (i) There exists a subsequence such that ˜i conα to a locally Lipschitz function ∞ : M∞ ×(α∞ , ω∞ ) → R. Recall that verges in Cloc the Arzela–Ascoli theorem says that if we have a sequence of Lipschitz functions on a compact metric space with Lipschitz constant L, then there exists a subsequence which converges uniformly to a Lipschitz function with Lipschitz constant L. Thus (i) follows from Lemmas 30.4, 30.7, and 30.8. (ii) Inequality (30.30) holds in the weak sense. I.e., for any nonnegative C 2 function ϕ on M∞ × [t1 , t2 ] with support in K × [t1 , t2 ], where α∞ < t1 < t2 < ω∞ and K is compact, we have (30.33)      t2 ∂∞ n 2 + |∇∞ | −Rg∞ + ϕ dμg∞ dt ≥ 0. ∇∞ , ∇ϕ+ − ∂t 2 (ω∞ − t) t1 M∞ Let g˜i  Φ∗i gi . By Lemma 7.129(i) in Part I, since the solutions gi (t) have bounded sectional curvatures (depending on i) and by pulling back quantities on Φi (Ui ) ⊂ Mi to Ui ⊂ M∞ , we have (30.34)

( '  t2 2 # $ ˜i  ∂  n  ∇˜i , ∇ϕ + ∇˜i  − Rg˜i + ϕ dμg˜i dt ≥ 0 + − ∂t 2 (ωi − t) g ˜i g ˜i t1 M∞ provided i is sufficiently large so that [t1 , t2 ] ⊂ (αi , ωi ) and K ⊂ Ui . Thus (30.33) shall follow from the convergence of each term on the lhs of (30.34) as i → ∞. We achieve this in several steps.   t  Step 1. Convergence of t12 M∞ −Rg˜i + 2(ωni −t) ϕ dμg˜i dt. By the Cheeger– Gromov convergence of gi (t) to g∞ (t), we clearly have 

t2

(30.35) t1



 n −Rg˜i + ϕ dμg˜i dt 2 (ωi − t) M∞    t2  n → −Rg∞ + ϕ dμg∞ dt. 2 (ω∞ − t) t1 M∞



Step 2. Convergence of

# $ ˜i , ∇ϕ − ∇  M∞

 t2  t1

g ˜i

∂ ˜i ∂t ϕ

 dμg˜i dt. By Lemmas

30.7 and 30.8, there exists C < ∞ such that ||˜i ||W 1,∞ (K×[t1 ,t2 ]) ≤ C, and hence (30.36)

||˜i ||W 1,2 (K×[t1 ,t2 ]) ≤ C

1. REDUCED DISTANCE OF TYPE A SOLUTIONS

143

independent of i. Using the Banach–Alaoglu theorem, there exists a subsequence  such that ˜i  converges weakly in W 1,2 (K × [t1 , t2 ]) to ∞ |K×[t1 ,t2 ] .3 Hence K×[t1 ,t2 ]

 t2  # $ ∂ ˜i ˜ ∇i , ∇ϕ ϕ dμg˜i dt (30.37) − ∂t g ˜i t1 M∞    t2  ∂∞ → ϕ dμg∞ dt. ∇∞ , ∇ϕg∞ − ∂t t1 M∞ t  Step 3. Convergence of t12 M∞ |∇˜i |2g˜i ϕ dμg˜i dt. We divide the proof into two substeps. t  Step 3a. Lower bound for lim inf i→∞ 2 |∇˜i |2 ϕ dμg˜ dt. We have 



t1 t2



t1

M∞  t2 

0≤ 

M∞

= −2

M∞

t1

t2

g∞



|∇∞ |2g∞ ϕ dμg∞ dt + # $ ∇∞ , ∇˜i

M∞

t1

g∞

M∞

t1

t1

t2



t1

By the convergence of g˜i to g∞ , we obtain   t2  2 ˜ |∇i |g˜i ϕ dμg˜i dt ≥ lim inf i→∞

M∞

|∇˜i |2g∞ ϕ dμg∞ dt

ϕ dμg∞ dt.

M∞

t1

i



t2

Taking the lim inf of this implies that   t2  |∇˜i |2g∞ ϕ dμg∞ dt ≥ (30.38) lim inf i→∞

g ˜i

 2   ∇∞ − ∇˜i  ϕ dμg∞ dt

t2

t1

M∞

|∇∞ |2g∞ ϕ dμg∞ dt.



 t2 

2

M∞

|∇∞ |g∞ ϕ dμg∞ dt.

Step 3b. Upper bound for lim supi→∞ t1 M∞ |∇˜i |2g˜i ϕ dμg˜i dt. Since ˜i converges weakly to ∞ in W 1,2 ,  t2   t2  2 (30.39) |∇˜i |2g∞ ϕ dμg∞ dt − |∇∞ |g∞ ϕ dμg∞ dt lim sup i→∞

t1

M∞



t2



= lim sup i→∞

t1

M∞

t1

M∞

# $ ∇˜i − ∇∞ , ϕ∇˜i

g∞

dμg∞ dt.

By Cheeger–Gromov convergence, we may replace g∞ by g˜i in the lhs of (30.39). Step 3b shall follow from showing that  t2  # $ ∇˜i − ∇∞ , ϕ∇˜i dμg˜i dt (30.40) lim sup i→∞

t1

M∞

g ˜i

is nonpositive. The argument below of Enders follows Lemma 9.21 in Morgan and Tian [251]. Recall from (7.92) in Part I that for each i sufficiently large we have the elliptic partial differential inequality: ˜i − n ≤ 0, 2Δg˜i ˜i − |∇˜i |2g˜i + Rg˜i + ωi − t 3 The

choice of metric on M∞ × (α∞ , ω∞ ) used to define W 1,2 (K × [t1 , t2 ]) is not essential.

144

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

which holds in the weak sense (see Lemma 7.129(ii) in Part I); i.e., we have

(  ' # $ ˜i − n 2 ˜ ˜ dμg˜i ≤ 0 (30.41) + ψi −|∇i |g˜i + Rg˜i + −2 ∇i , ∇ψi ωi − t g ˜i M for any nonnegative C 2 function ψi on M∞ with support in some compact subset K and for i sufficiently large. Recall from above (or Lemma 7.111 in Part I) that, on a Riemannian manifold, 1,∞ 1,2 and hence in Wloc . Moreover, any locally Lipschitz function is contained in Wloc 1,2 any nonnegative Lipschitz function on K, since it is in W (K), may be approximated in W 1,2 (K) by nonnegative C ∞ functions.4 Hence (30.41) holds assuming only that ψi ≥ 0 is Lipschitz with support in K. Now since ˜i converges in C 0 (K) to ∞ , there exists εi  0 such that ∞ − ˜i + εi ≥ 0 on K. So we are justified in taking ψi = (∞ − ˜i + εi )ϕ to obtain from (30.41) that   #  $  ˜ ˜ (30.42) lim sup − ∇i , ∇ (∞ − i + εi )ϕ dμg˜i ≤ 0 g ˜i

M

i→∞

  ˜   since ψi → 0 in C 0 (K) and since −|∇˜i |2g˜i + Rg˜i + ωii−n −t  ≤ C on K independent of i (using Lemmas 30.4 and 30.7). Now integrating (30.42) in time implies that  t2  # $ 0≥ lim sup ϕ ∇˜i , ∇(˜i − ∞ ) dμg˜i dt t1



+



lim sup i→∞

t1



g ˜i

M

i→∞ t2



t2

=

lim sup i→∞  t2

t1

≥ lim sup i→∞

t1

M



M

M

# $ (˜i − ∞ ) ∇˜i , ∇ϕ dμg˜i dt g ˜i

# $ dμg˜i dt ∇˜i − ∇∞ , ϕ∇˜i g ˜i

# $ ∇˜i − ∇∞ , ϕ∇˜i dμg˜i dt. g ˜i

Thus (30.40) is nonpositive. This completes Step 3b. In conclusion, we have shown that  t2   t2  2 ˜ (30.43) lim |∇i |g˜i ϕ dμg˜i dt = |∇∞ |2g∞ ϕ dμg∞ dt. i→∞

t1

M∞

t1

M∞

This proves that (30.30) holds in the weak sense. We leave it as an exercise for the reader to check that (30.30) also holds in the support sense.  2 Remark 30.12. Note that (30.43) says that |∇˜i |2g˜i → |∇∞ |g∞ in the sense of distributions. Since ˜i converges weakly in W 1,2 to ∞ on compact subsets of space-time, we also have that ˜i → ∞ in the sense of distributions. 4 This follows, for example, from the existence of a smoothing operator on W 1,2 (K) which preserves the nonnegativity of functions, such as convolution or the linear heat flow with Dirichlet boundary condition.

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS

145

2. Reduced volume at the singular time for Type I solutions It is well known that for complete solutions of the Ricci flow with bounded curvatures, the reduced volumes have the monotonicity property (see Corollary 8.17 in Part I for an exposition). Using estimates from the previous section, as well as estimates proved in this section, we shall establish the well-definedness and monotonicity of the reduced volume based at the singular time for Type I solutions. In the next section we shall use this monotonicity to obtain shrinking GRS as singularity models. Under the hypotheses of Theorem 30.10 and corresponding to the solution (Mn∞ , g∞ (t)) and ∞ , we define the reduced volume based at the singular time (p∞ , ω∞ ) by  −n/2 −∞ (x,t) (4π (ω∞ − t)) e dμg∞ (t) (x) . (30.44) V˜∞ (t)  V˜p∞ ,ω∞ (t)  M∞

We have the following reduced volume monotonicity based at the singular time for a sequence of solutions. Theorem 30.13. Assume the hypotheses of Theorem 30.10 with r = 1, i.e., a Type I assumption. Let V˜∞ (t) be a reduced volume based at the singular time corresponding to ∞ . We have the following: (1) (Bounded above by 1) V˜∞ (t) is well defined and V˜∞ (t) ≤ 1. d ˜ (2) (Monotonicity) V˜∞ (t) is differentiable and dt V∞ (t) ≥ 0. In particular, limt→ω∞ V˜∞ (t) ≤ 1. (3) (The equality case) If V˜∞ (t1 ) = V˜∞ (t2 ) for some t1 < t2 , both in (α∞ , ω∞ ), then (Mn∞ , g∞ (t) , ∞ (t) , − ω∞1−t ) is a shrinking GRS structure for all t ∈ [t1 , t2 ]; i.e., Rc g∞ (t) + ∇g∞ (t) ∇g∞ (t) ∞ (t) −

1 2 (ω∞ − t)

g∞ (t) = 0.

Since Theorem 30.10 is stated for a sequence of solutions to the Ricci flow, we reformulate Theorem 30.13 for a single Type I solution. Corollary 30.14 (Reduced volume monotonicity based at the singular time for a Type I solution). Let (Mn , g (t)), t ∈ [0, T ), be a complete Type I singular solution with T < ∞. Corresponding to p,T in (30.31), let V˜p,T (t) be the reduced volume based at the singular time. Then: (1) V˜p,T (t) is well defined and V˜p,T (t) ≤ 1. (2) V˜p,T (t) is differentiable and d ˜ Vp,T (t) ≥ 0. dt In particular, limt→T V˜p,T (t) ≤ 1. (3) If V˜p,T (t1 ) = V˜p,T (t2 ), where 0 < t1 < t2 < T , then (Mn , g (t) , p,T (t) , − T 1−t ) is a shrinking GRS structure for all t ∈ [t1 , t2 ]; i.e., Rc g(t) + ∇2 p,T (t) −

1 g (t) = 0. 2 (T − t)

146

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Before we prove Theorem 30.13 at the end of this section, we first discuss a more precise upper bound for the reduced distance  of Type I solutions due to Naber [273] (compare with the more elementary Lemma 7.13(iii) in Part I, which bounds  in terms of the maximum curvature). Lemma 30.15 (The reduced distance of a Type I solution has at most quadratic growth in space). Suppose that (Mn , g (t)), t ∈ [α, ω], where −∞ < α < ω < ∞, is a complete solution of the Ricci flow with bounded curvature and satisfying the Type I condition: (30.45)

  Rmg(t)  ≤ M ω−t

on M × [α, ω).

Then, for any p, q ∈ M and ε ∈ (0, ω−α 2 ), the reduced distance   p,ω satisfies (30.46)

 (x, t) ≤

Bd2g(t) (q, x) 2 (ω − t)

+

A + 2 (q, t) B

for all x ∈ M and t ∈ [α + ε, ω), where A < ∞ and B < ∞ are constants depending only on n, ε, and M . In particular, (30.47)

 (x, t) ≤

Bd2g(t) (p, x) 2 (ω − t)

+

A + 2n2 M. B

  Proof. Let (x, t) ∈ M × [α + ε, ω), let q ∈ M, and let β : 0, dg(t) (q, x) → M be a minimal geodesic from q to x with respect to g (t). Define (30.48)

u (s)   (β (s) , t) .

The idea of the proof is to establish, via a gradient estimate, a first-order odi for u (s) to control its growth. Let γ : [t, ω] → M be a minimal L-geodesic with γ (t) = z and γ (ω) = p. By (7.89) in Part I, (30.49)

|∇|2 (z, t) = −R (z, t) −

K (γ) (ω − t)

3/2

+

 (z, t) , ω−t

where (30.50)

   ω  3 ∂R     R K (γ)  ω − tˆ 2 − 2∇R · γ  tˆ + 2 Rc γ  tˆ , γ  tˆ − dtˆ. ∂t ω − tˆ t

 1. Bounding K (γ). First suppose the weaker Type A assumption that  Step Rmg(t)  ≤ M r on M × [α, ω). By (30.3), we have on M × [α + ε, ω) that (ω−t)    ∂R   Cn,ε,M   ˆ 2r  ∂t  y, t ≤  ω − tˆ

and

 Cn,ε,M |∇R| y, tˆ ≤  3r . ω − tˆ 2

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS

147

Therefore, applying this and the Type A assumption to (30.50), we obtain (30.51) |K (γ)| ⎛ ⎞    2      ω 2     ˆ ˆ  32 Cn,ε,M Cn,ε,M γ t nM γ t n M ⎠ˆ ω − tˆ ⎝ ≤ r +  2r +  r+1 dt 32 r +  ˆ ˆ ω − t t ω−t ω − tˆ ω − tˆ

  2  ω 2  32 (nM + Cn,ε,M ) γ  tˆ  M n 2C n,ε,M ≤ ω − tˆ +  r  2r + r+1 dtˆ ω − tˆ t ω − tˆ ω − tˆ     2   ω  32 (nM + Cn,ε,M ) R + γ tˆ  ω − tˆ ≤ dtˆ  r ω − tˆ t

 ω  32 2Cn,ε,M + (nM + Cn,ε,M ) n2 M n2 M ˆ ω−t + +  2r r+1 dtˆ t ω − tˆ ω − tˆ  ω   2   3 −r  ω − tˆ 2 ≤ (nM + Cn,ε,M ) R + γ  tˆ  dtˆ t 5 3 n2 M 2Cn,ε,M + (nM + Cn,ε,M ) n2 M −2r −r + 3 (ω − t) 2 (ω − t) 2 , + 5 − 2r − r 2 2

provided r < 54 . Now we consider the special case where r = 1 (i.e., we have a Type I singular solution) since this gives us a clear way to estimate the penultimate line of (30.51). In this case we have  1 (30.52) |K (γ)| ≤ 2 2Cn,ε,M + (nM + Cn,ε,M ) n2 M + n2 M (ω − t) 2 1

+ 2 (nM + Cn,ε,M ) (ω − t) 2  (z, t) . Step 2. Bounding |∇|. Again using r = 1, by (30.49) and (30.52), we have (30.53)

2

|∇| (z, t) ≤ |R| (z, t) +

|K (γ)| 3/2

(ω − t) A + B (z, t) ≤ ω−t

+

 (z, t) ω−t

on M × [α + ε, ω), where (30.54)

 A  4Cn,ε,M + 2 nM + Cn,ε,M + 32 n2 M,

(30.55)

B  2 (nM + Cn,ε,M ) + 1.

That is, we have the key inequality:  (30.56) on M × [α + ε, ω).

|∇| (z, t) ≤

A + B (z, t) ω−t

148

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Step 3. The odi for u (s) in (30.48). By taking z = β(s) in (30.56) we have 1  u (s) ≤ √ (30.57a) A + Bu (s), ω−t (30.57b) u (0) =  (q, t) . Define

2   Bs 1 A √ + A + B (q, t) − , B 2 ω−t B which is the solution to the corresponding ode 1  U  (s) = √ (30.58a) A + BU (s), ω−t (30.58b) U (0) =  (q, t) . U (s) 

  By the ode comparison theorem, we have u (s) ≤ U (s) for all s ∈ 0, dg(t) (q, x) . In particular, by taking s = dg(t) (q, x), we obtain 2  1 Bdg(t) (q, x)  A √  (x, t) ≤ + A + B (q, t) − B B 2 ω−t Bd2g(t) (q, x) A + + 2 (q, t) , ≤ 2 (ω − t) B

which is (30.46). Step 4. Proof of (30.47). Taking q = p and using R (p, t) ≤ (using the constant path as the test path for )  ω  1 (30.59) ω − tˆR p, tˆ dtˆ  (p, t) ≤ √ 2 ω−t t ≤ n2 M.

n2 M ω−t ,

we obtain

Applying this to (30.46) yields (30.47).



We also have bounds for the first derivatives of  in both space and time. Corollary 30.16 (Gradient and time derivative bounds for  of Type I solutions). Under the hypothesis of Lemma 30.15, there exist constants C, D, E, F < ∞ depending on n, ε ∈ (0, ω−α 2 ), and M such that (30.60) and (30.61)

D dg(t) (p, x) C |∇| (x, t) ≤ √ + ω−t ω−t   F d2g(t) (p, x)  ∂    (x, t) ≤ E + 2  ∂t  ω−t (ω − t)

on M × [α + ε, ω). Proof. (1) Applying (30.47) to (30.56), we obtain B 2 d2g(t) (p, x) 2A + 2Bn2 M + (30.62) |∇| (x, t) ≤ . ω−t 2 (ω − t)2 √ This proves (30.60) with C = 2A + 2Bn2 M and D = √B2 , where A and B are given by (30.54)–(30.55).

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS

149

(2) By (7.94) in Part I,    ∂  ||   (x, t) ≤ 1 |∇|2 + 1 |R| + .  ∂t  2 2 2 (ω − t) 2

M , and (30.47) to this yields Applying (30.60), |R| ≤ nω−t   2 D2 d2g(t) (p, x)  ∂  n2 M   (x, t) ≤ C + + 2  ∂t  ω−t 2 (ω − t) (ω − t)

+ Now (30.61) follows easily.

Bd2g(t) (p, x) 4 (ω − t)

2

+

A B

+ 2n2 M . 2 (ω − t) 

A Recall that by (30.53), we have  (x, t) + B ≥ 0 on M × [α + ε, ω). We have the following result of Naber [273] after Perelman.

Lemma 30.17 (Lower bound for the reduced distance of a Type I solution). Suppose that (Mn , g (t)), t ∈ [α, ω], where −∞ < α < ω < ∞, is a complete solution of the Ricci flow with bounded curvature and satisfying the Type I condition (30.45). Let p ∈ M and let  = p,ω . Then for any ε ∈ (0, ω−α 2 ) there exists a constant C < ∞ depending only on n, M , and ε such that for any x1 , x2 ∈ M and t ∈ [α + ε, ω) we have   dg(t) (x1 , x2 ) A A ≥  (x1 , t) + +  (x2 , t) + (30.63) 1 − C, 1 B B 4B 2 (ω − t) 2 where A and B are given by (30.54) and (30.55). In particular, we have the following lower bound for the reduced distance:  dg(t) (x, p) A ≥ (30.64)  (x, t) + 1 − C ≥ −C 1 B 4B 2 (ω − t) 2 on M × [α + ε, ω). Proof. The idea is to follow the proof of Lemma 19.46 in Part III on bounding the sum  (x1 , t) +  (x2 , t) of the reduced distance at two points at the same time, while now using the Type I condition. Given any x1 , x2 ∈ M and any t¯ ∈ [α + ε, ω), let γ1 : [t¯, ω] → M and γ2 : [t¯, ω] → M be minimal L-geodesics whose graphs join (x1 , t¯) and (x2 , t¯) to (p, ω), respectively. We have for t ∈ [t¯, ω],    ∂ d  dg(t) (γ1 (t) , γ2 (t)) = − dg(t) (γ1 (t) , γ2 (t)) (30.65) − dt ∂t / 2 . dγa (t) , − ∇a dg(t) (γ1 (t) , γ2 (t)) , dt a=1 where the vector field ∇a dg(t) ( · , · ) denotes the gradient of the two-variable function dg(t) ( · , · ) with respect to the a-th variable. Using (30.45), we can bound the term on the rhs of the first line of (30.65) by applying Perelman’s changing distances estimate. In particular, by (18.15)–(18.16)

150

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

in Part III with K =  −

(30.66)

M ω−t

and radius = K −1/2 , we have

 ∂ dg(t) (γ1 (t) , γ2 (t)) ≤ ∂t

10(n−1) 3

√ K=

10(n−1) 3

M 1/2 1

(ω − t) 2

.

 We can bound the term on the second line of (30.65) by using the equality ∇a dg(t)  = 1 as well as (7.54) in Part I. Namely, we have g(t)    ∇a dg(t) (γ1 (t) , γ2 (t)) , γa (t)  ≤ |γa (t)|

g(t)

= |∇|g(t) (γa (t) , t)  A + B (γa (t) , t) ≤ , ω−t where we have used (30.56) in the second inequality. By combining the above, we obtain −

(30.67)

 d  dg(t) (γ1 (t) , γ2 (t)) ≤ dt

10(n−1) 3

M 1/2 1

(ω − t) 2  2  A + B (γa (t) , t) + . ω−t a=1

Since γa is a minimal L-geodesic, for any t ∈ (t¯, ω) we have  ω      2  1  (γa (t) , t) = √ ω − tˆ R γa tˆ , tˆ + γa tˆ g(tˆ) dtˆ (30.68) 2 ω−t t  ω      2  1 ω − tˆ R γa tˆ , tˆ + γa tˆ g(tˆ) dtˆ ≤ √ 2 ω − t t¯  t  1 ω − tˆ|R|max tˆ dtˆ + √ 2 ω − t t¯   √ ω − t¯ n2 M   (xa , t¯) + √ ≤ √ ω − t¯ − ω − t ω−t ω−t  ω − t¯   (xa , t¯) + n2 M ≤ √ ω−t  2 since |R|max tˆ ≤ nω−Mtˆ . Hence, by applying (30.68) to (30.67), we have for any t ∈ (t¯, ω) (30.69)

−

 d  dg(t) (γ1 (t) , γ2 (t)) dt

  1/4 √ A A B 1/2 (ω − t¯) ≤ 2n M +  (x1 , t¯) + +  (x2 , t¯) + 3/4 B B (ω − t) const + 1 , (ω − t) 2

where const < ∞ depends only on n, M , and ε.

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS

151

Integrating (30.69) over [t¯, ω] yields dg(t¯) (γ1 (t¯), γ2 (t¯)) 1 2

≤ 4B (ω − t¯)

1 2

√ 2n M +



A  (x1 , t¯) + + B



A  (x2 , t¯) + B



1

+ 2 const (ω − t¯) 2 . Therefore estimate (30.63) follows since x1 = γ1 (t¯) and x2 = γ2 (t¯). Finally, (30.64) follows from taking x1 = p and x2 = x in (30.63) and from applying (30.59).  Now we can apply the estimate above to the sequence used to define the reduced distance ∞ . Proposition 30.18. Under the hypotheses of Theorem 30.13 we have that for  ∞ any ε ∈ 0, ω∞ −α 2 (30.70) (30.71) (30.72)

d2g∞ (t) (x, p∞ )

Cd2g∞ (t) (x, p∞ ) − C ≤ ∞ (x, t) ≤ + C, C (ω∞ − t) ω∞ − t   dg∞ (t) (p, x) 1 |∇∞ | (x, t) ≤ C √ + , ω∞ − t ω∞ − t

  d2g∞ (t) (p, x)  ∂∞  1  (x, t) ≤ C  2 + ω  ∂t  ∞−t (ω∞ − t)

on M∞ × (α∞ + ε, ω∞ ) for some C < ∞. Analogous estimates hold for a reduced distance based at the singular time of a single Type I solution. Proof. Applying inequalities (30.64) and (30.47) to (Mni , gi (t) , pi ), t ∈ [αi , ωi ], in the hypothesis of Theorem 30.10, we have that for any (x, t) ∈ M∞ ×[α∞ +ε, ω∞ ) d2g˜i (t) (x, p∞ ) C (ωi − t)

− C ≤ ˜i (x, t) ≤

Cd2g˜i (t) (x, p∞ ) ωi − t

+ C,

where C < ∞ is independent of both (x, t) and i and where i is sufficiently large. Hence, taking the limit as i → ∞, we obtain (30.70). Recall from Theorem 30.10(i) that ∞ is locally Lipschitz. By applying (30.60) and (30.61) to i defined in Theorem 30.10 and then taking the limit of ˜i , we obtain (30.71) and (30.72), respectively.  Now we can give the Proof of Theorem 30.13. (1) V˜∞ (t) ≤ 1. For each i the reduced volume of the solution gi (t) based at (pi , ωi ) is  −n ˜ (30.73) Vi (t)  (4π (ωi − t)) 2 e−i (x,t) dμgi (t) (x) . Mi

Since each solution gi (t), t ∈ [αi , ωi ], is complete with bounded curvature, by Lemma 8.16(ii) in Part I we have that each V˜i (t) is integrable. Moreover, by Theorem 8.20 and Corollary 8.17, both in Part I, we also have that each V˜i (t) is

152

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

d ˜ Vi (t) ≥ 0 and V˜i (t) ≤ 1. Fix any compact subset K ⊂ M∞ differentiable in t, dt and t ∈ [α∞ , ω∞ ). Then, by Fatou’s lemma,  −n/2 −∞ (x,t) (4π (ω∞ − t)) e dμg∞ (t) (x) K    ˜ lim inf (4π (ωi − t))−n/2 e−i (x,t) dμg˜i (t) (x) = K i→∞  ˜ ≤ lim inf (4π (ωi − t))−n/2 e−i (x,t) dμg˜i (t) (x) i→∞

K

≤ lim inf V˜i (t) i→∞

≤ 1. Hence V˜∞ (t) ≤ 1 is well defined for t ∈ [α∞ , ω∞ ). (Note that this holds as long as the limit ∞ exists, in particular, under the more general Type A assumption on gi (t).) (2) Monotonicity of V˜∞ (t). By the definition of derivative, (30.74)

V˜∞ (t + h) − V˜∞ (t) dV˜∞ (t) = lim h→0 dt h = (4π)− 2 lim n

h→0

M∞

Φ(q, t, h)dμg∞ (t) (q) ,

provided the limit exists and where

e−∞ (q,t+h) dμg∞ (t+h) (q) e−∞ (q,t) 1 − . Φ(q, t, h)  n n h (ω∞ − t − h) 2 dμg∞ (t) (q) (ω∞ − t) 2 ∂ dμ = −R dμ to this, we see that at any point Applying the standard formula ∂t (q, t) where ∂∞ /∂t exists (in particular, for each t, a.e. on M∞ ), we have   n ∂∞ e−∞ (q,t) − − R lim Φ(q, t, h) = n g∞ . h→0 ∂t (ω∞ − t) 2 2 (ω∞ − t)

Claim 1. We have   (30.75) lim Φ(q, t, h)dμg∞ (t) (q) = h→0

M∞

lim Φ(q, t, h)dμg∞ (t) (q) ,

M∞ h→0

so that the evolution of the reduced volume based at the singular time is given by (30.76)    dV˜∞ n e−∞ (q,t) ∂∞ −n (t) = (4π) 2 − − Rg∞ (t) dμg∞ (t) (q). n dt 2 (ω∞ − t) ∂t M∞ (ω∞ − t) 2 Proof of Claim 1. Clearly (30.76) follows from (30.75). Now, as in the proof of Theorem 8.20 in Part I, equation (30.75) shall follow from Lebesgue’s dominated convergence theorem and estimating Φ(q, t, h) by an integrable function on M∞ , independent of h sufficiently small. We proceed to do this. By 30.18, we have that e−∞ has quadratic exponential decay and  ∂Proposition  that  ∂t∞  has quadratic growth. This, together with the Bishop–Gromov volume comparison theorem, implies that Φ(q, t, h) is bounded by an integrable function on M∞ , independent of h sufficiently small. This proves Claim 1 (see pp. 396–398 in Part I for further details).

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS

We wish to use (30.76) to prove that

˜∞ dV dt

153

(t) ≥ 0.

Claim 2. The rhs of (30.76) is nonnegative. Proof of Claim 2. By (30.33) we have (30.77)      t2 ∂∞ n 2 −Rg∞ + ∇∞ , ∇ϕg∞ + |∇∞ |g∞− ϕ dμg∞ dt ≥ 0 ∂t 2 (ω∞ − t) t1 M∞ for any nonnegative C 2 function ϕ on M∞ × [t1 , t2 ] with compact support. Since we have (30.70), (30.71), and (30.72) and in particular e−∞ is Lipschitz and decays quadratic exponentially, inequality (30.77) holds by approximation for ϕ = e−∞ . Since ∇∞ , ∇e−∞ g∞ + |∇∞ |2g∞ e−∞ = 0, we conclude that (30.78) V˜∞ (t2 ) − V˜∞ (t1 ) = (4π)− 2



n

t2 t1

≥ 0.



M∞

n

2 (ω∞ − t)

−

 ∂∞ −Rg∞ e−∞ dμg∞ dt ∂t

This proves (2). (3) The equality case. Suppose that V˜∞ (t1 ) = V˜∞ (t2 ) for some t1 < t2 . Then from (30.78) and (30.30), we conclude that (30.79)

Q∞  −

∂∞ n − Δg∞ ∞ + |∇∞ |2g∞ − Rg∞ + =0 ∂t 2 (ω∞ − t)

in the sense of distributions on M∞ × [t1 , t2 ]. This implies that   n (30.80) ∗∞ (ω∞ − t)− 2 e−∞ = 0, ∂ where ∗∞  − ∂t − Δg∞ + Rg∞ , in the sense of distributions. Applying standard parabolic regularity theory to (30.80), we conclude that ∞ is C ∞ on M∞ × [t1 , t2 ]. Thus (30.80) holds in the classical sense. On the other hand, recall from (7.94) in Part I that we have

−2

∂ ˜i  ˜ 2 ˜i + ∇i  = 0. − Rg˜i (t) + ∂t ωi − t g ˜i (t)

Hence, by Remark 30.12, we obtain (30.81)

P∞  −2

∂∞ ∞ 2 + |∇∞ |g∞ (t) − Rg∞ (t) + =0 ∂t ω∞ − t

in the sense of distributions. By (30.79) and (30.81), we have     2 (30.82) v∞  (ω∞ − t) Rg∞ + 2Δg∞ ∞ − |∇∞ |g∞ + ∞ − n = (ω∞ − t) (P∞ − 2Q∞ ) = 0.

e−∞ (ω∞ − t)

n/2

e−∞ n/2

(ω∞ − t)

154

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Since u∞  (4π)− 2 (ω∞ − t)− 2 e−∞ satisfies ∗∞ u∞ = 0 by (30.80), we have by Lemma 6.8 in Part I (i.e., the original Proposition 9.1 in Perelman [312]) that 2    1 ∗ g∞ g∞  g∞  u∞ . 0 = ∞ v∞ = −2 (ω∞ − t) Rcg∞ +∇ ∇ ∞ − 2 (ω∞ − t) n

n

We conclude that Rcg∞ +∇g∞ ∇g∞ ∞ −

1 g∞ = 0 2 (ω∞ − t)

on M∞ × [t1 , t2 ]. Now Theorem 30.13 is proved.



3. Type I solutions have shrinker singularity models In this section, by applying the reduced volume monotonicity based at the singular time and Perelman’s pseudolocality theorem, we show that any Type I singular solution must have a nonflat shrinking GRS as a singularity model. Of course, this does not rule out the possibility of other types of singularity models associated to a Type I solution. Let (Mn , g (t)), t ∈ [0, ω), be a Type I singular solution on a closed manifold. Given any λi → ∞, we may rescale the solution by defining  (30.83) gi (t)  λi g ω + λ−1 i t for t ∈ [−λi ω, 0). This is equivalent to the “parabolic rescaling” corresponding to the sequence of times ti  ω − λ−1 i , which is defined by rescaling by the inverse of the time to blow up: g˜i (t)  (ω − ti )

−1

g (ti + (ω − ti ) t) .

In particular, we have the correspondence  g˜i (t + 1) = λi g ω + λ−1 i t = gi (t) . for some M < ∞,   M   |Rmgi | (x, t) = Rmλi g(ω+λ−1 t)  (x) ≤ i |t|

Note that since |Rm| (x, t) ≤ (30.84)

M ω−t

for t ∈ [−λi ω, 0). Hence, given any sequence pi ∈ M, by Perelman’s no local collapsing theorem and Hamilton’s Cheeger–Gromov compactness theorem, there exists a subsequence {(M, gi (t) , pi )} which converges to a complete ancient solution (30.85)

(Mn∞ , g∞ (t) , p∞ ) ,

t ∈ (−∞, 0) ,

with |Rmg∞ | (x, t) ≤ M |t| ; i.e., g∞ (t) is a Type I ancient solution. For this Cheeger– Gromov convergence, let {Ui } be the corresponding exhaustion of M∞ and let (30.86)

Φi : (Ui , p∞ ) → (M, pi )

be the corresponding embeddings, so that Φ∗i gi (t) → g∞ (t) in the C ∞ -topology on compact subsets of M∞ × (−∞, 0). We first obtain limits of Type I solutions which may possibly be flat. The rough idea is to exploit the equality case of reduced volume monotonicity.

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS

155

Proposition 30.19 (Limits of Type I solutions with fixed basepoints are (possibly flat) shrinkers). Let (Mn , g (t)), t ∈ [0, ω), be a Type I singular solution on a closed manifold. Then for any p ∈ M and for any λi → ∞, the limit (Mn∞ , g∞ (t) , p∞ ) in (30.85) with pi ≡ p is a complete shrinking GRS with bounded curvature. Proof. First let pi ∈ M be any sequence. By Corollary 30.11, for each i there exists a reduced distance function gpi ,ω and the reduced volume  −n/2 −gp ,ω (x,t) g ˜ Vpi ,ω (t)  (4π (ω − t)) e i dμg(t) (x) M

based at the singular time for the original solution g (t), t ∈ [0, ω). By Corollary 30.14(1)–(2), we have that d ˜g V (30.87) (t) ≥ 0 and V˜pgi ,ω (t) ≤ 1. dt pi ,ω The reduced distance function i  gpii ,0 and the reduced volume V˜i  V˜pgii,0 based at the singular time for the rescaled solution gi (t), t ∈ [−λi ω, 0), are related to gpi ,ω and V˜pgi ,ω by  (30.88) i (q, t) = gpi ,ω q, ω + λ−1 i t and

 V˜i (t) = V˜pgi ,ω ω + λ−1 i t ,

(30.89)

respectively. Now Proposition 30.18 implies that each i is locally bounded and locally Lipschitz, uniformly in i. Hence, from the Arzela–Ascoli theorem, by passing to a subsequence we have that ∞ (q, t)  lim i (Φi (q) , t) i→∞

exists and is locally Lipschitz on M∞ × (−∞, 0), where the Φi are as in (30.86). By (30.88) and (30.70), we have that d2g(ω+λ−1 t) (Φi (q) , pi )

Cd2g(ω+λ−1 t) (Φi (q) , pi )

i − C ≤ i (Φi (q) , t) ≤ −Cλ−1 −λ−1 i t i t  −1 ∗ Since λi Φi g ω + λi t → g∞ (t) and since Φi (p∞ ) = pi , we obtain

(30.90)

i

d2g∞ (t) (q, p∞ )

− C ≤ ∞ (q, t) ≤

+ C.

Cd2g∞ (t) (q, p∞ )

+ C, −Ct −t where the constant C ∈ [1, ∞) is the same as in (30.90). Now let pi ≡ p for all i. Then, by (30.89) and (30.87), for any fixed t ∈ (−∞, 0) and for i large enough,  g 1 ≥ V˜i (t) = V˜p,ω ω + λ−1 t (30.91)

i

is defined and nondecreasing in i. Define the reduced volume corresponding to ∞ by  −n/2 −∞ (x,t) V˜∞ (t)  (−4πt) e dμg∞ (t) (x) M∞

for t ∈ (−∞, 0). Similarly to the proof of Claim 1 in the proof of Theorem 30.13(2), we have on compact time intervals that the reduced volume integrands

156

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS −n/2

(−4πt) e−i (Φi (x),t) dμΦ∗i gi (t) (x) are bounded above on M∞ by an integrable g n-form independent of i and t. Thus, using the monotonicity of V˜p,ω (t) and the Lebesgue dominated convergence theorem, we obtain that the limits  (30.92) V˜∞ (t) = lim V˜i (t) = lim V˜p,ω ω + λ−1 i t ≡ const ∈ (0, 1] i→∞

i→∞

exist for t ∈ (−∞, 0). By the argument used to obtain Corollary 30.14(3), we conclude that 1 Rc g∞ (t) + ∇g∞ (t) ∇g∞ (t) ∞ (t) + g∞ (t) = 0; 2t that is, g∞ (t) is a gradient shrinker with C ∞ potential function ∞ (t).



Remark 30.20. It is possible that g∞ (t) in Proposition 30.19 is flat. For example, consider the case where n = 3. If there are {pi } and λi → ∞ such that  −1 lim λ−1 = 0, i |Rmg | pi , ω − λi i→∞

then the corresponding limit satisfies |Rmg∞ | (p∞ , −1) = lim |Rmgi | (pi , −1) i→∞  −1 = lim λ−1 i |Rmg | pi , ω − λi i→∞

= 0. Since n = 3, Rmg∞ ≥ 0 and, by the strong maximum principle, we conclude that g∞ (t) is flat for t ≤ −1. By Theorem 28.20, g∞ (t) has bounded curvature on compact intervals in (−∞, 0). Hence, Chen and Zhu’s uniqueness theorem for complete solutions to the Ricci flow with bounded curvature (see [63]) implies that g∞ (t) is flat for all t ∈ (−∞, 0). We wish to characterize those p ∈ M for which the corresponding shrinker ω, and in Proposition 30.19 is nonflat. Suppose that there exists p ∈ M, ti c > 0 such that (ω − ti ) |Rmg | (p, ti ) ≥ c for all i. Then any corresponding limit g∞ (t) is nonflat (using the rescaling factors λi  (ω − ti )−1 ). In this case we have established the existence of a nonflat gradient shrinker singularity model. It is not a priori clear whether there always exists such a point p. We know by the gap theorem (see Lemma 8.7 in [77]) that (ω − t) max |Rmg | (x, t) ≥ x∈M

This implies that for any ti

ω there exists {pi } such that (ω − ti ) |Rmg | (pi , ti ) ≥

(30.93)

1 . 8

1 . 8

Since M is compact and by passing to a subsequence, we may assume that pi → p∗ for some p∗ ∈ M. However, this does not immediately imply that (ω − ti ) |Rmg | (p∗ , ti ) ≥ c for i large enough and some c > 0. Note that it is possible that (30.94)

lim (ω − ti )−1/2 dg(ti ) (pi , p∗ ) = ∞.

i→∞

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS

157

Generalizing the above discussion, we have Definition 30.21 (Set of Type I singular points). Let (30.95) 0 1 ΣI  p∗ ∈ M : ∃ti → ω and pi → p∗ , lim inf (ω − ti ) |Rmg | (pi , ti ) > 0 . i→∞

That is, ΣI is the set of points p such that there exists a sequence of (Riemann curvature) Type I essential points approaching (p, ω). By (30.93), ΣI = ∅ for any singular solution on a closed manifold. Notwithstanding the possibility of (30.94), one can prove the following result, which is a consequence of the above discussion plus pseudolocality. Theorem 30.22 (Limits of Type I solutions with basepoints in ΣI are nonflat shrinkers). If (Mn , g (t)), t ∈ [0, ω), is a Type I singular solution on a closed manifold. Then for any p∗ ∈ ΣI , corresponding to pj ≡ p∗ and any λj → ∞, the bounded curvature shrinker limit (Mn∞ , g∞ (t) , p∞ ) in Proposition 30.19 is nonRicci flat. Proof. Aiming for a contradiction, we suppose that g∞ (t) ≡ g∞ is Ricci flat on M∞ for all t ∈ (−∞, 0). (This actually implies that (M∞ , g∞ (t)) is isometric to Euclidean space by Lemma 27.9.) Let ε, δ > 0 be as in Perelman’s pseudolocality Theorem 10.3 in [312] and fix r0 ∈ (0, injg∞ (p∞ )). By the pointed Cheeger–Gromov convergence of gj (t) = 2 λj g(ω + λ−1 j t) to g∞ (t) and since g∞ (− (εr0 ) ) is flat, we have that for j ≥ j0 , where j0 is sufficiently large, | Rmgj (−(εr0 )2 ) | ≤ r0−2 and

in Bgj (−(εr0 )2 ) (p∗ , r0 )

Volgj (−(εr0 )2 ) Bgj (−(εr0 )2 ) (p∗ , r0 ) ≥ (1 − δ) ωn r0n .

By Theorem 10.3 in [312] (see Proposition 1 in [211] by one of the authors for the version we use), we have that for all j ≥ j0 ,5   2 ∗ Rmgj  (x, t) ≤ (εr0 )−2 in B (30.96) gj (−(εr0 )2 ) (p , εr0 ) × [− (εr0 ) , 0). Since p∗ ∈ ΣI , there exists ti → ω and pi → p∗ such that c (30.97) |Rmg | (pi , ti ) ≥ ω − ti for some c > 0. Fix a j ≥ j0 . From (30.97), for i sufficiently large we have   c −2 Rmg  (pi , −λj (ω − ti )) ≥ ≥ 2 (εr0 ) (30.98) j λj (ω − ti ) and 2

−λj (ω − ti ) ∈ [− (εr0 ) , 0) since ω − ti → 0 as i → ∞. We obtain a contradiction to (30.96) since, for i sufficiently large, we have pi ∈ Bgj (−(εr0 )2 ) (p∗ , εr0 ) . Hence g∞ (t), t ∈ (−∞, 0), is not Ricci flat. 5 For

t = 0.



the application of Proposition 1 in [211] it is not necessary that gj (t) be defined at

158

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

The following says that bounded scalar curvature, finite-time singular solutions, if they exist, must be Type II. Theorem 30.23 (Type I singular solutions have unbounded scalar curvature). Any finite-time Type I singular solution on a closed manifold must have sup M×[0,T )

R = ∞.

Proof. By Theorem 30.22, there exists an associated singularity model (Mn∞ , g∞ (t)) which is a non-Ricci flat shrinking GRS. If supM×[0,T ) R < ∞, then by the proof of Lemma 28.16 (M∞ , g∞ (t)) must be Ricci flat, a contradiction.  In addition to ΣI defined by (30.95), we have the following. Definition 30.24 (Sets of singular points). Given a finite-time singular solution (Mn , g (t)), t ∈ [0, ω), define the following sets of singular points: 0 1 ΣR  p ∈ M : lim inf (ω − t) R (p, t) > 0 , t→ω

Σ  {p ∈ M : ∃ ti → ω and pi → p such that |Rm| (pi , ti ) → ∞} . From Definitions 30.21 and 30.24 it is immediately clear that ΣR ⊂ ΣI ⊂ Σ. In other words, Σ comprises the weakest notion of singular point, whereas ΣR comprises the strongest notion of singular point. The following surprising result, that for Type I solutions the above notions of singular point are the same, is due to Enders, M¨ uller, and Topping [104]. The proof uses the strong maximum principle and pseudolocality. Theorem 30.25 (Equality of sets of singular points). If (Mn , g (t)) is a Type I singular solution to the Ricci flow on a closed manifold, then (30.99)

ΣR = ΣI = Σ.

Proof. It suffices to show that ΣI ⊂ ΣR and Σ ⊂ ΣI . Roughly speaking, the issue is to obtain some local curvature control. As evidenced by the work of Hamilton and, especially, Perelman, such control is often obtainable indirectly by limiting arguments. (1) Let p ∈ M − ΣR . Then there exists cj → 0 and tj ∈ [ω − cj , ω) such that cj (30.100) Rg (p, tj ) < . ω − tj Define λj  (ω − tj )−1 . By Proposition 30.19, corresponding to pj ≡ p and λj , the limit (Mn∞ , g∞ (t) , p∞ ) in (30.85) is a complete shrinking GRS with bounded curvature. By (30.100) we have Rg∞ (p∞ , −1) = 0. By applying the strong maximum ∂R principle to the equation for ∂tg∞ , we obtain Rcg∞ (t) ≡ 0 for t ≤ −1. Then, by Chen and Zhu’s uniqueness theorem (see [63]), we conclude that Rcg∞ (t) ≡ 0 for t < 0. Since we are on a shrinker, this implies that Rmg∞ (t) ≡ 0. By Theorem 30.22, p ∈ M − ΣI . (2) Let p ∈ M − ΣI . Then lim (ω − t) |Rm| (p, t) = 0.

t→ω

Hence, for any λj → ∞, corresponding to pj ≡ p and λj , the limit shrinker (Mn∞ , g∞ (t) , p∞ ) in Proposition 30.19 is isometric to Euclidean space. By the

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

159

same reasoning used to obtain (30.96) (using Perelman’s pseudolocality theorem), we have   2 Rmgj  (x, t) ≤ (εr0 )−2 in B gj (−(εr0 )2 ) (p, εr0 ) × [− (εr0 ) , 0), 0 0

which is equivalent to |Rmg | (x, t) ≤

λj0 (εr0 )

2

in B 

g ω−

 p, (εr0 )2 λj

 εr0 1/2 λj 0

2 × ω−

(εr0 )2 λ j0 , ω

 .

  −1/2 This implies that Bg(ω−λ−1 (εr0 )2 ) p, λj0 εr0 ⊂ M − Σ, which in turn implies j0 that p ∈ M − Σ.  0

As a consequence, for Type I solutions, we have the following characterization of whether a point is singular or nonsingular. Corollary 30.26 (Distinguishing between singular points and nonsingular points). Let (Mn , g (t)), t ∈ [0, ω), be a Type I singular solution on a closed manifold. For each p ∈ M, either (1) lim inf t→ω (ω − t) R (p, t) > 0 or (2) there exists C < ∞ such that for every ti → ω and pi → p, lim sup |Rm| (pi , ti ) ≤ C. i→∞

Another direct consequence of Theorem 30.25 is the following strengthening of Theorem 30.22. Theorem 30.27 (Limits of Type I solutions with basepoints in Σ are nonflat shrinkers). Let (Mn , g (t)), t ∈ [0, ω), be a Type I singular solution on a closed manifold. Suppose that p ∈ M is such that there exist ti → ω and pi → p with |Rm| (pi , ti ) → ∞ (such a point always exists). Then for any ti → ω, the sequence (M, gˆi (t) , p) subconverges to a non-Ricci flat shrinking GRS, where   gˆi (t)  |Rm| (p, ti ) g ti + |Rm|−1 (p, ti ) t . Problem 30.28. For any Type A singular solution on a closed manifold, does there exist a corresponding singularity model which is a nonflat shrinking GRS? We are not aware of an example of a singular solution on a closed manifold which does not have a nonflat shrinking GRS singularity model. 4. Some results on Type I ancient solutions The general classification of κ-noncollapsed ancient solutions in high dimensions is a difficult problem. In this section we discuss some qualitative properties of Type I ancient solutions, including shrinkers. 4.1. Limits of κ-noncollapsed Type I ancient solutions. Let (Mn , g (t)), t ∈ (−∞, 0), be an ancient solution to the Ricci flow. Let + τi  0 and τi− ∞ be sequences of positive numbers. Define the corresponding forward (+) and backward (−) rescaled solutions of the Ricci flow by 1  (30.101) gi± (t)  ± g τi± t , t ∈ (−∞, 0) , τi where gi± denotes either gi+ or gi− (and similarly for τi± ). We shall prove a result regarding the limits of gi± (t).

160

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

We shall use the following elementary result. Lemma 30.29 (Characterization of canonical form for shrinkers). Let (Mn , g (t) , f (t) , 1t ), t ∈ (−∞, 0), be a complete shrinking Ricci soliton with bounded curvature.  2 if and only if this soliton is in canonical form; i.e., Then ∂f (t) = ∇g(t) f (t) ∂t

(30.102)

g(t)

g(t) = −t ϕ(t)∗ g (−1)

and

f (t) = f (−1) ◦ ϕ(t),

where ϕ (t) is defined by (30.104) below. Proof. By hypothesis, ∂ 1 (30.103) g (t) = −2 Rc g(t) = 2∇g(t) ∇g(t) f (t) + g (t) . ∂t t  g(t) 2 ∂f   (1) Suppose ∂t (t) = ∇ f (t) g(t) . From Corollary 27.7, we may define diffeomorphisms ϕ(t) : M → M, t ∈ (−∞, 0), by  ∂ 1  g(−1) ϕ (x, t) = ∇ (30.104a) f (−1) (ϕ (x, t)) , ∂t −t (30.104b) ϕ (−1) = idM . We then need to show that (30.102) holds. Define (30.105)

gˆ (t)  −t ϕ(t)∗ g (−1)

and fˆ (t)  f (−1) ◦ ϕ(t).

By Theorem 4.1 in [77] (or an easy calculation), we have ∂ 1 gˆ (t) = 2∇gˆ(t) ∇gˆ(t) fˆ (t) + gˆ (t) = −2 Rc gˆ(t) , (30.106) ∂t t   ∂ fˆ  gˆ(t) ˆ 2 (t) = ∇ f (t) (30.107) . ∂t g ˆ(t) Since gˆ (−1) = g (−1), by Chen and Zhu’s uniqueness theorem and Kotschwar’s backwards uniqueness theorem, we have gˆ (t) = g (t) for all t ∈ (−∞, 0). Thus (30.103) and (30.106) imply ∇g(t) ∇g(t) fˆ (t) = ∇g(t) ∇g(t) f (t) .    g(t) f (t)2 and (30.107), we obtain By ∂f ∂t (t) = ∇ g(t)   ∂ g(t) ∂f ∇ f (t) = ∇g(t) (t) + 2 Rcg(t) ∇g(t) f (t) ∂t    ∂t = 2 ∇g(t) ∇g(t) f (t) + Rcg(t) ∇g(t) f (t) 1 = − ∇g(t) f (t) t and

  ∂ g(t) ˆ ∂ fˆ ∇ f (t) = ∇g(t) (t) + 2 Rcg(t) ∇g(t) fˆ (t) ∂t ∂t 1 g(t) ˆ = − ∇ f (t) . t g(−1) From this and ∇ f (−1) = ∇g(−1) fˆ (−1), we obtain ∇g(t) fˆ (t) = ∇g(t) f (t). Thus  2 ∂f ∂ fˆ   (t) = ∇g(t) fˆ (t) (t) . = ∂t ∂t g(t)

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

161

Since fˆ (−1) = f (−1), we conclude that fˆ (t) = f (t). From (30.105) we conclude (30.102). (2) The “if” part of the lemma follows from Theorem 4.1 in [77].  Exercise 30.30. Show that if f is normalized so that R + |∇f |2 − f ≡ 0 at t = −1, then 1 R + |∇f |2 + f ≡ 0 t for t ∈ (−∞, 0). Solution. One computes that     ∂ 1 1 1 R + |∇f |2 + f = − R + |∇f |2 + f . ∂t t t t Analogous to Perelman’s “asymptotic shrinker” result for κ-solutions (see Theorem 28.24), we have the following application due to Naber [273] of the reduced volume monotonicity based at the singular time. Theorem 30.31 (Backward and forward asymptotic shrinkers for κ-noncollapsed Type I ancient solutions). Let (Mn , g (t)), t ∈ (−∞, 0), be a κ-noncollapsed (below all scales) ancient solution satisfying |Rm| (x, t) ≤

C |t|

on M × (−∞, 0)

for some C < ∞. Then, for any τi+  0 and τi− ∞, there  xn0 ∈±M and sequences exist subsequences such that M , gi (t) , (x0 , −1) , defined by (30.101), converges to a complete shrinking GRS  ± n ±  M∞ , g∞ (t) , x± . 0 , −1 + Furthermore, the forward asymptotic soliton (M+ ∞ , g∞ (t)) and the backward − − asymptotic soliton (M∞ , g∞ (t)) are κ-noncollapsed, are in canonical form, but are possibly isometric to Euclidean space.

Proof. By the Type I hypothesis, we have   C   Rmgi±  (x, t) = τi± |Rmg | (x, τi± t) ≤ |t| on M×(−∞, 0), for all i. Since g (t) is κ-noncollapsed for some κ > 0, by the scaleinvariance of this property, we have that the gi± (t) are all κ-noncollapsed for this same κ. Therefore we may apply Hamilton’s Cheeger–Gromov-type compactness  theorem to conclude that there exists a subsequence such that M, gi± (t) , (x0 , −1) converges to a complete limit solution   ± n ± , t ∈ (−∞, 0) , M∞ , g∞ (t) , x± 0 , −1     C ± ±  (x, t) ≤ (t) is κ-noncollapsed and where Rmg∞ where g∞ |t| . In particular, there exist smooth embeddings  ± ±  ± ± Φ± i : Ui , x0 ⊂ M∞ , x0 → (M, x0 ) ,  ± ∗ ±  ± gi (t) → g∞ (t) in the C ∞ where Ui± is an exhaustion of M± ∞ , such that Φi ± topology on compact subsets of M∞ × (−∞, 0).

162

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

By Corollary 30.11, the reduced distance functions g±

(xi 0 ,0) : M × (−∞, 0) → R for gi± (t) based at the singular time 0 exist. Moreover, Proposition 30.18 implies  g± that each function (xi 0 ,0) Φ± i ( · ) , · is locally bounded and locally Lipschitz, uniformly in i. Thus, by the Arzela–Ascoli theorem and (30.70), they subconverge to a locally Lipschitz continuous function ± ± ∞ : M∞ × (−∞, 0) → R

satisfying

 d2g± (t) q, x± 0

 Cd2g± (t) q, x± 0

∞ − C ≤ ± +C ∞ (q, t) ≤ −Ct −t on M± ∞ × (−∞, 0) for some C < ∞. We define the reduced volume corresponding to ± ∞ (based at the singular time 0) by  ± ± ˜ ± (−4πt)−n/2 e−∞ (t) dμg∞ V∞ (t)  (t) ∞

M± ∞

for t ∈ (−∞, 0). As in the proof of Proposition 30.19 (see (30.92)), we have that  ± g± g ± τi t ≡ C ± (t) = lim V˜(xi0 ,0) (t) = lim V˜(x (30.108) V˜∞ 0 ,0) i→∞

i→∞

for t ∈ (−∞, 0) and for some constants C (30.109)

±

±

∈ [0, 1]. From this we then obtain that

±

g∞ (t) g∞ (t) ± ± Rc g∞ ∇ ∞ (t) + (t) + ∇

1 ± g (t) = 0; 2t ∞

± i.e., g∞ (t) is a gradient shrinker. Furthermore, by (30.109) and (30.82), we have

(30.110) (30.111)

n = 0, 2t  ± 2 ± ∞ (t) − n ±   ± ± = 0, Rg∞ (t) + 2Δg∞ (t) ∞ (t) − ∇∞ (t) + −t ± ± ± Rg ∞ (t) + Δg∞ (t) ∞ (t) +

which imply (30.112) (30.113)

±  ±  ∇∞ (t)2 + ∞ (t) = 0, ± Rg∞ + (t) t n ±   (t) −  2 ∞ ± ± 2   ± ∇ = 0. Δg∞  (t) − (t) + ∞ (t) ∞ −t

By (7.94) in Part I, we have 2

 2 ∂± ± ∞ ∞ (t)  ± (t) − ∇± = 0. + ∞ (t) + Rg∞ (t) ∂t t ∂±

2

Combining this with (30.112), we obtain ∂t∞ = |∇± ∞ | . By Lemma 30.29, we  ± 1 ± conclude that the shrinkers M∞ , g∞ are in canonical form.  (t) , ± (t) , ∞ t Regarding the backwards limit in Theorem 30.31, we have the following general result for shrinkers. The idea is that we have good control backwards in time of the 1-parameter family of diffeomorphisms associated to shrinkers.

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

163

Proposition 30.32 (Backwards limits with fixed basepoint of a shrinker are  isometric to the shrinker). Let Mn , g (t) , f (t) , 1t , t ∈ (−∞, 0), be a complete shrinking GRS with bounded curvature and which is in canonical form. If x0 ∈ M ∞, then every Cheeger–Gromov convergent subsequence of (M, gi− (t) , and τi− (x0 , −1)) has limit isometric to (M, g (t) , (p0 , −1)) for some p0 ∈ M.  Proof. Since the shrinker M, g (t) , f (t) , 1t , t ∈ (−∞, 0), is in canonical 2 form, we have ∂f ∂t = |∇f | . The dilated solutions satisfy (30.114)

gi− (t) =

1  − g τi t = −tϕ(τi− t)∗ g (−1) , τi−

where the ϕ (t) are defined by (30.104a)–(30.104b). Thus, for  each t ∈ (−∞, 0) and each i, the pointed Riemannian manifolds M, gi− (t) , x0 and (M, −tg (−1) , ϕ(x0 , τi− t)) are isometric via the isometry ϕ(τi− t). By (27.46), f ( · , t) is a proper function with limx→∞ f (x, t) = ∞ for each t. Since the function t → ϕ (x0 , t) satisfies the ode (30.104a), we have that the set {ϕ (x0 , t) : t ∈ (−∞, −1]} is contained in the compact set {x : f (x, −1) ≤ f (x0 , −1)}. Hence, for each t ∈ (−∞, 0), the backwards limit of the basepoints (30.115)

lim ϕ(x0 , τi− t) =  lim ϕ(x0 , t )  p0

i→∞

t →−∞

exists, where p0 ∈ M is a critical point of f .6 It follows from (30.115) that the sequence of pointed complete Riemannian isom   manifolds M, gi− (−1), x0 = M, g(−1), ϕ(x0 , −τi− ) converges as i → ∞ to (M, g (−1) , p0 ) in the Cheeger–Gromov sense (using the global diffeomorphisms ϕ(−τi− )).7 Since we have convergence at t = −1 and bounded curvature, by Hamil- ton’s Cheeger–Gromov compactness theorem, the solutions Mn , gi− (t) , (x0 , −1) − (t) , (p0 , −1)) (also using the global diffeomorphisms converge to a solution (Mn , g∞ isom − − ϕ(−τi )). Since g∞ (−1) = g (−1), by the forward and backward uniqueness of isom

− (t) = complete solutions to the Ricci flow with bounded curvatures, we have g∞ g (t) for all t ∈ (−∞, 0). 

We now characterize an isometric case for Theorem 30.31. Theorem 30.33. If, in Theorem 30.31, the forward asymptotic soliton (M+ ∞, − (t)) and the backward asymptotic soliton (M− , g (t)) are isometric to each ∞ ∞ other as solutions, then they are each isometric to the original solution (M, g (t)). + g∞

+ − − Proof. Fix t ∈ (−∞, 0). Suppose that (M+ ∞ , g∞ (t)) and (M∞ , g∞ (t)) are isometric to each other. By pulling back one of the solutions by an isometry, we − + − may assume that M+ ∞ = M∞  M∞ and g∞ (t) = g∞ (t)  g∞ (t). Define − m∞ (t)  + ∞ (t) − ∞ (t) .

example, on the Gaussian shrinker, where M = Rn and f (x, t) = and p0 = 0.

6 For √x −t

|x|2 , −4t

we have ϕ (x, t) =

that the ϕ(−τi− ) converge to a map, which one would not expect to be a diffeomorphism. For example, if f has a unique critical point p0 , then the ϕ(−τi− ) converge to the constant map p0 . 7 Note

164

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Then by (30.109) we have ∇g∞ (t) ∇g∞ (t) m∞ (t) = 0, so that ∇g∞ (t) m∞ (t) is a parallel gradient vector field. Claim. + − V˜∞ (t) = V˜∞ (t) .

(30.116)

g∞ (t) − ∞ (t) and Proof of claim. (1) m∞ (t) is constant. Then ∇g∞ (t) + ∞ (t) = ∇ + − + Δg∞ (t) ∞ (t) = Δg∞ (t) ∞ (t). In this case, (30.111) implies ∞ (t) = − ∞ (t), which in turn yields (30.116). (2) m∞ (t) is not constant. Then ∇g∞ (t) m∞ (t) is nonzero and by the de Rham holonomy splitting theorem we have the following. The shrinker (M∞ , g∞ (t)) is  isometric to N n−1 , h(t) × R, ds2 , where (N , h(t)) is isometric to {x ∈ M∞ : m∞ (x, t) = 0} with the metric induced by g∞ (t). Moreover, there exist a, b ∈ R depending on t with a = 0 such that

(30.117)

m∞ (y, s, t) = as + b

for y ∈ N and s ∈ R. From (30.109) we can now show that there exists φ(t) : N → R such that s2 +a ˆs + ˆb 4t s 2 = φ (y, t) − −a ˆt + ˆb + a ˆ 2 t2 2t

− ∞ (y, s, t) = φ (y, t) −

for some a ˆ, ˆb ∈ R and Rch +∇h ∇h φ +

1 h = 0. 2t

By translating the coordinate on R and absorbing the constant ˆb + a ˆ2 t2 into φ, we may assume that a ˆ = ˆb = 0; i.e., − ∞ (y, s, t) = φ (y, t) −

s2 . 4t

Hence, by (30.112) and (30.117), we have  2 + 2 −  ∞ (t) ∞ (t)   0 = ∇+ − ∇− ∞ (t) + ∞ (t) − t/ t . ∂ as + b − 2 = 2a ∇∞ (t) , +a + ∂s t b = a2 + . t This implies m∞ (y, s, t) = as − a2 t.

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

We conclude that for t < 0  + (30.118) (t) = V˜∞

165

(−4πt)−n/2 e−∞ (x,t) dμg∞ (t) (x)   − −n/2 = (−4πt) e−∞ (x,t)−m∞ (y,s,t) dμh (y) ds R N  2 s −n/2 et( 2t −a) e−φ(y,t) dμh (y) ds = (−4πt) R N   s2 −n/2 4t = (−4πt) e e−φ(y,t) dμh (y) ds +

M+ ∞

− = V˜∞ (t) .

R

N

This completes the proof of the claim. By the claim, we obtain that  +  − g g τi t = lim V˜(x τi t . (30.119) lim V˜(x 0 ,0) 0 ,0) i→∞

i→∞

g V˜(x 0 ,0)

g By the monotonicity of (t), this implies that V˜(x (t) ≡ const for t ∈ 0 ,0) (−∞, 0). We conclude that g (t) is a gradient shrinker in canonical form with C ∞ potential function g(x0 ,0) (t). By Proposition 30.32, (M, g (t)) is isometric to −  (M− ∞ , g∞ (t)).

The following result, obtained by X. Cao and Q. Zhang (see [51] for a proof), − ensures that some backward limit g∞ (t) is nonflat. Theorem 30.34 (Existence of asymptotic shrinkers for Type I ancient solutions). For any κ-noncollapsed complete Type I ancient solution of the Ricci flow, there exists a backward limit which is a nonflat shrinker. Theorem 30.31 raises the following. Problem 30.35. Does there exist a backward asymptotic shrinker for any κnoncollapsed Type II ancient solution? The answer to this question is yes when n = 3. Theorem 30.36. For any 3-dimensional κ-noncollapsed Type II ancient solution with bounded curvature, there exists a backward asymptotic shrinker.  Proof. Let M3 , g (t) , t ∈ (−∞, 0], be a 3-dimensional κ-noncollapsed Type II ancient solution with bounded curvature. Then g (t) must have positive sectional curvature (see p. 374 of [77]). By the argument of Proposition 9.29 in [77] (here M may be either closed or noncompact since we have the κ-noncollapsed assumption), we have There exists (xi , ti ) ∈ M × (−∞, 0) with ti → −∞ such  the following. that M3 , gi (t) , xi , where  gi (t)  Ri g ti + Ri−1 t and Ri  Rg (xi , ti ) , ∞ converges in the pointed Cheeger–Gromov sense to a noncompact κ-noncollapsed  C 3 steady GRS M∞ , g∞ (t) , x∞ , t ∈ (−∞, ∞), with the property that Rg∞ achieves its space-time maximum (of 1) at (x∞ , 0) and that Rmg∞ > 0. Note that Rmg∞ cannot have a zero because there are no 2-dimensional nonflat κ-noncollapsed steady solitons.

166

30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Since M∞ is noncompact, by Theorem 9.66 in [77], there exists a sequence of points {x∞,i } in M∞ such that the pointed sequence of solutions  3 M∞ , g∞,i (t) , x∞,i , t ∈ (−∞, 0], where

  g∞,i (t)  Rg∞ (x∞,i , 0) g∞ Rg∞ (x∞,i , 0)−1 t ,

converges in C ∞ pointed Cheeger–Gromov sense to an ancient solution of the  the 3 Ricci flow M∞,∞ , g∞,∞ (t) , x∞,∞ , t ∈ (−∞, 0], which splits as the product of R with a constant curvature ancient solution on the 2-sphere. This is our asymptotic shrinker (recall by Lemma 8.26 in [77] that the limit of a limit is a limit).  4.2. Shrinking Ricci solitons are gradient and κ-noncollapsed. In this subsection we show that complete noncompact shrinking Ricci solitons with bounded curvatures must be gradient and κ-noncollapsed. We shall use the following estimate, which is related to (27.19). Lemma 30.37 (Bounds for vector fields of shrinking Ricci solitons). Let (Mn , g, X, −1) be a complete noncompact shrinking Ricci soliton structure with bounded Ricci curvature, so that (30.120)

2Rij + ∇i Xj + ∇j Xi − gij = 0.

Fix O ∈ M, and let r (x)  dg (x, O). Then there exists const < ∞ such that (30.121)

X, ∇r (x) ≥

r (x) − const 2

on M − {O}. Proof. Let K  supM |Rc| < ∞. By Proposition 18.8 in Part III, for any x ∈ M − B (O, 2) and any minimal unit speed geodesic γ : [0, r (x)] → M joining O to x, we have    r(x) 2   K +1 . Rc (γ (s) , γ (s)) ds ≤ 2 (n − 1) 3 0 Using (30.120), we obtain X, ∇r (x) − X (O) , γ  (0) =



r(x)

  ∇γ  (s) X, γ  (s) ds

0

  1   = − Rc (γ (s) , γ (s)) + ds 2 0   2 1 K +1 . ≥ r (x) − 2 (n − 1) 2 3 

r(x)

 Partially extending Corollary 27.7, we have Lemma 30.38 (Nongradient shrinking Ricci solitons and completeness of their vector fields). If (Mn , g, X, −1) is a complete noncompact shrinking Ricci soliton structure with bounded Ricci curvature, then the vector field X is complete.

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167

Proof. By (30.120), we have 1 2 2 Rc (X) + ∇X X + ∇ |X| − X = 0. 2 Thus

1 1 X |X|2 = ∇X X, X = −2 Rc (X, X) − X |X|2 + |X|2 , 2 2 so that X |X|2 = −2 Rc (X, X) + |X|2 . Then    2 2 X |X|  ≤ (1 + 2K) |X| .

It is not hard to see that this yields the lemma. For example, let σ be an integral curve of X and let s be an arc length parameter along σ. Then wherever |X| ≥ 1,8 we have d |X|2 ≤ (1 + 2K) |X|2 . ds Integrating this odi implies the completeness of X.  One may extend the canonical form for shrinkers, characterized in Lemma 30.29, to the nongradient case. Let (Mn , g, X, −1) be a complete shrinker with bounded Ricci curvature, so that 2 Rc +LX g − g = 0. = − 1t X (ϕ (x, t)) and ϕ (x, −1) = x. By Define ϕ (t) : M → M by Lemma 30.38, the 1-parameter family of diffeomorphisms ϕ (t) is defined for all ∗ ∗ −1 t ∈ (−∞, 0). Define g (t)  −tϕ (t) g and X (t)  − 1t ϕ (t) X = − 1t (ϕ (t) )∗ X. We compute   ∂  1 ∂  ∗ g (t) = g (t0 ) − t0 ϕ (t) g (−1) , ∂t t=t0 t0 ∂t t=t0   ∂  where ∂t ϕ (t)∗ g (−1) = LX(t0 ) ϕ (t0 )∗ g (−1) . Hence t=t ∂ ∂t ϕ (x, t)

0

∂ 1 g (t) = g (t) + LX(t) g (t) . ∂t t On the other hand, with g = g (−1) and X = X (−1), we have for t ∈ (−∞, 0) ∗

−2 Rc (g (t)) = ϕ (t) (−2 Rc (g)) ∗

= ϕ (t) (LX g − g)  = Lϕ(t)∗ X ϕ (t)∗ g − ϕ (t)∗ g 1 = LX(t) g (t) + g (t) . t  Thus Mn , g (t) , X (t) , 1t satisfies the shrinking gradient soliton equation for all t ∈ (−∞, 0). Exercise 30.39. Compute

∂ ∂t X

(t).

Regarding shrinking Ricci solitons with bounded curvature in general, we have the following result of Naber [273]. This is another application of the monotonicity of the reduced volume based at the singular time. 8 Note, as an aside, that if σ (s) ∈ M − B (O, ρ), where ρ  4 max {const, 1}, then by (30.121) we have |X| (σ (s)) ≥ 1.

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Theorem 30.40 (Shrinkers with |Rm| ≤ C must be gradient and κ-non collapsed). Let Mn , g (t) , X (t) , 1t , t ∈ (−∞, 0), be a complete noncompact shrinking Ricci soliton with bounded curvature and in canonical form, so that 1 2Rij + ∇i Xj + ∇j Xi + gij = 0. t Then:

 (1) There exists a function f : M → R such that M, g (t) , f (t) , 1t is a complete shrinking GRS (with f normalized ).9 (2) There exists κ > 0 depending only on n and e−f dμ such that (M, g) is κ-noncollapsed below all scales.

Proof. (1) Let g  g (−1), fix O ∈ M, and let r (x)  dg (x, O). By Lemma 30.38, the vector field X  X (−1) is complete. Let ϕ (t) : M → M, t ∈ (−∞, 0), be the 1-parameter family of diffeomorphisms generated by − 1t X with ϕ (−1) = id. By Lemma 30.37, there exists ρ < ∞ such that X satisfies (30.122)

∇r, X > 1 in M − Bg (O, ρ) .

Hence, for t ∈ (−∞, 0), . / dϕ 1 1 d r (ϕ (t)) = ∇r, (t) = − ∇r, X (ϕ (t)) > − dt dt t t whenever ϕ (t) ∈ M − Bg (O, ρ). Hence, for any y ∈ M, there exists t0 < 0 such that ϕ (y, t) ∈ Bg (O, ρ) for all t ≤ t0 . ∞, Similarly to the proof of Proposition 30.32, we conclude that for any τi−  − the sequence of pointed solutions M, gi (t) , (y, −1) subconverges to (M, g (t) , (z, −1)) for some z ∈ B (O, ρ) (with the property that X (z) = 0). Moreover, by the proof of Theorem 30.31 (where we do not need to assume κ-noncollapsed since we have convergence), a reduced distance  based at the singular time 0 yields a gradient shrinker structure for g (t). ∞. By Proposition 30.32, there exists p0 ∈ M (2) Let x0 ∈ M, and let τi − such that M, gi− (t) , (x0 , −1) subconverges to (M, g (t) , (p0 , −1)). By Theorem 30.31, we have that (M, g (t) , gp0 ,0 (t) , 1t ) is a GRS with normalized potential function being f (t) = gp0 ,0 (t), the reduced distance based at the singular time. Hence  g n V˜pg0 ,0 (t)  (−4πt)− 2 e−p0 ,0 (t) dμg(t) M −n = (−4πt) 2 e−f (t) dμg(t) ≡ const .

M

By Perelman’s weakened no local collapsing theorem (see Theorem 8.26 in Part I  for an exposition), there exists κ > 0 depending only on n and e−f dμ such that (M, g) is κ-noncollapsed below all scales.  9 Note

that this implies that X − ∇f is a Killing vector field.

5. NOTES AND COMMENTARY

169

Note the following: (a) A steady GRS may not necessarily be κ-noncollapsed for any κ > 0. For example, we may take the product of the cigar and Rn−2 . (b) An expanding Ricci soliton may be neither gradient nor κ-noncollapsed for any κ > 0 (see Lott [208]). 5. Notes and commentary §1. For Definition 30.1, see Definition 3.4 in Enders [103]. Theorem 30.10 is Theorem 3.3.1 of Enders [102] in the Type A case; see Naber [273] for the Type I case. Regarding the Type A condition, we remark that matched asymptotics calculations by Angenent and two of the authors [11] indicate that for any n, k ≥ 3 there should exist rotationally symmetric degenerate neckpinches with space-time S n × [0, T ) and curvature blow-up rate given by 2 −2+ k

sup |Rm| ( · , t) ∼ C (T − t) Sn

.

Note that when k = 3, such a solution is Type A. §2. Theorem 30.13 is in Naber [273] and is Corollary 4.3 in Enders [103] (there it is stated more generally for Type A solutions). See also the antecedent Proposition 11.2 in Perelman [312] (or the expository Theorem 19.53 in Part III). For Theorems 30.22 and 30.27, see Enders, M¨ uller, and Topping [104]. For related work, see Le and Sesum [184].

CHAPTER 31

Hyperbolic Geometry and 3-Manifolds We’ve got no future, we’ve got no past Here today, built to last. – From “West End Girls” by Pet Shop Boys

The purpose of this chapter is to prepare the reader for the discussion of nonsingular solutions to the Ricci flow on closed 3-manifolds in the subsequent chapters. The main aspect of nonsingular solutions related to this chapter is the occurrence therein of finite-volume hyperbolic limits and their corresponding pieces in the solutions. In §1 we review basic facts about hyperbolic space, its isometries, and its quotients. In §2 we recall some basic geometric and topological facts related to hyperbolic manifolds. In §3 we discuss the Margulis lemma and ends of finite-volume hyperbolic manifolds. In §4 we recall the Mostow rigidity theorem. In §5 we discuss Seifert fibered manifolds and graph manifolds in relation to Cheeger–Gromov collapse. 1. Introduction to hyperbolic space In this section we discuss models and isometries of hyperbolic space. 1.1. Models of hyperbolic n-space Hn . Let Hn , n ≥ 2, denote hyperbolic n-space, the simply-connected complete Riemannian n-manifold with constant sectional curvature equal to −1. By the Cartan–Ambrose–Hicks theorem, hyperbolic n-space is unique up to isometry. There are various concrete models of hyperbolic n-space; we note three of them. The disk model is Dn  {x = (x1 , . . . , xn ) ∈ Rn : |x| < 1} with the metric gD 

 4 dx21 + · · · + dx2n

. 2 (1 − |x| )2 The ideal boundary ∂Dn  {x : |x| = 1} is also called the sphere at infinity. The upper half-space model is U n  {x ∈ Rn : xn > 0} with the metric gU 

dx21 + · · · + dx2n . x2n

We call the ideal boundary ∂U n  {x : xn = 0} the hyperplane at infinity. With its standard conformal structure, the one point compactification of ∂U n by the point at infinity is conformally equivalent to the sphere at infinity. 171

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A beautiful model of hyperbolic space is the hyperboloid model. Let En,1 denote Minkowski space, i.e., Rn+1 with the (indefinite) Lorentzian metric dx21 + · · · + dx2n − dx2n+1 . The hyperboloid model is the submanifold 0 1 H n  x ∈ Rn+1 : x, xn,1  x21 + · · · + x2n − x2n+1 = −1 and xn+1 > 0 with the induced metric gH . Note that although the ambient metric is indefinite, the induced metric is positive definite. The equivalence of the above models of hyperbolic space may be seen by the following facts. (1) The diffeomorphism α : (H n , gH ) → (Dn , gD ) defined by

(x1 , . . . , xn ) x − xn+1 en+1 = , 1 + xn+1 1 + xn+1 where en+1  (0, . . . , 0, 1), is an isometry. (2) The diffeomorphism α (x) 

β : (Dn , gD ) → (U n , gU ) defined by β (x)  2

x + en |x + en |2

− en

is an isometry. Exercise 31.1. Show that α and β are isometries between the models. 1.2. Geodesics and the group of isometries of hyperbolic space. Let us begin with the 2-dimensional hyperbolic plane. Recall that the upper half-plane U 2 = {z = x + iy ∈ C |y = Im(z) > 0} has the hyperbolic metric dx2 + dy 2 −4dzd¯ z = . y2 (z − z¯)2 We can list all the orientation-preserving isometries of the metric as follows. First, from the definition of the metric, we see easily that f (z)  λz, where λ ∈ R>0 , and g(z)  z + a, where a ∈ R, are isometries. Next, we claim that h(z)  − z1 preserves the hyperbolic metric. Indeed, let w = h(z). Then, gU =

h∗ (gU ) = −

4d( z1 )d( z1¯ ) 4dwdw ¯ dzd¯ z = − = −4 . 1 1 2 2 (w − w) ¯ (z − z¯)2 ( z − z¯ )

It is well known that each M¨ obius transformation   az + b a b (31.1) F (z) = , where ∈ SL(2, R), c d cz + d of the upper half-plane U 2 is a composition of f ’s, g’s, and h above. Therefore, M¨ obius transformations z → az+b cz+d , where a, b, c, d ∈ R with ad − bc = 1, are orientation-preserving isometries of the hyperbolic upper half-plane U 2 . Lemma 31.2. All orientation-preserving isometries of U 2 are M¨ obius transformations of the form in (31.1).

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173

We leave it as an exercise for the reader to verify the lemma. Hint: Recall that an isometry is determined by its differential at a single point. Show that SL(2, R) acts transitively on the space of all unit tangent vectors. We can also find all geodesics in the hyperbolic plane as follows. Since we know a lot of isometries, it suffices to find one geodesic and then use isometries to find others. Example 31.3. The positive y-axis is a geodesic in the upper half-plane U 2 . Let γ : [0, 1] → U 2 be a smooth path from ia to ib, where 0 < a < b. The length in the hyperbolic metric of γ is  1 |γ  (t)| L(γ) = dt. 0 Im(γ(t)) Write γ(t) = x(t) + iy(t), where y(t) > 0. Then !  1    1 x (t)2 + y  (t)2  y (t)  b  dt ≥  dt = ln . L(γ) = y(t) a 0 0 y(t) Note that L(γ) = ln ab if and only if x(t) = 0 and y  (t) ≥ 0. This shows that the positive y-axis is a minimal geodesic and that the distance between ia and ib is dU (ia, ib) = ln ab . By using the isometrics z → az+b obius transformations cz+d and the fact that M¨ preserve angles and the set of all circles and lines, we obtain that all geodesics in U 2 are (portions of) vertical lines or circles perpendicular to the x-axis. All of the above computations can be carried out, without too much change, for the upper half-space model U n for n ≥ 3. The counterpart of the map z → − z1 for U 2 is the inversion. Recall that the inversion ιy,r about the sphere centered at y ∈ Rn of radius r is the bijection of Rn ∪ {∞} defined by x−y + y. ιy,r (x) = r 2 |x − y|2 Inversions of Rn ∪ {∞} preserve angles, i.e., are conformal. The inversion about the unit sphere centered at the origin is ι0,1 (x) = |x|x 2 , which is the orientation-reversing isometry z → z1¯ in dimension 2. Obviously, ι0,1 preserves U n . A M¨ obius transformation of Rn is defined to be a composition of the inversion ι0,1 with x → λBx + b, B ∈ O(n), b ∈ Rn , λ > 0. We denote the group of all ob(n). The subgroup of orientation-preserving M¨obius transformations of Rn by M¨ M¨obius transformations is denoted by M¨ob+ (n). Lemma 31.4. The inversion ι0,1 preserves the hyperbolic metric gU on U n . We leave the verification of the lemma as an exercise; this is a straightforward generalization of the calculation that we performed for the upper half-plane above. Consider Rn−1 as the subspace Rn−1 × {0} ⊂ Rn . Then each M¨ obius transformation T of Rn−1 extends naturally to a M¨obius transformation T ∗ of Rn so that T ∗ (U n ) = U n . Here is how it works. The extension of the inversion ι0,1 is the inversion ι0,1 about the unit sphere in Rn ; the extension of x → λx is given by the same formula; and the extension of x → Ax + b, A ∈ O(n − 1), is given by sending (x, t) to (Ax + b, t) for (x, t) ∈ Rn−1 × R. In short, the extension map produces an injective group homomorphism φ from M¨ob(n − 1) to M¨ ob(n) so that the image

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31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

φ(M¨ ob(n − 1)) is exactly the subgroup of M¨obius transformations of Rn leaving U n invariant. By inspecting the metric gU , we see that isometries of U n include f (x) and g(x) defined by (1) f (x) = kx, where k is a positive real number, (2) (31.2)

g(w, xn ) = (Aw + a, xn ),

where w = (x1 , . . . , xn−1 ) ∈ Rn−1 , xn ∈ R>0 , A ∈ O(n − 1), and a ∈ Rn−1 . It can be shown that the set of all isometries of U n is the set of compositions of the functions f , g, and ι0,1 described previously. So the group M¨ob(n − 1) may be identified with the isometry group of hyperbolic space. Let Isom(Hn ) and Isom+ (Hn ) be the groups of isometries and orientationpreserving isometries of hyperbolic n-space Hn . We can summarize the discussion as Lemma 31.5. Using the U n model of hyperbolic space, we see that ∼ M¨ Isom (Hn ) = ob(n − 1) 0 1 = x → λA (i (x)) + b : λ ∈ R>0 , A ∈ O (n − 1) , i = id or ιy,r , b ∈ Rn−1 . Moreover, M¨ ob+ (n − 1) ∼ = Isom+ (Hn ) consists of the elements of M¨ob(n − 1) where i = id and A ∈ SO(n − 1) or i is an inversion about a sphere and A ∈ O(n − 1) − SO(n − 1). We now give a more detailed description of the isometry group of 3-dimensional obius transformations can be expressed hyperbolic space H3 . Two-dimensional M¨ nicely in terms of the linear fractional transformations and reflection z → z¯; i.e., each orientation-preserving M¨obius transformation of the complex plane is of the form az + b , φM : z −→ cz + d   a b where M = ∈ SL (2, C); i.e., a, b, c, d ∈ C satisfy ad − bc = 1. Note that: c d (i) φM  = φM if and only if M  = ±M . (ii) The map from PSL (2, C)  SL (2, C) / {± id} to M¨ ob+ (2) defined by [M ] −→ φM is a Lie group isomorphism. As a consequence, we have the isomorphisms of groups  ob+ (2) ∼ (31.3) Isom+ H3 ∼ = M¨ = PSL (2, C) , where matrix multiplication in SL (2, C) induces multiplication in its Z2 -quotient PSL (2, C). Similarly, our earlier discussion has shown that we have the isomorphisms  ob+ (1) ∼ (31.4) Isom+ H2 ∼ = M¨ = PSL (2, R)  SL (2, R) / {± id} , where PSL(2, R) acts on the upper half-plane by linear fractional transformations z → az+b cz+d , a, b, c, d ∈ R and ad − bc = 1.

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An interesting property of inversions of Rn is that they preserve the set of all (codimension 1) hyperspheres and hyperplanes in Rn . Since any inversion is a composition of ι0,1 , a scaling kx, and conjugation by a translation of Rn , it suffices to establish this property for ι0,1 . To see this, note that the equation of a hyperplane or a hypersphere in Rn is of the form Ax·x +Ba·x +C = 0 where A, B, C ∈ R, a ∈ R , and a · x is the standard Euclidean inner product. x in this equation, we obtain, Replacing x by ι0,1 (x) = x·x n

A + B a · x + C x · x = 0. This is again an equation of a hypersphere or hyperplane. By taking intersections of hyperplanes and hyperspheres, we conclude that inversion preserves the set of all circles and lines in Rn . Finally, we discuss the geodesics in U n . Using the same calculation as in Example 31.3, which works in any dimension, we see that the positive xn -axis is a geodesic in U n . Applying the isometries of U n just obtained and the basic properties of inversion, we obtain Lemma 31.6. All geodesics in the upper half-space model U n of the hyperbolic geometry are (portions of ) vertical lines or circles perpendicular to the hyperplane ∂U n at infinity. Proof. Using translations g(x) = x + a, we conclude that all vertical lines (i.e., lines perpendicular to ∂U n ) are geodesics. Using the inversion ι0,1 and scaling f (x) = kx for k > 0, we see that all semicircles perpendicular to ∂U n are geodesics. On the other hand, given any nonzero tangent vector v in U n , there exists a unique semicircle or vertical line tangent to v. The lemma follows.  1.3. Types of isometries of hyperbolic space. Isometries of Hn are classified according to the locations of their fixed points. Consider the disk model of hyperbolic space, so that the ideal boundary ∂Hn ∼ = ¯ n  Hn ∪ ∂Hn S n−1 . If ϕ ∈ Isom (Hn ), then ϕ extends to a homeomorphism of H which is topologically a closed ball. Hence by the Brouwer fixed point theorem, ¯ n. ϕ has a fixed point in H Definition 31.7. An orientation-preserving isometry ϕ of Hn is (1) elliptic if there exists a fixed point of ϕ in Hn , (2) parabolic if there are no fixed points of ϕ in Hn and exactly one fixed point of ϕ on ∂Hn , (3) hyperbolic if there are no fixed points of ϕ in Hn and exactly two fixed points of ϕ on ∂Hn . Clearly these conditions are mutually exclusive. What is nontrivial is to show that if ϕ ∈ Isom+ (Hn ) has at least three fixed points on ∂Hn , then ϕ is elliptic. See Ratcliffe’s textbook [334] for a proof. Example 31.8. (1) In the disk model, if ϕ ∈ SO (n, R), then ϕ is an elliptic isometry. (2) In the upper half-space model U n , given b ∈ Rn − {0} with bn = 0, the translation x −→ x + b

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31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

is a parabolic isometry. When n = 2, 3, every parabolic isometry is conjugate to an isometry of this form. (3) In the upper half-space model, if A ∈ SO (n − 1, R) and λ > 0 with λ = 1, then x −→ (λA (x1 , . . . , xn−1 ) , λxn ) is a hyperbolic isometry. In all dimensions any hyperbolic isometry is conjugate to an isometry of this form. (4) The composition of two inversions about disjoint (n − 1)-spheres is a hyperbolic isometry. When n = 3, we can give criteria for when [M ] ∈ PSL (2, C) is elliptic, parabolic, or hyperbolic.   a b Lemma 31.9. Given M = ∈ SL (2, C) − {± id}, corresponding to c d  [M ] ∈ PSL (2, C) ∼ = Isom+ H3 , we have that (1) [M ] is elliptic if and only if tr (M ) ∈ (−2, 2) ⊂ R, (2) [M ] is parabolic if and only if tr (M ) = ±2, (3) [M ] is hyperbolic if and only if tr (M ) ∈ (−∞, −2) ∪ (2, ∞) ⊂ R or tr (M ) ∈ C − R. As a special case, if [M ] fixes ∞ ∈ S 2 ∼ = R2 ∪ {∞}, then c = 0, a = 0, and −1 d = a . We have, as an isometry of the upper half-space U 3 , gU , (31.5)

φM (z, t) = (a2 z + ab, |a|2 t)

for z ∈ C and t ∈ (0, ∞). Since



tr (M ) = a + a−1 = a 1 −



1 |a|

2

+2

Re (a) |a|

2

,

we have (1) M is elliptic if and only if |a| = 1 and a = ±1, (2) M is parabolic if and only if a = ±1, (3) M is hyperbolic if and only if |a| = 1. Since any parabolic isometry fixing the point at infinity is of the form (z, t) −→ (z + b, t) , where b ∈ C,  the subgroup of Isom+ H3 of parabolic isometries fixing a given point on the sphere at infinity is isomorphic to R2 . Recall the following. Definition 31.10 (Nilpotent group). Let G be a group. Given subgroups H and J of G, let [H, J] denote the subgroup of G generated by the subset  hjh−1 j −1 : h ∈ H, j ∈ J . Define inductively Gk  [G, Gk−1 ] for k ∈ N, where G0  G. We say that G is nilpotent if G = {e} for some  ∈ N. The following result applies to abelian subgroups, e.g., subgroups isomorphic to Z × Z. In particular, it is useful in the study of embedded tori in complete hyperbolic 3-manifolds (see the proof of Lemma 31.27).

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177

Lemma 31.11 (Nilpotent subgroups of Isom (Hn )). If G ⊂ Isom (Hn ) is a nilpotent subgroup, then either (1) G = {id}, (2) there exists ϕ ∈ G − {id} which is elliptic, (3) every element of G − {id} is hyperbolic and has the same two fixed points in ∂Hn , or (4) every element of G − {id} is parabolic and has the same fixed point in ∂Hn . In the study of cusps of hyperbolic manifolds the following is useful. Definition 31.12. A horoball in Hn , using the disk model Dn , is an intersection Dn ∩ B n , where B n is a closed ball in Rn tangent to ∂Dn . The boundary of a horoball is called a horosphere. A horoball is the closure of the limit of balls in hyperbolic space as the radii tend to infinity. Topologically, if n > 2, a horosphere is diffeomorphic to a Euclidean space and hence is simply connected. In the upper half-space model U n , a horoball either is an intersection U n ∩ B n , where B n is a closed ball in Rn tangent to ∂U n , or is the region above a horizontal hyperplane: {x : xn ≥ c}, where c > 0. 1.4. Hyperbolic manifolds and some examples. If (Hn , h) is a complete hyperbolic manifold, then its universal covering space ˜ with the lifted metric is a complete, simply-connected Riemannian manifold ˜ , h) (H ˜ is isometric to Hn . Let p : Hn → ˜ h) with constant sectional curvature −1. Hence (H, H denote the projection map and let n

(31.6)

Γ  Aut (Hn , p) ⊂ Isom (Hn )

denote the group of covering transformations. For any x ∈ H, the fundamental group π1 (H, x) is naturally isomorphic to Γ. The group Γ acts freely and properly discontinuously on Hn ; that is, for any compact sets K1 , K2 ⊂ Hn , the set {γ ∈ Γ : γ (K1 ) ∩ K2 = ∅} is finite and γ(x) = x for all x ∈ Hn and for all γ ∈ Γ − {id}. (In fact, Γ is a torsion-free discrete subgroup.) Hence (H, h) is isometric to Hn /Γ. Thus the study of hyperbolic manifolds is the same as the study of discrete torsion-free subgroups of Isom (Hn ). Henceforth, a finite-volume hyperbolic manifold is assumed to be connected, complete, and noncompact. Finite-volume hyperbolic manifolds in dimension at least 3 are very difficult to construct. All known examples before the work of Thurston are either expressed in the form Hn /Γ by describing the discrete group Γ explicitly or by taking convex hyperbolic polyhedra and gluing their faces isometrically. We conclude with some examples. (1) If S is a Riemann surface with nonabelian fundamental group, then the uniformization theorem says there exists a unique complete hyperbolic metric g on S conformal to the complex structure. This fundamental theorem provides the key link between geometry, analysis, and algebraic geometry for surfaces.

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(2) Take the Riemann surface S = C − {0, 1}, the twice-punctured plane. Then S is biholomorphic to H2 /Γ(2), where Γ(2) is the principal congruence subgroup of level 2; i.e., Γ(2) = {[A] ∈ PSL(2, Z) | A ≡ id mod 2}. The biholomorphism is given by a modular function arising from classical elliptic function theory. (3) Since PSL(2, Z) is a discrete subgroup of PSL(2, R), any torsion-free subgroup Γ of PSL(2, Z) produces a complete hyperbolic surface H/Γ. This includes the example above. (4) One can generalize the construction to dimension 3 by taking the Bianchi group. Take a positive square-free integer d (i.e., d is not divisble by any perfect square except for √ 1) and let Od be the ring of integers in the quadratic imaginary number field Q( −d). Then PSL(2, Od ) is a discrete subgroup of PSL(2, C), so that H3 / PSL(2, Od ) has finite volume. Therefore, any finite-index torsion-free subgroup Γ of PSL(2, Od ) gives rise to a finite-volume hyperbolic 3-manifold H3 /Γ. (5) In Thurston’s famous 1978–1980 notes [401], he was able to see that the figure-eight knot complement in S 3 has a complete finite-volume hyperbolic metric. He achieved this by decomposing the knot complement as a union of two tetrahedra without vertices (the so-called ideal tetrahedra), realizing each of the tetrahedron by the regular ideal hyperbolic tetrahedron and gluing them by isometries. The associated discrete subgroup of PSL(2, C) is a subgroup of the Bianchi group PSL(2, O3 ). This example of Thurston initiated the geometrization conjecture in dimension 3. (6) The first known closed hyperbolic 3-manifold is called the Seifert–Weber space, constructed in 1933 by H. Seifert and C. Weber. It is motivated by Poincar´e’s construction of the Poincar´e homology sphere, which is to glue each face of a regular dodecahedron to its opposite face by a π/5 rotation. (A picture of the identification can be found in Chapter 9 of Seifert and Threlfall’s classical book on topology [361].) As realized by Poincar´e, if one chooses the dihedral angle of a regular spherical dodecahedron carefully, then the gluing maps can be realized by spherical isometries, so that the quotient Poincar´e homology sphere has a Riemannian metric of constant sectional curvature 1. In the case of Seifert and Weber, they take the regular dodecahedron and glue each face to its opposite face by a 3π/5 rotation and realize that such gluing maps can be realized by hyperbolic isometries on a regular dodecahedron. The quotient space is then a closed hyperbolic 3-manifold. 2. Topology and geometry of hyperbolic 3-manifolds In this section we discuss aspects of the topology and geometry of 3-manifolds, especially hyperbolic 3-manifolds. For simplicity, we shall assume that all the 3manifolds that we consider in the rest of this chapter are orientable. 2.1. The loop theorem and its consequences. Recall that a 3-manifold is irreducible if every embedded 2-sphere bounds an embedded 3-ball. By the Sch¨ onflies theorem, R3 and H3 are irreducible. As a consequence, we have Lemma 31.13. Any complete hyperbolic 3-manifold is irreducible.

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Let S 2 be an embedded 2-sphere in a complete hyperbolic manifold  3 Proof. H , h and let p : H3 → H be the universal covering. Since S 2 is simply connected, 2 given x ∈ S 2 and a choice of x ˜ ∈ p−1 (x), there exists a unique lift S˜ ⊂ H3 with 2 2 2 x ˜ ∈ S˜ , where S˜ is an embedded 2-sphere. By the Sch¨ onflies theorem, S˜ bounds ˜ 3 ⊂ H3 . We claim that B 3  p(B) ˜ is an embedded 3-ball in H an embedded 3-ball B ˜ → B is a local diffeomorphism with p| ˜2 : S˜2 → S 2 a bounding S 2 . Indeed, p|B˜ : B S homeomorphism. Hence, by degree theory, p|B˜ is 1-1 and hence a diffeomorphism; the claim follows.  Recall that a surface of genus g ≥ 1 embedded in a 3-manifold by ι : Σ2 → M3 is incompressible if the map induced on fundamental groups ι∗ : π1 (Σ) → π1 (M) is injective. If Σ is not incompressible in M, then we say that Σ is compressible in M. In the case where Σ ∼ = S 2 , one says that an embedded S 2 ⊂ M is compressible if it bounds a 3-ball in M. (We have actually used an equivalent definition of incompressibility; see pp. 31–34 of Jaco [158].) Let D2 denote the closed unit ball in R2 . Theorem 31.14 (Loop Theorem). Let M3 be a compact 3-manifold, let Σ2 be a connected surface in ∂M, and let N ⊂ π1 (Σ) be a normal subgroup. If ker (i∗ ) − N is nonempty, where i : Σ → M is the inclusion map, then there exists an embedding j : D2 → M such that / N, j (∂D) ⊂ Σ and [ j|∂D ] ∈ where [ j|∂D ] ∈ π1 (Σ) denotes the homotopy class of j|∂D . This result, due to Papakyriakopoulos, is useful for studying the compressibility and incompressibility of embedded surfaces in 3-manifolds. Taking N = {1}, we have the following. Corollary 31.15 (Compressible boundary components). If M3 is a compact 3-manifold and Σ2 is a closed connected two-sided compressible surface of genus ≥ 1 in ∂M, then there exists an element 1 = α ∈ ker (i∗ ), where i : Σ2 → M is the inclusion map, and an embedding j : D2 → M such that j (∂D) ⊂ Σ and [ j|∂D ] = α. In particular, α may be represented by a simple closed curve in Σ, which implies α is a primitive element (i.e., it is not a nontrivial power of another element). In the last sentence we used the following. Lemma 31.16. Any element of the fundamental group of an oriented closed surface which is represented by a simple closed curve is a primitive element. Proof. Suppose that σ, with [σ] ∈ π1 (Σ, p) − {id}, is a simple loop in an oriented closed surface Σ2 . By the classification of surfaces, Σ is homeomorphic to the compact plane region bounded by a 4g-sided polygon with sides labeled −1 −1 −1 and identified accordingly, where g ≥ 1. In Σ a1 , b1 , a−1 1 , b1 , . . . , ag , bg , ag , bg each of these sides, ai or bj , identifies to a nonseparating loop; i.e., Σ − ai and Σ − bj are connected for all i, j = 1, . . . , g. (1) Suppose that σ is not separating. We claim that there exists a selfhomeomorphism of Σ mapping σ to a1 . To see this, let Σσ and Σa1 be the connected compact surfaces with two boundary circles obtained by cutting Σ along σ and a1 ,

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respectively. By the classification of surfaces with boundary, there exists a homeomorphism φ : Σ σ → Σ a1 mapping the two boundary copies of σ to the two boundary copies of a1 , with both copies mapped in the same way. This homeomorphism induces a self-homeomorphism of Σ which maps σ to a1 . Since [a1 ] is primitive, we conclude that [σ] is also primitive. (2) Suppose that σ is separating. Cut Σ along σ to obtain two connected compact surfaces Σ21 and Σ22 , each with boundary consisting of the single circle σ. Since [σ] = 1, there exists 1 ≤ k ≤ g − 1 such that Σ1 has genus k and Σ2 has genus g − k. Now consider Σ again as a 4g-sided polygon with sides −1 −1 −1 a1 , b1 , a−1 1 , b1 , . . . , ag , bg , ag , bg . Define the commutator [a, b]  aba−1 b−1 . Cut Σ along a simple loop σ ¯ which is homotopic to the product [a1 , b1 ] [a2 , b2 ] · · · [ak , bk ] to obtain two connected compact ¯ 21 and Σ ¯ 22 , each with boundary consisting of σ ¯ 1 has genus surfaces Σ ¯ . We have that Σ ¯ i , i = 1, 2, ¯ 2 has genus g −k. Hence there exist homeomorphisms φi : Σi → Σ k and Σ ¯ for i = 1, 2 (and we may assume they map σ to σ ¯ the such that φi maps σ to σ same way). Hence we can glue the homeomorphisms φ1 and φ2 together to obtain a homeomorphism φ1  φ2 : Σ → Σ which maps σ to σ ¯ . Since [¯ σ ] is primitive, we  again conclude that [σ] is primitive.1 Remark 31.17. When the genus of the surface is at least two, another way to prove that [σ] is primitive is to endow the surface with a hyperbolic metric. Recall that if α is a homotopically nontrivial loop in a closed hyperbolic surface, then α is freely homotopic to a unique closed geodesic α∗ . Furthermore, if α is a simple loop, then so is α∗ ; see for instance Chapter 3 in Casson and Bleiler’s book [54] for a proof. Now suppose otherwise that σ is an essential simple loop that is homotopic to β k for some loop β and k > 1. Since σ is essential, β is essential. Then σ ∗ = (β ∗ )k by the uniqueness of the geodesic representative. This contradicts the fact that σ ∗ is simple. In view of the fact that we are interested in incompressible tori, we consider the (opposing) compressible case. Lemma 31.18 (Map on π1 induced by inclusion for compressible torus). Let M3 be a compact 3-manifold (with possibly nonempty boundary) such that π1 (M) contains no torsion element (e.g., M is a closed hyperbolic 3-manifold or the truncation of a finite-volume hyperbolic 3-manifold ). If i : Σ2 → M is a compressible torus, then either (1) ker (i∗ ) = π1 (Σ) or (2) ker (i∗ ) is an infinite cyclic subgroup of π1 (Σ) generated by a primitive element. Remark 31.19. The lemma is not true for lens spaces. Proof. Since i : Σ → M is compressible, it follows that ker (i∗ ) = {0}. Suppose ker (i∗ ) = π1 (Σ) ∼ = Z × Z. We claim that ker (i∗ ) must be an infinite cyclic subgroup of π1 (Σ). Since π1 (M) contains no torsion element, the quotient group π1 (Σ) / ker (i∗ ) ∼ = image (i∗ ) ⊂ π1 (M) contains no torsion element. Thus 1 That

[¯ σ ] is primitive is a nontrivial fact, unlike the case of [a1 ] above.

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 π1 Σ2 / ker (i∗ ) must be isomorphic to either 0, Z, or Z × Z.2 By the classification of abelian groups, ker (i∗ ) is isomorphic to Z provided ker (i∗ ) = π1 (Σ). On the other hand, by Corollary 31.15, a nontrivial element of ker (i∗ ) is represented by a simple closed curve in Σ. Since this element must be primitive, it  generates ker (i∗ ) ∼ = Z. For the statement and proof of the next result, all homology and cohomology coefficients are R. Lemma 31.20 (Induced map on H1 for the boundary inclusion). Let M3 be a compact orientable 3-manifold with boundary Σ2 = ∂M, which may be disconnected. Let i : Σ → M be the inclusion map. The rank of i∗ : H1 (Σ) → H1 (M) is equal to half the dimension of H1 (Σ). R

m

Proof. Recall the following two basic facts from linear algebra. If A : Rn → is a linear map and if A∗ : Rm → Rn denotes its dual, then

(31.7)

dim (image (A)) = dim (image (A∗ )) = rank (A)

and (31.8)

dim (ker (A)) + dim (image (A)) = n.

We have the short exact (co)homology sequence j

(31.9)

i

H2 (M, Σ) −−−−→ H1 (Σ) −−−∗−→ H1 (M) ⏐ ⏐ ⏐ ⏐∼ D 4∼ = 4= H 1 (M)

i∗

−−−−→ H 1 (Σ) ,

where D is the isomorphism from Lefschetz duality (this requires Σ = ∂M) and where i∗ is the dual of i∗ . Thus, by (31.7), we have dim (image (j)) = dim (image (i∗ )) = dim (image (i∗ )) . Now the exactness of the sequence in (31.9) says dim (image (j)) = dim (ker (i∗ )) , so that dim (image (i∗ )) = dim (ker (i∗ )) . Finally, by (31.8), we have dim (ker (i∗ )) + dim (image (i∗ )) = dim (H1 (Σ)) , so that dim (ker (i∗ )) =

1 2

dim (H1 (Σ)).3



We have the following standard results from geometric topology, which we shall use in the next subsection. is, we have ruled out the possibilities Zn , Zn × Zm , and Z × Zn . that the above information on the kernel of i∗ : H1 (Σ) → H1 (M), on the level of homology, says nothing about the kernel of i∗ : π1 (Σ) → π1 (M). 2 That 3 Note

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Lemma 31.21 (Separating surfaces and incompressibility). Suppose that Σ2 is a closed surface of genus ≥ 1 in a closed 3-manifold M3 separating M into two compact submanifolds M1 and M2 with ∂M1 = ∂M2 = M1 ∩ M2 = Σ. If Σ is incompressible in both M1 and M2 , then Σ is incompressible in M. Moreover, both π1 (M1 ) and π1 (M2 ) inject into π1 (M). The lemma is an easy consequence of the following: Theorem 31.22 (Seifert and Van Kampen). If a topological space X = X1 ∪X2 is a union of two path-connected open sets X1 and X2 so that X1 ∩ X2 is path connected, then π1 (X, x0 ) ∼ = (π1 (X1 , x0 ) ∗ π1 (X2 , x0 )) /H, where H is the normal subgroup generated by (i1 )∗ (α) [(i2 )∗ (α)]−1 for all α ∈ π1 (X1 ∩ X2 , x0 ) and where i1

X1 j1

X1 ∩ X2 # i2 X2  #j2 X1 ∪ X2

are the inclusion maps. Furthermore, if (i1 )∗ and (i2 )∗ are injective, then (j1 )∗ and (j2 )∗ are injective. Remark 31.23. Many books do not state the last fact in the theorem, which turns out to be very useful in practice. It is a consequence of the normal form theorem for amalgamated free products of groups; see for instance Theorem 25 in Chapter 1 of D. Cohen’s book [85]. Proof of Lemma 31.21. Let Xa be the union of Ma and an open collar of Σ2 in M for a = 1, 2. Note that Σ is a deformation retract of X1 ∩ X2 . Let x0 ∈ Σ. Since Σ is incompressible in both M1 and M2 , we have that (ia )∗ : π1 (X1 ∩ X2 , x0 ) → π1 (Xa , x0 ) is injective for a = 1, 2. So, by Theorem 31.22, (ja )∗ : π1 (Xa , x0 ) → π1 (X1 ∪ X2 , x0 ) = π1 (M, x0 ) is injective for a = 1, 2. The lemma easily follows.  Remark 31.24. For another, “cut and paste”, proof of Lemma 31.21, see the notes and commentary. 2.2. Incompressibility of tori in hyperbolic 3-manifolds. In the next section (see Theorem 31.44 below) we shall show that each topological end of a finite-volume hyperbolic 3-manifold H3 , h is isometric to [0, ∞) × V 2 with the metric gcusp = dr 2 + e−2r gflat , where (V, gflat ) is a flat torus. Each slice Vr  {r} × V is an embedded flat torus (with constant second fundamental form) in (H, h). In this subsection we discuss topological properties of embedded tori, including Vr , in H. 31.25 (Boundary tori of hyperbolic 3-manifolds are incompressible). Lemma Let H3 , h be a finite-volume hyperbolic 3-manifold. The 2-torus Vr is incompressible in H.

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Proof. Let i : Vr → H denote the inclusion map. If the lemma is false, then there exists a homotopically nontrivial loop f : S 1 → Vr such that i ◦ f : S 1 → H is null-homotopic. Thus the loop i ◦ f can be lifted to the universal cover π : H3 → H; we denote the lift by f˜ : S 1 → H3 , so that π ◦ f˜ = i ◦ f . Let P  π −1 (Vr ) ⊂ H3 . Since π is a Riemannian covering map and since the hypersurface Vr is complete (in fact, compact), flat, and totally umbillic with principal curvatures identically −1, we have that its inverse image P is a complete, flat, totally umbillic surface P is a horosphere;4 with principal curvatures identically −1 embedded in H3. Hence in particular, P is diffeomorphic to R2 . The image f˜ S 1 lies in P . Since P is simply connected, the lifted map f˜ : S 1 → P , when considered as a map into P , is null-homotopic in P . Thus f = π ◦ f˜ is null-homotopic in Vr . This contradicts the assumption of f being homotopically nontrivial.  Remark 31.26. It is a well-known fact that if π : Y˜ → Y is a universal covering space and if X ⊂ Y is a path-connected subset, then the inclusion map i : X → Y induces an injective homomorphism in π1 if and only if each component of π −1 (X) is simply connected. The proof is the same as the above. Now we recall the topological properties of embedded tori in finite-volume hyperbolic 3-manifolds. 2 Lemma 31.27 (Embedded tori in hyperbolic 3-manifolds).  3 Let T be an embedded 2-torus in a finite-volume hyperbolic 3-manifold H , h . Then either (I) T is incompressible and T is isotopic to a standard torus slice in a hyperbolic cusp, (II) T is compressible and T bounds an embedded solid torus B 2 × S 1 , or (III) T is compressible and T bounds a compact 3-manifold lying inside an embedded 3-ball in H.

Proof. Clearly we have one of the following two possibilities: (A) T is incompressible or (B) T is compressible. If (A), then i∗ : π1 (T ) → π1 (H) is injective and we have  ∼ i∗ (π1 (T )) ⊂ Γ ⊂ Isom H3 , Z×Z= where we have used the natural isomorphism between the group of covering transformations Γ and π1 (H). We claim that the abelian subgroup i∗ (π1 (T )) is generated by two parabolic isometries. First, there are no elliptic elements in Γ since otherwise, by definition, we would have a contradiction to the fact that Γ acts freely on H3 . Hence, by Lemma 31.11, either (A1) every element of i∗ (π1 (T )) − {id} is hyperbolic and has the same two fixed points in ∂H3 or (A2) every element of i∗ (π1 (T )) − {id} is parabolic and has the same fixed point in ∂H3 . 4 This

is a well-known fact (see for example [55] or [156]).

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We claim that case (A1) cannot happen. Suppose φ1 and φ2 are two hyperbolic isometries that generate i∗ (π1 (T )) ∼ = Z × Z and that fix the same two points on ∂H3 ∼ = C ∪ {∞}. We may assume, without loss of generality, that the two fixed points on ∂H3 are 0 and ∞. Then, as special cases of (31.5), (31.10)

φi (z, t) = (ci z, |ci | t) ,

where 0 < |ci | = 1, for i = 1, 2, and where for all r, s ∈ Z − {0} we have (31.11)

cr1 cs2 = 1.

 Let L be the geodesic in the upper half-space U 3 , gU joining the two fixed points 0 and ∞; i.e., let L be the positive z-axis. Then L is invariant under φ1 and 2 φ2 , so that φi , i = 1, 2, acts by translation on L with respect to the metric dz z 2 on L induced by the hyperbolic metric gU . Let ai denote the translation distance on L for φi , i = 1, 2. By the incompressibility assumption, we see that the subgroup of (R, +), i.e., of the real numbers under addition, generated by a1 and a2 , is isomorphic to Z × Z. Now it is well known that a subgroup of R isomorphic to Z × Z is dense. Thus we see that the subgroup generated by φ1 and φ2 is not discrete. This completes the proof of ruling out case (A1). Hence we have that case (A2) holds and that i∗ (π1 (T )) is generated by a pair of parabolic isometries having the same fixed point in ∂H3 . By Theorem 31.44, the quotient space H3 /i∗ (π1 (T )) is isometric to a hyperbolic cusp (R × V 2 , dr 2 + e−2r gflat ). From the discreteness of Γ we see that for r0 large enough,  [r0 , ∞) × V, dr 2 + e−2r gflat is isometric to a subset of (H, h). In particular, i∗ (π1 (T )) is equal to the j∗ (π1 (V)) for some standard torus slice j : V → H embedded in a hyperbolic cusp of H. We conclude, from Waldhausen’s theorem that homotopic incompressible surfaces are isotopic in Haken manifolds (see Corollary 5.5 in [426]), that i (T ) ⊂ H is isotopic to j (V) ⊂ H. That is, case (I) in the statement of the lemma holds. This finishes the analysis of possibility (A). If (B), then there exists a nontrivial element in ker (i∗ : π1 (T ) → π1 (H)) which is represented by an embedded loop α in T . By the Loop Theorem (Corollary 31.15), there exists an embedded disk D1 ⊂ H whose boundary is α and such that D1 ∩ T = α (the last property follows from a cut and paste argument similar to that in the alternate proof of Lemma 31.21 given in the notes and commentary at the end of this chapter). Push the disk D1 to one side to obtain a nearby parallel embedded disk D2 with ∂D2 ⊂ T and such that the region in H bounded by D1 , D2 , and the annulus in T between ∂D1 and ∂D2 is a compact 3-manifold homeomorphic to D2 × I, where D is a 2-ball and I is a closed interval. Now in T cut out the annulus between ∂D1 and ∂D2 and glue in D1 and D2 to obtain an embedded 2-sphere S 2 . Since H3 is irreducible (see Lemma 31.13), S 2 bounds an embedded 3-ball B 3 . Now there are two cases: (B1) (D × I) ∩ B = D1 ∪ D2 , (B2) D × I ⊂ B. In case (B1) (D × I) ∪ B is an embedded solid torus with boundary T ; that is, case (II) holds.

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In case (B2), the torus T lies in B. By the Jordan–Brouwer separation theorem, we have that B − T is the disjoint union of two connected open 3-dimensional submanifolds U1 and U2 , where U 1 is compact and ∂U 1 = T . Thus case (III) holds. This completes the proof of the lemma.  The following, which pertains to case (III) of Lemma 31.27, is an example of an embedded compressible torus in any hyperbolic 3-manifold not bounding a solid torus. Example 31.28. Take a knotted torus T 2 in S 3 bounding a solid torus X in S . Now take an open 3-ball Y inside X and let B 3 = S 3 − Y. Inside B lies T . Now embedding this B into any hyperbolic 3-manifold H3 results in a compressible torus T in H. This T does not bound a solid torus. 3

Motivated by the noncompact hyperbolic limit case of nonsingular solutions on 3-manifolds, we consider the following. Definition 31.29. We say that a compact submanifold Hc3 of a closed 3manifold is a topological hyperbolic piece5 if ∂Hc is a disjoint union of embedded tori and the interior of Hc admits a complete finite-volume hyperbolic metric. Let Hc3 be a topological hyperbolic piece in a closed 3-manifold M3 . By Lemma 31.25, ∂Hc is incompressible in Hc . From this and Lemma 31.21 we immediately obtain the following, which is relevant to 3-dimensional nonsingular solutions of Ricci flow (see Proposition 33.9 in Chapter 33). Corollary 31.30. If ∂Hc is incompressible in M − Hc , then ∂Hc is incompressible in M. 3. The Margulis lemma and hyperbolic cusps The main topics in this section are the Margulis lemma and its consequences for the geometry of ends of finite-volume hyperbolic 3-manifolds. 3.1. Topological ends. Let Mn be a noncompact differentiable manifold. Given a compact set K ⊂ M, define   E is a connected component of E (K)  E : . ¯ is noncompact M − K and E If K1 and K2 are compact sets with K1 ⊂ K2 , then we have a natural map (31.12)

φ : E (K2 ) → E (K1 )

defined by φ (E2 )  E1 , where E1 is the connected component of M−K1 containing E2 ∈ E (K2 ).6 Definition 31.31 (Topological end). We say that a compact set K ⊂ M is end-complementary if for every compact set K ⊃ K the map φ : E (K ) → E (K) 5 Topologists

call such a manifold a “simple 3-manifold with torus boundary”. ¯1 is noncompact since any set containing a noncompact closed set is nonthat E compact. Thus E1 ∈ E (K1 ), so that φ is well defined. 6 Observe

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is a bijection. If K is an end-complementary compact set, then we call each element of E (K) a topological end. The set of topological ends is well defined. If K1 and K2 are end-complementary compact sets in M, then we have a natural bijection φ2 ◦ φ−1 1 : E (K1 ) → E (K2 ) , where φ1 : E (K ) → E (K1 ) and φ2 : E (K ) → E (K2 ) are defined for K ⊃ K1 ∪ K2 in the same way as φ in (31.12). The bijection φ2 ◦ φ−1 1 is independent of the choice of K ⊃ K1 ∪ K2 . In this sense the definition of the set of topological ends E (K) is independent of the choice of end-complementary compact set K. So, when we speak of a topological end, we mean an element E of E (K) for some end-complementary compact set K. Roughly speaking, one can think of a topological end as a connected component of a neighborhood of infinity. Example 31.32 (Cylinder). Let Mn = N n−1 × R, where N is a closed manifold. The compact set N × [0, 1] is end-complementary and M has two topological ends. Exercise 31.33. Show that the number of topological ends is not necessarily nondecreasing under pointed Cheeger–Gromov limits. Hint. One can easily construct a sequence of pointed Riemannian manifolds {(Mni , gi , xi )} of the same dimension, where Mi has i topological ends and whose pointed Cheeger–Gromov limit is Euclidean n-space. In this case the limit has only one topological end, whereas the lim inf of the number of ends of Mi is equal to infinity. 3.2. The Margulis lemma and hyperbolic cusps. Consider the horoball Un  {x ∈ Rn : xn ≥ 1} contained in the upper halfspace model and let r = ln xn . The hyperbolic metric may be written as  dx2 + · · · + dx2n ghyp  1 = dr 2 + e−2r dx21 + · · · + dx2n−1 . 2 xn Let L be a lattice in Rn−1 which is isomorphic to Zn−1 and which acts by translation. Then dx21 + · · · + dx2n−1 induces a flat metric gflat on V n−1  Rn−1 /L. The hyperbolic metric ghyp on Un induces the hyperbolic metric gcusp  dr 2 + e−2r gflat on Un /L = [0, ∞)×V, where L acts on the first n−1 components. The Riemannian manifold ([0, ∞) × V, gcusp ) is called a hyperbolic cusp. Example 31.34. When n = 3, V 2 is a torus and there exist two parabolic isometries of the hyperbolic disk H3 which fix a point x ∈ ∂H3 at infinity and a horoball B tangent to ∂H3 at x such that ([0, ∞) × V, gcusp ) is isometric to the quotient of B by the group generated by the two parabolic isometries (see Lemma 31.25 above). If a closed subset U with the induced metric of a complete hyperbolic manifold (Hn , h) is isometric to a hyperbolic cusp, we call (U, h|U ) a hyperbolic cusp end (or cusp end) of (H, h). We call (U, h|U ) a maximal cusp end if it is a cusp end which is not properly contained in another cusp end. Note that the pre-image of a maximal cusp under the projection map p : Hn → n H consists of the union of horoballs whose interiors are disjoint and are such that

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187

some of the horoballs are tangent to each other. In a maximal cusp end, for each r > 0, the slice {r} × V n−1 ⊂ Hn is an embedded hypersurface, whereas the slice {0} × V ⊂ H has self-intersections. However, different maximal cusp ends may intersect. If this happens, then any torus slice T of one cusp end cannot be completely contained in another cusp end E. This can be seen by lifting T and E to the universal cover Hn . Indeed, if T is contained in E, then consider the intersection of the pre-image of T under the covering map p with a horoball B projecting onto E. The surface p−1 (T ) ∩ B must be a horosphere inside B. Since all horospheres in B are tangent to ∂B at the infinity of Hn , we have that p−1 (T ) ∩ B must be one such horosphere. This implies that T is a torus slice of the cusp end E, which contradicts the assumption. Exercise 31.35 (Slices in a cusp are totally umbillic). Show that the slices Vrn−1  {r}×V ⊂ [0, ∞)×V are totally umbillic with respect to gcusp . In particular, ∂ for each r, with respect to the unit normal vector ν  ∂r , the second fundamental form II of Vr with respect to gcusp is equal to the negative of the induced metric: (31.13)

II = − gcusp |Vr .

In other words, for the compact truncated cusp [0, r] × V, the boundary component Vr is concave, whereas for the noncompact truncated cusp [r, ∞) × V, the boundary Vr is convex. Given a subset of a group S ⊂ G, S ⊂ G denotes the subgroup generated by S. We have the following fundamental result. Theorem 31.36 (Margulis lemma—algebraic version). For any n ∈ N there exists a constant εn > 0 such that if Γ ⊂ Isom (Hn ) acts properly discontinuously and if x ∈ Hn , then the group Γεn (x)  {γ ∈ Γ : d (γ (x) , x) ≤ εn } is virtually nilpotent; i.e., Γεn (x) has a subgroup of finite index which is nilpotent, where the index is bounded by a constant depending only on n. The constant εn is called the (n-dimensional) Margulis constant. Remark 31.37. When n = 3, the group Γεn (x) is virtually abelian; i.e., Γε3 (x) has a subgroup of finite index which is abelian. We have the following local geometric consequence of the algebraic Margulis lemma. Theorem 31.38 (Margulis lemma—local consequence). Let (Hn , h) be a complete hyperbolic manifold so that (H, h) = Hn /Γ, where Γ ⊂ Isom (Hn ) is a discrete subgroup acting freely and properly discontinuously on Hn . Given ε ∈ (0, εn ] and ¯ ε (x) denote the subgroup of π1 (H, x) ∼ x ∈ H, let Γ = Γ generated by piecewise smooth loops based at x with length ≤ ε. Then: ¯ ε (x). (1) B (x, ε/2) is isometric to a geodesic ball of radius ε/2 in Hn /Γ  ¯ ε (x) is isomorphic to either {1} , Z, or a discrete subgroup of Isom En−1 (2) Γ acting freely on En−1 .

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Now we turn to study the ε-thin and ε-thick parts of a hyperbolic manifold. Definition 31.39. The ε-thin part M(0,ε] is the set of points x ∈ M such that there exists a piecewise smooth loop α based at x such that [α] ∈ π1 (M, x) − {1}

and

Lg (α) < ε.

The ε-thick part M[ε,∞) is the set of points x ∈ M such that for every piecewise smooth loop α based at x with [α] ∈ π1 (M, x) − {1} we have Lg (α) ≥ ε. The ε-thick-thin decomposition of a Riemannian manifold (M, g) is M = M(0,ε] ∪ M[ε,∞) . Example 31.40. (1) (Simply-connected Riemannian manifolds are all thick.) If we have π1 (M, x) = {1} , then for every ε > 0 we have M(0,ε] = ∅ and

M[ε,∞) = M.

(2) (Closed Riemannian manifolds are all ε-thick for ε small enough.) If (Mn , g) is closed and if π1 (M, x) = {1} , then for ε > 0 sufficiently small, we have M(0,ε] = ∅ and M[ε,∞) = M. We may simply take any ε < ε0 , where ε0  inf Lg (α) α

and where the infimum is taken over all piecewise smooth loops α in M with [α] ∈ π1 (M) − {1}.  (3) (Hyperbolic cusp ends are eventually ε-thin.) If H3 , h is a finite-volume hyperbolic 3-manifold and if [0, ∞) × V 2 ⊂ H is a cusp end with h = dr 2 + e−2r gflat , then for every ε > 0 there exists r < ∞ such that ([r, ∞) × V, h) ⊂ H(0,ε] . Note that, by Lemma 31.25, the tori {r} × V ⊂ H are incompressible. Note that if α is a geodesic loop, which is smooth except possibly at its basepoint, in a complete hyperbolic manifold (Hn , h), then [α] ∈ π1 (H, x) − {1}. To see this, suppose that [α] = 1. Then α lifts to a geodesic α ˜ in Hn with a selfintersection, which is a contradiction. As a consequence, we obtain part (1) of the following; see Proposition D.2.6 of [24] for the proof of part (2). Lemma 31.41. Let (Hn , h) be a complete hyperbolic manifold. (1) Then the injectivity radius at any point in the ε-thick part H[ε,∞) is at least ε/2. (2) If (H, h) has finite volume, then for any ε > 0, the ε-thick part H[ε,∞) is compact. Given a complete hyperbolic manifold (Hn , h) and ε ∈ (0, εn ], where εn is the Margulis constant, an ε-thin end of H is defined as the closure of a connected component of H − H[ε,∞) . An ε-thin end is not always a topological end; i.e., the

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set H[ε,∞) is not always end-complementary. This happens when there is a compact connected component of H − H[ε,∞) . On the other hand, we have the following. Lemma 31.42 (Topological ends are essentially ε-thin). Let (Hn , h) be a finitevolume hyperbolic manifold. Then we have the following. (1) Let K be an end-complementary compact set in H. If E ∈ E (K) is a topological end, then for any ε > 0 there exists E  ⊂ E with E − E  compact such that E  is an ε-thin end. (2) There exists ε0 > 0 depending on h such that if ε ≤ ε0 , then any ε-thin end is a topological end. The above result is actually a consequence of Theorem 31.44 below, which we shall prove from the following geometric consequence of the Margulis lemma. Theorem 31.43. Let (Hn , h) , n ≥ 3, be a finite-volume hyperbolic manifold and let ε ∈ (0, εn ], where εn is the Margulis constant. The ε-thin part H(0,ε] is the disjoint union of subsets, positive distances apart, of the following forms: (1) (Hyperbolic cusp end) A subset S homeomorphic to [0, ∞) × V n−1 , where V is a closed manifold admitting a flat metric, with the interior being diffeomorphic to (0, ∞) × V. Moreover there exists a closed set S  ⊂ S with S − S  compact and such that (S  , h|S  ) is isometric to  [0, ∞) × V, dr 2 + e−2r gflat , where gflat is a flat metric on V. (2) (Tubular neighborhood of geodesic loop—Margulis tube) A subset S home¯ n−1 (1) × S 1 whose interior is diffeomorphic to B n−1 (1) × omorphic to B 1 S . Moreover there exists a unique smooth geodesic loop γ ⊂ S such that for δ > 0 sufficiently small, (a) the δ-neighborhood Nδ (γ) of γ is contained in S, (b) the submanifold (Nδ (γ) , h|Nδ (γ) ) is isometric to B n−1 (r) × S 1 () with the metric n−1  2 (31.14) gtube  xi ds + dxi + ds2 i=1

for some r,  > 0, where B n−1 (r) denotes the Euclidean (n − 1)-ball of radius r and S 1 () denotes the circle of length . (3) (Circle) A smooth geodesic loop of length ε. Regarding part (2), see Example 31.45 below. 3.3. Geometry of finite-volume hyperbolic ends. The next classical fact follows from the above geometric consequence of the Margulis lemma. Theorem 31.44 (Finite-volume hyperbolic ends are standard cusps). Let (Hn , h), n ≥ 2, be a finite-volume hyperbolic manifold and let K be an end-complementary compact set in H. For every topological end E ∈ E (K) there exists a closed subset ¯ − int (U ) is compact and (U, h| ) is isometric to [0, ∞) × V n−1 U ⊂ E such that E U with a metric of the form gcusp = dr 2 + e−2r gflat , where (V, gflat ) is a closed flat manifold.

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Proof. Let (Hn , h) be a complete hyperbolic manifold. Since its fundamental group is countable, the length spectrum of (H, h), i.e., the set of lengths of smooth geodesic loops, is countable.7 Therefore we may choose a “generic” ε ∈ (0, εn ] in Theorem 31.43 so that case (3) does not occur for this ε, i.e., so that there are no smooth geodesic loops of length ε in H. Thus each component of the ε-thin part H(0,ε] for such an ε is either a Margulis tube or a cusp end. Now assume that (H, h) has finite volume. The ε-thick part H[ε,∞) is compact due to the volume bound and the injectivity radius lower bound on H[ε,∞) (see Lemma 31.41 for the latter fact). Since we have ruled out case (3) of Theorem 31.43, the boundary of the ε-thick part H[ε,∞) is equal to the boundary of the ε-thin part H(0,ε] , which is a smooth embedded surface. Therefore H[ε,∞) is a smooth compact manifold with boundary. This implies that H[ε,∞) has only a finite number of boundary components, which in turn implies that the ε-thin part H(0,ε] has only a finite number of components. Thus there are only a finite number of Margulis tubes in H(0,ε] . Now choose ε to be smaller than the shortest closed geodesic in this finite set of Margulis tubes. Then each component of H(0,ε] is a cusp end. It follows that each topological end is a cusp end; i.e., Theorem 31.44 is proved.  Example 31.45 (Margulis tube metric as a quotient). Consider the diffeomorphism Φ : Rn−1 × R → Rn−1 × R+ defined by Φ (x, s)  (es x, es )  (y 1 , . . . , y n−1 , y n ). The hyperbolic metric on the upper half-space is n−1   s i 2 n  i 2 n−1  2 + (d (es ))2 i=1 dy i=1 d e x xi ds + dxi + ds2 , = = 2 2 n s (y ) (e ) i=1 which is the same formula as (31.14). Define the ball  B (r)  z ∈ Rn−1 : |z| < r and the cone

 C (r)  (y , y n ) ∈ Rn−1 × R+ : |y | < ry n ,  where y  y 1 , . . . , y n−1 . We have Φ : B (r) × R → C (r) . Given any  ∈ R+ , consider the action of Z on Rn−1 × R generated by the isometry (x, s) → (x, s + ). We then have that (B (r) × R) /Z is the Margulis tube (31.14). We have the following quantitative version of the Margulis lemma in relation to 3-dimensional hyperbolic cusp ends due toColin Adams [3]. 7 An aside: If (Hn , h) is geometrically finite (in particular, if (H, h) has finite volume), then the length spectrum is discrete.

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31.46 (Volume is bounded below by the number of cusp ends). If  3 Theorem H , h is a finite-volume hyperbolic 3-manifold with m cusp ends, then Vol (H, h) ≥ mv, where v ≈ 1.0149416 is the volume of the regular ideal tetrahedron. Moreover, there are m disjoint √ embedded cusp ends in H such that each of these cusp ends has volume at least 3/4. Thus the volume of a complete hyperbolic 3-manifold bounds the number of cusp ends that it may contain. Note that the theorem is true even though maximal cusp ends may intersect. 3.4. Small almost flat 2-tori. Since the complement of the cusp ends of a finite-volume hyperbolic manifold is compact, the following is a consequence of Theorem 31.44 and the fact that Riemannian manifolds are geometrically locally Euclidean.  Corollary 31.47 (Small almost flat 2-tori are in the cusp ends). Let H3 , h be a finite-volume hyperbolic 3-manifold. For any C < ∞ there exists a positive constant ε˜ > 0 depending only on (H, h) and on C such that if T 2 ⊂ H is an embedded 2-torus, where (1) the induced metric h|T satisfies |sect ( h|T )| ≤ ε˜, (2) Area (T , h|T ) ≤ ε˜, (3) | IIhT |h ≤ C, where IIhT is the second fundamental form of T with respect to h, then T is contained in one of the maximal cusp ends of H. Proof. If the corollary is not true, then there exists a sequence of embedded 2-tori Ti2 ⊂ H3 where Area(Ti , h|Ti ) ≤ i−1 , | sect( h|Ti )| ≤ i−1 , | IITi | ≤ C, and Ti is not contained in any of the maximal cusp ends of H. Let K denote the closure of the complement of the union of the maximal cusp ends of H. We then have Ti ∩ K = ∅ for all i and by Theorem 31.44, K is a compact set. Choose any sequence of points xi ∈ Ti ∩ K. Since the xi are all contained in a compact subset of H, the corresponding pointed sequence Riemannian  of (rescaled) manifolds {(H, ih, xi )} converges to Euclidean space R3 , gE , 0 .8 Since | IIih | ≤  3 T ih C and Area(T , ih| ) ≤ 1, with respect to the limit (H, ih, x ) → R , g , 0 the i i E Ti i1/2 2 tori Ti converge to a complete totally geodesic (and flat) hypersurface T∞ ⊂ R3 with area ≤ 1 (and passing through 0). Here, in essence, we have applied a standard compactness theorem for surfaces in R3 with bounded (actually limiting to zero) second fundamental form and with bounded area (see Langer [180] for instance). This is a contradiction.  Remark 31.48. We shall apply the corollary in the study of noncompact hyperbolic limits of nonsingular solutions (see Lemma 33.18 in Chapter 33). 8 Note that, on the other hand, if the T were contained in one of the maximal cusp ends i   [0, ∞) × V, gcusp = dr 2 + e−2r gflat ⊂ (H, h), then it would be possible that {(H, ih, xi )} converges to (V, const ·gflat ) × R and that Ti converges to a (totally geodesic) torus slice.

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4. Mostow rigidity In this section we state a special case of the Mostow rigidity theorem and some important consequences. Theorem 31.49 (Mostow rigidity). If (H1n , h1 ) and (H2n , h2 ), n ≥ 3, are finitevolume hyperbolic manifolds and if φ : π1 (H1 ) → π1 (H2 ) is an isomorphism, then there exists a unique isometry ι : (H1 , h1 ) → (H2 , h2 ) such that ι∗ = φ. This implies the volume of a finite-volume hyperbolic manifold is a topological invariant. Recall that two topological spaces X and Y are homotopy equivalent if there exist continuous maps f : X → Y and g : Y → X such that g ◦ f is homotopic to idX and f ◦ g is homotopic to idY . Corollary 31.50 (Homotopy equivalent ⇔ diffeomorphic ⇔ isometric). Let (H1n , h1 ) and (H2n , h2 ) be finite-volume hyperbolic manifolds, where n ≥ 3. Then the following conditions are equivalent: (1) (H1 , h1 ) and (H2 , h2 ) are isometric, (2) H1 and H2 are diffeomorphic, (3) H1 and H2 are homotopy equivalent. Next we give a description of the isometry group of a finite-volume hyperbolic manifold, which is a consequence of the Mostow rigidity theorem. Given a group G, let Aut (G) denote the group of automorphisms of G, i.e., the group (where multiplication is defined by composition) of isomorphisms of G onto itself. Recall that the inner automorphism group of G is the normal subgroup of Aut (G) consisting of conjugations  Inn (G)  g → aga−1 : a ∈ G and the outer automorphism group is the quotient Out (G)  Aut (G) / Inn (G) . This group is a quantitative measure of how many automorphisms there are which are not given by conjugation. Corollary 31.51 (Isometry group of a hyperbolic manifold). If (Hn , h) , n ≥ 3, is a finite-volume hyperbolic manifold, then the isometry group of (H, h) is finite and Isom (H) ∼ = Out (π1 (H)) .

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5. Seifert fibered manifolds and graph manifolds In this section we give some topological definitions related to collapsing sequences of compact 3-manifolds, which we shall study in the next chapter. Definition 31.52 (Foliation). A smooth k-dimensional foliation of an ndimensional differentiable manifold Mn is an atlas {(Uα , xα )} of M such that for any α, β, the transition function ϕβ ◦ ϕ−1 α : ϕα (Uα ∩ Uβ ) → ϕβ (Uα ∩ Uβ ) has the first n−k dependent variables  depending only on thefirst n−k independent variables. That is, if we let x♥ = x1 , . . . , xn−k and x♠ = xn−k+1 , . . . , xn , then ϕβ ◦ ϕ−1 α (x♥ , x♠ ) = (Φ (x♥ ) , Ψ (x♥ , x♠ )) for some functions Φ and Ψ mapping into Rn−k and Rk , respectively. The submanifolds



x ∈ Uα : (ϕ1α , . . . , ϕn−k ) = const , α  1 where ϕα  ϕα , . . . , ϕnα , are called plaques. Note that each plaque is a kdimensional submanifold of M. A leaf is a k-dimensional submanifold of M which is a maximal union of plaques. Definition 31.53 (Seifert fibered manifold). A Seifert fibered manifold is a compact 3-manifold which admits a foliation whose leaves are circles. Examples of Seifert fibered manifolds are a product of a surface with the circle, the unit tangent bundle of a Riemannian surface, and a 3-manifold with a circle action which has no global fixed points. The simplest example of a Seifert fibered manifold is the solid torus B 2 × S 1 with the product foliation whose leaves are {p} × S 1 . It turns out that the solid torus B × S 1 has many different S 1 foliations Fp,q , where p, q ∈ Z are relatively prime integers. These foliations on the solid torus are the building blocks for constructing all Seifert fibered manifolds. Here is the description of the foliation Fp,q . Consider the solid torus B × S 1 as the quotient of B × R by the action of Z whose generator τ sends (z, t) to (e2pπi/q z, t + 1). The diffeomorphism τ preserves the product foliation {p} × R on B × R. Furthermore, each leaf {p} × R becomes a circle in the quotient solid torus. Thus the quotient foliation, denoted by Fp,q , is a foliation whose leaves are circles. A Seifert fibered 3-manifold is said to have an orientable foliation if the circle fibers can be coherently oriented. A simple topological argument shows that each Seifert fibered 3-manifold either has an orientable foliation or has a two-fold cover which has an orientable foliation. All Seifert fibered 3-manifolds with orientable fibrations are obtained as follows. Take a compact surface X with nonempty boundary and consider the product foliation on X ×S 1 . Now glue solid tori with foliations Fpi ,qi to the boundary components of X × S 1 by diffeomorphisms preserving the S 1 fibers. Seifert fibered 3-manifolds have been classified up to diffeomorphism. The classification is given by the invariants (pi , qi ), the topology of the surface X, and the Euler characteristic class of the circle fibration. It can be shown that each Seifert fibered 3-manifold has a finite cover which is either the product space X × S 1 for some surface X or the principal S 1 bundle over a closed surface of Euler class 1.

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Graph manifolds are generalizations of Seifert fibered manifolds and can be defined as follows. A finite graph G = (V, E) is a finite collection V of vertices and a collection E of edges joining pairs of vertices. If an edge e joins vertices u and v, we say that e is incident to u and v. In this definition we allow for multiple edges to be incident to the same pair of vertices and we also allow for an edge to join a vertex to itself. The following is due to Waldhausen [427]. Definition 31.54 (Graph manifold). A graph manifold M3 is a closed 3manifold modeled on a finite graph G = (V, E), where to each vertex v ∈ V there corresponds a Seifert fibered manifold Mv whose boundary components are tori and to each edge e ∈ E there corresponds a product manifold Me ∼ = T 2 × [0, 1] such that if an edge e is incident to a vertex v, then there corresponds a gluing of a boundary component of Mv to a boundary component of Me via a torus diffeomorphism. Based on the classification of Seifert fibered manifolds discussed above, graph manifolds have been classified (see [427]). Graph manifolds are relevant to nonsingular solutions, which are discussed in the next few chapters, because of the following result (see [58] and [59]). Theorem 31.55 (Cheeger and Gromov). Let M3 be a compact 3-manifold whose boundary components consist of 2-tori or whose boundary is empty. If M admits a sequence of collapsing Riemannian metrics {gi }, i.e., metrics where   2 (31.15) lim max inj gi (x) · max | Rm gi | = 0, i→∞

x∈M

M

then M admits an F -structure and hence is a graph manifold. 9

6. Notes and commentary For the proofs of most of the theorems stated in this chapter we refer the reader to Thurston [401] and to Benedetti and Petronio [24] and the references on p. 1 therein. See Ratcliffe [334] for a comprehensive and detailed treatment of hyperbolic manifolds in any dimension. §1. For a proof of the Cartan–Ambrose–Hicks theorem, see Theorem 1.37 in Cheeger and Ebin [57]. See Chapter A of [24] or see Cannon, Floyd, Kenyon, and Parry [42] for detailed discussions of the models of hyperbolic n-space. For Lemma 31.5, see also Corollary A.3.8 in [24]. For the facts about Isom (Hn ) in (31.3) and (31.4), see also Theorem A.4.1, Theorem A.4.2, and Proposition A.5.13, all in [24]. For an exposition of the Brouwer fixed point theorem, see for example pp. 13–15 of Milnor [234]. For the classification of isometries of Hn into elliptic, parabolic, and hyperbolic types, see for example Propositions A.5.14 and A.5.18 in [24]. For a proof of Lemma 31.11, see Lemma D.2.4 in [24]. §2. For a proof of Theorem 31.14, see Theorem I.1 and Corollary I.2, both in Jaco [158]. For the Seifert and Van Kampen theorem, Theorem 31.22, see for example p. 372 of Rolfsen [337]. 9 See

[58] for the definition of an F -structure.

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195

For a proof of the Jordan–Brouwer separation theorem, see p. 89 of Guillemin and Pollack [131]. Using a standard low-dimensional topological “cut and paste” argument, we give a self-contained proof of Lemma 31.21. We successively replace a map f of a disk into a 3-manifold by homotopic maps with better properties (see for example Scott and Wall [360]). Suppose that  ker i∗ : π1 (Σ2 ) → π1 (M3 ) = {1} . Then, by Corollary 31.15, there exists α ∈ ker {i∗ : π1 (Σ) → π1 (M)} − {1} such that there exists a smooth embedding of a closed disk f : D2 → M such that10 f (∂D) ⊂ Σ and [ f |∂D ] = α. By deforming f in a thin collar of ∂D in D, we may assume that f is “normal” (in particular, transversal) to Σ in this collar of ∂D. Then, by applying the Transversality Homotopy Theorem (see p. 70 of [131]) to f restricted to D minus a thinner collar of ∂D, we may assume that the map f |int(D) is transversal to Σ without changing f in this thinner collar. This implies that f −1 (Σ) ⊂ D is a codimension 1 submanifold.11 Since f is transversal to Σ in a collar of ∂D, we have f −1 (Σ) is a finite disjoint union of loops, where each loop is contained in int (D) unless the loop is equal to ∂D itself. Now consider an “innermost” loop C ⊂ f −1 (Σ), i.e., a loop C whose interior does not contain any loops of f −1 (Σ). Then f (C) ⊂ Σ and C bounds a disk B 2 ⊂ D whose image under f is contained in either M1 or M2 . In either case, by our assumption that Σ is incompressible in both M1 and M2 , we can replace f |B by a homotopic map sending B entirely into Σ. Keeping f the same outside of B, this yields a new map from D homotopic to the original, which we still call f . We may then slightly push f in a neighborhood of B to the other side (of f |B ) to remove C from f −1 (Σ). Repeating this process at most a finite number of times eventually yields no loops in f −1 (Σ) except ∂D. That is, we may assume that either f (D) ⊂ M1 or f (D) ⊂ M2 . Again using that Σ is incompressible in both M1 and M2 , we conclude that f |∂D must be null-homotopic in Σ. This yields a contradiction and hence shows that π1 (Σ) injects into π1 (M). A similar argument shows that both π1 (M1 ) and π1 (M2 ) inject into π1 (M). We leave it as an exercise for the reader to verify this. §3. For Theorem 31.36, see Theorem 5.10.1 in [401] or Theorem D.1.1 in [24]. For a proof of Theorem 31.38, see Proposition D.2.1 and Theorem D.2.2 of [24]. For a proof of Lemma 31.42, see Proposition D.3.2 of [24]. For a proof of Theorem 31.43, see Theorem D.3.3 and Proposition D.3.12 of [24]. For a proof of Theorem 31.44, see Proposition D.3.12 on pp. 151–152 of [24]. For Example 31.45, see p. 151 of [24].

10 Note

that Corollary 31.15 holds in the smooth category. general, if F : X → Y is transversal to Z ⊂ Y, then F −1 (Z) is a submanifold of X , where the codimension of F −1 (Z) in X is equal to the codimension of Z in Y (see the theorem on p. 28 of [131]). 11 In

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§4. For the proof of the Mostow rigidity theorem, see Mostow [258] and Prasad [332]; see also Theorem C.0 on p. 83 and Theorem C.5.2 on p. 123 of [24] and Theorem 5.7.2 and Corollaries 5.7.3 and 5.7.4 on p. 101 of [401]. For the proof of Corollary 31.51, see Theorem C.5.6 on p. 125 and Remark C.5.7 on p. 126 of [24].

CHAPTER 32

Nonsingular Solutions on Closed 3-Manifolds Maybe this time is forever... – From “Father Figure” by George Michael

Beginning with this chapter we give an exposition of Hamilton’s visionary paper [143] and thus consider an important special class of solutions to the normalized Ricci flow, namely those which both exist for infinite forward time and have uniformly bounded curvatures, i.e., nonsingular solutions. Regarding the role of nonsingular solutions in his program for the Ricci flow on 3-manifolds, in [143] Hamilton wrote: “. . . one may hope to produce nonsingular solutions after a finite number of surgeries.” To wit, one may hope that for any closed 3-manifold M and any initial metric g0 on M there exists a solution g (t), t ∈ [0, ∞), with g (0) = g0 to the volume normalized Ricci flow with surgery,1 where the surgery times are only finitely many and where the solution after the last surgery time is a nonsingular solution to the Ricci flow. In short, this is Hamilton’s program to prove Thurston’s geometrization conjecture. On the other hand, Perelman’s work indicates that, in order to prove the geometrization conjecture, the global in time existence of the unnormalized Ricci flow with surgery, without necessarily a uniform curvature bound nor necessarily a finite number of surgeries (an infinite number of surgery times tending to infinity is allowed), is evidently sufficient; see the second to last paragraph of §1 of Perelman [313]. Regarding the completeness of Perelman’s proofs in [312] and [313], building on Hamilton’s earlier body of work, of the Poincar´e and geometrization conjectures, please see Cao and Zhu [49], Kleiner and Lott [161], Morgan and Tian [251], [252], and Bessi`eres, Besson, Maillot, Boileau, and Porti [29]. Throughout this chapter we shall assume that the 3-manifolds under consideration are orientable.

1. Introduction In this section we expand upon the above motivation for the study of nonsingular solutions. At the end of this section we give an outline of the chapter.

1 For

the original definition of 4-dimensional Ricci flow with surgery, see [142]. 197

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1.1. The flip side—singular solutions and singularity models. Nonsingular solutions, by definition, do not form singularities in finite time. In contrast, much of the study of the Ricci flow, besides the quest for proving convergence theorems, is focused on understanding the behavior of singular solutions. This has been one the main focuses of this book series. In the case of the Ricci flow on closed 3-manifolds, the nature of finite-time singularities is more controlled and restricted than in higher dimensions. Recall that a singularity model is defined as a limit of rescalings centered at space-time points approaching the singularity time (see Definition 28.7). In dimension 3, a finite-time singularity model is a κ-solution. All finite-time singularities of the Ricci flow on closed orientable 3-manifolds either shrink to “round points” (in particular, the solution asymptotically approaches constant curvature as it shrinks and the underlying 3-manifold is diffeomorphic to a spherical space form) or exhibit weak neckpinches; i.e., there exists a singularity model which is a round shrinking cylinder or its orientable Z2 -quotient. In the nonspherical space form case, the existence of such weak neckpinches follows from Hamilton’s work together with Perelman’s no local collapsing theorem (see for example Theorems 9.68 and 9.70 in [77]). Much more strongly and partly based on his results on κ-solutions (some of which are described in Chapters 19 and 20 of Part III), Perelman proved a canonical neighborhood theorem (see §12.1 of [312]), which gives a very good description of the high curvature regions of solutions to the Ricci flow on closed 3-manifolds. However, even in dimension 3, much of the fine structure of singularities is still unknown; for example, a priori one could conceivably have a Cantor set of 2-spheres collapsing simultaneously.2 This possibility also holds for a similar situation in the study of the mean curvature flow (even in dimension 2). The reader may see §5 of Chapter 2 of Volume One for a discussion of (strong) rotationally symmetric neckpinches as well as Chapter 36 of this book for a discussion of rotationally symmetric degenerate neckpinches, the latter being an example of a Type IIa singularity. The more general singularity theory together with the understanding of solutions in high curvature regions, which are particularly effective in dimension 3, are developed by Hamilton and Perelman in [138], [142], [312], and [313]. In the case of degenerate neckpinches, the geometric-topological surgery that one performs does not change the topological type of the 3-manifold. On the other hand, in the case of nondegenerate neckpinches, surgery may change the topological type of the 3-manifold. When this occurs, we call the geometric-topological surgery topologically essential. A topologically essential surgery does one of the following: (1) It splits the 3-manifold into two disjoint 3-manifolds, neither of which is diffeomorphic to S 3 and whose connected sum is diffeomorphic to the original 3-manifold. (2) It produces a new 3-manifold whose connected sum, either with (a) S 2 ×S 1 or with (b) RP 3 , is diffeomorphic to the original 3-manifold. The former case arises when a topological handle is pinched along an S 2 , whereas the latter arises when one has a neckpinch whose singularity model is the orientable Z2 -quotient of the round shrinking cylinder. 2 We

thank John Lott for pointing this out to us.

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By the work of Angenent and one of the authors [13], one may have for example a “rotationally symmetric” RP 3 #RP 3 (considered as a quotient of S 2 × R, i.e., as S 2 ×[0, 1] with each 2-sphere at the ends identified to an RP 2 ) where we have either (1) a neckpinch at one end or (2) two simultaneous neckpinches at opposite ends. 1.2. Relation of nonsingular solutions to Ricci flow with surgery. In Ricci flow with surgery, after performing surgery, whether topologically essential or not, one continues the solution until the next singularity. One then repeats this process of forming a singularity, performing surgery, and continuing the Ricci flow. Hamilton’s program and Perelman’s implementation, seek to obtain either (1) finite-time extinction or (2) infinite-time existence. By finite-time extinction, we mean that after some finite time the solution is empty; i.e., all components contract until their volumes tend to zero. As examples of this, we note topological spherical space forms contracting to points and products S 2 × S 1 contracting to circles. In either case (1) or (2), one hopes and expects to infer the existence of a geometric decomposition, in the sense of Thurston, of the original closed orientable 3-manifold. The most difficult aspect of Hamilton’s program, as well as Perelman’s implementation, is to show that surgery times cannot accumulate. The techniques of Perelman (see [312] and [313]), building on Hamilton’s body of work, address this important issue. The subject of this chapter, nonsingular solutions, corresponds to the infinitetime existence case. Although singular solutions and nonsingular solutions are complementary, they share a common technique in their study, namely Hamilton’s Cheeger–Gromov-type compactness theorem (see Theorem 3.10 in Part I). 1.3. Outline of the rest of the chapter. In §2 we present examples and the statement of the main result of this chapter and the next, that is, Hamilton’s theorem that nonsingular solutions of the normalized Ricci flow on closed 3-manifolds admit geometric decompositions in the sense of Thurston. We also indicate a very brief outline of the proof; the detailed proof, including related background material, occupies the rest of the chapter and the next. In §3 we separate the study of nonsingular solutions into the so-called positive, zero, and negative cases according to the asymptotic behavior of Rmin (t). If a nonsingular solution sequentially collapses, then it is diffeomorphic to a graph manifold. So we assume that the solution is noncollapsed. By applying Hamilton’s Cheeger–Gromov compactness theorem, we provide a criterion for when M3 is diffeomorphic to a space form with nonnegative curvature. In §4 we prove by applying the above criterion that in the positive and zero cases, M is diffeomorphic to a space form of positive and zero curvature, respectively. In §5 we prove that in the negative case any sequential limit must be a complete hyperbolic manifold with finite volume, which may be either compact or noncompact. If the limit is compact, then M itself admits a hyperbolic metric. Otherwise we obtain a hyperbolic piece of M.

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The next chapter is devoted to the proof that the inclusion maps of such pieces induce injections of their fundamental groups. 2. The main result on nonsingular solutions In this section we present the statement of Hamilton’s classification theorem for 3-dimensional nonsingular solutions and give a brief outline of Hamilton’s proof. 2.1. Definition and examples of nonsingular solutions. We consider the normalized Ricci flow (NRF) on a closed manifold Mn : 2 ∂ gij = −2Rij + r gij , ∂t n 5  where r (t) = M Rg(t) dμg(t) M dμg(t) denotes the average scalar curvature. This flow preserves the volume of g (t) and is obtained from the Ricci flow by rescaling space and time (see §9.1 of Chapter 6 of Volume One).

(32.1)

Definition 32.1 (Nonsingular solution). A solution (Mn , g (t)) to the NRF on a closed manifold is called nonsingular if it is defined for all time t ∈ [0, ∞) and Rmg(t)  ≤ C for some C < ∞. The NRF on a closed manifold is nonsingular when the initial metric is any one of the following: (1) Ricci solitons—these are the fixed points of the NRF modulo the group of diffeomorphisms, in particular, Einstein metrics. (2) A metric on a 3-manifold with positive Ricci curvature (see Hamilton [135]). (3) A metric on a surface (see Hamilton [137] and [67]). (4) A locally homogeneous metric on a 3-manifold (see Jackson and one of the authors [154]). (5) A metric on an n-manifold with 2-positive curvature operator, in dimension 4 by Hamilton [136] and H. Chen [65], and in dimensions greater than 4 by B¨ohm and Wilking [30].3 (6) A metric with positive complex sectional curvature, proving the 1/4pinched spherical space form conjecture (see Brendle and Schoen [35]). (7) A K¨ahler metric on a complex n-manifold with c1 = 0 or c1 < 0 and whose K¨ahler class is proportional to c1 (see H.-D. Cao [43]). In each of the above cases, except for case (4) and for non-Einstein solitons in case (1), we have convergence to an Einstein metric as t → ∞. On the other hand, the Ricci flow on a closed manifold blows up in finite time when the initial metric is any of the following: (8) Any metric on a surface with χ > 0 (this follows from (3) above). (9) A metric on an n-manifold with positive scalar curvature (see [135]). (10) A locally homogeneous metric on a 3-manifold of class (a) SU(2) or (b) S 2 × R (see Jackson and one of the authors [154]). (11) A K¨ahler metric on a complex n-manifold with c1 > 0. 3 See earlier work of Huisken [152], Margerin [217], [218], and Nishikawa [294], [295] for a metric on an n-manifold with sufficiently pointwise-pinched positive sectional curvatures.

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(12) A rotationally symmetric neckpinch (see Angenent and one of the authors [13]). (13) A rotationally symmetric degenerate neckpinch (see Gu and Zhu [127]). 2.2. Statement of the main result on 3-dimensional nonsingular solutions. The following is Hamilton’s classification of 3-dimensional nonsingular solutions [143]. Theorem 32.2 (Nonsingular solutions on 3-manifolds admit geometric decompositions). If a closed 3-manifold M3 admits a nonsingular solution of the NRF, then M is diffeomorphic to one of the following: (A1) a graph manifold,4 (A2) a spherical space form S 3 /Γ, (A3) a flat manifold, (A4) a hyperbolic manifold, (A5) the union along incompressible 2-tori of finite-volume hyperbolic manifolds and graph manifolds. In particular, in all cases M admits a geometric decomposition in the sense of Thurston.5 We say that a nonsingular solution of the NRF sequentially collapses if there exists a sequence of times ti → ∞ such that the metrics g(ti ) collapse; i.e., lim ρ (ti )2 max |Rm| = 0,

(32.2)

i→∞

M×{ti }

where ρ (ti )  max inj g(ti ) (x) x∈M

is the maximum injectivity radius of any point for the metric g(ti ) (compare with (31.15)). Otherwise, we say that the solution g(t) is noncollapsed. As we shall exposit, Hamilton proved that, under the hypothesis of Theorem 32.2, exactly one of the following cases occurs, corresponding to the five cases in the statement of the above theorem: (B1) (Sequentially collapses) It then follows from Cheeger–Gromov theory that M admits an F -structure and is diffeomorphic to a graph manifold (see Theorem 31.55). In the rest of the chapter we shall assume that g(t) is noncollapsed. Thus there exists a constant δ > 0 such that for each time t ∈ [0, ∞) there exists a point xt ∈ M such that injg(t) (xt ) ≥ δ. This enables us to apply Hamilton’s Cheeger–Gromov-type compactness theorem to obtain a limit solution g∞ (t) (see Theorem 32.5 below). (B2) (Exponential convergence to a spherical space form) The solution g(t) converges exponentially fast in every C k -norm as t → ∞ to a static constant positive sectional curvature metric g∞ on M. the definition and topological classification of graph manifolds, see §5 in Chapter 31. the definition of geometric decomposition, see Thurston [402]. This is also briefly discussed in Chapter 1 of Volume One and Chapter 9 of Part I. 4 For 5 For

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(B3) and (B4) (Sequential convergence to a flat or hyperbolic manifold) There exist a sequence of times ti → ∞ and diffeomorphisms φi : M → M such that the sequence of pulled-back metrics φ∗i g(ti ) converges to a constant (either zero for (B3) or negative for (B4)) sectional curvature metric g∞ on M as i → ∞. (B5) (Toroidal decomposition into hyperbolic and graph manifold pieces) There m exists a finite collection of finite-volume hyperbolic 3-manifolds  3 Hα , hα α=1 and there exist smooth 1-parameter families of embeddings ψα (t) : Hα → M such that the pulled-back metrics ψα (t)∗ g(t) converge to hα as t → ∞ for each α. Moreover, M can be decomposed into two time-dependent 3-manifolds M1 (t) and M2 (t), where Mj (t) and Mj (t ) are ambient isotopic6 in M for all t and t , j = 1, 2, whose intersection is their mutual boundary and consists of a finite disjoint union of 2-tori. Geometrically, the metrics pulled back from M1 (t) converge to the hyperbolic pieces; i.e.,   (Hα , hα ) ψα (t)−1 M1 (t), ψα∗ (t)g(t) converge to α

as t → ∞, while (M2 (t), g(t)) collapses. Topologically, the boundary tori are all incompressible, and all of the homomorphisms induced by the inclusion maps i∗ : π1 (N ) → π1 (P) are injective, whenever we have both (1) N = M1 (t) ∩ M2 (t), M1 (t), or M2 (t), (2) P = M1 (t), M2 (t), or M are such that N ⊂ P. 2.3. Brief outline of Hamilton’s proof of Theorem 32.2. The first idea is to separate the study of noncollapsed nonsingular solutions of the NRF on closed 3-manifolds into cases, depending on the asymptotic behavior of the minimum scalar curvature. Let Rmin (t)  minx∈M R (x, t). We have three cases: (I) Rmin (t) > 0 at some time t, 0 as t → ∞, and (II) Rmin (t) (III) Rmin (t) increases to a negative limit as t → ∞. By considering these sequential limits of noncollapsed nonsingular solutions, one shows that in cases (I) and (II), M3 admits a metric with constant nonnegative sectional curvature. This leaves us with case (III). In this case, unless M itself admits a hyperbolic metric, all asymptotic limits are complete noncompact hyperbolic 3-manifolds with finite volume (see Proposition 32.12 below). Corresponding to each hyperbolic limit H3 there is a time-dependent almost hyperbolic piece in M which is immortal, i.e., defined for all large enough time. The interior of this compact 3-dimensional submanifold of M is diffeomorphic to H and is metrically as close as we like to the truncation of H along constant mean curvature tori with sufficiently small area A > 0. Importantly, one shows that each of these hyperbolic pieces is incompressible in M. 6 That is, there exists a continuous 1-parameter family of homeomorphisms F of M with t F0 = id and F1 (Mj (t)) = Mj (t ).

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203

Now, the closure C 3 of the complement of the union of all the immortal almost hyperbolic pieces in M is a compact 3-manifold with boundary consisting of a disjoint union of tori, each with area A, such that the maximum injectivity radius of (C, g (t)|C ) can be made as small as we like by taking A sufficiently small and t sufficiently large. By Theorem 31.55, C is a graph manifold. 3. The three cases of nonsingular solutions In this section we begin by observing that the noncollapsed nonsingular assumption allows us to take limits of the solution as time tends to infinity. We then divide the study of these limits according to whether the minimum scalar curvature is positive at some finite time, tends to zero, or tends to a negative constant. 3.1. Evolution of R under the NRF. Let (Mn , g (t)), t ∈ [0, T ), be a solution to the NRF on a closed manifold. Recall that the evolution of the scalar curvature of g (t) is given by (32.3)

∂R 2 2 2 = ΔR + 2 |Rc| − rR ≥ ΔR + R (R − r) , ∂t n n

where r is the average scalar curvature. Applying the weak maximum principle to equation (32.3) yields   ∂R d− (32.4) Rmin (t) = inf (x, t) : R (x, t) = Rmin (t) dt ∂t 2 ≥ Rmin (t) (Rmin (t) − r (t)) n for the lower Dini derivative of the Lipschitz function Rmin (t). Consequently, we have the following. Lemma 32.3 (Monotonicity of Rmin (t)). Let (Mn , g (t)) be a solution to the NRF on a closed manifold. (1) If Rmin (t0 ) > 0 for some t0 , then Rmin (t) > 0 for all t ≥ t0 . (2) If Rmin (t) ≤ 0 for all t, then Rmin (t) is nondecreasing. If, in addition, g (t) is not Einstein, then Rmin (t) is strictly increasing. Proof. (1) At any time where Rmin > 0, we have d 2 2 ln Rmin ≥ (Rmin − r) > − r. dt n n This shows that if Rmin > 0 holds at some time, then it holds at all later times. (2) The first statement in this part follows from (32.4) since in general Rmin (t)− r (t) ≤ 0. To prove the second statement, suppose Rmin (t1 ) = Rmin (t2 )  c ≤ 0 for some t1 < t2 . Then Rmin (t) ≡ c for t ∈ [t1 , t2 ]. By applying the strong maximum principle to equation (32.3), we conclude that R (t) ≡ c and that g (t) is Einstein. 

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3.2. Compactness theorem for the NRF. The first main tool we shall use in the proof of Theorem 32.2 is the compactness theorem. As in Definition 3.6 in Part I, we have Definition 32.4 (Pointed C ∞ Cheeger–Gromov convergence). We say that a sequence {(Mni , gi (t) , xi )}i∈N , t ∈ (α, ω), of pointed solutions to the NRF on closed manifolds converges to a complete pointed family of metrics (Mn∞ , g∞ (t) , x∞ ) , t ∈ (α, ω) , if there exist (1) an exhaustion {Ui }i∈N of M∞ by open sets with x∞ ∈ Ui and (2) a sequence of diffeomorphisms Φi : Ui → Vi  Φi (Ui ) ⊂ Mi with Φi (x∞ ) = xi    such that Ui , Φ∗i gi (t)|Vi converges in C ∞ to (M∞ , g∞ (t)) uniformly on compact sets in M∞ × (α, ω). Let rg denote the average scalar curvature of g. The compactness theorem for the NRF is as follows (this is completely analogous to Theorem 3.10 in Part I). Theorem 32.5 (Hamilton’s Cheeger–Gromov-type compactness). Let {(Mni , gi (t) , xi )}i∈N , t ∈ (α, ω) % 0, be a sequence of complete pointed solutions to the NRF on closed manifolds such that (1) (uniformly bounded curvatures) |Rm gi |gi ≤ C0

on Mi × (α, ω)

for some constant C0 < ∞ independent of i and (2) (injectivity radius estimate at t = 0) injgi (0) (xi ) ≥ ι0 for some constant ι0 > 0. Then there exists a subsequence such that {(Mni , gi (t) , xi )}i∈N converges as i → ∞ to a complete pointed solution (Mn∞ , g∞ (t) , x∞ ), t ∈ (α, ω), to the Ricci flow with cosmological constant: 2 ∂ (g∞ )ij = −2 (Rc g∞ )ij + r∞ · (g∞ )ij , ∂t n where the limit r∞ (t)  limi→∞ rgi (t) exists and the limit solution has uniformly bounded curvature. (32.5)

Proof. First observe that rgi (t) is uniformly bounded because Rgi (t) is uniformly bounded. Applying the change of scale between Ricci flow and NRF (see (32.9)–(32.10) below) to Shi’s derivative estimates, for any ε > 0 we have bounds   on all derivatives of the curvature ∇k Rmgi  ≤ Ck on Mi × (α + ε, ω), independent of i. This implies, for each time derivative of the average scalar curvature, the  dk r  estimate  dtkgi  ≤ Ck . For example, using (32.3), we compute that for a solution g (t) to the NRF, 6   dr d = Rdμ dμ dt dt M M    1 2 2  = 2 |Rc| − rR + R (r − R) dμ. n dμ M M

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205

Each higher time derivative of r (t) is an integral of a polynomial expression of Rm and its space derivatives. In view of this, the proof of the compactness theorem in [139] goes through without much change. In particular, in the limit one obtains a solution to (32.5).  Exercise 32.6. (1) Show that for any n ≥ 3 there exists a sequence of closed pointed Riemannian manifolds {(N n , hi , xi )} with average scalar curvatures rhi → +∞ and which converge to a complete noncompact pointed Riemannian manifold n , h∞ (t) , x∞ ) with bounded curvature. Note that this is not possible (N∞ for n = 2 by the Gauss–Bonnet formula. (2) Show that for any n ≥ 3 there exists a sequence of closed pointed Riemannian manifolds {(N n , hi , xi )} with uniformly bounded curvature which conn , h∞ , verges to a complete noncompact pointed Riemannian manifold (N∞ x∞ ) and for which the sequence {rhi } does not converge; in particular, there exist two subsequences for which the average scalar curvatures converge to two different numbers.  Hint: (1) Let part of (N n , hi ) be isometric to S n−1 ( 1i ) × 0, i3 , which drifts off to infinity relative to xi as i → ∞. (2) Consider the set of points in Rn+1 of distance 1 from the disk Dn (i) × {0}. (Pancake) For i odd, let (N n , hi ) be a smoothing of this C 1 hypersurface. (Umbrella pole and base) For i even, let (N n , hi ) be the same attached to a smoothing of S n−1 (1) × [1, in ] capped off at the top. Remark 32.7. Corollary 3.18 in Part I, which was intended to be a consequence of Hamilton’s compactness theorem, is incorrect as stated. We would like to thank John Lott and Peter Topping for pointing this out to us. For correct statements, see Corollaries E.2 and E.4 in [161] and see [418]. 3.3. Limits as ti → ∞ via the compactness theorem.  Let M3 , g (t) , t ∈ [0, ∞), be a noncollapsed nonsingular solution to the NRF flow on a closed 3-manifold. Then there exists δ > 0 such that for any ti → ∞ there exist points xi ∈ M such that (32.6)

inj g(ti ) (xi ) ≥ δ.

To such a sequence {(xi , ti )}, consider the corresponding sequence of pointed solutions (obtained by translating backward in time)  3 (32.7) M , gi (t) , xi , where gi (t)  g (ti + t) . By Theorem 32.5, there exists a subsequence which converges to a complete pointed solution  3 (32.8) M∞ , g∞ (t) , x∞ , t ∈ (−∞, ∞) , of equation (32.5), where the limit solution has uniformly bounded curvature and finite volume Vol (g∞ (t)) ≤ Vol (g (0)). We call (M∞ , g∞ (t) , x∞ ) a limit solution corresponding to {(xi , ti )}.

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3.4. Division of the study of nonsingular solutions into three cases. We shall show that for any noncollapsed nonsingular solution to the NRF on a closed 3-manifold, there exists such a limit which admits a metric with constant curvature.  The properties of any limit M3∞ , g∞ (t) , x∞ in (32.8) depend on the asymptotic behavior of Rmin (t). Accordingly, we divide the analysis into three cases. As we shall see, the more positive the curvature, the easier the analysis. CASE I. (Positive) Rmin (t0 ) is positive for some t0 ∈ [0, ∞). CASE II. (Zero) limt→∞ Rmin (t) = 0. CASE III. (Negative) limt→∞ Rmin (t) < 0. In Cases  I and II, it is useful to consider the corresponding solution to the Ricci ˜ ij with g˜ (0) = g (0). The two flow. Let M3 , g˜ t˜ be the solution to ∂∂t˜g˜ij = −2R solutions g and g˜ differ by the following change of scale in space and time:  t˜   (32.9) g (t) = ψ t˜ g˜ t˜ , t= ψ (τ ) dτ, 0

where

 ˜

  t  2 2 t (32.10) r˜ (˜ τ ) d˜ τ = exp r (τ ) dτ . 3 0 3 0   ˜ max t˜  maxx∈M Rg˜ x, t˜ . Let R  ψ t˜ = exp

 Lemma 32.8 (Criterion for g (t) to have nonnegative curvature). Let M3 , g (t) , t ∈ [0, ∞), be a noncollapsed nonsingular solution to the NRF flow on a closed 3manifold. Let [0, T˜ ) be the maximal time interval for the corresponding solution    t˜ g˜ t˜ to the Ricci flow. Suppose that ti  0 i ψ (τ ) dτ , where t˜i satisfies  (32.11) t˜i → T˜ and t˜i ψ t˜i → ∞.  Then for any {xi } satisfying (32.6) and limit solution M3∞ , g∞ (t), x∞ , t ∈ (−∞, ∞), to the NRF corresponding to {(xi , ti )}, we have that M∞ is diffeomorphic to M and g∞ (t) is either flat or has positive sectional curvature. Thus M is diffeomorphic to a space form with nonnegative curvature. Proof. Step 1. g∞ (t) has nonnegative sectional curvature. Let νg˜ denote the smallest eigenvalue of Rmg˜ . We may assume that νg˜ ( · , 0) is negative somewhere (otherwise we are done). Let C  − inf x∈M νg˜ (x, 0) > 0. Then the Hamilton–Ivey estimate says that at any point and time where νg˜ < 0, we have   |νg˜ | ln |νg˜ | + ln C −1 + t˜ − 3 ≤ Rg˜ . Thus

 R    R −1   − 3.   gi (t) = g˜(ti +t) ≥ ln ψ(t ν + t)(C + t + t) i i gi (t) νg (t)  |νg˜(t | i i +t) Now, by (32.11), we have limi→∞ ψ(t˜i )(C −1 + t˜i ) = ∞. By (32.10),     ψ(t  2  ti +t   i + t)    ≤ C |t| . r (τ ) dτ ln =      ψ(t˜i )  3 ti

−1 + t Hence, we also have limi→∞ ψ(t i + t)(C i + t) = ∞ for all t ∈ (−∞, ∞). This implies that sect(g∞ (t)) ≥ 0.

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207

Step 2. M is diffeomorphic to a space form. Since g∞ (t) has finite volume, M∞ must be compact (a complete noncompact Riemannian manifold with nonnegative Ricci curvature must have at least linear volume growth and, in particular, infinite volume; see Yau [446]). This implies that M∞ is diffeomorphic to M and that r∞ (t) = rg∞ (t) . Hence g∞ (t) is a solution to the NRF. Now, by the classification of solutions to the Ricci flow on closed 3-manifolds with nonnegative sectional curvature (see Hamilton [136] and Shi [371]), we have that (M∞ , g∞ (t)) is either (i) a flat manifold, (ii) the compact quotient of the product of a positively curved 2-sphere and R, or (iii) has positive sectional curvature. In this case, by Hamilton’s theorem, M∞ is diffeomorphic to a spherical space form. We now rule out case (ii). Suppose that a finite cover of (M∞ , g∞ (t)) is isometric to S 2 , h∞ (t) × S 1 (ρ∞ (t)), where h∞ (t) is a positively curved solution S 2 of   2 ∂ h∞ = −R (h∞ ) + r∞ h∞ , (32.12) ∂t 3 where ρ∞ (t) is the length of S 1 and where r∞ (t) = rg∞ (t) = rh∞ (t) . Because of the factor 23 = 1 in (32.12), we have by the Gauss–Bonnet formula that d 1 8 Area (h∞ ) = − r∞ Area (h∞ ) = − π. dt 3 3 Hence h∞ (t) only exists up to a finite singular time, contradicting that g∞ (t) exists for all time.  Remark 32.9. Alternatively, case (ii) may be ruled out by Perelman’s no local collapsing theorem applied to g∞ (t) with t → −∞ or by the result in [153] by Ilmanen and one of the authors. 4. The positive and zero cases of nonsingular solutions In this section we prove the following:  (1) Positive case: If Rmin (t) is positive after some finite time, then M3 , g (t) converges to a spherical space form under the NRF. (2) Negative case: If Rmin (t) tends to zero as t → ∞, then for some sequence ti → ∞ we have that g (ti ) converges to a flat metric on M3 . 4.1. The positive case—M3 is a spherical space form. For Case I, we now prove the following.

 Proposition 32.10. If a noncollapsed nonsingular solution M3 , g (t) , t ∈ [0, ∞), to the NRF on a closed manifold satisfies Rmin (t0 ) > 0 for some t0 , then M is diffeomorphic to a spherical space form. Proof. Since Rmin (t0 ) > 0, we have T˜ < ∞. So, by the Hamilton–Ivey ˜ max t˜i → ∞. Now, estimate, there exists a sequence of times t˜i → T˜ such that R since the t˜i satisfy    ˜ max t˜i = ψ t˜i Rmax (ti ) ≤ Cψ t˜i , R by Lemma 32.8 we have that M is diffeomorphic to a space form with nonnegative curvature.

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32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS

On the other hand, since M admits a metric with positive scalar curvature, by a result of Gromov and Lawson [125] and Schoen and Yau [354], M does not admit a flat metric.  4.2. The zero case—M3 admits a flat metric. The borderline Case II is a bit more delicate because only asymptotically does Rmin (t) approach zero. In particular, for t < ∞, it is not clear whether the solution g (t) behaves more like it has positive, zero, or negative curvature.  Proposition 32.11. If a noncollapsed nonsingular solution M3 , g (t) , t ∈ [0, ∞), to the NRF on a closed manifold satisfies limt→∞ Rmin (t) = 0 for some t0 , then M admits a flat metric.  We shall use three subcases to prove this. Again, let g˜ t˜ be the solution to ˜ the Ricci  flow with g˜ (0) = g (0). If the maximal time interval [0, T ) of existence ˜ ˜ of g˜ t is finite, then supM×[0,T˜) Rmax = ∞. By the proof of Proposition 32.10, there exists a limit solution g∞ (t) on M∞ ∼ = M which is either flat or has positive sectional curvature. The latter cannot happen since then Rg(t) would be positive for some t ∈ [0, ∞), a contradiction. For the rest of the proof of Proposition 32.11, we shall assume that T˜ = ∞. Recall that the evolution of the volume for the Ricci flow is    d˜ (32.13) Vol g˜(t˜) = −˜ r t˜ Vol g˜(t˜) . ˜ dt This motivates dividing the analysis of Case II with T˜ = ∞ into the following three subcases: A. (Positive) There exists a sequence t˜i → ∞ such that  (32.14) Vol g˜(t˜i ) → 0. B. (Negative) There exists a sequence t˜i → ∞ such that  (32.15) Vol g˜(t˜i ) → ∞. C. (Zero) There exist constants c > 0 and C < ∞ such that  (32.16) c ≤ Vol g˜(t˜) ≤ C for all t˜ ∈ [0, ∞).   Subcase A: Recall that g (ti ) = ψ t˜i g˜ t˜i . By (32.14), we have    −3/2  −3/2 0 ← Vol g˜ t˜i = ψ t˜i Vol (g (ti )) = C · ψ t˜i .  Thus ψ t˜i → ∞. By Lemma 32.8 and since Rg(t) cannot be positive, the limit solution g∞ (t) is flat. Subcase B: By (32.15) and (32.13), we have  t˜i    g (0)) → −∞. Rg˜(τ ) dμg˜(τ ) dτ = −Vol g˜ t˜i + Vol (˜ 0

M

Hence there exists a sequence of times ti → ∞ with  (32.17) Rg˜(t ) dμg˜(t ) < 0. i i M

4. THE POSITIVE AND ZERO CASES OF NONSINGULAR SOLUTIONS

209

 t  Let ti  0 i ψ (τ ) dτ , let xi satisfy (32.6), and let M3∞ , g∞ (t), x∞ , t ∈ (−∞, ∞), be the limit solution of (32.5) corresponding to {(xi , ti )}. Since limt→∞ Rmin (t) = 0, we have Rg∞ (t) ≥ 0 for all t ∈ (−∞, ∞). We shall show that the limit solution (M∞ , g∞ (t)) is compact and flat. We have by (32.17) that  Rg(ti ) dμg(ti ) ≥ C Rmin (ti ) → 0 0> M

as i → ∞. Thus

 M

 Rg(ti ) − Rmin (ti ) dμg(ti ) → 0

as i → ∞. Since Rg(ti ) − Rmin (ti ) ≥ 0 and converges to Rg∞ (0) uniformly on compact sets, we have  Rg∞ (0) dμg∞ (0) = 0. M∞

Since Rg∞ (0) ≥ 0, we conclude from this that Rg∞ (0) ≡ 0. Since Rg∞ (t) ≥ 0 and ∂ 2 2 Rg∞ = Δg∞ Rg∞ + 2 |Rcg∞ | − r∞ Rg∞ ∂t 3 for all t ∈ R, by the strong maximum principle we have that Rcg∞ (t) ≡ 0 for t ≤ 0. This implies that M∞ is compact and since dim(M∞ ) = 3 we conclude that g∞ (t) is flat for all t ∈ R.  Subcase C: We have T˜ = ∞ and ψ t˜ ≥ c for some c > 0. Hence, by Lemma 32.8 and since Rg(t) cannot be positive, the limit solution (M∞ , g∞ (t)) is flat and closed. This completes the proof of Proposition 32.11. we may finish the proof of Subcase C as follows. =  Alternatively,   Since g (t) ˜i → ∞ such that M3 , g˜ t˜i + t˜ ψ t˜ g˜ t˜ , where C1 ≤ ψ t˜ ≤ C, there exist t  converges to an eternal solution M3∞ , g˜∞ t˜ , t˜ ∈ (−∞, ∞), with M∞ ∼ = M. We claim that the monotonicity of Perelman’s energy functional F implies that      −f ˜ t˜  inf{F g˜ t˜ , f : ˜ = 1}, we have for e d˜ μ t g˜∞ t˜ is flat. Defining λ M the immortal solution g˜ t˜ to the Ricci flow (see (5.56) in Part I),   2 d ˜ ˜ ˜ ˜ j f˜ e−f˜d˜ ˜ i∇ λ t ≥2 μ ≥ 0, Rij + ∇ dt˜ M       ˜ ∞ t˜  inf f F g˜∞ t˜ , f . Since where f˜ t˜ is the minimizer of F g˜ t˜ , · . Let λ (by (5.53) in Part I)  ˜ t˜ ≤ R ˜ (t) λ max ≤ C < ∞, we have    ˜ ∞ t˜ = lim λ ˜ t˜ + t˜i = sup λ ˜ t˜ ≡ const . λ

(32.18)

i→∞

t˜∈[0,∞)

On the other hand, we have   2 d ˜ ˜ 7 ˜ ∞ f˜∞  e−f˜∞ d˜ ˜ ∞∇ λ∞ t ≥ 2 μ∞ ≥ 0, Rc∞ + ∇ dt˜ M∞    where f˜∞ t˜ is the minimizer of F g˜∞ t˜ , · . Hence ˜ ∞ f˜∞ ≡ 0, 7∞ + ∇ ˜ ∞∇ Rc 7 ∞ ≡ 0 since steady Ricci solitons on closed manifolds are which implies that Rc Ricci flat (see Proposition 1.66 in Part I).

210

32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS

5. The negative case—sequential limits must be hyperbolic In this section, by just analyzing the evolution of the scalar curvature, we prove that, in Case III, Any asymptotic limit is hyperbolic. Recall that by Lemma 32.3 if limt→∞ Rmin (t) < 0, then Rmin (t) is a strictly increasing function of time unless g (t) is hyperbolic. By rescaling the initial metric, we may assume without loss of generality that lim Rmin (t) = −6 < 0.

(32.19)

t→∞

The following result, although elementary in that it is a consequence of the evolution equation for the scalar curvature, is fundamental to understanding Case III.  Proposition 32.12 (Case III: Any limit is hyperbolic). Let M3 , g(t) be a noncollapsed nonsingular solution of the NRF satisfying (32.19). Let ti → ∞ and  let {xi } satisfy (32.6). Then any limit solution M3∞ , g∞ (t), x∞ , t ∈ (−∞, ∞), to the NRF corresponding to {(xi , ti )} is a complete (either compact or noncompact) hyperbolic manifold with Vol (g∞ (t)) ≤ Vol (g(t)). Moreover, for all t ∈ R, r∞ (t) = lim rgi (t) ≡ −6. i→∞   3 We call M∞ , g∞ (t), x∞ a hyperbolic limit of the solution M3 , g(t) . (32.20)

Proof. Recall from (32.4) that d 2 Rmin ≥ − Rmin (r − Rmin ) . dt 3 Since Rmin (t) ≤ −6 for all t ∈ [0, ∞), we have d Rmin ≥ 4 (r − Rmin ) ≥ 0. dt Integrating this yields  ∞ (32.21) (r(t) − Rmin (t)) dt ≤ −6 − Rmin (0)  C < ∞. 0

Since the integrand r(t) − Rmin (t) is nonnegative, for every ε > 0, there exists a time tε < ∞ such that for every t ∈ [tε , ∞),  t+1 0≤ (r(τ ) − Rmin (τ )) dτ ≤ ε. t

Taking the limit, we obtain for every t ∈ R,  t+1  t+1  rgi (τ ) − (Rgi )min (τ ) dτ (r∞ (τ ) + 6) dτ = lim i→∞

t

t



ti +t+1

= lim

i→∞

ti +t

 rg(τ ) − (Rg )min (τ ) dτ

≤ ε. Since r∞ (τ ) + 6 ≥ 0, rg(τ ) − (Rg )min (τ ) ≥ 0, and ε > 0 is arbitrary, we obtain (32.22) which implies (32.20).

lim Rmin (t) = lim r (t) = −6,

t→∞

t→∞

6. NOTES AND COMMENTARY

211

Next we compute that   |R − r| dμ ≤ ((R − Rmin ) + (r − Rmin )) dμ = 2A (r − Rmin ) , M

M

where A ≡ Vol (g(t)). Thus, from (32.21), we have  ∞  ∞ |R − r| dμdt ≤ 2A (r (t) − Rmin (t)) dt ≤ 2AC. 0

Hence  t+1  t

M∞

M

0

  Rg (τ ) + 6 dμ∞ dτ ≤ lim ∞

i→∞

 t

t+1

 M

 Rg

i (τ )

 − rgi (τ )  dμgi (τ ) dτ ≡ 0

for all t ∈ R. Therefore Rg∞ (x, t) ≡ −6 for all x ∈ M∞ and t ∈ R. From ∂ 2 0 = Rg∞ = ΔRg∞ + 2 |Rc g∞ |2 − r∞ Rg∞ = 2 |Rc g∞ + 2g∞ |2 , ∂t 3 we conclude that (32.23)

Rc g∞ ≡ −2g∞ .

Since dim(M∞ ) = 3, this implies that sect (g∞ (t)) ≡ −1. Finally, note that Vol (g∞ (t)) ≤ A from C ∞ Cheeger–Gromov convergence.



By Proposition 32.12, if the limit solution (M∞ , g∞ (t)) is on a compact manifold, then M∞ ∼ = M and so M admits a metric with constant negative sectional curvature −1. This is case (B4) in Subsection 2.2 of this chapter. The next chapter is devoted to the analysis of the most difficult case where M3∞ is noncompact and to finishing the proof of Theorem 32.2 by showing that case (B5) must hold. 6. Notes and commentary For work on nonsingular solutions in dimension 4, see Fang, Y. Zhang, and Z. Zhang [106]. For the finiteness of the number of surgeries of the Ricci flow on 3-manifolds, see Bamler [19], [20], [21], [22], [23]. Definition 32.1 is Definition 1.1 in [143]. Theorem 32.2 is Theorem 1.3 of Hamilton [143]. For (32.3), see Corollary 6.64(5) in Volume One. For Lemma 32.3, see Theorem 2.1 in [143]. Subsection 4.1 follows §5 of [143]. Subsection 4.2 follows §6 of [143]. Proposition 32.12 is Lemma 7.1 in [143].

CHAPTER 33

Noncompact Hyperbolic Limits And when your run is over just admit when it’s at its end. – From “ ‘Till I Collapse” by Eminem featuring Nate Dogg

In this chapter we discuss what happens when noncompact hyperbolic limits occur in Case III for 3-dimensional nonsingular solutions to the normalized Ricci flow (NRF) (see Proposition 32.12 and the comments after its proof). The outline of this chapter is as follows. In §1 we state the main results that shall be proved in  this chapter. In §2 we show that a noncompact hyperbolic limit H3 , h , after truncating the cusp ends, corresponds to a time-dependent almost hyperbolic piece of the non singular solution M3 , g (t) via harmonic diffeomorphisms which are approximate isometries. In §3 we prove the stability of any hyperbolic limit H. This result says that H has a corresponding (time-dependent) stable asymptotically hyperbolic submanifold of M which exists for all time and limits to H as t → ∞. This proof uses foliations of the ends of almost hyperbolic pieces by constant mean curvature (CMC) tori. In §4 we prove the crucial result that almost hyperbolic pieces are necessarily incompressible in the original manifold M. We remark that there is an alternate proof of the stability of hyperbolic limits, which is presented in §90 of Kleiner and Lott [161]. This proof avoids the use of CMC surfaces and makes stronger use of the Mostow rigidity theorem. There are also alternate proofs of the incompressibility of the cuspidal tori. In §8 of Perelman [313], for solutions to the unnormalized Ricci flow, his proof uses 2/3 the scale-invariant quantity λ (t) Vol (g (t)) , where λ (t) is the lowest eigenvalue of the elliptic operator −4Δg(t) + Rg(t) . In §93.1 of Kleiner and Lott [161], a 2/3 is given. Both of these proofs use the simpler argument using Rmin (t) Vol (g (t)) existence of Ricci flow with surgery. In this chapter we shall say that a 3-dimensional nonsingular solution (M3 , g(t)), t ∈ [0, ∞), to the NRF satisfies Condition H if it is in Case III with the existence of at least one complete noncompact hyperbolic limit and if M is connected, closed, and orientable. Since M is closed, any hyperbolic limit has finite-volume. Throughout this  chapter, each finite-volume hyperbolic limit H3 , h will be assumed to be connected, orientable, complete, and noncompact. We also adopt the notations of the previous chapter. 213

214

33. NONCOMPACT HYPERBOLIC LIMITS

1. Main results on hyperbolic pieces In this section, after recalling some basic facts about finite-volume hyperbolic manifolds, we define what it means for a noncompact hyperbolic limit of a nonsingular solution on a closed 3-manifold to be stable. We then state the main results on (1) the stability of hyperbolic limits in the nonsingular solution, (2) the incompressibility of the boundary tori of corresponding immortal asymptotically hyperbolic pieces in the underlying 3-manifold. 1.1. Elementary properties of hyperbolic limits.  Let H3 , h, x∞ be a pointed complete noncompact finite-volume hyperbolic 3-manifold. First recall what we know from §3 of Chapter 31 about (H, h, x∞ ) and its truncations. Mainly as a consequence of the Margulis lemma, we have the following (see Theorems 31.44, 31.46, and 31.36). There exists an end-complementary compact set K in H such that: ¯ h| ¯ ) is (h1) For each topological end E ∈ E (K) of H, the submanifold (E, E isometric to a standard hyperbolic cusp ([0, ∞) × TE2 , gcusp ), E E where gcusp = dr 2 + e−2r gflat for some flat metric gflat on a 2-torus TE . Henceforth ¯ we shall identify E ⊂ H with [0, ∞) × TE by an isometry. ¯ : E ∈ E (K)} is a collection of disjoint subsets. (h2) {E √ ¯ ≥ 3/4 for each E ∈ E (K). E (TE ) = Volh (E) (h3) Areagflat % ¯ and H− % ¯ (h4) H is diffeomorphic to H− E∈E(K) E E∈E(K) E is a deformation retract of H. √Before stating the next property we define the truncations of H. Given A ∈ (0, 3/4], let HA denote the compact 3-manifold with boundary obtained from H by truncating each cusp end at the torus slice with area equal to A:  (33.1) HA  H − {(rE,A , ∞) × TE } . E∈E(K)

That is, rE,A ∈ (0, ∞) is defined by E ) = Area({rE,A } × TE , gcusp |{rE,A }×TE ). A = e−2rE,A Area(gflat % Note that ∂HA = E∈E(K) {rE,A } × TE and that int (HA ) is diffeomorphic to H. √ (h5) There exists A˜ ∈ (0, 3/4] depending only on (H, h, x∞ ) such that: (a) x∞ ∈ int(HA˜ ) (see Proposition 33.13). (b) For each E ∈ E (K) the cusp [rE,A˜ , ∞) × TE is contained in the ε3 /2-thin part H(0,ε3 /2] of H (see Definition 31.39), where ε3 is the 3-dimensional Margulis ˜ constant. In other words, H − HA ⊂ H(0,ε3 /2] for A ∈ (0, A].  3 Now let M , g (t) , t ∈ [0, ∞), be a nonsingular solution of the NRF on a closed 3-manifold. Suppose that for some xi ∈ M and for some ti → ∞ the sequence of solutions {(M, gi (t), xi )}, where gi (t) = g (ti + t), converges in the C ∞ pointed Cheeger–Gromov sense to a static (i.e., time-independent) finite-volume hyperbolic

(33.2)

1. MAIN RESULTS ON HYPERBOLIC PIECES

215

 manifold H3 , h, x∞ . By the definition of convergence and as a consequence of the structure of hyperbolic ends, this implies the following: ˜ there Elementary properties of hyperbolic limits. For any A ∈ (0, A], exists iA ∈ N such that for each i ≥ iA : (1) There exists a compact submanifold M3i,A ⊂ M with tori boundary such −1

that (Mi,A , g (t)) is A-close in the C A  -topology to the truncated hyperbolic manifold (HA , h|HA ) for t ∈ [−A−1 + ti , A−1 + ti ].1 We call Mi,A an almost hyperbolic piece on this time interval. (2) There exist disjoint compact submanifolds Ui,A,E ⊂ M, for each E ∈ −1 E (K), such that (Ui,A,E , g(t)) is A-close in the C A  -topology to ([A, A−1 ] E ) for t ∈ [−A−1 + ti , A−1 + ti ]. Note that for A small, × TE , dr 2 + e−2r gflat Mi,A overlaps with each Ui,A,E . We call Ui,A,E a truncated almost cusp end. However, this does not a priori mean that a piece of (M, g (t)), such as Mi,A , needs to remain geometrically close to (HA , h|HA ) for all sufficiently large times t even if we allow the piece to change continuously with time. The reason for this is that we are dealing with sequential limits. In particular, a priori, it is conceivable that a piece of (M, g (t)) can be close to hyperbolic for a long time, then move away from being close to hyperbolic (perhaps also for a long time), and return to being close to hyperbolic for a long time, then move away again, etc.2 If this is the case, then we informally say that the almost hyperbolic piece jumps around. Fortunately, as we shall see from Proposition 33.5 below, this does not happen. The following notion, when combined with the harmonic map condition, shall provide us with a canonical way to parametrize almost hyperbolic pieces in nonsingular solutions.  Definition 33.1 (CMC boundary conditions). Let H3 , h be a finite-volume hyperbolic 3-manifold. An embedding F : (HA , h|HA ) → M3 , g , where A ∈ ˜ is said to satisfy the CMC boundary conditions if (0, A], (1) F (∂HA ) ⊂ M is a CMC hypersurface, (2) the area of each component of F (∂HA ) is equal to A, (3) F∗ (N ) is normal to F (∂HA ) with respect to g, where N is the unit outward normal vector to ∂HA with respect to h. 1.2. Stability of noncompact hyperbolic limits—definitions and statements. Since we wish to rule out the possibility of an almost hyperbolic piece jumping around as described in the previous subsection, we make the following definition. √ Let A˜ ∈ (0, 3/4] be as in (h5) in Subsection 1.1 of this chapter.

  1 That is, there exist diffeomorphisms F i,A (t) : (HA , h|HA ) → Mi,A , g (t) such that ≤ A for t ∈ [−A−1 + ti , A−1 + ti ], where · denotes the Fi,A (t)∗ (g (t)) − h|HA A−1

C

(HA )

floor function. 2 Compare this with the nonuniqueness of tangent cones or the nonuniqueness of asymptotic cones (see for example pp. 315–316 of [77]).

216

33. NONCOMPACT HYPERBOLIC LIMITS

 Definition 33.2 (Stable hyperbolic limit). Let M3 , g (t) be a nonsingular solution satisfying Condition H. We say that a (connected, complete, and noncompact) finite-volume hyperbolic limit (H3 , h) is a stable hyperbolic limit of (M, g (t)) as t → ∞ provided the following conditions hold. There exist ˜ with (1) T0 < ∞ and a smooth nonincreasing function A : [T0 , ∞) → (0, A] limt→∞ A(t) = 0, (2) a smooth 1-parameter family of compact submanifolds M3A(t) (t) ⊂ M with tori boundary defined for t ∈ [T0 , ∞), (3) a smooth 1-parameter family of harmonic diffeomorphisms  F (t) : (HA(t) , h|HA(t) ) → MA(t) (t) , g (t) , defined for t ∈ [T0 , ∞) and satisfying the CMC boundary conditions, with the property that for each m ∈ N " " " " ∗ (33.3) lim "F (t) g (t) − h|HA(t) " m = 0. t→∞ C (HA(t) ,h)  We shall call MA(t) (t) , g (t) , t ∈ [T0 , ∞), a stable asymptotically hyperbolic submanifold in (M, g (t)) corresponding to (H, h). Remark 33.3 (Regarding the definition of stable hyperbolic limit). (1) The 1-parameter family of submanifolds MA(t) (t) being smooth is the same as ∂MA(t) (t) depending smoothly on t and MA(t) (t) staying on the same side of ∂MA(t) (t).  (2) As t → ∞ the 1-parameter family MA(t) (t) , g (t) of Riemannian man∞ ifolds converges √ to (H, h) in the C pointed Cheeger–Gromov sense. (3) For A ∈ (0, 3/4] and for t sufficiently large with A (t) ≤ A, we have that F (t) (∂HA ) is a disjoint union of embedded concave tori in M and tends to totally umbillic as t → ∞, all with respect to g (t). (4) Even if we fix the function A(t), a corresponding stable asymptotically hyperbolic submanifold may not be unique. For example, we could imagine the existence of a Z2 -symmetric nonsingular solution to NRF obtained by smoothly doubling a truncated hyperbolic manifold, which may have two disjoint isometric stable asymptotically hyperbolic submanifolds. Can one prove that such an example is possible? A useful notion to start with, which is a priori weaker than that of stable hyperbolic limit, is the following.  Definition 33.4 (Immortal almost hyperbolic piece). Let M3 , g (t) be a √ nonsingular solution satisfying Condition H. Fix A ∈ (0, 3/4] and k ∈ N. We say that a family of compact 3-dimensional submanifolds MA (t), defined for t ∈ [T, ∞) for some T ≥ 0, of (M, g (t)) is an immortal (A,k)-almost hyperbolic piece if there exists a finite-volume hyperbolic manifold H3 , h and there exist harmonic diffeomorphisms (33.4)

F (t) : (HA , h|HA ) → (MA (t) , g (t))

satisfying the following properties: (1) The maps F (t) depend smoothly on t ∈ [T, ∞) and satisfy the CMC boundary conditions. Hence each connected component of ∂MA (t) depends smoothly on t and has area equal to A.

1. MAIN RESULTS ON HYPERBOLIC PIECES

(2) For each t ∈ [T, ∞), " " "F (t)∗ g (t) − h| " k HA C (H

A ,h)

≤

217

1 . k

Here, the immortal almost hyperbolic piece MA (t) remains close—by a fixed distance in C k —to being hyperbolic for all time, whereas in the definition of stability the piece MA(t) (t) asymptotically limits to a hyperbolic manifold. Note that the diffeomorphism types of MA(t) (t) and MA (t) are both independent of t. In Definition 33.4, we say that the immortal (A, k)-almost hyperbolic piece corresponds to the hyperbolic 3-manifold (H, h). By the Mostow rigidity theorem (see Corollary 31.50), the corresponding hyperbolic manifold (H, h) is unique. The preliminary version of the result that almost hyperbolic pieces cannot jump around in M is the following. 33.5 (Immortality of almost hyperbolic pieces). Suppose that  3Proposition M , g (t) is a nonsingular solution satisfying Condition H. If {(xi , ti )} is a sequence of points and times with ti → ∞ such that the sequence (M,  gi (t) , xi ), where gi (t) = g (t + ti ), converges to a finite-volume hyperbolic limit H3 , h, x∞ ˜ and for any k,3 there exist T < ∞ and an imas i → ∞, then for any A ∈ (0, A] mortal (A, k)-almost hyperbolic piece MA (t) defined for t ∈ [T, ∞) corresponding to (H, h). Using that the parameters (A, k) may be chosen to be arbitrarily good, we shall show that any immortal (A0 , k0 )-almost hyperbolic piece MA0 (t) corresponding to (H, h), with (A0 , k0 ) sufficiently good, must be eventually contained in a stable asymptotically hyperbolic submanifold MA(t) (t) corresponding to (H, h). This improved version of Proposition 33.5 is summarized in the first paragraph of §12 of [143], where Hamilton wrote: For large t the metric is as close as we like to hyperbolic; not just on HB but as far beyond as we like.  Proposition 33.6 (Stability of hyperbolic limits). Let M3 , g (t) be a nonsingular solution satisfying Condition H. If {(xi , ti )} is a sequence of points and times with ti → ∞ such that the sequence {(M, gi (t) , xi )}, where gi (t) = g (t + ti ), converges to a finite-volume hyperbolic limit H3 , h, x∞ as i → ∞, then (H, h) is a stable hyperbolic limit of (M, g (t)). The following notion, which lies between those of stable hyperbolic limit and immortal (A, k)-almost hyperbolic piece, shall be useful when we consider the incompressibility of tori in §4 of this chapter.  Definition 33.7. Let M3 , g (t) be a nonsingular solution satisfying Con√ dition H. Given A ∈ (0, 3/4], we say that a smooth family of submanifolds M3A (t) ⊂ M, t ∈ [tA , ∞), where tA < ∞, is an immortal asymptotically  hyperbolic piece corresponding to a hyperbolic limit H3 , h if (1) ∂MA (t) is comprised of CMC tori with area A, (2) (MA (t) , g(t)) converges to (HA , h) in C ∞ as t → ∞. 3 Here, A ˜ is defined to satisfy both (h5) in Subsection 1.1 of this chapter and the condition in the paragraph containing (33.18) below.

218

33. NONCOMPACT HYPERBOLIC LIMITS

In particular, int(MA (t)) is diffeomorphic to H and for any k ∈ N there exists tA,k ∈ [tA , ∞) such that MA (t) is an immortal (A, k)-almost hyperbolic piece for t ≥ tA,k . In §3 of this chapter we shall first prove the special cases of Propositions 33.5 and 33.6 where the limits H3 , h have the minimal number of cusp ends among all limits. We then prove both of these propositions by induction on the number of cusp ends and concurrently with the following result. This additional result says that M can be decomposed along tori into immortal almost hyperbolic pieces and collapsed pieces (where the injectivity radius is everywhere small). 33.8 (Decomposition into hyperbolic and collapsed pieces). Let  3Proposition M , g (t) be a nonsingular solution satisfying Condition H. Then there exists a fi m  3 , hα α=1 nite collection of complete hyperbolic 3-manifolds with finite volume Hα such that√for any ε > 0 sufficiently small, there exist a time Tε < ∞, a number Aε ∈ (0, 3/4] with corresponding truncations (Hα )Aε of Hα , and harmonic embeddings Fα (t) : ((Hα )Aε , hα ) → (M, g (t)) ,

t ∈ [Tε , ∞),

with the following properties. The maps Fα (t) depend smoothly on t and satisfy both the CMC boundary conditions and Fα (t)∗ g(t) − hα C 1/ε ((Hα )A

ε

,hα )

< ε.

Here the submanifolds Fα (t)((H α )Aε ) are mutually disjoint in M and the maximum % injectivity radius of M − α Fα (t) ((Hα )Aε ) with respect to g(t) is less than ε for all t ∈ [Tε , ∞). 1.3. Hyperbolic pieces have incompressible boundary. A main use of the continuous dependence on time of the decomposition in Proposition 33.8 is to help prove that boundary tori of immortal almost hyperbolic pieces in 3-dimensional nonsingular solutions are incompressible in the ambient 3-manifold (see §4 of this chapter for a proof).  Proposition 33.9 (Incompressibility of boundary tori). Let M3 , g (t) be a nonsingular solution satisfying Condition H. For any t (sufficiently large) for which an immortal asymptotically hyperbolic piece M3A (t) ⊂ M is defined, its boundary ∂MA (t) is incompressible in M. That is, the inclusion map ι : ∂MA (t) → M induces an injection ι∗ : π1 (∂MA (t)) → π1 (M) on fundamental groups. By combining all three of Propositions 32.12 (on the existence of hyperbolic limits), 33.8, and 33.9 for Case III and Theorem 31.55, we conclude that M may be decomposed along incompressible tori into graph manifolds and hyperbolic pieces, which implies that M admits a geometric decomposition in the sense of Thurston. This completes the proof of Hamilton’s Nonsingular Solutions Theorem 32.2 modulo the proofs of Propositions 33.11 and 33.12 below to be given in the next chapter. Both of these propositions shall be used in the study of harmonic parametrizations of almost hyperbolic pieces in §2 below.

2. HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES

219

1.4. Results on CMC surfaces and harmonic maps. When we have n-dimensional manifolds with boundary, whose ends (collars of the boundaries) are geometrically close to hyperbolic cusps, we shall apply the implicit function theorem (IFT) to prove the existence of constant mean curvature tori in these ends. In particular, we shall show that a metric on a manifold [a, b] × V n−1 , which is close to a finite-volume hyperbolic cusp metric, may be “swept out” by constant mean curvature hypersurfaces which are close to the slices {r} × V. Definition 33.10. Given a region U ⊂ M we say that a family % of hypersurfaces {Ss }s∈I in M, where I is an interval, sweeps out U if U ⊂ s∈I Ss . The following result is on the existence of a CMC sweep-out in almost hyperbolic cusps; the result will be proved as Proposition 34.1. Proposition 33.11 (Existence of a CMC sweep-out in almost hyperbolic cusps). Given any [a, b] ⊂ R and ε0 > 0, if a C ∞ Riemannian metric g on [a, b] × V n−1 is sufficiently close in the C 2,α -topology to a hyperbolic cusp metric gcusp = dr 2 + e−2r gflat for some α ∈ (0, 1), then there exists a smooth 1-parameter family of C ∞ CMC (with respect to g) hypersurfaces which sweep out (a + ε0 , b − ε0 ) × V and which are close in the C 2,α -norm to the standard slices {r} × V (see Definition K.9 in Appendix K). The following result is on the existence of harmonic maps near the identity; for k = 2 the result will be proved by the IFT as Proposition 34.13. We say that a map F : (M, ∂M, g) → (M, ∂M, g˜) satisfies the normal boundary condition if F∗ (N ) is normal to ∂M with respect to g˜, where N is the unit outward normal vector to ∂M with respect to g. Proposition 33.12 (Existence of harmonic maps near the identity). Suppose that (Mn , g) is a compact C ∞ Riemannian manifold with negative Ricci curvature and concave boundary ∂M; i.e., II (∂M) ≤ 0. For any C ∞ metric g˜ sufficiently close to g in the C 2,α -topology with α ∈ (0, 1), there exists a unique C ∞ harmonic diffeomorphism F : (M, g) → (M, g˜) satisfying the normal boundary condition and C 2,α -close to the identity map. Furthermore, if we replace g˜ by a family g˜(t) depending smoothly on t, then the corresponding harmonic diffeomorphisms F (t) depend smoothly on t. 2. Harmonic maps parametrizing almost hyperbolic pieces In this section we apply Propositions 33.11 and 33.12 to prove the existence of harmonic parametrizations of almost hyperbolic pieces at a given (large) time and we then show that we can continue them forward in time in the nonsingular solution as long as they stay near isometries. This last result will be used in the next section to prove the stability of hyperbolic limits. 2.1. Existence of harmonic embeddings. If we have a complete noncompact finite-volume hyperbolic limit H3 of a sequence of Riemannian manifolds M3i , then we can parametrize the corresponding almost hyperbolic pieces in Mi by almost isometric harmonic embeddings satisfying the CMC boundary conditions in Definition 33.1.

220

33. NONCOMPACT HYPERBOLIC LIMITS

Proposition 33.13 of harmonic parametrizations of almost hy (Existence perbolic pieces). Let M3i , gi , xi i∈N be a sequence of complete Riemannian 3Cheeger–Gromov manifolds converging in the C ∞ pointed √sense to a noncompact  finite-volume hyperbolic manifold H3 , h, x∞ . If A ∈ (0, 3/8] is such that x∞ ∈ int (HA ), then for each i sufficiently large there exists a harmonic embedding Fi : (HA , h|HA ) → (Mi , gi ) satisfying the CMC boundary conditions. Moreover, we have " " lim "Fi∗ gi − h|HA "C k (H ,h) = 0 i→∞

A

for all k ∈ N and we have limi→∞ dgi (Fi (x∞ ) , xi ) = 0. The idea of the proof is simply the following. By the definition of Cheeger– Gromov convergence, there exist embeddings Φi : (HA/2 , h|HA/2 ) → (Mi , gi ) , " " " " which approach isometries in the sense that "Φ∗i gi − h|HA/2 "

→ 0 for each √ k as i → ∞. For each topological end E of H and each A ∈ (0, 3/8] there exists E a CMC torus Ti,A ⊂ int(HA/2 ), with respect to Φ∗i gi and with area equal to A. The almost hyperbolic piece (Hi,A , Φ∗i gi ), defined to be the submanifold bounded E , limits to (HA , h|HA ). by the union of the Ti,A Essentially by Proposition 33.12 we have that for i large enough there exists a harmonic diffeomorphism Gi : (HA , h|HA ) → (Hi,A , Φ∗i gi ) such that Φi ◦ Gi : (HA , h|HA ) → (Mi , gi ) satisfies the CMC boundary conditions. These embeddings Φi ◦ Gi limit to isometries as i → ∞. C k (HA/2 )

∞ Proof. As a consequence √ of the definition of C pointed Cheeger–Gromov convergence, for each A ∈ (0, 3/8] there exists a sequence of embeddings

Φi : (HA/2 , h|HA/2 ) → (Mi , gi ) , defined for i large enough, such that Φi (x∞ ) = xi and for all k ∈ N, " " " ∗ " → 0. "Φi gi − h|HA/2 " k C (HA/2 )

By Proposition 33.11, for i sufficiently large, there exists a smooth 1-parameter % family { E∈E(K) Ts (E, i, A)}s∈I of unions of CMC tori in HA/2 , with respect to Φ∗i gi . Here, the CMC tori are close to the standard tori slices sweeping out H2A/3 − H3A/2 and the subset K ⊂ H is a suitable end-complementary compact set. The areas of the standard CMC slices in each component of H2A/3 −H3A/2 , with respect to h, take on all values in [2A/3, 3A/2] and the areas of the CMC tori Ts (E, i, A) in the sweep-out depend continuously on s. Hence, for i large enough and each E in the sweep-out which is CMC and which has E ∈ E (K), there exists a torus Ti,A ∗ area A, both with respect to Φi gi . Let Hi,A denote the unique compact 3-dimensional submanifold of H with CMC boundary  E Ti,A . ∂Hi,A = E∈E(K)

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221

Since Φ∗i gi converges to h|HA/2 in C ∞ , we have that  E Ti,A → ∂HA , lim i→∞

E∈E(K)

where this convergence is also in C ∞ (compare with (34.70) in the next chapter). Hence there exist diffeomorphisms Ψi : (HA , h|HA ) → (Hi,A , Φ∗i gi ) which are close to isometries in the sense that for all k ∈ N ∪ {0} ,4 " ∗ ∗ " "Ψi (Φi gi ) − h| " k (33.5) HA C (H ,h) → 0. A

Φ∗i gi ,

Since ∂Hi,A is CMC with respect to we have that ∂HA is also CMC with respect to Ψ∗i Φ∗i gi . Since ∂HA is totally umbillic and strictly concave in (HA , h|HA ), by (33.5) and Proposition 33.12, for i sufficiently large there exists a unique harmonic diffeomorphism Gi : (HA , h|HA ) → (HA , Ψ∗i Φ∗i gi ) close to the identity map satisfying Gi (∂HA ) = ∂HA and (Gi )∗ (N ) is normal to ∂HA with respect to Ψ∗i Φ∗i gi . Here N is the unit outward normal vector to ∂HA with respect to h. We then have that Fi  Φi ◦ Ψi ◦ Gi : (HA , h|HA ) → (Mi , gi ) is a harmonic embedding satisfying boundary conditions. Note that % the CMC E ). Fi (∂HA ) = (Φi ◦ Ψi ) (∂HA ) = Φi ( E∈E(K) Ti,A By (33.5) and since the harmonic diffeomorphism Gi is obtained by the implicit function theorem, we have for any k ∈ N, " ∗ ∗ ∗ " ∗ " " " "Fi gi − h| " k " HA C (H ,h) = Gi Ψi Φi gi − h|HA C k (H ,h) → 0 A

A

as i → ∞. Finally, since Φi (x∞ ) = xi , we have that dgi (Fi (x∞ ) , xi ) = dgi (Φi ((Ψi ◦ Gi ) (x∞ )) , Φi (x∞ )) → 0 as i → ∞ since Ψi and Gi both tend to the identity. This completes the proof of Proposition 33.13.  2.2. Continuing harmonic parametrizations of almost hyperbolic pieces. The result of this subsection states that if we are given a family of metrics g (t), t ∈ [α, ω], on a 3-manifold M3 (such as a solution to the NRF) and an almost isometric harmonic embedding Fα into (M, g (α)) from a truncated hyperbolic 3manifold satisfying the CMC boundary conditions, then we may extend Fα to a 1-parameter family of almost isometric harmonic embeddings F (t) into (M, g (t)) for t ≥ α satisfying the CMC boundary conditions and defined so long as F (t) remains sufficiently close to an isometry. 4 To accomplish this, we only need to perturb Ψ away from the identity in a collar of ∂H i A in HA . Hint: Write ∂Hi,A as a graph over ∂HA .

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33. NONCOMPACT HYPERBOLIC LIMITS

Proposition 33.14 (Continuing almost isometric harmonic parametrizations of hyperbolic pieces). Suppose that √ (H3 , h) is a noncompact finite-volume hyperbolic 3-manifold. Then for any A ∈ (0, 3/8] there exists an integer k0 ≥ 4 depending only on (H, h) and A such that the following are true: If g (t), t ∈ [α, ω], is a smooth 1-parameter family of metrics on a closed 3manifold M3 and if for some k ≥ k0 we have that Fα : (HA , h|HA ) → (M, g (α)) ∞

harmonic embedding satisfying the CMC boundary conditions with " ∗ " 1 "F g (α) − h| " k (33.6) , α HA C (HA ,h) ≤ k then there exists β ∈ [α, ω] such that [α, β] is the maximum interval on which there exists a unique smooth 1-parameter family of C ∞ harmonic embeddings is a C

(33.7)

F (t) : (HA , h|HA ) → (M, g (t))

satisfying the CMC boundary conditions with F (α) = Fα and " " 1 "F (t)∗ g (t) − h| " k (33.8) HA C (HA ,h) ≤ k for all t ∈ [α, β]. Furthermore, either β = ω or " " 1 "F (β)∗ g (β) − h| " k . (33.9) HA C (HA ,h) = k Proof. We use a version of the continuity method to prove the existence of F (t). (1) “Openness”. We shall show √ Claim 1. Given (H3 , h) and A ∈ (0, 3/8], there exists k0 ∈ N with the following property. Suppose that t0 ∈ [α, ω] and k ≥ k0 − 1 are such that there exists a harmonic embedding Ft0 : (HA , h|HA ) → (M, g (t0 )) satisfying the CMC boundary conditions with " ∗ " "Ft g (t0 ) − h| " k (33.10) HA C (H 0

2 . k Then for t sufficiently close to t0 there exists a unique C ∞ harmonic embedding A ,h)


0 with the following property. There exists an embedding Ft0 ,ε0 : (HA−ε0 , h|HA−ε ) → (M, g (t0 )) which extends the map Ft0 and 0 which satisfies " " 2 " ∗ " < . "Ft0 ,ε0 g (t0 ) − h|HA−ε0 " k k C (HA−ε0 ,h) Note that Ft0 ,ε0 (HA−ε0 ) contains some ε1 -neighborhood of MA (t0 ), where ε1 > 0. Now, by Proposition 33.11 and provided k0 is chosen large enough, there exists ε2 > 0 such that for each A ∈ (A − ε2 , A + ε2 ) and for each end of H there exists a unique CMC torus in Ft0 ,ε0 (HA−ε0 ) with area A , where the CMC torus with area A is a component of Ft0 (∂HA ) = ∂MA (t0 ). Statement A now follows from Proposition 33.11 by taking t sufficiently close to t0 . Proof of Claim 1. Let K (t) ⊂ M be the unique smooth 1-parameter family of compact 3-dimensional submanifolds with ∂K (t) = Tt (where Tt is constructed in Statement A) and with K (t) close to Ft0 (HA ). We now can apply Proposition 33.12 (more precisely, the proof of Proposition 33.13), again assuming k0 is chosen large enough, to conclude that, for t sufficiently close to t0 , there exists a C ∞ harmonic diffeomorphism F (t) : (HA , h|HA ) → (K (t) , g (t)|K(t) ) ⊂ (M, g (t)) , which is close enough to Ft0 so that (33.11) holds, while also satisfying the conditions that F (t) (∂HA ) = Tt = ∂K (t) and that F (t)∗ (N ) is normal to Tt with respect to g (t). The diffeomorphisms F (t) depend smoothly on t. Since each component of Tt is a CMC torus with respect to g (t) and with area equal to A, we have that F (t) satisfies the CMC boundary conditions. By the smooth dependence of F (t) on t, this completes the proof of Claim 1. (2) “Closedness”. Let k ≥ k0 , where k0 is as in part (1). Suppose that γ ∈ (α, ω] is such that we have a smooth 1-parameter family of C ∞ harmonic embeddings F (t) : (HA , h|HA ) → (M, g (t)) , defined for t ∈ [α, γ), which start at Fα , which satisfy the CMC boundary conditions, and which also satisfy the condition (33.8); i.e., (33.12) for all t ∈ [α, γ).

" " "F (t)∗ g (t) − h| " k HA C (H

A ,h)

≤

1 k

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33. NONCOMPACT HYPERBOLIC LIMITS

The following convergence, which proves closedness, is essentially a consequence of the proof of Corollary 4.11 in Part I.6 Claim 2. There exists a sequence ti → γ such that F (ti ) converges in C ∞ to Fγ  lim : (HA , h|HA ) → (M, g(γ)), i→∞

∞

where Fγ is a C harmonic embedding satisfying the CMC boundary conditions and (33.8) with respect to g (γ). ∗ Proof of Claim 2. Since, by (33.12), supHA |F (t) g (t) − h|HA |h ≤ k1 , we have in HA that   1 − k−1 h ≤ F (t)∗ g (t) ≤ 1 + k−1 h for all t ∈ [α, γ). This implies that if |X|h = 1, then (33.13)

1 − k−1 ≤ |F (t)∗ X|g(t) ≤ 1 + k−1 . 2

Since the metrics g (t) are uniformly equivalent, there exists C < ∞ such that   (33.14) C −1 1 − k−1 ≤ |F (t)∗ X|2g(α) ≤ C 1 + k−1 for all t ∈ [α, γ). In particular, both |dF (t)|g(α),h and | (dF (t))−1 |g(α),h are bounded. By inequality (33.14) and using the assumption that M is compact,7 we have that the family of maps {F (t)}t∈[α,γ) is uniformly equicontinuous. By the Arzela–Ascoli theorem, there then exists a sequence ti → γ such that F (ti ) converges in C 0 to a continuous map Fγ : HA → M. We now consider the higher covariant derivatives of F (t) with respect to h and g (t). Suppressing the dependence on t in our notation, we compute that ∇h (F ∗ g) = ∇h (g (dF ⊗ dF )) (33.15)

= g(∇g,h dF ⊗ dF ) + g(dF ⊗ ∇g,h dF ),

where the covariant derivative ∇g,h is defined analogously to (K.12) in Appendix  K. In local coordinates xi on the domain and {y α } on the range, this says   ∂F γ ∂F β g,h γ (∇ dF ) (∇h (F ∗ g))ijk = gβγ (∇g,h dF )βij k + ik . ∂x ∂xj From this equation we may express ∇g,h dF as a polynomial function of ∇h (F ∗ g), g −1 , and (dF )−1 , which all have bounded C 0 -norms. Since (33.12) implies that  ∗ |(∇h )j F (t) g (t) |h ≤ Cj is bounded for 0 ≤ j ≤ k, by induction we can bound |(∇g(t),h )j dF (t) |g(α),h ≤ Cj 6 In Subsection 2.2 on “Compactness of maps” in Chapter 4 of Part I, the Arzela–Ascoli theorem applied to maps is discussed. There, the proof assumes that we have an isometry f from (U, g) to (V, h) and then shows how to bound all the derivatives of the map. Suppose that f is an ε-approximate isometry instead of an isometry. Then f is still an isometry from f ∗ h to h and the difference between f ∗ h and g is ε. Thus we may do the same computations as in equation (4.4) of Subsection 2.2 in Chapter 4 of Part I but then add in the differences between f ∗ h and g (and also the derivatives for higher terms) to get a bound. Here, we give an equivalent proof. 7 We do not actually need to assume that M is compact since the assumption that the maps F (t) are harmonic embeddings satisfying the CMC boundary conditions implies that they are uniformly bounded; i.e., there exists a compact set K ⊂ M such that F (t) (HA ) ⊂ K.

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225

for 0 ≤ j ≤ k. To see this, note that analogously to (33.15) we obtain formulas of the form   (∇h )j (F ∗ g) = g (∇g,h )j dF ⊗ dF + g dF ⊗ (∇g,h )j dF + J, where J is a polynomial expression in g and at most j − 1 covariant derivatives of dF . By the above, we conclude that there exists a subsequence such that F (ti ) converges to Fγ in C k . Since the maps F (ti ) : (HA , h|HA ) → (M, g (ti )) are harmonic, we conclude that the limit map Fγ : (HA , h|HA ) → (M, g (γ)) is harmonic. We next show that Fγ is an embedding. By (33.14) and by the convergence of F (ti ) to Fγ , we have  2 C −1 |X|2h ≤ (Fγ )∗ X g(α) ≤ C |X|2h for some C < ∞. In particular, Fγ is an immersion. On the other hand, the maps F (ti ), which limit to Fγ , are embeddings (in particular, the F (ti ) are injective) for all i. Hence, the only possibility for Fγ not to be injective is that Fγ (∂HA ) has a self-intersection (clearly Fγ |int(HA ) is injective). However, this is impossible since Fγ (∂HA ) is strictly concave (with respect to the outward normal to Fγ (HA )).8 We conclude that Fγ is an embedding. From the convergence in C k we have that Fγ satisfies the CMC boundary conditions. We now show that (33.8) holds for t = γ. Since F (ti ) converges to Fγ in C k , we have that " ∗ " " " ∗ "Fγ g (γ) − h| " k−1 = lim "F (ti ) g (ti ) − h| " k−1 , HA C

(HA ,h)

i→∞

HA C

(HA ,h)

which is one less derivative than we need. On the other hand, we claim that we actually have C ∞ convergence of F (ti ) to Fγ , so that by (33.12) we have " " " " 1 ∗ (33.16) "Fγ∗ g (γ) − h|HA "C k (H ,h) = lim "F (ti ) g (ti ) − h|HA "C k (H ,h) ≤ . A A i→∞ k To see that the convergence of F (ti ) to Fγ is in C ∞ , first note that by standard regularity theory, Fγ is actually a C ∞ harmonic map because k ≥ k0 ≥ 4. Since " ∗ " 1 1 "Fγ g (γ) − h| " k−1 ≤ < , HA C (HA ,h) k k−1 where k ≥ k0 , there exists a smooth family of C ∞ harmonic embeddings Fˆ (t) : (HA , h| ) → (M, g (t)) HA

satisfying the CMC boundary conditions for t ∈ (γ − ε, γ + ε) for some ε > 0 close to isometries such that Fˆ (γ) = Fγ . By the uniqueness of these harmonic embeddings, we have Fˆ (t) = F (t) for all t ∈ (γ − ε, γ]. Hence F (ti ) = Fˆ (ti ) converges to Fˆ (γ) = Fγ in C ∞ as claimed. This completes the proof of Claim 2. Finally, to see the uniqueness of F (t) in (33.7), suppose that we have two families of harmonic embeddings Fi (t) : (HA , h|HA ) → (M, g (t)), t ∈ [α, ti ], where t1 < t2 , both starting at Fα and satisfying the CMC boundary conditions and 8 Note that, at a point of self-intersection of F (∂H ), if it were to exist, the two unit outward γ A normals are pointing in opposite directions.

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33. NONCOMPACT HYPERBOLIC LIMITS

(33.11). Then, by the fact that F2 (t) (∂HA ) = F1 (t) (∂HA ) from Statement A and by the uniqueness part of Proposition 33.12, we have that F2 (t) = F1 (t) for t ∈ [α, t1 ]. Proposition 33.14 is now proved.  3. Proof of the stability of hyperbolic limits In this section we discuss the proof of Proposition 33.6 on the stability of (finite-volume hyperbolic) asymptotic limits of 3-dimensional nonsingular solutions satisfying Condition H. The difficulty is that we only have sequential convergence of the corresponding time-translated pointed solutions. The facts which we wish to exploit to overcome the aforementioned difficulty are (1) in the “negative” Case III all noncollapsed asymptotic limits are hyperbolic and (2) the Mostow rigidity theorem (see the discussion surrounding (33.33) below). Throughout this section  3 M , g (t) shall denote a nonsingular solution satisfying Condition H. We shall prove Proposition 33.6 by induction on the number of cusp ends of the asymptotic limits. In view of this, we start with the following. Definition 33.15 (Space of hyperbolic limits). Let Hyp(M3 , g (t)) denote the set of all complete hyperbolic 3-manifolds (H3 , h) which occur as the pointed limit of the solutions {(M, g (t + ti ) , xi )} for some sequence {(xi , ti )} with ti → ∞. We call (xi , ti ) a sequence corresponding to the limit (H, h). Note that since Vol (g (t)) is constant, if (H, h) ∈ Hyp(M, g (t)), then by the Cheeger–Gromov convergence the limit has finite volume: (33.17)

Vol (h) ≤ Vol (g (0)) < ∞.

Hence, by Theorem 31.46, the number of cusp ends of each (H, h) is bounded above by Vol (g (0)). Now clearly there is an asymptotic limit (H, h) ∈ Hyp(M, g (t)) such that the number of cusp ends of (H, h) is minimal among all elements of Hyp(M, g (t)). In the next subsection we prove Propositions 33.5 and 33.6 for such an (H, h). Then in the following subsection we prove the general case by induction on the number of cusp ends. 3.1. Stability of hyperbolic limits with a minimal number of cusp ends. Let (H3 , h) ∈ Hyp(M3 , g (t)) be a hyperbolic limit. Recall that we have the following properties: (1) There exist a sequence {(xi , ti )}i∈N with ti → ∞ and x∞ ∈ H such that the solutions {(M, g (t + ti ) , xi ) : t ∈ [0, ∞)}i∈N converge to the static eternal solution (H, h, x∞ ) of the NRF in the C ∞ pointed Cheeger–Gromov sense. (2) Let A˜ be as in (h5) in Subsection 1.1 of this chapter and suppose that ˜ Then the truncation HA in (33.1) is well defined. A ∈ (0, A]. ˜ there exists  = (H, A) ∈ N with (3) By Proposition 33.12, given A ∈ (0, A], the property that if g˜ is a metric on HA which is close to h in the sense that ||˜ g − h|HA ||C  (HA ,h) ≤ 1 , then there exists a unique harmonic diffeomorphism F : (HA , h|HA ) → (HA , g˜) close to the identity map such that F (∂HA ) = ∂HA and F∗ (N ) is normal to ∂HA with respect to g˜, where N is the unit outward normal vector to ∂HA with respect to h.

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS

227

By property (1), the sequence of pointed closed Riemannian 3-manifolds {(M, g (ti ) , xi )}i∈N

(33.18) ∞

converges in the C pointed Cheeger–Gromov sense to (H, h, x∞ ). Assume that ˜ and recall that x∞ ∈ int(H ˜ ). By Proposition 33.12, for i sufficiently A ∈ (0, A] A large, there exist harmonic embeddings Fi : (HA , h|HA ) → (M, g (ti )) ,

(33.19)

where Fi (∂HA ) = ∂(Fi (HA )) is a disjoint union of CMC tori, each with area equal to A, and where (Fi )∗ (N ) is normal to ∂(Fi (HA )), all with respect to g (ti ). Furthermore, " " (33.20) lim "Fi∗ g (ti ) − h| " k =0 HA C (HA ,h)

i→∞

for each k ∈ N ∪ {0} and lim dg(ti ) (Fi (x∞ ) , xi ) = 0.

i→∞

˜ and k ∈ N, there exists i0 = i0 (A, k) such that for In particular, given A ∈ (0, A] any i ≥ i0 we have " " ∗ 1 "Fi g (ti ) − h| " k ; HA C (HA ,h) ≤ k that is, (Fi (HA ), g (ti )) is an (A, k)-almost hyperbolic piece at time ti . We first prove that given a hyperbolic limit with a minimal number of cusp ends and given (A, k), for each i sufficiently large the (A, k)-almost hyperbolic piece (Fi (HA ), g (ti )) can be smoothly continued to an immortal almost hyperbolic piece. This is a special case of Proposition 33.5 and the use of Mostow rigidity is key to its proof. Proposition 33.16 (Almost hyperbolic pieces with minimal cusp ends are immortal). Suppose (H3 , h) ∈ Hyp(M, g (t)) is a hyperbolic limit with a minimal ˜ where A˜ is as in (h5), and let k ∈ N. Then number of cusp ends. Let A ∈ (0, A], there exist ¯ı = ¯ı (A, k) ∈ N, submanifolds M3A,k (t) ⊂ M, and a smooth 1-parameter family of harmonic diffeomorphisms (33.21a) (33.21b)

F¯ı (t) : (HA , h|HA ) → (MA,k (t) , g (t)) , MA,k (t¯ı ) = F¯ı (HA ) ,

F¯ı (t¯ı ) = F¯ı ,

defined for t ∈ [t¯ı , ∞), where F¯ı is defined by (33.19) and where (i) ∂MA,k (t) = F¯ı (t) (∂HA ) is a disjoint union of CMC tori, each with area equal to A, and F¯ı (t)∗ (N ) is normal to ∂MA,k (t), all with respect to g (t), and (ii) " " 1 "F¯ı (t)∗ g (t) − h| " k HA C (HA ,h) ≤ k for all t ∈ [t¯ı , ∞). Hence MA,k (t), t ∈ [t¯ı , ∞), is an immortal (A, k)-almost hyperbolic piece. ˜ let  be given by property (3) above and let k () ∈ N Proof. Given A ∈ (0, A], be as in Theorem 34.22 in the next chapter. Then define k˜0  max{k () , k0 }, where k0 is as in Proposition 33.14.9 9 The

˜0 depends only on H and A. natural number k

228

33. NONCOMPACT HYPERBOLIC LIMITS

˜ and some k. Now suppose that the proposition is false for some A ∈ (0, A] ˜ Without loss of generality we may assume that k ≥ k0 . By Proposition 33.14, we then have that for each i ∈ N there exists a finite maximal time interval [ti , βi ] on which there exists a smooth 1-parameter family of harmonic embeddings Fi (t) with Fi (ti ) = Fi satisfying properties (i) and (ii) in the proposition, with t = βi being the first time at which " " 1 "Fi (t)∗ g (t) − h| " k . (33.22) HA C (HA ,h) = k Consider the sequence of pointed solutions of the NRF translated in time by βi : {(M, g (t + βi ) , Fi (βi ) (x∞ ))}i∈N . Choosing k˜0 even larger if necessary, it follows from (33.22) that (with x∞ ∈ int(HA )) inj g(βi ) (Fi (βi ) (x∞ )) is uniformly bounded from below by a positive constant. From the Cheeger– Gromov-type compactness theorem for pointed solutions of the NRF, i.e., Theorem 32.5, there exists a subsequence {(M, g (t + βi ) , Fi (βi ) (x∞ ))}i∈N converging to a limit solution  3 M∞ , g∞ (t) , y∞ , t ∈ (−∞, ∞) . Since we are in the negative Case III and βi → ∞, we may apply Proposition 32.12 to obtain that ˜ x ˜ 3 , h, (33.23) (M∞ , g∞ (t) , y∞ ) ≡ (H ˜∞ ) is a finite-volume hyperbolic 3-manifold. Since we have chosen (H, h) to have the ˜ has ˜ h) minimal number of cusp ends in the space Hyp(M, g (t)), we have that (H, at least as many cusp ends as (H, h). By the definition of C ∞ pointed Cheeger–Gromov convergence at t = 0, there ˜i }i∈N of H ˜ by open sets with x ˜i and there exists a exists an exhaustion {U ˜∞ ∈ U ˜ i → V i  Φi ( U ˜i ) ⊂ M with sequence of diffeomorphisms Φi : U (33.24)

Φi (˜ x∞ ) = Fi (βi ) (x∞ ) ˜ uniformly on compact sets ˜i , Φ∗ [ g (βi )| ])} converges in C ∞ to (H, ˜ h) such that {(U i Vi ˜ in H. Using this, we now construct a harmonic embedding from (HA , h|HA ) into ˜ ˜ (H, h). Consider the sequence of embeddings −1 ˜i (33.25) Φ−1 (Vi ) → U i ◦ Fi (βi )|Fi (βi )−1 (Vi ) : Fi (βi ) ∩ ∩ ˜ H. HA We shall show below that for i sufficiently large, (33.26)

Fi (βi ) (HA ) ⊂ Vi ,

so that (33.27) In particular,

Fi (βi )−1 (Vi ) = HA .

˜ Φ−1 i ◦ Fi (βi ) : HA → H is an embedding for i sufficiently large.

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS

229

To see (33.26), first note that by (33.24) we have (33.28)

˜∞ Φ−1 i ◦ Fi (βi ) (x∞ ) = x

for all i. Moreover, for i large enough " " ∗ " " −1 ˜ − h| −1 " Φi ◦ Fi (βi ) h Fi (βi ) (Vi ) " k C (Fi (βi )−1 (Vi ),h) " " ≤ "Fi (βi )∗ g (βi ) − h|HA "C k (H ,h) A " " " " ∗  −1 ∗ ˜ + "Fi (βi ) ( Φi h − g (βi ))" k C (Fi (βi )−1 (Vi ),h) " " ∗ 1 " ˜ − g (βi )" h ≤ + " Φ−1 " k i k C (Vi ∩(Fi (βi )(HA )),(Fi (βi )−1 )∗ h) 2 ≤ (33.29) k ˜ → (Vi , g (βi )| ) is as close to an isometry as we ˜i , h) by (33.22) and since Φi : (U Vi ˜ ˜ like in any C k -norm.10 Thus Φ−1 i ◦ Fi (βi ), as measured from h to h, is close to an isometry in its domain. Since HA is compact, it follows from (33.28) and (33.29) for our given k, that x∞ , C) Φ−1 i ◦ Fi (βi ) (HA ) ⊂ B (˜ ˜i }i∈N exhausts H, ˜ we have for some C < ∞ independent of i. Since the sequence {U ˜ ˜ that for i large enough, Φ−1 ◦ F (β ) (H ) ⊂ U ⊂ H; that is, i i A i i Fi (βi ) (HA ) ⊂ Vi ⊂ M, which is (33.26). Since Fi (βi )−1 (Vi ) = HA , we have that (33.29) says " " ∗ 2 " −1 ˜ − h| " (33.30) ≤ . " Φi ◦ Fi (βi ) h HA " k k C (HA ,h) Now consider the embeddings HA Fi (βi )

Φ−1 i ◦Fi (βi ) −−−−−−−−→

˜i ⊂ H ˜ U

 #Φi M.

By (33.30) and the “Arzela–Ascolitheorem for maps” (see Claim 2 in the proof of k−1 (HA , h) to Proposition 33.14), we have that Φ−1 i ◦ Fi (βi ) subconverges in C ˜ a map Ψ∞ : HA → H. Moreover, by (33.22) we have " " ∗ 1 " −1 " (33.31) = . " Φi ◦ Fi (βi ) (Φ∗i g (βi )) − h|HA " k k C (HA ,h) ˜ in C ∞ on compact sets, while we only know that Now Φ∗i g (βi ) converges to h −1 Φi ◦ Fi (βi ) converges to Ψ∞ in C k−1 (HA , h). However, we claim that by passing to a subsequence, the latter convergence is also in C ∞ (HA ). Because of this, we 10 Note

∗

that Fi (βi )−1 h is close to g (βi ).

230

33. NONCOMPACT HYPERBOLIC LIMITS

˜ the map Ψ∞ is a harmonic embedding conclude that with respect to h and h satisfying both " " 1 " " ∗ ˜ = (33.32) "Ψ∞ h − h|HA " k k C (HA ,h) and the CMC boundary conditions. In particular, Ψ∞ is an almost isometry, but not an isometry. To see the claim, recall that  −1 ∗ ∗ ˜ Φ−1 i ◦ Fi (βi ) : (HA , h) → Φi (Fi (βi ) (HA )), Φi g (βi ) ⊂ (Ui , Φi g (βi )) is a harmonic diffeomorphism satisfying the CMC boundary conditions and (33.31). ˜ in C ∞ as Φ−1 (Fi (βi ) (HA )) converges to H ˜ A in C ∞ Since Φ∗i g (βi ) converges to h i −1 (since ∂(Φi (Fi (βi ) (HA ))) consist of CMC tori of area A with respect to Φ∗i g (βi )), by the regularity theory for elliptic boundary value problems (for the linear theory, see the proof of Proposition 34.13), we obtain uniform estimates for the derivatives ◦ Fi (βi ) up to arbitrary order. Hence, by passing to a subsequence, the of Φ−1 i ∞ Φ−1 ◦ F i (βi ) converge to Ψ∞ in C (HA ) as claimed. i ˜ has at least as many cusp ends as (H, h), we may apply Theorem ˜ h) Since (H, 34.22 (a consequence of the Mostow rigidity theorem). Hence, for any  ∈ N and by choosing k˜0 larger if necessary, we have that if k ≥ k˜0 , then (H, h) is isometric ˜ and there exists an isometry ˜ h) to (H, ˜ ˜ h) I : (H, h) → (H,

(33.33)

with dC  (HA ,h) (Ψ∞ , I) < 1 , where this distance is defined by (K.13). Now, by the uniqueness part of Proposition 33.12 and by choosing  large enough, we have that Ψ∞ = I. However this contradicts (33.32), which completes the proof of Proposition 33.16.  Building on Proposition 33.16, we next prove that hyperbolic limits with minimal cusp ends are stable. Theorem 33.17 (Stability of hyperbolic limits with minimal cusp ends). A hyperbolic 3-manifold (H3 , h) contained in Hyp(M, g (t)) which attains the minimal number of cusp ends is necessarily a stable hyperbolic limit (in the sense of Definition 33.2) of (M, g (t)) as t → ∞. ˜ we have Proof. By Proposition 33.16, for each k ∈ N such that A  k−1 ≤ A, the following. Abbreviate by t(k)  t¯ı(k−1 ,k) and by F (k) (t)  F¯ı(k−1 ,k) (t). Then each (33.34)

F (k) (t) : (H1/k , h|H1/k ) → (M, g (t)) ,

t ∈ [t(k) , ∞),

is a smooth family of harmonic embeddings. Moreover, F (k) (t)(∂H1/k ) is a disjoint union of CMC tori, each with area equal to 1/k, where F (k) (t)∗ (N ) is normal to F (k) (t)(∂H1/k ), all with respect to g (t), and where " " 1 " " (k) ∗ ≤ (33.35) "F (t) g (t) − h|H1/k " k k C (H1/k ,h) for all t ∈ [t(k) , ∞).

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS

231

Step 1. Reduction to a claim. Theorem 33.17 is a consequence of the following. Claim. There exists a subsequence {kr }r∈N such that {t(kr ) }r∈N is a strictly increasing sequence and such that (33.36)

F (kr ) (t) (H1/kr ) ⊂ F (ks ) (t) (H1/ks )

for all t ∈ [t(ks ) , ∞) and 1 ≤ r < s < ∞. Indeed, assume that the claim is true. Without loss of generality, we may ˜ Choose a smooth nonincreasing function A (t), assume that k1 satisfies k21 ≤ A. defined for t sufficiently large, satisfying A(t(kr ) ) =

2 kr−1

for r ≥ 2.

Given any t > t(k1 ) , there exists a unique r (t) ∈ N − {1} such that t ∈ (t(kr(t)−1 ) , 2 t(kr(t) ) ]. We then have that A (t) ≥ kr(t)−1 and by (33.35) that each harmonic embedding   F (kr(t)−1 ) (t) : (HA(t) , h) → (M, g (t)) , t ∈ [t(kr(t)−1 ) , ∞), HA(t)

 1 is a kr(t)−1 -almost isometry with the image F (kr(t)−1 ) (t)H (∂HA(t) ) of the boundA(t) ary consisting of a disjoint union of almost totally umbillic tori with respect to g (t). By Proposition 33.12, for each t > t(k1 ) sufficiently large, close to the map  (kr(t)−1 )  F (t) H (∂HA(t) ) there exists a unique almost isometric harmonic emA(t) bedding F (t) : (HA(t) , h) → (M, g (t)) , where F (t) (∂HA(t) ) is a disjoint union of CMC tori, each with area equal to A (t), and where F (t)∗ (N ) is normal to F (t) (∂HA(t) ), all with respect to g (t). Since " " " (k " 1 "F r(t)−1 ) (t)∗ g (t) − h| " ≤ H1/kr(t)−1 " " kr(t)−1 C k (H1/k ,h) r(t)−1

and

1 kr(t)−1

≤ 12 A (t), one easily concludes that " " " " lim "F (t)∗ g (t) − h|HA(t) " t→∞

C  (HA(t) ,h)

=0

for each  ∈ N. Moreover, by (33.36) and by Proposition 33.12, it follows that the maps F (t) depend smoothly on t. In particular, by the uniqueness of harmonic diffeomorphisms near the identity, we have the important fact that there is no discontinuity of F (t) at any t = t(kr ) . The maps F (t), defined for all t sufficiently large, satisfy all the conditions of Definition 33.2. Hence Theorem 33.17 is proved assuming the claim. Step 2. Proof of the claim. First observe the following. Lemma 33.18 (Almost flat 2-tori with small area). If T 2 is an embedded 2-torus in an almost hyperbolic piece in M3 corresponding to H3 , with bounded intrinsic and extrinsic curvature and sufficiently small area, then T is contained in an almost hyperbolic cusp region of the almost hyperbolic piece.

232

33. NONCOMPACT HYPERBOLIC LIMITS

This simple fact follows directly from Corollary 31.47 since the image of such a 2-torus under an almost isometry is a 2-torus with the same properties in H, which must then be contained in one of the maximal cusp ends of H. ˆ if F (k ) (t(k ) )(H1/k ) Subclaim. There exists kˆ ∈ N such that for any k > k ≥ k, 

∩ F (k) (t(k ) )(H1/k ) = ∅, then 

F (k) (t)(H1/k ) ⊂ F (k ) (t) (H1/k )

(33.37) 

for all t ∈ [t(k ) , ∞). Proof of the subclaim. To prove the subclaim, it suffices to show that the same statement holds with the conclusion replaced by (33.38)







F (k) (t(k ) )(H1/k ) ⊂ F (k ) (t(k ) )(H1/k ); 

i.e., it suffices to show that (33.37) holds for t = t(k ) . To see this, recall that F (k) (t)(∂H1/k ) depends continuously on t and is a finite disjoint union of almost flat CMC almost umbillic tori, with each torus component having area 1/k. Because of this and by adjusting kˆ larger if necessary to ensure that (F (k) (t)(H1/k ), g (t)) ˆ it follows from is sufficiently close to hyperbolic for any t ∈ [t(k) , ∞) and k ≥ k,  Lemma 33.18 that we have (33.37) for all t ∈ [t(k ) , ∞). Now we prove (33.38). Let 

H(k)  F (k) (t(k ) )(H1/k )





 and H(k)  F (k ) (t(k ) )(H1/k ),

 ∩ H(k) = ∅. Now ∂H(k) = T1 ∪ · · · ∪ Tm is so that by the subclaim hypothesis, H(k) a disjoint union of embedded 2-tori. Essentially by Lemma 33.18, for each i, either  Ti ⊂ int(H(k) ) or

 Ti ∩ H(k) = ∅.

 . We In the former case, Ti is contained in an almost hyperbolic cusp region of H(k) shall show that the latter case is not possible. Suppose that there exists at least one 2-torus; without loss of generality assume   that it is Tm , such that Tm ∩ H(k) = ∅. Let x0 ∈ H(k) ∩ H(k) . Choose any x1 ∈ Tm 

and let α : [0, L] → (M, g(t(k ) )) be a geodesic joining x0 to x1 , minimizing length among all paths in H(k) . By the geometry of the almost hyperbolic cusp regions,  α ⊂ int(H(k) ). Now α intersects H(k) . Indeed, define  / H(k) } < L. u0  inf{u ∈ [0, L] : α (u) ∈   and α|(u0 ,L] does not intersect H(k) . Let T  be the 2-torus Then α (u0 ) ∈ ∂H(k)  boundary component of H(k) which contains α (u0 ). We then have, essentially by  Lemma 33.18, that T ⊂ H(k) . This is a contradiction since then T  must be an almost hyperbolic cusp region of H(k) but it has too small area 1/k since k > k. This completes the proof of the subclaim.

Finally we finish the proof of the claim. By the subclaim, it is sufficient to show that there exists a subsequence {kr }r∈N such that {t(kr ) }r∈N is increasing and F (kr ) (t(ks ) )(H1/kr ) ∩ F (ks ) (t(ks ) )(H1/ks ) = ∅ for all 1 ≤ r < s < ∞. We leave it as an exercise for the reader to verify that this follows from the immortality of almost hyperbolic pieces (see Proposition 33.16),

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS

233

 the fact that the volumes of F (k) t(k) (H1/k ) are uniformly bounded from below, and the fact that the volume of (M, g(t)) is finite and constant. This completes the proof of Theorem 33.17.  3.2. Disjointness of hyperbolic pieces and the stability of all hyperbolic limits. From the proof of the subclaim containing (33.37), one can show that two immortal almost hyperbolic pieces are either equivalent or are disjoint. We leave it to the reader to deduce the following lemma essentially from the fact that topological ends of finite-volume hyperbolic 3-manifolds are hyperbolic cusps (see Theorem 31.44). Lemma 33.19 (Disjointness of almost hyperbolic pieces). There exists A∗ ∈ ˜ where A˜ is as in (h5), such that for any 0 < A1 ≤ A2 ≤ A∗ there exists (0, A],  k0 ∈ N with the following properties. Let M3 , g (t) , t ∈ [t1 , t2 ), where t1 < t2 ≤ ∞, be a smooth family of closed Riemannian 3-manifolds. If the Ha3 , ha are finite-volume hyperbolic 3-manifolds and Fa (t) : (Ha )Aa → M are embeddings with " " 1 " " ≤ "Fa (t)∗ g(t) − ha |(Ha )Aa " k k0 C 0 ((Ha )Aa ,ha ) for a = 1, 2 and t ∈ [t1 , t2 ), then for all t ∈ [t1 , t2 ) either (1) F1 (t) ((H1 )A1 ) and F2 (t)((H2 )A2 ) are disjoint or (2) we have the inclusions F2 (t)((H2 )2A2 ) ⊂ F1 (t) ((H1 )A1 )

and

F1 (t)((H1 )2A2 ) ⊂ F2 (t)((H2 )A2 ).

As we shall see from the discussion in the rest of this section, in the case of a nonsingular solution to the NRF satisfying √ Condition H we may refine the above statement as follows. There exist A ∈ (0, 3/4] and k ∈ N such that if MA (t), t ∈ [tA,k , ∞), is an immortal (A, k)-almost hyperbolic piece, then MA (t) is contained in a stable asymptotically hyperbolic submanifold. In particular, MA (t) is actually an immortal asymptotically hyperbolic piece. Moreover, any two immortal (A, k)almost hyperbolic pieces are either disjoint or equal on their mutual time interval of existence. In the remainder of this section, we shall simultaneously prove, by an induction argument based on the number of cusp ends, all three of Propositions 33.5, 33.6, and 33.8.  Let H03 , h0 ∈ Hyp0 (M, g (t))  Hyp(M, g (t)) with x0∞ ∈ H0 be a pointed complete noncompact hyperbolic limit with a minimal number of cusp ends. Let A∗ be as in Lemma 33.19. By Theorem 33.17, there exist T0 < ∞, A0 : [T0 , ∞) → (0, A∗ ], and a corresponding stable asymptotically hyperbolic submanifold (in the sense of Definition 33.2) M30,A0 (t) (t) ⊂ M,

t ∈ [T0 , ∞).

In the following, (33.39)

ε
ε

for all i.

Let Hyp1 (M, g (t)) denote the space of pointed hyperbolic limits corresponding  / M0,A0 (ti ) (ti ). Let H13 , h1 ∈ Hyp1 (M, g (t)) be a to sequences {(xi , ti )} with xi ∈ limit with the minimal number of cusp ends. In view of (33.39) and by adjusting the basepoints in the sequence if necessary, we have that there exist x1∞ ∈ H1 and   M − M0,A0 (t) (t) × {t} (x1i , t1i ) ∈ t∈[T0 ,∞)

  converges in the C ∞ > ε and such that M, g t + t1i , x1i with  pointed Cheeger–Gromov sense to H1 , h1 , x1∞ . We have the following, whose verification shall closely follow the proof of Theorem 33.17, which in turn shall depend on the proof of Proposition 33.16.  Claim. Corresponding to H13 , h1 , x1∞ there exist T1 ∈ [T0 , ∞), a function A1 : [T1 , ∞) → (0, A∗ ] decreasing to zero, and a stable asymptotically hyperbolic submanifold M31,A1 (t) (t) ⊂ M − M0,A0 (t) (t) , t ∈ [T1 , ∞). injg(t1i ) (x1i )

Proof of the claim. Step 1. Existence of an immortal almost hyperbolic from 1disjoint  piece 1 , x converges (t). Let A ∈ (0, A ). Since the sequence M, g t M0,A ∗ i i  0 (t) to H1 , h1 , x1∞ and by (33.19), for each i sufficiently large we have that t1i ≥ T0 and that there exists a harmonic diffeomorphism   F1,i : ((H1 )A , h1 |(H1 )A ) → F1,i ((H1 )A ), g t1i −1 1 satisfying the CMC boundary conditions and limi→∞ dh1 (F1,i (xi ), x1∞ ) = 0. Moreover, by (33.20) we have " " " ∗  1 " lim "F1,i g ti − h1 |(H1 )A " k =0 i→∞

C ((H1 )A ,h1 )

for each k ∈ N ∪ {0}. Define T1  t1i , where i is to be chosen below. Denote M31,A (T1 )  F1,i ((H1 )A ) and F1 (T1 )  F1,i . In this new notation, we have that F1 (T1 ) : ((H1 )A , h1 |(H1 )A ) → (M1,A (T1 ), g(T1 ))

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS

235

is a harmonic diffeomorphism satisfying the CMC boundary conditions and that " " 1 " " ≤ , (33.41) "F1 (T1 )∗ g(T1 ) − h1 |(H1 )A " k k C ((H1 )A ,h1 ) where k ∈ N is as large as we like, by taking i sufficiently large. By applying Lemma 33.19 to (33.41) and to M0,A0 (t) (t) being a stable asymptotically hyperbolic submanifold, we have for i sufficiently large that M0,A0 (T1 ) (T1 ) ∩ M1,A (T1 ) = ∅. We shall complete Step 1 by establishing the following statement. For any k ∈ N, by choosing i sufficiently large, one can continue F1 (T1 ) as harmonic diffeomorphisms F1 (t) : ((H1 )A , h1 |(H1 )A ) → (M1,A (t), g(t)), where M1,A (t)  F1 (t)((H1 )A ), " " " " (33.42) "F1 (t)∗ g(t) − h1 |(H1 )A "

C k ((H

1 )A ,h1 )

t ∈ [T1 , ∞),

≤

1 , k

and (33.43)

M0,A0 (t) (t) ∩ M1,A (t) = ∅.

Hence M1,A (t) is an immortal almost hyperbolic piece disjoint from the submanifold M0,A0 (t) (t). The statement is a consequence of the proof of Proposition 33.16 for the following reasons: (1) By Lemma 33.19 and by taking k sufficiently large, as long as (33.42) remains true, so does (33.43). To wit, as long as the pieces are close to hyperbolic, they are disjoint. (2) As long as (33.43) remains true, we may apply the proof of Proposition 33.16 to derive (33.42). The reason for (2) is that if we ever have equality in (33.42) at some times βi ∈ [t1i , ∞) for a sequence of i tending to ∞ (the dependence on i therein is implicit since F1 (T1 ) = F1,i ), we can then apply the argument in the part of the proof of Proposition 33.16 between the displays (33.22) and (33.33) to obtain a new ˜ 1 ) contained in Hyp1 (M, g (t)) (corresponding to (33.23)). ˜ 31 , h hyperbolic limit (H Since (H1 , h1 ) has the minimal number of cusp ends in Hyp1 (M, g (t)), we have ˜ 1 ) has at least as many cusp ends as (H1 , h1 ). We may now argue (using ˜ 1, h that (H the Mostow rigidity theorem) as in the paragraph containing (33.33) to obtain a contradiction. Step 2. Existence of a stable asymptotically hyperbolic piece disjoint from M30,A0 (t) (t). Now, by applying the proof of Theorem 33.17 to the immortal almost hyperbolic piece M1,A1 (t) ⊂ M − M0,A0 (t) (t) corresponding to (H1 , h1 ), we have that (H1 , h1 ) is a stable hyperbolic limit. Moreover, we have a corresponding stable asymptotically hyperbolic submanifold M31,A1 (t) (t), t ∈ [T1 , ∞), with M1,A1 (t) ⊂ M1,A1 (t) (t) ⊂ M − M0,A0 (t) (t) . This completes the proof of the claim. It is now evident that we may proceed inductively as follows. Assume that we have defined for b = 0, 1, . . . , a the spaces Hypb (M, g (t)), the sequences {(xbi , tbi )}i∈N ,

236

33. NONCOMPACT HYPERBOLIC LIMITS

the corresponding hyperbolic limits (Hb3 , hb ) ∈ Hypb (M, g (t)), and the corresponding stable asymptotically hyperbolic submanifolds {M3b,Ab (t) (t)}t∈[Tb ,∞) , where T0 ≤ T1 ≤ · · · ≤ Ta and Ab (t) decreases to zero. Case (Aa ). If there exists T¯a ∈ [Ta , ∞) such that sup

t∈[T¯a ,∞) x∈M−

inj g(t) amax b=0 Mb,Ab (t) (t)

(x) ≤ ε,

then we let N = a and stop here. For sufficiently large times t, the complement of %N ¯ b=0 Mb,Ab (t) (t) is ε-collapsed. So, for t ≥ TN , (M, g (t)) is the union of N stable asymptotically hyperbolic submanifolds and an ε-collapsed piece, whose intersections are comprised of tori. Case (Ba ). Otherwise, we define Hypa+1 (M, g (t)) to be the %aspace of pointed / b=0 Mb,Ab (ti ) (ti ). hyperbolic limits corresponding to sequences {(xi , ti )} with xi ∈ 3 There exists (Ha+1 , ha+1 ) ∈ Hypa+1 (M, g (t)) with the least number of cusp ends , ta+1 )}i∈N . By arguing as in the claim and with corresponding sequence {(xa+1 i i above, we have a corresponding stable asymptotically hyperbolic submanifold {M3a+1,Aa+1 (t) (t)}t∈[Ta+1 ,∞) , where Ta+1 ≥ Ta . In this case we continue our induction. We now show that there exists N ≤ 2 Volg(t) (M) < ∞ such that Case (AN ) holds. For otherwise, we have that {M3b,Ab (t) (t)}t∈[Tb ,∞) is defined for 1 ≤ b ≤ a, where a > 2 Volg(t) (M). Recall from Theorem 31.46 that Vol(Hb , hb ) ≥ 1 for each b = 0, 1, . . . , a. Hence, by choosing t large enough, we have for each b = 0, 1, . . . , a that Volg(t) (Mb,Ab (t) (t)) ≥ 12 . Since

a  1 Mb,Ab (t) (t) ≥ (a + 1) , Volg(t) (M) ≥ Volg(t) 2 b=0

we have a contradiction. In summary, we have obtained (1) a finite collection of sequences N {(x0i , t0i )}i∈N , {(x1i , t1i )}i∈N , . . . , {(xN i , ti )}i∈N ,

(2) associated hyperbolic limits 3 , hN ), (H03 , h0 ), (H13 , h1 ), . . . , (HN

(3) corresponding time-dependent pairwise-disjoint 3-dimensional compact submanifolds of M with boundary {M30,A0 (t) (t)}t∈[T0 ,∞) , {M31,A1 (t) (t)}t∈[T1 ,∞) , . . . , {M3N,AN (t) (t)}t∈[TN ,∞) , where 0 ≤ T0 ≤ T1 ≤ · · · ≤ TN < ∞ and Aj : [Tj , ∞) → [0, A∗ ) for 0 ≤ j ≤ N , which are stable asymptotically hyperbolic submanifolds satisfying x∈M−

inj g(t) Nmax a=0 Ma,Aa (t) (t)

(x) ≤ ε

for t ≥ T¯N ; that is, for sufficiently large times, the complement of the union of the immortal almost hyperbolic pieces has small injectivity radius,

4. INCOMPRESSIBILITY OF BOUNDARY TORI

237

(4) harmonic diffeomorphisms Fa (t) : (Ha )Aa (t) → Ma,Aa (t) (t) satisfying the CMC boundary conditions and " " " " =0 lim "Fa (t)∗ g (t) − ha |(Ha ) " m A (t) t→∞

a

C

((Ha )Aa (t) ,ha )

for each m ∈ N and each 0 ≤ a ≤ N . So the hyperbolic limits in (2) are stable. Furthermore, each of the hyperbolic limits has the minimal number of cusp ends “amongst the remaining limits”. In particular, the number of cusp ends of Hc is greater than or equal to the number of cusp ends of Hb for c ≥ b. In conclusion, we have proved all three propositions.  Exercise 33.20. Show that x∈M−

inj g(t) Nmax a=0 Ma,Aa (t) (t)

(x) → 0

as t → ∞. 4. Incompressibility of boundary tori of hyperbolic pieces The works of Meeks, Schoen, Simon, Yau, and others on the applications of minimal surface theory to the geometry and topology of 3-manifolds are integral to some developments in low-dimensional topology in the late 1970s and early 1980s. For example the first proof of the equivariant loop theorem, exploiting the canonical nature of minimal surfaces, is due to Meeks and Yau [224], [225] and forms one of the components of the proof of the Smith conjecture. Minimal surfaces are also used by Meeks, Simon, and Yau [223] to prove that closed 3-manifolds with positive Ricci curvature are prime (soon thereafter, this result was subsumed by Hamilton’s classification). The stability of minimal surfaces is used by Schoen and Yau [354] to classify closed 3-manifolds with positive scalar curvature essentially up to homotopy; see Gromov and Lawson [125] for an independent proof. In particular, they prove that in the prime decomposition of such 3-manifolds, there are no K (Π, 1)’s.11 Related to these developments, in [353] Schoen and Yau proved the positive mass theorem of general relativity in sufficiently low dimensions; see Witten [437] for an elegant proof for spin manifolds. In this section we see an application of minimal surface theory to Ricci flow, i.e., the use of Theorem 33.25 in the proof of Theorem 33.21 below. Throughout this section M3 , g (t) shall again denote a nonsingular solution satisfying H.  Condition Let H3 , h ∈ Hyp(M, g (t)) be an asymptotic limit and let A˜ be as in (h5) in Subsection 1.1 of this chapter. By Proposition 33.6, (H, h) is a stable hyperbolic limit of (M, g (t)). As a consequence of this, we have that there exist a function A → ˜ and corresponding immortal asymptotically hyperbolic tA , defined for A ∈ (0, A), pieces M3A (t) ⊂ M, t ∈ [tA , ∞), which converge to (HA , h) in C ∞ as t → ∞. By examining the proof of Proposition ˜ 33.6, we may assume that (1) A → tA is nonincreasing and (2) if 0 ≤ A2 < A1 < A, then MA1 (t) ⊂ MA2 (t) for t sufficiently large. In particular, we may assume that 11 A connected topological space X is called an Eilenberg–MacLane space of type K (Π, 1) if π1 (X) ∼ = Π and all of its higher homotopy groups vanish; i.e., πk (X) = 0 for k ≥ 2.

238

33. NONCOMPACT HYPERBOLIC LIMITS

˜ is strictly decreasing in the definition of stability the function A : [T0 , ∞) → (0, A) and then define tA = A−1 (t). 4.1. Reducing incompressibility to an area-decreasing property for minimal disks. we shall prove the following fundamental incompressibility result In this section for M3 , g (t) . Recall that the boundaries ∂M3A (t) ⊂ M of immortal asymptoti˜ and t ∈ [tA , ∞). cally hyperbolic pieces are isotopic to each other for all A ∈ (0, A) ˜ and t ∈ Theorem 33.21 (Incompressibility of boundary tori). For A ∈ (0, A) [tA , ∞) we have that ∂MA (t) is incompressible in M. That is, the inclusion map ι : ∂MA (t) → M induces an injection ι∗ : π1 (∂MA (t)) → π1 (M) on fundamental groups. Throughout this section D2 = D2 (1) shall denote the Euclidean unit 2-disk. First we make the following definition. ˜ let FA (t) denote the space of all smooth Definition 33.22. Given A ∈ (0, A), 2 maps f from the unit disk D into M − MA (t) such that f (∂D) ⊂ ∂MA (t) and such that the map f |∂D represents a nontrivial element of π1 (∂MA (t)). Let (33.44)

A FA (t) 

inf

f ∈FA (t)

Area f (D)

denote the least area with respect to g (t) of all maps in FA (t). Remark 33.23. In the rest of this chapter we shall often suppress the dependence on A in our notation. When f ∈ F (t) is an embedding, we shall often abuse notation and write f (D) ∈ F (t). Proof of Theorem 33.21. Note that we may assume that A is sufficiently small and t is sufficiently large. It follows from Lemma 31.25 that ∂MA (t) is incompressible in MA (t). By Lemma 31.21, it suffices to prove that ∂MA (t) is incompressible in M − MA (t). We shall prove this by contradiction. Suppose that ∂MA (t) is compressible in M − MA (t). Then choose any ε ∈ + (0, 2π). By Proposition 33.24 below, we have that d dtA F (t) ≤ −2π + ε for all t ∈ [tε , ∞). However, this contradicts the fact that A F (t) > 0 for all t ∈ [tε , ∞).  Proposition 33.24 (Compressible tori yield disks whose areas decrease at  least linearly). Let M3 , g (t) be a nonsingular solution satisfying Condition H. Suppose that some torus boundary component TA2 (t) of an immortal asymptotically hyperbolic piece M3A (t) is compressible in M − MA (t). Then: (1) The set FA (t) is nonempty and the function A FA (t) > 0 exists. (2) For every ε > 0, there exists a time tε,A < ∞ such that for all t ≥ tε,A , (33.45)

A FA (t + Δt) − A FA (t) d+ A FA (t)  lim sup ≤ −2π + ε. dt Δt Δt→0+

Hamilton’s proof of Proposition 33.24 exploits the following result of Meeks and Yau (see Theorem 1 on p. 443 of [224]), which also gives a geometrically canonical proof of the loop theorem (compare Corollary 31.15). We digress to discuss this result.

4. INCOMPRESSIBILITY OF BOUNDARY TORI

239

 Theorem 33.25 (Meeks and Yau). Let N 3 , g be a compact Riemannian 3manifold with boundary. Suppose that ∂N is convex, i.e., the second fundamental form of ∂N ⊂ N is positive with respect to the unit outward normal, and suppose that ∂N is compressible in N . Let F denote the space of all smooth maps f from the unit disk D2 into N such that f |∂D : ∂D → ∂N represents a nontrivial element of π1 (∂N ). Then there exists a conformal harmonic map f0 : D → N in F of least area among all maps in F.12 Moreover, f0 satisfies the following properties: (1) The map f0 is an embedding and hence [ f0 |∂D ] is a primitive element of ker {i∗ : π1 (∂N ) → π1 (N )}. (2) For z ∈ ∂D, the tangent map df0 : Tz D → Tf0 (z) N takes the unit inner normal of ∂D to a nonzero inner normal to ∂N (i.e., the embedded disk f0 (D) is normal to ∂N ). (3) If the boundary component containing f0 (D) is a torus T ⊂ ∂N , then [ f0 |∂D ] ∈ π1 (T ) is a primitive element generating the infinite cyclic group ker {i∗ : π1 (T ) → π1 (N )} ∼ = Z. Remark 33.26. More generally (see p. 462 of [224]), if S ⊂ ∂N consists of some of the components of ∂N , then in the space of all smooth maps f from the unit disk D into N such that f |∂D : ∂D → S represents a nontrivial element of π1 (S), provided this space is nonempty, there exists a conformal harmonic map f0 : D → N which minimizes area. Here we give a proof of part (3) of Theorem 33.25. We first show that the homomorphism induced by inclusion j∗ : H1 (T , R) → H1 (N , R) is not the zero map. Suppose on the contrary that j∗ is the zero map. Then by attaching handlebodies to all boundary components of N other than T , we obtain a compact orientable 3-manifold W 3 with boundary ∂W = T . We have that the map k∗ : H1 (T , R) → H1 (W, R) , induced by the inclusion k : T → W, is also the zero map. This contradicts Lemma 31.20. We claim that ker {i∗ : π1 (T ) → π1 (N )} is not isomorphic to Z×Z. Otherwise, by the well-known fact that a subgroup G of Z × Z with G ∼ = Z × Z has finite index, it follows that ker (j∗ ) = ker (i∗ ) ⊗ R = H1 (T , R). This contradicts j∗ not being the zero map. Since a subgroup G of Z×Z is isomorphic to either 0, Z, or Z×Z, it follows that ker {i∗ : π1 (T ) → π1 (N )} ∼ = Z. Now c = [ f0 |∂D ] is in ker (i∗ ) and c is primitive. Thus c is a generator of ker (i∗ ). This completes the proof of part (3). The rest of this section is devoted to the proof of Proposition 33.24. 4.2. Bounding the time derivative of the area of minimal disks in terms of length and area. In this subsection, as a first step in proving Proposition 33.24, we shall derive + an upper bound for d dtA F in terms of the length and area of a minimal disk (see inequality (33.58) below). 12 By

conformal harmonic map we mean a harmonic map which is also a conformal map.

240

33. NONCOMPACT HYPERBOLIC LIMITS

Fix any time t0 > 0, which will be chosen sufficiently large. Since the boundary torus TA (t0 ) in the hypothesis of Proposition 33.24 is convex and is assumed to be compressible in M − MA (t0 ), we may apply Meeks and Yau’s theorem, Theorem 33.25. Hence there exists a smooth embedded minimal disk f0 : D → M − MA (t0 ) normal to TA (t0 ) such that f0 |∂D : ∂D → TA (t0 ) represents a nontrivial primitive element of ker (i∗ ) ⊂ π1 (TA (t0 )), where i : TA (t0 ) → M − MA (t0 ) denotes the inclusion map and where the area of D(t0 )  f0 (D)

(33.46)

is least among all maps in F (t0 ). + For the sake of a comparison argument in estimating d dtA F (t0 ), we shall now define smoothly varying embedded disks Disk0 (t) ⊂ M − MA (t) for t − t0 sufficiently small. First, extend the embedding f0 to a smooth embedding f¯0 defined in an η-neighborhood D2 (1 + η) of D = D2 (1), where η > 0 is small. Then define the embedded comparison disk at time t to be  (33.47) Disk0 (t)  f¯0 D2 (1 + η) ∩ M − MA (t). By definition, (33.48)

  d  d+ A F (t0 ) ≤ Area g(t) Disk0 (t) .  dt dt t=t0

Note that Disk0 (t0 ) = f0 (D) = D(t0 ). For t−t0 sufficiently small, the disk Disk0 (t) bounds an embedded loop Loop0 (t)  ∂ Disk0 (t) ⊂ TA (t) and it is easy to see that Disk0 (t) ∈ F (t).

immortal asymptotically hyperbolic piece at time t0 MA(t0)

D(t0) TA(t0)

M−MA(t0)

Figure 33.1. The disk D(t0 ) in M − MA (t0 ).

4. INCOMPRESSIBILITY OF BOUNDARY TORI

D(t0)

MA(t0)

...

241

M−MA(t0)

...

...

f˜0(D2(1+η)) TA(t0)

Loop0(t) Disk0(t)

MA(t)

M−MA(t)

...

f˜0(D2(1+η)) TA(t)

Figure 33.2. Disk0 (t) and Loop0 (t). +

The bound for d dtA F (t0 ) involves the normal velocity of Loop0 (t0 ), which we first discuss. Let N (t0 ) be a choice of smooth unit normal vector field to TA (t0 ), with respect to g(t0 ). Since the embedded torus TA (t) depends smoothly on t, for ε > 0 small enough and for each t ∈ (t0 − ε, t0 + ε), TA (t) can be uniquely written as an exponential normal graph over TA (t0 ). That is, there exists a unique smooth function φ : TA (t0 ) × (t0 − ε, t0 + ε) → R such that each map Φt : TA (t0 ) → TA (t) defined by 0) (φ (x, t) N (t0 )) Φt (x) = expg(t x

is a diffeomorphism. We call ∂φ ∂t ( · , t0 ) the normal velocity function of TA (t0 ). Since MA (t0 ) converges to (HA , h|HA ), we have that inside MA (t0 ) ⊂ M,        k ∂  k   ∇ ∇ 2 Rc − 2 r g  (x, t0 ) → 0  ) = g (x, t 0     ∂t 3 uniformly in x as t0 → ∞ for each k ≥ 0; here r is the average scalar curvature. So, since the boundary component TA (t0 ) is a CMC torus of fixed area A, we conclude that the normal velocity function ∂φ ∂t of TA (t0 ) in M tends to zero as t0 → ∞. Let V 0 (t0 ) : Loop0 (t0 ) → R denote the normal velocity function of Loop0 (t0 ) with respect to the outward unit normal of the boundary of Disk0 (t0 ) ⊂

242

33. NONCOMPACT HYPERBOLIC LIMITS

 f¯0 D2 (1 + η) . Since D (t0 ) is normal to TA (t0 ), we have that     ∂φ (33.49) |V 0 (t0 )| =  (t0 )|Loop0 (t0 )  → 0 as t0 → ∞. ∂t

 Let x ∈ int (D (t0 )). By definition (33.47) we have that x ∈ int Disk0 (t) for t near t0 . If {e1 , e2 } is a basis for Tx Disk0 (t), then the induced area element on Disk0 (t) is  dA (t) = det (J (t)) ω 1 ∧ ω 2 , where

 J (t) 

 and where ω i

2 i=1

g (t) (e1 , e1 ) g (t) (e1 , e2 ) g (t) (e2 , e1 ) g (t) (e2 , e2 )



is the dual basis of 1-forms; i.e., ω j (ei ) = δij . Hence   ∂J ∂ 1 dA (t) = trace J(t) (t) dA (t) . ∂t 2 ∂t

Choosing {e1 , e2 } to be orthonormal at time t0 , we obtain       ∂g ∂  ∂g 1 (t0 ) (e1 , e1 ) + (t0 ) (e2 , e2 ) dA (t0 ) . dA (t) = ∂t  2 ∂t ∂t t=t0

Hence, under the NRF the evolution of the area is given by (33.50)       d  ∂ 0 dA (t0 ) + Area g(t) Disk (t) = V 0 (t0 )ds (t0 ) dt t=t0 ∂t D(t0 ) Loop0 (t0 )    2 = r − Rc (e1 , e1 ) − Rc (e2 , e2 ) dA (t0 ) 3 D(t0 )  + V 0 (t0 )ds (t0 ) , Loop0 (t0 )

where ds (t0 ) is the induced arc length element with respect to g(t0 ). Similarly to an argument in Schoen and Yau [354], we may bound the rhs. In particular, at time t0 we have (33.51)

Rc (e1 , e1 ) + Rc (e2 , e2 ) = sect (e1 ∧ e2 ) + sect (e1 ∧ e3 )

+ sect (e2 ∧ e1 ) + sect (e2 ∧ e3 ) 1 = R + sect (e1 ∧ e2 ) , 2 where sect (e1 ∧ e2 ) denotes the sectional curvature of the plane spanned by e1 and e2 . Let II denote the second fundamental form. It follows from the Gauss equations that the components of the Riemann curvature tensor of the disk D (t0 ) are given by D M Rijk = Rijk + IIi IIjk − IIik IIj . Thus the Gauss curvature K of D satisfies (33.52)

D M K = R1221 = R1221 + II11 II22 − (II12 )2

= sect (e1 ∧ e2 ) + det (II) .

4. INCOMPRESSIBILITY OF BOUNDARY TORI

243

Let κ1 and κ2 denote the eigenvalues of II, i.e., the principal curvatures of D (t0 ). Since D (t0 ) is minimal, we have κ1 + κ2 = 0 so that det (II) = −κ21 ≤ 0. Hence, by the Gauss–Bonnet formula and since χ(D (t0 )) = 1,    (33.53) sect (e1 ∧ e2 ) dA (t0 ) ≥ K dA (t0 ) = 2π − k ds (t0 ) , D(t0 )

D(t0 )

∂D(t0 )

where k denotes the geodesic curvature of the curve ∂D (t0 ) in the surface D (t0 ). From (33.48), (33.50), (33.51), and (33.53), we conclude that Lemma 33.27. Under the hypotheses of Proposition 33.24 and for t0 sufficiently large, we have      2 1 d+ A F 0 k + V ds (t0 ) + (t0 ) ≤ −2π + r − R dA (t0 ) , (33.54) dt 3 2 ∂D(t0 ) D(t0 ) where the area of D (t0 ) ∈ F (t0 ) equals A F (t0 ). We want to estimate the last two terms on the rhs of (33.54). It follows from (32.22), which says limt→∞ Rmin (t) = limt→∞ r (t) = −6, that given any ε > 0, there exists tε (1) < ∞ such that for all t0 ≥ tε (1),       2 2 1 1 r − R dA (t0 ) ≤ r − Rmin dA (t0 ) (33.55) 3 2 3 2 D(t0 ) D(t0 ) ≤ − (1 − ε) Area g(t0 ) (D (t0 )) 

= − (1 − ε) A F (t0 ) .

To understand the term ∂D(t0 ) k ds (t0 ) in (33.54), we first consider the model  case of an exact hyperbolic cusp [0, ∞) × V 2 , dr 2 + e−2r gflat , where (V, gflat ) is a flat 2-torus. Suppose that we have a loop L which lies inside a slice {r} × V and which bounds a surface Σ2 ⊂ [r, ∞) × V that intersects the slice at right angles; for example, take Σ = [r, ∞) × L, where we consider L as contained in V. We claim that the geodesic curvature k of the loop L inside Σ is identically equal to 1. To see this, let T and N denote the unit tangent and unit outward normal vector fields of L as a loop in Σ, respectively. Since Σ is normal to the slice {r} × V, we have N = −∂/∂r. The geodesic curvature of L is defined by k = DT N, T , where D denotes the induced covariant derivative on the surface Σ. Let B denote a choice of unit normal vector field to Σ, so that {T, N, B} forms an orthonormal frame along L. Since the second fundamental form II of the slice {r} × V satisfies II = e−2r gflat , we have13 DT N = ∇T N − ∇T N, B B = II (T ) − II (T, B) B = T, where ∇ denotes the covariant derivative of the ambient cusp. Therefore k ≡ 1 for the loop L in Σ. Now since (MA (t) , g(t)) converges to (HA , h|HA ) and since TA (t) is a torus of constant mean curvature of fixed area A, for any ε > 0, there exists tε (2) < ∞ such that for t0 ≥ tε (2),  (33.56) k ds (t0 ) ≤ (1 + ε) L g(t0 ) (∂D (t0 )) , ∂D(t0 )

where L g denotes the length with respect to g. 13 As

a (1, 1)-tensor, II = id.

244

33. NONCOMPACT HYPERBOLIC LIMITS

Since V 0 (t0 ) → 0 as t0 → ∞, there exists tε (3) < ∞ such that for t0 ≥ tε (3),  (33.57) V 0 (t0 ) ds (t0 ) ≤ ε L g(t0 ) (∂D (t0 )) . ∂D(t0 )

˜ and ε > 0, from (33.54), (33.55), (33.56), and (33.57) we Hence, given A ∈ (0, A) have shown for t0 ≥ max {tε (1) , tε (2) , tε (3)} that d+ A F (t0 ) ≤ −2π + (1 + 2ε) L g(t0 ) (∂D (t0 )) − (1 − ε) A F (t0 ) . dt To summarize, we have Lemma 33.28 (Evolution of the area of a minimizing disk). Assume the hy˜ and ε > 0, there exists tε < ∞ potheses of Proposition 33.24. For any A ∈ (0, A) such that if t ≥ tε , then d+ A F 1/2 −1/2 (t) ≤ −2π + (1 + ε) L g(t) (∂D (t)) − (1 + ε) A F (t) , dt where the area of D (t) ∈ F (t) equals A F (t).

(33.58)

Thus we have reduced the proof of Proposition 33.24 to bounding the length of ∂D (t) by the area of D (t); this is accomplished by Lemmas 33.30 and 33.41 in Subsection 4.4. 4.3. Disks in hyperbolic and almost hyperbolic cusps. The purpose of this subsection is to present a couple of elementary facts which help motivate and prepare for the proof of Proposition 33.40 below, which in turn is used to prove the main Proposition 33.24.  Consider a hyperbolic cusp V 2 × R, gcusp , where gcusp = dr 2 + e−2r gflat and gflat is a flat metric on a 2-torus V. Let γ ⊂ V be a path. Then γ × {ρ} has length e−ρ Lgflat (γ) and hence (33.59)

d ρ (e Lgcusp (γ)) = 0. dρ

Moreover, the area of the surface γ × [0, ρ¯] ⊂ V × [0, ∞) is given by  ρ¯  Area gcusp (γ × [0, ρ¯]) = e−ρ Lgflat (γ) dρ = 1 − e−ρ¯ Lgflat (γ) . 0

Thus the quantity (33.60)

1 Area gcusp (γ × [0, ρ¯]) = Lgflat (γ) 1 − e−ρ¯

is independent of ρ¯. Note also that Area gcusp (γ × [0, ∞)) = Lgflat (γ). Next we shall observe that the intersection of a smooth embedded disk with tori slices in an almost hyperbolic piece is almost always a finite disjoint union of smooth embedded loops. Let N 3 be a compact 3-manifold, let Σ2 be a boundary component, and let C be an open collar of Σ, which we identify via a diffeomorphism with Σ × [a, b), where Σ is identified with Σ × {a}. Suppose that f : D2 → N is an embedding of a disk with f (∂D) ⊂ Σ. Let U 2  f −1 (C) and write f  (f1 , f2 ) : U → Σ × [a, b),

4. INCOMPRESSIBILITY OF BOUNDARY TORI

245

so that f2 (∂D) = a. By Sard’s theorem, almost every r ∈ (a, b) is a regular value of f2 . For such r we have that the map f is transversal to the slice Σ × {r} and hence f −1 (Σ × {r}) = f2−1 (r) is a 1-dimensional submanifold of D, which must be a disjoint union of loops. Since f is an embedding, for such r, the set f (D) ∩ (Σ × {r}) is a disjoint union of smooth embedded loops. In particular, we have the following: Lemma 33.29 (Embedded disks intersecting tori slices). Let f be a smooth embedding of a disk D2 into a differentiable 3-manifold containing a boundary collar T 2 ×[a, b). Suppose that f (∂D) ⊂ T ×{a} and write f  (f1 , f2 ) on f −1 (T ×[a, b)). Then almost every z ∈ (a, b) is a regular value of f2 . For such z the map f is transversal to the torus slice T × {z} and hence f2−1 ({z}) = f −1 (T × {z}), as well as f (D) ∩ (T × {z}), is a disjoint union of smooth embedded loops. We shall apply the above fact to the part of a minimal disk D(t) in an almost hyperbolic piece of a nonsingular solution. The main idea in what follows is to use comparison disks (see (33.72) below) to understand the area of the minimal disk and the length of its boundary circle. 4.4. Bounding length by area for minimal disks. ˜ be an immortal asymptotically hyperbolic Let MA (t), where A ∈ (0, A), piece corresponding to a hyperbolic limit (H, h, x∞ ), as given by Proposition 33.6. Throughout this subsection we shall assume, as in the hypothesis of Proposition 33.24, that i : TA (t) → M − MA (t) satisfies ker(i∗ ) = {1}; that is, TA (t) is compressible in M − MA (t). As in (33.46) we have a smooth embedded minimal disk D(t) ⊂ M − MA (t) with Area g(t) (D(t)) = A F (t). Observe that by Lemma 33.28 we have ˜ and ε > 0. If t sufficiently large is such that Lemma 33.30. Let A ∈ (0, A) L g(t) (∂D(t)) ≤ (1 + ε)

−1

A FA (t), then

d+ A FA (t) ≤ −2π. dt So the proof of Proposition 33.24 is reduced to the complementary case where

(33.61)

(33.62)

L g(t) (∂D(t)) > (1 + ε)−1 A F (t) .

The rest of this subsection is devoted to this case. We shall show that by choosing ˜ small enough and t sufficiently large, we may assume that the length A ∈ (0, A) L g(t) (∂D(t)) is as small as we like (see (33.86) below). Recall that for any B ∈ (0, A) and for t large enough, an immortal asymptotically hyperbolic piece MB (t) corresponding to the hyperbolic limit (H, h, x∞ ) satisfies MA (t) ⊂ MB (t). 3 Definition 33.31. Let CA,B (t) ⊂ M denote the almost hyperbolic cusp region bounded by both TA (t) ⊂ ∂MA (t) and the corresponding boundary torus in ∂MB (t).

For part of the following discussion, we shall suppress the dependence of t, which shall be assumed to be sufficiently large, in our notation. Let N TA denote the normal bundle of TA , and let ν be the unit normal field to TA pointing into CA,B . Given C0 > 0, define Σ0  {ρν(x) : x ∈ TA , ρ ∈ [0, C0 ]} ⊂ N TA .

246

33. NONCOMPACT HYPERBOLIC LIMITS

Restricting the exponential map to the normal bundle we have the map expν   g(t)  : N TA → M. Since CA,B is an almost hyperbolic cusp, by choosing C0 exp N TA sufficiently large and then choosing t sufficiently large, we have that expν |Σ0 : Σ0 → expν (Σ0 ) is a diffeomorphism onto a set which contains CA,B . We may also assume that the geodesics in expν (Σ0 ) emanating from TA and perpendicular to TA , besides having the property of not intersecting with each other, are minimal. Define the coordinate function r on expν (Σ0 ) to be the length along these geodesics normal to TA . We have that r (x) = dg(t) (x, TA )

(33.63) for x ∈ exp (Σ0 ). Define ν

(33.64)

TA,ρ (t)  {x ∈ expν (Σ0 ) : r (x) = ρ} ,

for ρ ∈ [0, C0 ]. The tori TA,ρ are parallel hypersurfaces; i.e., for each ρ1 , ρ2 ∈ [0, C0 ] with ρ1 < ρ2 and % for each x ∈ TA,ρ2 , we have dg(t) (x, TA,ρ1 ) = ρ2 − ρ1 . Moreover, we have that ρ∈[0,C0 ] TA,ρ = expν (Σ0 ) ⊃ CA,B and that TA,ρ is an embedded almost CMC torus for each ρ ∈ [0, C0 ] and t sufficiently large. Definition 33.32. Define ρA,B (t) to be the maximum number σ such that TA,ρ (t) ⊂ CA,B (t) for all ρ ∈ [0, σ]. ˜ and N < ∞, for B ∈ (0, A) sufficiently Remark 33.33. Given any A ∈ (0, A) small and t sufficiently large, we have ρA,B (t) ≥ N . Define (33.65)

SA,ρ (t)  D(t) ∩ TA,ρ (t).

Again we shall suppress t in our notation. Note that SA,0 = ∂D. It follows from Lemma 33.29 that SA,ρ is a finite disjoint union of smooth embedded loops for almost every ρ ∈ [0, ρA,B ]. Before discussing geometric considerations, in this paragraph we note the following topological considerations. The tori TA,ρ are isotopic to each other so that Defintheir fundamental groups π1 (TA,ρ ) are naturally isomorphic to each other.  ing the inclusion maps iA,ρ : TA,ρ → M, we find that the kernels ker (iA,ρ )∗ are naturally isomorphic as well. Recall that [SA,0 ] = [∂D] ∈ ker (i∗ ) − {1}, where i : TA (t) → M − MA (t). Hence, for ρ such that SA,ρ is a finite disjoint union of smooth embedded loops, among the loops comprising SA,ρ there exists a loop S˜A,ρ representing a nontrivial element of ker (i∗ ). Since SA,ρ is embedded, by Lemma 31.16 we have that [S˜A,ρ ] ∈ ker (i∗ ) is primitive. For ρ ∈ [0, ρA,B ] define AA,ρ  D ∩ {x ∈ CA,B : 0 ≤ r (x) ≤ ρ} ⊂ CA,B . Note that for a.e. ρ we have that ∂AA,ρ = SA,ρ ∪ SA,0 and that this set is a finite disjoint union of smooth embedded loops. Let (33.66)

L (ρ)  Length g(t) (SA,ρ ),

A (ρ)  Area g(t) (AA,ρ ) .

The function ρ → L (ρ) is positive, integrable, and defined almost everywhere. Note also that ρ → A (ρ) is positive and increasing.

4. INCOMPRESSIBILITY OF BOUNDARY TORI

247

S˜A,ρ SA,ρ SA,0 = ∂ D

CA,Β

TA,ρ

...

D

...

MA(t)

TA,ρ

TA

CMC tori with area B

MB(t)

... CMC torus with area A

Figure 33.3. The submanifolds CA,B , TA,ρ , SA,ρ , and S˜A,ρ . Let rD denote the restriction of r to D and let DrD denote its gradient. Since |DrD | ≤ |∇r| = 1, we have by the co-area formula (see Proposition 27.30) that for a.e. ρ¯ ∈ [0, ρA,B ] (33.67)     ρ¯

A (¯ ρ) = AA,ρ¯

dμ ≥

AA,ρ¯

|DrD | dμ =

ρ¯

L (AA,¯ρ ∩ {r = ρ}) dρ = 0

L (ρ) dρ 0

since AA,¯ρ ∩ {r = ρ} = SA,ρ . Given a.e. ρ¯ ∈ [0, ρA,B ] and a loop S˜A,¯ρ ⊂ SA,¯ρ as above, there exists an embedded loop S˜A,¯ρ (0) ∈ TA such that (expν )

−1

(S˜A,¯ρ ) = {¯ ρν(x) : x ∈ S˜A,¯ρ (0)}.

Consider the cylinder (topologically an embedded annulus) (33.68)

A8A,¯ρ  { expν ({rν(x) : x ∈ S˜A,¯ρ (0), r ∈ [0, ρ¯]})}.

This set is the union of the normal geodesics to TA of length ρ¯ ending in S˜A,¯ρ . Let S˜A,¯ρ (ρ)  A8A,¯ρ ∩ TA,ρ , ρ ∈ [0, ρ¯], which is a loop. Define (33.69)

8 ρ¯ (ρ)  Length g(t) (S˜A,¯ρ (ρ)) L

248

33. NONCOMPACT HYPERBOLIC LIMITS

S˜A,ρ(0) S˜A,ρ (ρ)

...

S˜A,ρ

...

SA,ρ

˜A,ρ A AA,ρ

TB

CA,B

TA,ρ TA,ρ

TA

Figure 33.4. The annulus A8A , ρ¯ and the loop S˜A , ρ¯ (ρ). and 8 (¯ A ρ)  Area g(t) (A8A,¯ρ ) =

(33.70)



ρ¯

8 ρ¯ (ρ) dρ. L

0

Clearly 8 ρ¯ (¯ L ρ) ≤ L (¯ ρ)

(33.71) since S˜A,¯ρ ⊂ SA,¯ρ .

Represent D(t) by an embedding ft : D2 → M − MA (t). Regarding the loop ˜ S A,¯ρ , we may assume that among the disjoint union of embedded loops comprising ft−1 (SA,¯ρ ) we have that ft−1 (S˜A,¯ρ ) is an innermost loop in D. In other words, the ˜ 2 in D bounded by the loop ft−1 (S˜A,¯ρ ) does not intersect any of the other disk D loops in ft−1 (SA,¯ρ ). Define the immersed comparison disk ˜ ∪ A8A,¯ρ . ˜  ft (D) D(t)

(33.72)

TA,0 = TA

...

S˜A,ρ−(ρ)

TA,ρ

S˜A,ρ−

comparison disk D

˜

TA,ρ–

˜ ft(D) ...

˜A,ρ– A original disk D

CA,B

˜ Figure 33.5. The original disk D(t) and the comparison disk D(t).

4. INCOMPRESSIBILITY OF BOUNDARY TORI

249

˜ Since the area of D(t) is less than or equal to the area of D(t),  ρ¯  ρ¯ 8 ρ¯ (ρ) dρ. 8 (¯ (33.73) L L (ρ) dρ ≤ A (¯ ρ) ≤ A ρ) = 0

0

In view of (33.59) and since we are in an almost hyperbolic cusp, we have that ˜ there exists tη < ∞ such that for t ≥ tη given any η > 0 and 0 < B < A ≤ A,  d  ρ−ρ 8 ρ¯ (ˆ 8 ρ¯ (¯ eˆ L ρ) ≤ ηeρˆL ρ) for ρˆ ∈ [ρ, ρ¯] ⊂ [0, ρA,B (t)). dˆ ρ Integrating this inequality yields    8   ρ−ρ ¯ 8 L L (ρ) − e (¯ ρ )  ρ¯ = ρ¯

ρ¯

ρ



≤η

  d  ρ−ρ 8 ρ¯ (ˆ ρ) dˆ ρ eˆ L dˆ ρ

ρ¯

8 ρ¯ (¯ eρˆL ρ) dˆ ρ

ρ

 8 ρ¯ (¯ = η eρ¯ − eρ L ρ) , which implies that

  ¯ ¯ 8 ρ¯ (ρ) ≤ eρ−ρ 8 ρ¯ (¯ L 1 + ηe2ρ−ρ¯ eρ−ρ −1 L ρ) .

Thus for any ε > 0, by choosing η = 2εe−ρA,B so that ηe2ρ−ρ¯ ≤ 2ε, we have   ¯ ¯ 8 ρ¯ (¯ 8 ρ¯ (ρ) ≤ eρ−ρ 1 + 2ε eρ−ρ −1 L ρ) . (33.74) L Motivated by (33.60), we have the following property for the almost hyperbolic cusp CA,B ⊂ M. ˜ Lemma 33.34 (Monotonicity-type formula for length). Let 0 < B < A ≤ A, where A˜ is as in (h5) in Subsection 1.1 of this chapter. Then for any ε > 0 sufficiently small, there exists tε < ∞ such that for t ≥ tε and ρ¯ ∈ (0, ρA,B (t)] we have that the function L (ρ) in (33.66) satisfies    ρ¯ eερ¯ d (33.75) L (ρ) dρ ≥ 0. d¯ ρ 1 − e−ρ¯ 0 Proof. In view of the calculation    ρ¯ eερ¯ d e−ρ¯ L (¯ ρ) ln (33.76) +ε− L (ρ) dρ =  ρ¯ , − ρ ¯ d¯ ρ 1−e 1 − e−ρ¯ L (ρ) dρ 0 0 we desire to bound L (¯ ρ) from below. As a consequence of (33.74) and (33.71), we have that for 0 ≤ ρ ≤ ρ¯ < ρA,B ,  ¯ 8 ρ¯ (ρ) ≤ eρ¯ (1 − 2ε) e−ρ + 2εeρ−2ρ L L (¯ ρ) .  ρ¯ 8 ρ¯ (ρ) dρ ≤ (eρ¯ − 1) (1 + ε (eρ¯ − 1)) L (¯ Integrating this, we obtain 0 L ρ). Hence, it follows from (33.73) that  ρ¯ eρ¯ − 1 L (¯ ρ) . L (ρ) dρ ≤ 1 − ε (eρ¯ − 1) 0 This implies that the rhs of (33.76) is nonnegative. Observe that ε > 0 was introduced to enable this nonnegativity. 

250

33. NONCOMPACT HYPERBOLIC LIMITS

Since

 ρ¯ eερ¯ lim L (ρ) dρ = L (0) , ρ→0 ¯ 1 − e−ρ¯ 0 we deduce from Lemma 33.34 and (33.67) the following. ˜ For ε > 0 sufficiently small there Corollary 33.35. Let 0 < B < A ≤ A. exists tε < ∞ such that for all t ≥ tε and all ρ¯ ∈ (0, ρA,B (t)] we have that  ρ¯ eερ¯ eερ¯ L (ρ) dρ ≤ A (¯ ρ) , (33.77) L g(t) (∂D(t)) = L (0) ≤ 1 − e−ρ¯ 0 1 − e−ρ¯ where A (¯ ρ) is defined by (33.66). In particular, L g(t) (∂D(t)) ≤

eερA,B (t) A (ρA,B (t)) . 1 − e−ρA,B (t)

An immediate consequence is that area almost bounds length. Namely,   Corollary 33.36. Let A ∈ 0, A˜ . For any δ > 0 there exist B ∈ (0, A) and Tδ < ∞ depending also on B such that at any t ≥ Tδ we have both ρA,B (t) ≥ 1 − ln(1 − √1+δ )  N and (33.78)

L g(t) (∂D(t)) ≤ (1 + δ) A (N ) ≤ (1 + δ) A (ρA,B (t)) .

Proof. By Remark 33.33, there exist Tδ < ∞ and B ∈ (0, A) such √ that εN ρA,B (t) ≥ N for t ≥ Tδ . Now choose ε > 0 small enough so that e ≤ 1 + δ. √ Then, by Corollary 33.35 and since 1−e1−N = 1 + δ, there exists Tδ ≥ Tδ such that for t ≥ Tδ we have that L g(t) (∂D(t)) ≤

eεN A (N ) ≤ (1 + δ) A (N ) . 1 − e−N



Since A FA (t) = Area g(t) (D (t)) ≥ A (ρA,B (t)) , it follows from (33.78) that for any δ > 0 there exists Tδ < ∞ such that for t ≥ Tδ , L g(t) (∂D(t)) ≤ (1 + δ) A F (t) . So the length is almost less than the area. The following says that somewhere far into the cusp, the length of the intersection of the minimal disk with a torus slice is not too large. ˜ and given ε > 0 sufficiently small, let tε be Lemma 33.37. Given A ∈ (0, A) given by Lemma 33.28. Let B ∈ (0, A) and T2ε < ∞ be as in Corollary 33.36. Suppose there exists t♦ ≥ max{tε , T2ε } such that (33.62) holds. Then there exists Λ ∈ (0, ∞) depending only on ε with the following property. 1 ) and ρ∗ ∈ [ρ# + Λ, ρA,B (t♦ )] there exists ρ♠ ∈ For any ρ# ≥ − ln(1 − √1+2ε ∗ [ρ# , ρ ] such that SA,ρ♠ (t♦ ) = D(t♦ )∩TA,ρ♠ (t♦ ) is a finite disjoint union of smooth embedded loops and (33.79)

Lg(t♦ ) (ρ♠ ) ≤ (1 + 2ε) e−(ρ

∗

−ρ♠ )

Lg(t♦ ) (0) .

Remark 33.38. Note that for ρ# + Λ ≤ ρA,B (t♦ ) to hold, we need B to be sufficiently small.

4. INCOMPRESSIBILITY OF BOUNDARY TORI

251

Proof. All of the following discussion is at time t♦ . There exists ρ♠ ∈ [ρ# , ρ∗ ] such that SA,ρ♠ is a finite disjoint union of smooth embedded loops (ignoring isolated points) and  e−ρ♠ L (ρ♠ ) = inf ∗ e−ρ L (ρ)  0 . ρ∈[ρ# ,ρ ]

We then have

  ∗  ρ ρ# e −e 0 ≤

ρ∗

L (ρ) dρ ≤ A (ρ∗ ) ≤ (1 + ε) L (0) ,

ρ#

where the middle inequality follows from the equivalent of (33.67) and the last inequality is by (33.62). Hence L (ρ♠ ) ≤ eρ♠

(1 + ε) L (0) 1 + ε −(ρ∗ −ρ♠ ) ≤ e L (0) eρ∗ − eρ# 1 − e−Λ

since ρ∗ ≥ ρ# + Λ. Taking Λ < ∞ sufficiently large (i.e., Λ ≥ ln(ε−1 + 2)), we obtain inequality (33.79).  Once again, all of the following discussion is at time t♦ . Let ρ♠ be as in Lemma 33.37. Recall that SA,ρ♠ (t♦ ) = D(t♦ ) ∩ TA,ρ♠ (t♦ ) is a disjoint union of circles and that S˜A,ρ♠ ⊂ SA,ρ♠ is a loop representing a nontrivial primitive element  of ker iρ♠ ∗ , where iρ♠ : TA,ρ♠ → M is the inclusion map. Let S9A,ρ♠ denote an  embedded geodesic loop in TA,ρ♠ representing the same element in π1 TA,ρ♠ as S˜A,ρ♠ and which has minimal length among all such loops. (1) Let AA,ρ♠ be an immersed annulus in TA,ρ♠ bounded by S˜A,ρ♠ ∪ S9A,ρ♠ . Later we shall choose AA,ρ♠ to satisfy the area estimate (33.83) below. (2) Let BA,ρ♠ be the union of those minimal geodesic segments starting from points in TA (t♦ ) and ending at points in S9A,ρ♠ which are also normal to TA (t♦ ). For t♦ sufficiently large we have that BA,ρ♠ ⊂ CA,B is a smooth embedded annulus. (Compare with the more formal definition (33.68) and Figure 33.4.) Length = Lg(t )(0) ♦

sum of Lengths = Lg(t )(ρ♠)

TA,ρ*

♦

...

TA,ρ TA,ρ

TA,ρ

+Λ #

♠

#

TA,0 = TA(t♦)

Figure 33.6. There exists ρ♠ satisfying (33.79).

TA,ρ

A,B

(t♦)

252

33. NONCOMPACT HYPERBOLIC LIMITS

−1 ˜ 2 ˜2 (3) Let D A,ρ♠ ⊂ D be the disk bounded by the loop ft♦ (S A,ρ♠ ). Define the immersed comparison disk 2 ˜ A,ρ (33.80) DA,ρ  ft (D ) ∪ AA,ρ ∪ BA,ρ . ♠

♦

♠

♠

♠

Here we have taken license with the way we defined DA,ρ♠ since this disk is immersed and may not be embedded. One may easily make this definition technically correct. S˜A,ρ

♠

D − CA,ρ

AA,ρ

♠

♠

...

TA,ρ

BA,ρ

SˆA,ρ

♠

♠

♠

Figure 33.7. The comparison disk DA,ρ♠ . By definition, A F (t♦ ) ≤ Area(DA,ρ♠ ).

(33.81)

Now we proceed to estimate Area(DA,ρ♠ ) from above. The following coarse estimate is used to bound the area of AA,ρ♠ . Lemma 33.39. Let T 2 be a flat torus and let [α] ∈ π1 (T ). If in T an embedded geodesic loop L and a smooth embedded loop S 1 both represent [α], then there exists a geodesic loop L parallel to L and an immersed annulus A bounded by S 1 ∪ L with14 Area (A) ≤ Length (L) · Length(S 1 ). a

T2

S1

b L L′

A

a

Figure 33.8. 14 Note

that Length (L ) = Length (L).

b

4. INCOMPRESSIBILITY OF BOUNDARY TORI

253

Proof. The geodesic loop L : [0, 1] → T defines a translation TL (deck transformation) of the universal covering space π : R2 → T given by the vector L˜ (1) − L˜ (0) ∈ R2 , where the path (geodesic line segment) L˜ : [0, 1] → R2 is the 2 lift of L. The quotient R / [TL ] is a cylinder of circumference L0  Length (L)   ˜ L (1) − L˜ (0). That is, R2 / [TL ] = (R/L0 Z) × R. We have a natural covering π ¯ : R2 / [TL ] → T so that π ˆ

R2 −→ R2 / [TL ] #π¯ π  T, where π ˆ is also a projection. The loop L lifts to a “meridian circle” L¯ in the cylinder R2 / [TL ], i.e., a geodesic loop which is perpendicular to each geodesic line in R2 / [TL ]. Since S 1 ⊂ T is a smooth embedded loop representing the same homotopy 1 class as L, it lifts to a smooth embedded loop S¯ ⊂ R2 / [TL ] of the same length 1 L1  Length(S¯ ) = Length(S 1 ). 1 Clearly S¯ is contained in a set of the form

Ω  (R/L0 Z) × [a, a + L1 ] ⊂ (R/L0 Z) × R, where a ∈ R. The area of this set is equal to L0 L1 . Clearly there exists a geodesic loop L , parallel to L, such that it has a lift L which is contained in Ω (in fact, we can prescribe any meridian circle in R2 / [TL ] as a lift). Thus there exists an 1 immersed annulus in R2 / [TL ] bounded by S¯ ∪ L with area at most L0 L1 . This immersed annulus projects to an immersed annulus in T bounded by S 1 ∪ L with  area at most L0 L1 . Recall that the pointed hyperbolic limit is denoted by (H, h, x∞ ). Corresponding to TA (t♦ ) is a topological end E in H and an exact hyperbolic cusp E ([0, ∞) × TE2 , dr 2 + e−2r gflat ). Corresponding to S9A,ρ♠ , let L denote the embedded geodesic loop in the flat torus TA = {rE,A } × TE of area A; let L0  E (L). Since we are in an almost hyperbolic piece and by (33.59), e−rE,A Lengthgflat for any ε > 0 there exists t1ε < ∞ such that the length of S9A,ρ satisfies ♠

(33.82)

Lengthg(t♦ ) (S9A,ρ♠ ) ≤ (1 + ε) e

−ρ♠

L0

provided t♦ ≥ t1ε .

Hence, by Lemma 33.39, (33.83)

Areag(t♦ ) (AA,ρ♠ ) ≤ (1 + ε)2 e−ρ♠ L0 L (ρ♠ )

provided t♦ ≥ t1ε .

Using again that we are in an almost hyperbolic piece and now using (33.60), we obtain the estimate (33.84)  2 2 Areag(t♦ ) BA,ρ♠ ≤ (1 + ε) 1 − e−ρ♠ L0 ≤ (1 + ε) L0 provided t♦ ≥ t1ε . Hence, by (33.81),

 A (ρ♠ ) ≤ (1 + ε)2 1 + e−ρ♠ L (ρ♠ ) L0 .

On the other hand, since ρ♠ ≥ ρ# , by (33.78) we also have that at time t♦ , (33.85)

L (0) ≤ (1 + 2ε) A (ρ♠ ) .

254

33. NONCOMPACT HYPERBOLIC LIMITS

It follows from Lemma 33.37 that

 L (0) ≤ (1 + ε)2 (1 + 2ε) 1 + e−ρ♠ L (ρ♠ ) L0   ∗ 2 ≤ (1 + ε) (1 + 2ε) 1 + (1 + 2ε) e−ρ L (0) L0 .

By choosing A small, we can make L0 as small as we like. This implies that L (0) is as small as we like: ˜ Proposition 33.40 (Length is small). For any ε1 > 0, there exists A0 ∈ (0, A) such that for any A ≤ A0 we have the following. Let ε be chosen sufficiently small and let tε and T2ε be as in Lemma 33.37. Choose t1ε < ∞ so that the inequalities (33.82), (33.83), and (33.84) hold for t ≥ t1ε . If t♦ ≥ max{tε , T2ε , t1ε } is such that (33.62) holds, then any area minimizing disk D (t♦ ) in FA (t♦ ) satisfies (33.86)

L g(t♦ ) (∂D (t♦ )) ≤ ε1 .

We are now ready to prove the following. Lemma 33.41. Let ε > 0 be sufficiently small. If A is sufficiently small and t♦ ≥ max{tε , T2ε , t1ε } is such that (33.62) holds, then d+ A FA (t♦ ) ≤ −2π + ε. dt Proof. Applying Proposition 33.40 with ε1 = ε/3 to (33.58) yields d+ A F (t♦ ) ≤ −2π + (1 + 2ε) L g(t♦ ) (∂D(t♦ )) − (1 − ε) A F (t♦ ) dt ≤ −2π + ε.



By this and Lemma 33.30, we obtain Proposition 33.24. At this point, to complete the proof of Hamilton’s theorem, Theorem 32.2, it only remains to prove Propositions 33.11 and 33.12. We shall accomplish this in the next chapter using the inverse function theorem. 5. Notes and commentary §1. Definition 33.1 is Definition 9.2 in [143]. Proposition 33.12 is Theorem 9.1 in Hamilton [143]. §2. Proposition 33.13 is Theorem 9.4 from [143]. Proposition 33.14 is Theorem 9.3 in [143]. §3. We followed §10 of [143] closely in proving the stability of hyperbolic limits of 3-dimensional nonsingular solutions of the Ricci flow. Another, albeit one more complicated to justify, way to obtain (33.26) is as follows. Observe that for i sufficiently large, we have that every totally umbillic ˜ → (Vi , g (βi )| ) ˜i . Since Φi : (U ˜i , h) ˜ 3 is contained in U CMC torus of area A/2 in H Vi is an almost isometry, we have that (Vi , g (βi )|Vi ) is an almost hyperbolic piece in  3 M , g (βi ) and is bounded by almost totally umbillic CMC tori of area approximately A/2. Since   Fi (βi ) (HA ), g (βi )|M3 (βi ) i,A

is also an almost hyperbolic piece in (M, g (βi )), now bounding almost totally umbillic CMC tori of area approximately A, we conclude that Fi (βi ) (HA ) ⊂ Vi .

5. NOTES AND COMMENTARY

255

§4. This section follows §11 and §12 of Hamilton [143]. For background and more advanced topics on minimal surfaces we refer the reader to Colding and Minicozzi [86], Lawson [183], and Osserman [300]. The well-known Smith conjecture says that the fixed point set of a periodic orientation-preserving diffeomorphism of S 3 is either empty or an unknotted circle (see p. 4 of Morgan and Bass [250]). A different proof of the most important case of Schoen and Yau’s positive energy theorem, using spinors, was given soon thereafter by Witten [437] (see also Parker and Taubes [305]). As we mentioned in the introduction to this chapter, see §8 of Perelman [313] and §93.1 of Kleiner and Lott [161] for other approaches to the proof of the incompressibility of the cuspidal tori. Proposition 33.24 is Theorem 11.1 in [143]. We have the following correspondences with results in §12 of [143]. Lemma 33.34 is Theorem 12.1 in [143]. For Corollary 33.35, see Corollary 12.2 in [143]. For Corollary 33.36, see Corollary 12.4 in [143]. For Lemma 33.37 see Lemma 12.5 in [143]. Lemma 33.39 is Corollary 12.7 in [143].

CHAPTER 34

Constant Mean Curvature Surfaces and Harmonic Maps by IFT The mist across the window hides the lines. – From “Steppin’ Out” by Joe Jackson

The goal of this chapter is to prove the existence of constant mean curvature (CMC) surfaces (see Proposition 34.1 below) and harmonic maps (see Proposition 34.13 below). These results were used in Chapter 33 for the case of nonsingular solutions forming complete noncompact hyperbolic limits. The proofs we give below rely on the implicit function theorem (IFT). In the realm of differential geometry, the IFT often enables one to obtain canonical geometric structures from almost canonical geometric structures. One of the keys for the successful use of the IFT is to understand the kernel of the linearization of the geometric equation that one is considering. In §1 we obtain CMC sweep-outs by tori of almost hyperbolic cusps. In §2 we consider harmonic maps near the identity map of the unit n-sphere. In §3 we consider a compact manifold (Mn , g) with negative Ricci curvature and concave boundary. For any metric g˜ sufficiently close to g we prove the existence of a harmonic diffeomorphism from (M, g) to (M, g˜) near the identity map. In §4 we prove the existence of isometries near almost isometries of truncated finite-volume hyperbolic 3-manifolds.

1. Constant mean curvature surfaces In this section we consider two similar applications of the IFT: (1) the existence of CMC hypersurfaces in almost hyperbolic cusps, (2) the existence of CMC hyperspheres in almost standard cylinders. 1.1. Sweep-outs of almost hyperbolic cusps by CMC tori. The following is a restatement of Proposition 33.11. Proposition 34.1 (Existence of a CMC sweep-out in almost hyperbolic cusps). Given any [a, b] ⊂ R and ε0 > 0, if a C ∞ Riemannian metric g on [a, b] × V n−1 is sufficiently close in the C 2,α -topology to a hyperbolic cusp metric gcusp = dr 2 + e−2r gflat for some α ∈ (0, 1), then there exists a smooth 1-parameter family of C ∞ CMC (with respect to g) hypersurfaces which sweep out (a + ε0 , b − ε0 ) × V and which are close in the C 2,α -norm to the standard slices {r} × V (see Definition K.9 in Appendix K). 257

258

34. CMC SURFACES AND HARMONIC MAPS BY IFT

Proof. Step 1. Linearization of the mean curvature of graphs in a cusp. Recall that the embedded slices {r} × V ⊂ (a, b) × V are CMC hypersurfaces with respect to gcusp . Consider those hypersurfaces which are graphs over V. That is, we associate to a function ϕ : V → (a, b) the graphical hypersurface Vϕ  {(ϕ (x) , x) : x ∈ V} in (a, b) × V. When ϕ = r is a constant function, we write Vr  {r} × V. Let g˜ be a C 2 metric on [a, b] × V and let ϕ ∈ C 2 (V; (a, b)). Let Hg˜ (Vϕ ) denote the mean curvature of the graph Vϕ , with respect to the metric g˜, where Hg˜ is  ∂  > 0.  to Vϕ satisfying N, defined using the choice of unit normal N ∂r 2 For any f ∈ C (V), corresponding to the normal deformation of the hypersur , the variation of the mean curvature of Vϕ is face Vϕ with velocity vector field f N given by 2

 ) = Δg˜,ϕ f + (|II| + Rc g˜ (N,  N  ))f, D (Hg˜ (Vϕ )) (f N

(34.1)

where Δg˜,ϕ denotes the Laplacian on Vϕ with respect to the induced metric, where II denotes the second fundamental form of Vϕ with respect to g˜. Equation (34.1) is a straightforward generalization of equation (B.17) for the mean curvature flow in Appendix B of Part I. Let r ∈ (a, b). When g˜ = gcusp and Vϕ = Vr , (34.1) yields   ) = Δr f, (34.2) D Hgcusp (Vr ) (f N where Δr is with respect to the metric e−2r gflat on Vr since in this case  = ∂ , II = − gcusp | , and sect (gcusp ) ≡ −1. N Vr ∂r Step 2. Suitable map between Banach spaces to apply the IFT. The existence of CMC hypersurfaces amounts to solving Hg (Vϕ ) + c = 0 for some constant c. So as to be able to invoke the IFT, we now address the issue of the kernel of Δr in (34.2), which comprises the constant functions. Let B denote the Banach space of C 2,α symmetric 2-tensors on (a, b) × V with the usual C 2,α -norm, with respect to gcusp (see §2 of Appendix K for the definition of H¨ older spaces). Let Met be the open cone of all positive-definite tensors in B. Denote    2,α   2,α   ˆ C (V; (a , b ))  ϕ ∈ C (V; (a , b )) : ϕ (x) dσgflat (x) = 0 (34.3)

V









for a , b ∈ R ∪ {−∞, ∞} with a < b . Let ε0 ∈ (0, b−a 2 ). Define the map Φ : Met × (a + ε0 , b − ε0 ) × Cˆ 2,α (V; (−ε0 , ε0 )) × R → C α (V; R) by (34.4)

Φ(g, r, ϕ, c)  Hg (Vr+ϕ ) + c.

The domain of Φ is an open subset of a product Banach space and Φ is continuously differentiable. Note also that Φ(gcusp , r, 0, n − 1) = 0.

1. CONSTANT MEAN CURVATURE SURFACES

259

By (34.2), the linearization of Φ at (gcusp , r, 0, n − 1), with respect to the variables ϕ and c, is given by the continuous map (34.5) LΦ : Cˆ 2,α (V; R) × R → C α (V; R) , where LΦ (f, u)  D(ϕ,c) Φ(gcusp , r, 0, n − 1) (f, u) = Δr f + u.  The kernel of LΦ is trivial. Indeed, if Δr f + u = 0, then V Δr f dσgflat = 0 implies  that u = 0 and Δr f = 0; since V f dσgflat = 0, we have that f ≡ 0. Given h ∈ C α (V; R), by Hodge theory and Schauder theory, there exists a unique f ∈ Cˆ 2,α (V; R) such that (34.6)

Δr f + havg = h, where havg denotes the average of h on (V, gflat ). Thus LΦ is onto. By the standard Schauder estimate, we have f C 2,α (V;R) + |havg | ≤ hC α (V;R) = LΦ (f, havg )C α (V;R) . Thus LΦ and (LΦ)−1 are continuous bijections of Banach spaces. Step 3. Applying the IFT to prove the existence of CMC hypersurfaces. Given any sufficiently small ε0 > 0 and any r ∈ (a + ε0 , b − ε0 ), by Theorem K.4, there exists a neighborhood U of gcusp in Met such that for all g ∈ U, there exist a unique function ϕr ∈ Cˆ 2,α (V; (−ε0 , ε0 )) and a unique constant cr ∈ R close to n − 1 such that Vr+ϕr is a CMC (with respect to g) hypersurface with mean curvature equal to −cr . We have that Vr+ϕr is near the slice {r} × V and that (ϕr , cr ) depends smoothly on r and g (see Theorem K.3). Moreover, since r is contained in an interval with compact closure, we may assume that U is independent of r. Step 4. Finishing the proof. Now suppose that g ∈ U is C ∞ . Since ϕr ∈ 2,α and Vr+ϕr is a CMC hypersurface, we conclude that ϕr ∈ C ∞ by applying C elliptic theory to Hg (Vr+ϕr ) + cr = 0. Thus we have obtained a family of C ∞  CMC hypersurfaces {Vr+ϕr }r∈(a+ε0 ,b−ε0 ) in [a, b] × V. Since V ϕr dσgflat = 0, the hypersurfaces Vr+ϕr are distinct. Now, for r close enough to a+ε0 we have r+ϕr ≤ a+2ε0 and for r close enough to b−ε0 we have r +ϕr ≥ b−2ε0 . Thus, for each x ∈ V, the function r → r +ϕr (x), r ∈ (a + ε0 , b − ε0 ), takes every value in the interval [a + 2ε0 , b − 2ε0 ]. We conclude that the Vr+ϕr sweep out [a + 2ε0 , b − 2ε0 ] × V. This completes the proof of the proposition.  Remark 34.2. Let A ∈ (0, ∞) be the area of one of the slices of ((a, b)×V, gcusp ). Let g (t) be a smooth family of metrics on [a, b] × V with each metric sufficiently close to gcusp . Let St denote the corresponding 1-parameter family of C ∞ CMC hypersurfaces with area A (with respect to g (t)) given by Proposition 34.1. From ∂ of the area (with respect to gcusp ) of the slices {r} × V is negative, the fact that ∂r it follows that St depends smoothly on t. Remark 34.3 (Is the family Vr+ϕr a foliation?). One should be able to prove that the family Vr+ϕr is a foliation (in this regard, see Ye [448] for related work); however, this is not necessary for the applications to Ricci flow. Problem 34.4. Prove that for g sufficiently close to gcusp , a neighborhood of [a + 2ε0 , b − 2ε0 ] × V is foliated by {Vr+ϕr }r∈(a+ε0 ,b−ε0 ) .

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1.2. Existence of CMC spheres in necks. Analogous to Proposition 33.11 we have the following consequence of the IFT for necks. This result may be of interest for studying neck formation in Ricci flow. Proposition 34.5 (Existence of CMC spheres in necks). Given [a, b] ⊂ R and in the C 2,α -topology ε > 0, if a C ∞ metric g on [a, b] × S n−1 is sufficiently close  n−1 2 , gsph is the unit to the standard cylinder metric gcyl = dr + gsph , where S sphere, then there exists a smooth 1-parameter family of CMC (with respect to g) hypersurfaces which sweep out [a + ε, b − ε] × S n−1 and which are C 2,α -close to the standard slices. Proof. The proof is essentially exactly the same as in Proposition 34.1, using Theorem K.4, except for the following:   (1) V n−1 , gflat is replaced by S n−1 , gsph . (2) The hyperbolic cusp metric gcusp = dr 2 + e−2r gflat is replaced by the standard cylinder metric gcyl = dr 2 + gsph . (3) We have for gcyl that   ∂ ∂ (34.7) Rc , = 0 and |II|2 = 0 ∂r ∂r for the slices {r} × S n−1 .  Let Sϕ  (ϕ (x) , x) : x ∈ S n−1 and define Φ(g, r, ϕ, c)  Hg (Sr+ϕ ) + c analogously to (34.4). Note that Φ(gcyl , r, 0, 0) = 0. Now using (34.1), we find that just as in (34.6), we have D(ϕ,c) Φ(gcyl , r, 0, 0) (f, u) = Δf + u. There are no other essential differences in the proof; we leave it to the reader to verify this.  2. Harmonic maps near the identity of S n Isometries of Riemannian manifolds are harmonic maps. However, for the unit sphere, there are other harmonic self-maps. For example, Smith [387] proved that for n ≤ 7 there exists a harmonic map of S n to itself in every homotopy class. For maps near the identity, we have the following. Proposition 34.6 (Harmonic maps of the unit S n near id). For n ≥ 3 a harmonic map of the unit n-sphere S n to itself which is sufficiently close to the identity in the C 2,α -norm must be an isometry. A harmonic map of the round 2sphere to itself which is sufficiently close to the identity in the C 2,α -norm must be a conformal diffeomorphism. In the rest of this section we shall prove this result using the IFT. Note that when n = 2, this is a classical result since any nonconstant harmonic map from a 2-sphere into a Riemannian manifold of any dimension at least 2 is a conformal map (in fact, such a map is a branched conformal minimal immersion by Corollary 1.7 in Sacks and Uhlenbeck [342]).

2. HARMONIC MAPS NEAR THE IDENTITY OF S n

261

2.1. Linearization and its kernel of the map-Laplacian on S n . Let g and g˜ be C ∞ Riemannian metrics on a manifold Mn and consider a diffeomorphism F : (M, g) → (M, g˜). Its map-Laplacian Δg,˜g F , defined by (K.19), is a section of the vector bundle F ∗ (T M) → M. To remove the dependence of this bundle on F , we pull back the map-Laplacian to get a vector field on M:  (34.8) F ∗ (Δg,˜g F )  F −1 ∗ (Δg,˜g F ) . In local coordinates, the map-Laplacian is given by b

c

˜ abc ◦ F ) ∂F ∂F , (Δg,˜g F )a = Δg (F a ) + g ij (Γ ∂xi ∂xj where Δg denotes the Laplacian, with respect to g, acting on functions, and where ˜ a denote the Christoffel symbols of g˜. Hence, its pull-back is the Γ bc 

−1 k  −1 ∂ F ( F (34.10) (Δg,˜g F ))k = ◦ F Δg (F a ) ∗ ∂y a

 ∂ F −1 k  ∂F b ∂F c ij ˜ a Γbc ◦ F +g ◦F . a ∂y ∂xi ∂xj (34.9)

The linearization Lg : C ∞ (T M) → C ∞ (T M) at the identity map of the oper −1 ator F → F (Δg,g F ) is given by ∗ Lg (V )  Δg V + Rcg (V ) ,

(34.11)

where Δg is the rough Laplacian (see (K.43) in Appendix K). We next investigate the obstacles to inverting Lgsph for the unit n-sphere (S n , gsph ). Let Δd  − (dδ + δd) denote the Hodge–de Rham Laplacian acting on differential forms. We say that λ is an eigenvalue of Δd with eigenform α if [1] (Δd + λ) α = 0.1 Let λk (S n ), k ≥ 0, denote the k-th eigenvalue of Δd acting on n 1-forms on (S , gsph ). By identifying vector fields with their metrically dual 1-forms on (S n , gsph ), we see that the eigenvalues of the linearization Lgsph : C ∞ (T ∗ S n ) → C ∞ (T ∗ S n ) satisfy (34.12) [1] [1] λk (Lgsph ) = λk (Δgsph + Rcgsph ) = λk (Δd + 2 (n − 1)) = λk (S n ) − 2 (n − 1) , where we used that Rcgsph = (n − 1) gsph and that Δg α = Δd α + Rc (α) for any 1-form α. [1] [1] Let c λk (S n ) and cc λk (S n ) denote the k-th eigenvalue of Δd acting on closed and co-closed 1-forms on (S n , gsph ), respectively. Regarding the rhs of (34.12), by taking p = 1 in (K.54), we have that c [1] λk

(34.13)

(S n ) = (k + 1) (n + k) ,

so that the two lowest eigenvalues of Δd acting on closed 1-forms are (34.14)

c [1] λ0

(S n ) = n

and

c [1] λ1

(S n ) = 2 (n + 1) .

On the other hand, taking p = 1 in (K.55), we have that (34.15) 1 For

cc [1] λk

(S n ) = (k + n − 1) (k + 2) ,

eigenvalues of the Laplacian, the geometer’s convention is opposite that of the analyst’s.

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so that the two lowest eigenvalues of Δd acting on co-closed 1-forms are (34.16)

cc [1] λ0

(S n ) = 2 (n − 1)

and

cc [1] λ1

(S n ) = 3n.

Thus, by the Hodge decomposition theorem, Theorem K.21, we conclude that for Δd acting on 1-forms on (S n , gsph ), the three lowest eigenvalues are n, 2 (n − 1), and 2 (n + 1), in increasing order. In terms of the linearization Lgsph = Δd + 2 (n − 1), this says that (34.17a)

λ0 (Lgsph ) = 2 − n,

(34.17b)

λ1 (Lgsph ) = 0,

(34.17c)

λ2 (Lgsph ) = 4

(not counting multiplicity and where λ1 = λ0 when n = 2). Note the following: Remark 34.7. (1) The eigenspace of Lgsph with eigenvalue 2 − n is the eigenspace of Δd for closed 1-forms with eigenvalue n. (2) The eigenspace of Lgsph with eigenvalue 0 is the eigenspace of Δd for co-closed 1-forms with eigenvalue 2 (n − 1). (3) All other eigenspaces of Lgsph have positive corresponding eigenvalue. We now have the following. Lemma 34.8 (The kernel of Lgsph ). For (S n , gsph ) we have the following: (1) If n ≥ 3, then ker(Lgsph ) is the space of 1-forms dual to the Killing vector fields of gsph . (2) If n = 2, then ker(Lgsph ) is the space of 1-forms dual to the conformal Killing vector fields of gsph . Proof. For (the dual 1-form of) any vector field W on a closed manifold (M, g) we may integrate by parts and commute covariant derivatives to obtain Yano’s formula:   1 (34.18) |∇i Wj + ∇j Wi |2 dμ = (|∇i Wj |2 + ∇i Wj ∇j Wi )dμ 2 M M  = (−Wj ∇i ∇i Wj − Wj ∇i ∇j Wi ) dμ M  Δg W + Rc(W ), W dμ =− M  + |div(W )|2 dμ. M

It follows from this that 2      1 2  − (34.19) Lg (W ) , W  dμ = ∇i Wj + ∇j Wi − div (W ) gij  dμ  2 M n M   2 |div (W )|2 dμ. − 1− n M Regarding the rhs, recall that ∇i Wj + ∇j Wi − n2 div (W ) gij = 0 if and only if W is a conformal Killing vector field, i.e., an infinitesimal conformal diffeomorphism.

2. HARMONIC MAPS NEAR THE IDENTITY OF S n

263

First, we prove part (2). If n = 2, then equation (34.19) implies   1 Lg (W ) , W  dμ = |∇i Wj + ∇j Wi − div (W ) gij |2 dμ, − 2 M M so that the linearized operator Lg is nonpositive with its kernel equal to the space of conformal Killing vector fields. Second, we prove part (1). Suppose that n ≥ 3 and V ∈ ker(Lgsph ). By Remark 34.7(2), we have that V is a co-closed 1-form, so that div (V ) = 0. So by (34.19) we have 2 sph div (V ) (gsph )ij = 0. (LV gsph )ij = ∇sph i Vj + ∇j Vi = n Conversely, since isometries are harmonic maps, we have the more general fact that an infinitesimal isometry satisfies the linearized harmonic map equation at the identity. We can also see this by a short calculation. Suppose that V is a Killing vector field on a Riemannian manifold (Mn , g). Then div (V ) = 12 trg (LV g) = 0. Since (34.20) 0 = div (LV g)j = ∇i ∇i Vj + ∇i ∇j Vi = (Δg V )j + ∇j div (V ) + Rc (V )j , we have that Lg in (34.11) satisfies (34.21)

Lg (V ) = Δg V + Rc (V ) = 0; 

i.e., V ∈ ker (Lg ).

2.2. Parametrizing self-maps of S n near the identity. Now we set up suitable Banach spaces for the IFT argument. Define the open subset (34.22)

O  {V ∈ T S n : |V | < π} ⊂ T S n .

For each x ∈ S n , the restriction of the exponential map (34.23)

g

expxsph : O ∩ Tx S n → S n − {−x}

is a diffeomorphism. Let C k,α (T S n ) denote the Banach space of C k,α sections of the tangent bundle of S n , where k ≥ 0 and α ∈ (0, 1). Define the open subset  (34.24) C k,α (O)  σ ∈ C k,α (T S n ) : |σx | < π for x ∈ S n . Let C k,α (S n , S n ) denote the space of C k,α self-maps of S n . We define Ψ : C k,α (O) → C k,α (S n , S n ) by (34.25)

g

Ψ (σ) (x)  expxsph (σx ) .

The map Ψ parametrizes the space of all C k,α maps of S n which do not take any point to its antipodal point. The Lie group of isometries of the unit n-sphere (S n , gsph ) is given by the orthogonal group  Isom (S n ) ∼ = O (n + 1) = A ∈ GL (n + 1) : At = A−1 . The real vector space KV (S n ) of Killing vector fields is its Lie algebra of infinitesimal isometries; i.e.,  ∼ o (n + 1) = B ∈ gl (n + 1) : B t = −B . KV (S n ) =

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Of course, KV (S n ) ⊂ C ∞ (T S n ) ⊂ C k,α (T S n ) for any k and α. Define a Banach subspace of C k,α (T S n ) by  ⊥ (34.26) KV (S n )k,α  σ ∈ C k,α (T S n ) : σ, τ L2 = 0 for all τ ∈ KV (S n ) ,  where σ, τ L2  S n σ, τ gsph dμgsph . Define the map 2,α (S n , S n ) Φ : Isom (S n ) × KV (S n )⊥ 2,α → C

by (34.27)

 g Φ (f, σ) (x) = expfsph (x) σf (x) .

Note that if |σ| < π, then Φ (f, σ) (x) = Ψ (σ) (f (x)). Since Φ (f, 0) = f and g Φ (idS n , σ) (x) = expxsph (σx ), the linearization of Φ at (idS n , 0) is equal to the “identity map”; i.e., 2,α DΦ(idS n ,0) : KV (S n ) × KV (S n )⊥ (T S n ) 2,α → C

is given by (34.28)

DΦ(idS n ,0) (τ, υ) = τ + υ.

In particular, DΦ(id   S2n ,0) is invertible. Now let Conf S be the Lie group of conformal diffeomorphisms of S 2 , gsph ,  2 let CK S be the real vector space of conformal Killing vector fields on S 2 , gsph , and let    ⊥ CK S 2 k,α  σ ∈ C k,α T S 2 : σ, τ L2 = 0 for all τ ∈ CK(S 2 ) . With this set-up, by the inverse function theorem, we have Lemma 34.9 (Parametrization of maps near the identity). (1) If n ≥ 2, then there exists a neighborhood of (idS n , 0) in Isom (S n ) × ⊥ KV (S n )2,α that is mapped diffeomorphically by Φ onto a neighborhood of idS n in C 2,α (S n , S n ). (2) If n = 2, we also have the following. There exists a neighborhood  of (idS 2 , 0) which is mapped diffeomorphically by the map Φ : Conf S 2 ×   ⊥ CK S 2 2,α → C 2,α S 2 , S 2 defined by (34.27) onto a neighborhood of idS 2 . We leave it as an exercise to prove part (2). 2.3. Proof of Proposition 34.6. Let F : (Mn , g) → (N n , h) be a diffeomorphism between closed Riemannian manifolds. Define I : T M → T M to be the unique bundle isomorphism such that V, W F ∗ h = I(V ), W g for all V, W ∈ Tx M, x ∈ M. In local coordinates, we have I(V )i = g ij (F ∗ h)jk V k . We shall use the following result: Lemma 34.10. If F : (Mn , g) → (N n , h) is a diffeomorphism between closed Riemannian manifolds, then for any Killing vector field X on (M, g) we have that    −1   I (F )∗ (Δg,h F ) , X L2 (g) = I (F −1 )∗ (Δg,h F ) , Xg dμg = 0. M

2. HARMONIC MAPS NEAR THE IDENTITY OF S n

265

Furthermore, if n = 2, then the above formula is true for any conformal Killing vector field X. id

F

Proof. By factoring F as (M, g) → (M, F ∗ h) → (N , h) and by the naturality of the map-Laplacian, we have (F −1 )∗ (Δg,h F ) = Δg,F ∗ h id .

(34.29)

Denoting g˜ = F ∗ h, we claim that

  1 Δg,F ∗ h id = divg g˜ − d (tr g g˜) , 2

(34.30)

where α denotes the dual vector field of a 1-form α, with respect to g˜. To see this, we may apply Corollary 3.19 in Volume One to obtain in local coordinates that   1 g )kij = g ij g˜k (∇i g˜j + ∇j g˜i − ∇ g˜ij ) . (Δg,˜g id)k = g ij −Γ (g)kij + Γ (˜ 2 By (34.29), (34.30), and integrating by parts,     1 (F −1 )∗ (Δg,h F ), XF ∗ h dμg = divg g˜ − d (tr g g˜) (X)dμg 2 M M  1 =− ˜ g , LX g − div (X) gg dμg . 2 M If n ≥ 3 and X is a Killing vector field, then the rhs is zero since LX g = 0 and div (X) = 0. If n = 2 and X is a conformal Killing vector field, the rhs is zero  since LX g − div (X) g = 0. With the aid of the above lemmas, we now complete the Proof of Proposition 34.6 when n ≥ 3. Consider the map ⊥

⊥

F : Isom (S n ) × KV (S n )2,α → KV (S n )0,α defined by

   F (f, σ)  I (Φ (f, σ)−1 )∗ Δgsph ,gsph (Φ (f, σ)) ,

where I satisfies V, W Φ(f,σ)∗ gsph = I(V ), W gsph for all V, W ∈ Tx S n . We leave it to the reader to check that F (f, σ) ∈ C α (T S n ); Lemma 34.10 tells us that it is ⊥ an element of KV (S n )0,α . For each f ∈ Isom (S n ), we have F (f, 0) = 0. Let ⊥

⊥

D2 F(f,σ) : KV (S n )2,α → KV (S n )0,α denote the linearization of F at (f, σ) with respect to the second factor. By Lemma 34.8(1) and Δgsph ,gsph idS n = 0, we have that  D2 F(id n ,0) = Lg  = (Δd + 2 (n − 1))| n ⊥ n ⊥ S

sph

KV(S )2,α

KV(S )2,α

is injective. Since D2 F(idS n ,0) is self-adjoint with respect to the appropriate Hilbert spaces, by Fredholm theory, we have that D2 F(idS n ,0) is invertible. Hence, by the IFT, for each f ∈ Isom (S n ) sufficiently close to idS n , we have that σ ≡ 0 is the unique vector field near 0 and in KV (S n )⊥ such that F (f, σ) = 0. Since F (f, σ) = 0 is equivalent to Φ (f, σ) being a harmonic self-map of (S n , gsph ) and by Lemma 34.9, we conclude that any harmonic map sufficiently close to idS n must be an isometry. 

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34. CMC SURFACES AND HARMONIC MAPS BY IFT

Remark 34.11. By applying the above proof to the n = 2 case, one can show that any harmonic map sufficiently close to idS 2 must be a conformal diffeomorphism, provided we ⊥

(1) replace Isom (S n ), KV (S n ), and KV (S n )2,α by Conf(S 2 ), CK(S 2 ), and  ⊥ CK S 2 2,α , respectively, and (2) use Lemma 34.8(2) instead of Lemma 34.8(1). 3. Existence of harmonic maps near the identity of manifolds with negative Ricci curvature In this section, using the IFT, we prove the existence of a harmonic map near an approximate isometry of Riemannian manifolds with negative Ricci curvatures and concave boundaries. 3.1. Statement of the main result. Let (Mn , g) and (N m , h) be compact C ∞ Riemannian manifolds with nonempty boundaries ∂M and ∂N , respectively. Definition 34.12. We say that a harmonic map F : (M, g) → (N , h) satisfies the normal boundary condition if (1) F (∂M) ⊂ ∂N and (2) F∗ (Ng ) is normal to ∂N with respect to h, where Ng denotes the unit inward normal to ∂M with respect to g. In this section we shall prove the following result (stated earlier as Proposition 33.12), which is used to study noncompact hyperbolic limits. Proposition 34.13 (Existence of harmonic maps near the identity). Suppose that (Mn , g) is a compact C ∞ Riemannian manifold with negative Ricci curvature and concave boundary ∂M; i.e., II (∂M) ≤ 0. For any C ∞ metric g˜ sufficiently close to g in the C 2,α -topology with α ∈ (0, 1), there exists a unique C ∞ harmonic diffeomorphism F : (M, g) → (M, g˜) satisfying the normal boundary condition and C 2,α -close to the identity map. Furthermore, if we replace g˜ by a family g˜(t) depending smoothly on t, then the corresponding harmonic diffeomorphisms F (t) depend smoothly on t. We wish to solve the equation Δg,˜g F = 0 with the normal boundary condition, where F is a diffeomorphism from (M, ∂M) to itself. Given a tangent vector W to M at a point in ∂M, let W and W⊥ denote the tangential and normal components of W with respect to g˜, respectively. Define the map   (34.31) Φ (˜ g , F )  F −1 ∗ (Δg,˜g F ) , F −1 ∗ (F∗ (N )) ∈ Ξ(M) × Ξ(∂M), where Ξ(M) and Ξ(∂M) denote spaces of vector fields on M and ∂M, respectively. We have that F : (M, g) → (M, g˜) is a harmonic map satisfying the normal boundary condition if and only if (34.32)

Φ (˜ g , F ) = (0, 0) .

We shall apply the IFT to solve this equation and prove the proposition.

3. HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Rc < 0

267

3.2. The linearization of Φ and the function spaces for the IFT argument. We first assume that all quantities, such as maps and sections of vector bundles, are smooth enough to obtain the formulas. By (K.43) and (K.44) in Appendix K, the linearization of Φ (˜ g , F ) at (g, id) and with respect to the second variable F is given by (34.33) ((D2 )(g,id) Φ) (V ) = (Δg V + Rc (V ) , (∇N V ) − II(V )) ∈ Ξ(M) × Ξ(∂M) for vector fields V on M with V⊥ = 0 on ∂M. Note that for vector fields V and W on M with V⊥ = 0 and W⊥ = 0 on ∂M, we have by integrating by parts that   (34.34) (Δg V + Rc(V )) · W dμ − ((∇N V ) − II(V )) · W dσ M ∂M    =− ∇V, ∇W  dμ + II(V, W )dσ + Rc(V, W )dμ, M

M

∂M

where we have used (∇N V ) · W = (∇N V ) · W and where dσ is the induced volume form on ∂M. Hence, if V ∈ ker((D2 )(g,id) Φ), then by taking W = V in the above equation, we have    2 (34.35) − |∇V | dμ + II(V, V )dσ + Rc(V, V )dμ = 0. M

∂M

M

Next, we define the  function spaces for the IFT argument that we shall apply to 2 ∗ the map Φ. Let C k,α S+ T M denote the Banach manifold of C k,α Riemannian metrics on M. Let C k,α (M; ∂M) denote the Banach manifold of C k,α diffeomork,α phisms of (M, ∂M) to itself. Let C (T M) denote the Banach space of C k,α vector fields U on M with U⊥ = 0 on ∂M. Let C k,α (T (∂M)) denote the Banach space of C k,α vector fields on ∂M. (See §2 of Appendix K for more details on the definitions of these spaces.) Consider the map  2 ∗ T M × C 2,α (M; ∂M) → C α (T M) × C 1,α (T (∂M)) (34.36) Φ : C 2,α S+ defined by (34.31). The linearization of Φ at (g, id) and with respect to the second variable F , (34.37)

2,α (D2 )(g,id) Φ : C (T M) → C α (T M) × C 1,α (T (∂M)),

is given by (34.33). Now assume that Rc < 0 on M and II (∂M) ≤ 0. In the next two subsections we shall show that (D2 )(g,id) Φ in (34.37) is an invertible operator (see Lemma 34.17 below). To motivative this, we observe some formal calculations. First, if V satisfies (34.35), then V ≡ 0. That is, the formal kernel (34.38)

forker((D2 )(g,id) Φ) = 0

is trivial. Second, by (34.34) we have the following formal self-adjointness property for the operator Lg (V ) = Δg V + Rc (V ): (34.39)

Lg (V ) , W L2 (T M) = V, Lg (W )L2 (T M)

provided (∇N V ) − II(V ) = 0 and (∇N W ) − II(W ) = 0 on ∂M. By (34.39), we have that V ∈ forker (Lg ) and (∇N V ) − II(V ) = 0 on ∂M if and only if V, Lg (W )L2 (T M) = 0 for all W satisfying (∇N W ) − II(W ) = 0

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34. CMC SURFACES AND HARMONIC MAPS BY IFT

on ∂M. In particular, since forker (Lg ) = 0, we have that the formal cokernel is ⊥ trivial: (forimage (Lg )) = 0. 3.3. Weak solutions of the linearized equation and regularity. Let (Mn , g) be a compact Riemannian manifold with boundary ∂M. By the trace theorem (see Taylor [400]),2 there exists a bounded linear operator, called a trace operator, T : W 1,2 (T M) → W 1/2,2 ( T M|∂M ) such that T (U ) = U |∂M for U continuous on M (more generally, we have that T : W k,2 (T M) → W k−1/2,2 ( T M|∂M ) is bounded for k > 1/2). See §2 of Appendix K for the definitions of these spaces. We also have the boundary tangent and normal projection maps 

: W 1/2,2 ( T M|∂M ) → W 1/2,2 (T (∂M)) ,

⊥

: W 1/2,2 ( T M|∂M ) → W 1/2,2 (N (∂M)) ,

respectively, where N (∂M) → ∂M is the normal line bundle. In particular, we have the composite maps  ◦ T and ⊥ ◦ T . Define the Banach space  W1,2 (T M)  U ∈ W 1,2 (T M) : T (U )⊥ = 0 . Motivated by (34.34) regarding the linearization of Φ, define the bilinear form I : W1,2 (T M) × W1,2 (T M) → R by



 I (U, V ) 

(34.40)

M

(∇U, ∇V  − Rc(U, V )) dμ −

II(U, V )dσ, ∂M

where II(U, V )  II(T (U ), T (V )). By the trace theorem, I is bounded. Now assume that g has negative Ricci curvature and ∂M is concave. Then there exists a constant δ > 0 such that we have the coercivity estimate     2 2 2 |∇U | + |U | dμ (34.41) I (U, U ) ≥ δ U W 1,2 (T M)  δ M

W1,2

for U ∈ (T M). In Theorem K.1, take A = B = W1,2 (T M) and f = I. Then, by (34.41), we have Lemma 34.14 (Weak solutions of the linearized equation). Let (Mn , g) be a compact manifold with negative Ricci curvature and concave boundary ∂M. For any Q ∈ L2 (T M) and any f ∈ L2 (T (∂M)), there exists a unique U ∈ W1,2 (T M) such that for all V ∈ W1,2 (T M) we have   (34.42) I (U, V ) = Q, V  dμ + f, V  dσ. M

∂M

2 In Proposition 4.5 on p. 287 of [400] the trace theorem is stated for functions. One may extend this to sections of vector bundles by locally trivializing the bundles.

3. HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Rc < 0

269

That is, U solves the equation −ΔU − Rc (U ) = Q

(34.43a)

in M,

(∇N U ) − II(U ) = f

(34.43b)

on ∂M

in the weak sense, where N is the unit inward normal to ∂M. We have the following regularity result for weak solutions of the elliptic boundary value problem (34.43). Lemma 34.15. Under the hypothesis of Lemma 34.14, if in addition we have Q ∈ C ∞ (T M) and f ∈ C ∞ (T (∂M)), then the unique solution U to (34.43) is in C ∞ (T M). The idea of the proof is as follows (for another proof, see Subsection 3.6 at the end this section). In a neighborhood of a point on the boundary of M, let   i of be local coordinates satisfying U, x {xn = 0} = U ∩ ∂M ⊂ ∂U and where N = ∂x∂n is the unit inward normal to ∂M. Write a vector field locally  ∂ as U = ni=1 U i ∂x i . On ∂U we have (∇N U )i =

 ∂U i − Γinj U j n ∂x

for i ≤ n − 1

j≤n−1

when U n = 0, where the Γinj denote the Christoffel symbols. Then the boundary value problem (34.43), with U⊥ = 0, locally takes the form (34.44a) L(U )i 

 k,≤n

(34.44b)

ak

j   ∂2U i ki ∂U + b + cij U j = φi j k  k ∂x ∂x ∂x j,k≤n

B(U )i 

in U for i ≤ n,

j≤n

 ∂U i + dij U j = ψ i n ∂x

on U ∩ ∂M for i ≤ n − 1,

j≤n−1 n n

B(U )  U = 0 on U ∩ ∂M,

(34.44c)

i i i i where ak , bki j , cj , φ for i, j, k,  ≤ n are smooth functions in U and dj , ψ for i, j ≤ n − 1 are smooth functions on U ∩ ∂M. Thus the boundary conditions are of Neumann type for the components i ≤ n − 1 and are of Dirichlet type for the component i = n. We now show, following Agmon, Douglis, and Nirenberg [4], that the regularity theory of boundary value problems for elliptic systems applies to this situation. The issue is to establish certain compatibility between the elliptic system and the boundary operator. The principal symbol for the elliptic system in (34.44a) is  ak (x) ξk ξ , (34.45) σL (x) (ξ) = A (ξ) id Tx M : Tx M → Tx M, A (ξ) 

for ξ =



k,≤n i ξi dx

i

∈

Tx∗ M.

The adjoint of σL (x) (ξ) is given by

adj (σL (x) (ξ)) = A (ξ)n−1 id Tx M .

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34. CMC SURFACES AND HARMONIC MAPS BY IFT

Let x ∈ ∂M and let ζ ∈ Tx ∂M − {0}. The characteristic equation det (σL (x) (ζ + τ N )) = 0 has unique conjugate roots τ + (ζ) and τ − (ζ) of multiplicity n with positive and negative imaginary parts, respectively. This is because the characteristic equation is equivalent to the quadratic equation   ak (x) ζk ζ + 2τ akn (x) ζk + ann (x) τ 2 . 0 = A (ζ + τ N ) = k,≤n−1

Define M

±

k≤n−1

: T (∂M) × R → R by

 n M ± (ζ, τ ) = τ − τ ± (ζ) ,

so that M + (ζ, τ ) M − (ζ, τ ) = det (σL (x) (ζ + τ N )). The principal symbol σB (x) (ξ) : Tx M → Tx M of the boundary operator B in (34.44b)–(34.44c) is given by σB (x) (ξ) (V ) = ξ (N ) V + V⊥ . We compute that (34.46)

[σB (x) (ζ + τ N ) ◦ adj (σL (x) (ζ + τ N ))] (V ) = |ζ + τ N |2(n−1) (τ V + V⊥ ) n−1  n−1  = τ − τ + (ζ) (τ V + V⊥ ) . τ − τ + (ζ)

Therefore the row vectors of the matrix corresponding to the transformation on the lhs of (34.46) are linearly independent modulo M + (ζ, τ ). This verifies the Complementing Boundary Condition on pp. 42–43 of [4]. Regarding this condition, in the introduction to [4], the authors write: This Complementing Condition is necessary and sufficient in order that it be possible to estimate all the derivatives occurring in the system, without loss of order, i.e., that inequalities of “coercive” type be valid. More precisely, by Theorem 9.3 in [4] we conclude that if Q ∈ C ∞ (T M) and f ∈ C ∞ (T (∂M)), then the solution U to (34.43) satisfies U ∈ C ∞ (T M). 3.4. Schauder estimates for the linearization. Now we consider the related Schauder theory. Note that the boundary conditions U⊥ = 0 and (∇N U ) − II(U ) = 0 have different orders of derivatives. In view of this, we invoke Theorem 9.3 of [4] (or Theorem 5 on p. 406 of Simon [381]) to obtain the following estimate. For any U ∈ C 2,α (T M), (34.47)    2  ∇ U α ≤ C −ΔU − Rc (U )C α (T M) + (∇N U ) − II(U )C 1,α (T (∂M))   + C U⊥ C 2,α (T (∂M)) + U C 0 (T M) . By an interpolation estimate (see §6.8 of [122]), one can show that for any ε > 0 there exists Cε < ∞ such that for any U ∈ C 2 (T M), (34.48)

U C 2 (T M) ≤ ε[∇2 U ]α + Cε U C 0 (T M) .

3. HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Rc < 0

271

On the other hand, we have the following elementary result. Claim. For any ε > 0 there exists Cε < ∞ such that for any U ∈ C 1 (T M), (34.49)

U C 0 (T M) ≤ ε∇U C 0 + Cε U L2 (T M) .

Proof of the claim. Suppose that the claim is false for some ε > 0. Then there is a sequence of vector fields Uk ∈ C 1 (T M) such that (34.50)

Uk C 0 (T M) > ε∇Uk C 0 + kUk L2 (T M) .

By scaling, we may assume that ∇Uk C 0 = 1, so that Uk C 0 (T M) > ε. Fix a point p ∈ M and let d  diam(M). Then |Uk (p)| − d ≤ |Uk (q)| ≤ |Uk (p)| + d

for any q ∈ M.

So, by (34.50), we have |Uk (p)| + d > ε + kUk L2 (T M) ≥ ε + k(|Uk (p)| − d) Vol(M)1/2 . Hence {|Uk (p)|} is a bounded sequence. Now we can apply the Arzela–Ascoli theorem to Uk to conclude that Uk → U∞ in C 0 (T M). Then (34.50) implies that U∞ L2 (T M) = 0; i.e., U∞ ≡ 0. The claim follows since this contradicts that Uk C 0 (T M) > ε for all k. Now, by combining (34.47), (34.48), and (34.49), we obtain U C 2,α (T M) ≤ C −ΔU − Rc (U )C α (T M) + C (∇N U ) − II(U )C 1,α (T (∂M))   + C U⊥ C 2,α (T (∂M)) + U L2 (T M) . In particular, for any U ∈ C 2,α (T M) with U⊥ = 0 on ∂M, we have (34.51)

U C 2,α (T M) ≤ C −ΔU − Rc (U )C α (T M)   + C (∇N U ) − II(U )C 1,α (T (∂M)) + U L2 (T M) .

By (34.41), (34.34), and the Cauchy–Schwarz inequality, we have 1 2 U W 1,2 (T M) ≤ −ΔU − Rc (U )L2 (T M) U L2 (T M) δ 1 + (∇N U ) − II(U )L2 (T (∂M)) U L2 (T (∂M)) . δ Applying U L2 (T (∂M)) ≤ C U W 1,2 (T M) from the trace theorem, there exists a constant C < ∞ such that U L2 (T M) ≤ U W 1,2 (T M)   ≤ C −ΔU − Rc (U )L2 (T M) + (∇N U ) − II(U )L2 (T (∂M)) . In conclusion, we apply this estimate to (34.51) to obtain Lemma 34.16 (Schauder estimate). For any U ∈ C 2,α (T M) with U⊥ = 0 on ∂M, (34.52)

U C 2,α (T M) ≤ C −ΔU − Rc (U )C α (T M) + C (∇N U ) − II(U )C 1,α (T (∂M)) .

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34. CMC SURFACES AND HARMONIC MAPS BY IFT

Now we can apply (34.52) and Lemma 34.15 to obtain Lemma 34.17. The linear operator (D2 )(g,id) Φ in (34.37) is invertible. Proof. Given any Q∞ ∈ C α (T M) and f∞ ∈ C 1,α (T (∂M)), we choose sequences Qk ∈ C ∞ (T M) and fk ∈ C ∞ (T (∂M)) such that Qk → Q∞ in C α (T M) and fk → f∞ in C 1,α (T (∂M)). By Lemma 34.15, for each k there exists a smooth solution Uk to (34.43) with Q = Qk and f = fk . Applying estimate (34.52) to Uk − U we obtain   Uk − U C 2,α (T M) ≤ C Qk − Q C α (T M) + fk − f C 1,α (T (∂M)) . From the C α convergence of Qk and the C 1,α convergence of fk , we conclude that Uk is a Cauchy sequence in C 2,α (T M) and its limit U∞ ∈ C 2,α (T M) is the unique solution to (34.43) with Q = Q∞ and f = f∞ . The estimate (34.52) implies that  the inverse operator of (D2 )(g,id) Φ is bounded. 3.5. Proof of Proposition 34.13. By Lemma 34.17, we may apply the IFT to the map Φ in (34.36) to obtain a C 2,α harmonic diffeomorphism F satisfying the normal boundary condition. To finish the proof of Proposition 34.13, we now apply a standard bootstrap argument to prove that F is smooth when g˜ is smooth. Since F is harmonic and C 2,α and g and g˜ are C ∞ , we can view the equation for F being harmonic as   b c 2 a ∂F c ˜ abc ◦ F ∂F ∂F = aij ∂ F + bak , (34.53) 0 = Δg (F a ) + g ij Γ c i j i j ∂x ∂x ∂x ∂x ∂xk   b ij k a ik ˜ a Γbc ◦ F ∂F where aij = g ij and bak c = −g Γij δc + g ∂xi , as a linear system for F with C 1,α coefficients. Hence, by Theorem 9.3 of [4], we obtain that F is C 3,α in the interior of M. Continuing this way, we conclude that F is C k,α in the interior for all k ≥ 4. We now  address  the boundary regularity of F . In a neighborhood of p ∈ ∂M, again let U, xi be local coordinates satisfying {xn = 0} = U ∩ ∂M ⊂ ∂U and ∂ N is the unit inward normal of U ∩∂M. Then the normal boundary condition ∂xn = ∂F a is ∂xn = 0 for a ≤ n − 1 and F n = 0 on U ∩ ∂M. Since the principal symbols of the operators in (34.53) and this boundary condition are the same as for (34.44), the complementing boundary condition holds. Therefore we can apply Theorem 9.3 of [4] to conclude that F is C ∞ up to the boundary. This completes the proof of Proposition 34.13.  Exercise 34.18. Formulate and prove a corollary of Proposition 33.12 which states that there exists a harmonic diffeomorphism close to an approximate isometry. 3.6. An alternative approach to solving equation (34.43). We may use the general results in §20.1 of H¨ormander [150]. There, one considers a compact manifold M with boundary and an m-th order linear elliptic partial differential operator P : C ∞ (M, E) → C ∞ (M, F ) of sections of complex3 vector 3 In our case we have real vector bundles, but it is easily seen that H¨ ormander’s theory holds in this case as well; alternatively, given a real vector bundle H, we may consider the complex vector bundle H ⊗R C.

4. APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES

273

bundles E and F over M together with a linear elliptic4 boundary partial differential operator B : C ∞ (M, E) → C ∞ (∂M, G), where G is a complex vector bundle over ∂M. Given p ∈ C ∞ (M, F ) and h ∈ C ∞ (∂M, G), one considers the elliptic boundary value problem (34.54a)

P (U ) = p in M,

(34.54b)

B (U ) = h on ∂M.

Under assumptions which are much weaker than in our situation, we have the following in Theorem 20.1.2 on p. 234 of H¨ormander [150]: if P is second order and B is first order, then 1

(P, B) : W k,2 (M, E) → W k−2,2 (M, F ) × W k−1− 2 ,2 (∂M, G) is a Fredholm operator for each k ≥ 2; for the definition of W k,p , see (K.11) in Appendix K. To give another proof of Lemma 34.15, we take E = F = T M, G = T (∂M), and consider (34.43): P (U ) = ΔU + Rc (U ) = −Q, B (U ) = (∇N U ) − II(U ) = f,

where U⊥ = 0.

Since Rc < 0 and II ≤ 0, we have (34.41). Hence ker (P, B) = 0, which implies that (P, B) is onto since it is Fredholm. Now since Q ∈ W k−2,2 (M, F ) and f ∈ 1 W k−1− 2 ,2 (∂M, G) for each k ≥ 2, there exists U ∈ W k,2 (M, E) solving equation (34.43). Because all of these U are the same, we obtain a unique C ∞ solution. 4. Application of Mostow rigidity to the existence of isometries In this section we discuss applications of the Mostow rigidity for finding isometries of hyperbolic manifolds assuming the existence of “almost isometries” (see Theorem 34.22 below). This result is encapsulated by the following quote of Hamilton (see Theorem 8.1 (“rigidity of hyperbolic manifolds”) in [143]): If a map of a large enough part of one complete hyperbolic manifold with finite volume into another with no fewer cusps is close enough to being an isometry, then there exists an actual isometry between the manifolds. 4.1. Dependence of the geometry of a hypersurface on the ambient metric. First we discuss how small changes in the ambient metric affect the geometry of a hypersurface. Let Mn be an oriented C ∞ manifold  and let g and g˜ be Riemannian metrics on M. Given local coordinates U, xi on M with C1 δij ≤ gij ≤ Cδij , ˜ k denote the Christoffel symbols of g and g˜, respectively. On U we let Γkij and Γ ij have  1    ˜k g − gC 1 (M,g) . Γij − Γkij  = g k (∇i g˜j + ∇j g˜i − ∇ g˜ij ) ≤ C ˜ 2 4 Here,

elliptic is in the sense of Definition 20.1.1(ii) in [150].

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34. CMC SURFACES AND HARMONIC MAPS BY IFT

Hence, if X and Y are vector fields on M, we have      i j k ∂  ˜  k  ˜ ∇X Y − ∇X Y  = X Y (Γij − Γij ) k  ∂x g g ≤ C |X|g |Y |g ˜ g − gC 1 (M,g) .

(34.55) Equivalently,

    ˜ g − gC 1 (M,g) . (∇ − ∇)Y  ≤ C |Y |g ˜

(34.56)

g

⊂ Mn be an oriented C ∞ hypersurface and let ν and Lemma 34.19. Let N ∞ ν˜ be C unit normal vector fields to N with respect to g and g˜, respectively, where ν and ν˜ are on the same side of N in M. We then have the following: (1) n−1

g − gC 0 (M,g) , |˜ ν − ν|g ≤ C ˜

(34.57)

in particular, |˜ ν |g ≤ C ˜ g − gC 0 (M,g) + 1. (2) (34.58)

˜ ν |g ≤ C, |∇˜

|∇ (˜ ν − ν)|g ≤ C ˜ g − gC 1 (M,g) .

(3) The second fundamental form satisfies (34.59)

8 (X, Y ) − II (X, Y ) | ≤ C |X| |Y | ˜ |II g g g − gC 1 (M,g) .

(4) ˜ II 8 (X, Y ) − ∇ II (X, Y ) | ≤ C |X| |Y | ˜ |∇ g g g − gC 2 (M,g) .   i n Proof. (1) Choose local coordinates U, x i=1 in a neighborhood of a point in N so that N ∩ U = {p ∈ U : xn (p) = 0} and ∂x∂n is on the same side as ν. On ∂ n N ∩ U let N = N j ∂x = 1. Then for j < n j be the normal field to N satisfying N ∂ we have that g(N, ∂xj ) = 0 and (34.60)

Nj = −

(34.61)

n−1 

Gij gin ,

i=1

 n−1 where Gij i,j=1 denotes the inverse matrix to (gij )n−1 i,j=1 . Using (34.61) we compute that n−1  2 |N |g = gnn − gin Gki gkn . i,k=1

Now we obtain (34.62)

n−1 ij ∂ ∂ N i=1 G gin ∂xj ∂xn − ν= =! .  |N |g ki g gnn − n−1 g G in kn i,k=1

We also have the analogous formula for ν˜. From this we may derive (34.57). (2) We obtain (34.58) from differentiating the formula (34.62) for both ν and ν˜. (3) By the definition of the second fundamental form of N , with respect to g and g˜, (34.63)

II (X, Y ) = g (∇X ν, Y )

and

8 (X, Y ) = g˜(∇ ˜ X ν˜, Y ) II

4. APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES

275

for X, Y ∈ Tx N . We then have 8 (X, Y ) − II (X, Y ) | |II ˜ − ∇)X ν˜, Y )| + |g (∇X (˜ ˜ X ν˜, Y )| + |g((∇ ν − ν) , Y )| ≤ | (˜ g − g) (∇   ˜ ν |g + |(∇ ˜ − ∇)˜ ≤ |X|g |Y |g ˜ ν |g + |∇ (˜ ν − ν)|g g − gC 0 (M,g) |∇˜ ≤ C |X|g |Y |g ˜ g − gC 1 (M,g) , where we have used (34.56), (34.57), and (34.58). ˜ II 8 (X, Y ) − ∇ II (X, Y ) | simi(4) By differentiating (34.63) and estimating |∇ larly to part (3), we obtain (34.60).  Furthermore, the curvatures of a hypersurface with respect to the two induced metrics are almost the same. g˜N

Lemma 34.20. Let N n−1 ⊂ M be an oriented C ∞ hypersurface. Let gN and denote the metrics on N induced by g and g˜ on M, respectively. Then

(34.64)

|RmgN − Rmg˜N | ≤ C ˜ g − gC 2 (M,g) .

Proof. The Levi-Civita connections for gN and g are related by ∇N X Y = ∇X Y + II (X, Y ) ν ; there is the analogous formula for g˜N and g˜. We then obtain (34.64) by the definition of the Riemann curvature tensor and Lemma 34.19.  Remark 34.21. One can prove that the following for any k ≥ 1: (1) 8 (X, Y ) − ∇k II (X, Y ) | ≤ Ck |X| |Y | ˜ ˜ k II |∇ g g g − gC k+1 (M,g) . (2) ˜ N )k Rmg˜ | ≤ Ck ˜ g − gC k+2 (M,g) . |(∇N )k RmgN −(∇ N (3) All of the above discussion still holds when M has nonempty boundary and when N is a component of the boundary of M. 4.2. Existence of isometries near almost isometries. The following describes when an “almost isometry” between hyperbolic manifolds is close to an isometry.  Theorem 34.22 (Existence of isometries near almost isometries). Let H3 , h be a finite-volume hyperbolic 3-manifold. Let HA be the truncation of H defined ˜ in (33.1), where A ∈ (0, A/2] and A˜ is as in condition (h5) in Subsection 1.1 of Chapter 33. For each  ∈ N there exists k = k() ∈ N satisfying the following property. ¯ is a finite-volume hyperbolic 3-manifold with at least as many cusp ¯ 3 , h) If (H ¯ is an embedding which is close to an isometry ends as (H, h) and if F : HA → H in the sense that " ∗ " 1 ¯ − h" k "F h (34.65) < , C (HA ,h) k

276

34. CMC SURFACES AND HARMONIC MAPS BY IFT

¯ which is close to F in the sense ¯ h) then there exists an isometry I : (H, h) → (H, that 1 (34.66) dC  (HA ,h) (F, I) < ,  ¯ are iso¯ h) where this distance is defined by (K.13). In particular, (H, h) and (H, metric. Proof. A main idea is to use the Mostow rigidity theorem. ¯ are isometric. ¯ h) Step 1. The hyperbolic manifolds (H, h) = (H, ¯ be an embedding. Then F (HA ) ⊂ H ¯ Proof of Step 1. Let F : HA → H 2 is a 3-dimensional compact submanifold and each boundary component TE,A   F {rE,A } × TE2 of F (HA ), where rE,A is given by (33.2) and where E ∈ E (K) is ¯ only at those 2-tori TE,A that are isotopic to an embedded 2-torus. We truncate H ¯ ˜ We then have that standard tori in cusp ends of H; call the resulting manifold H. ˜ ¯ F (HA ) ⊂ H ⊂ H. By Lemma 31.27, each TE,A either ˜ or (a) is contained in ∂ H ˜ which is either a solid torus or lies (b) bounds a compact 3-manifold NE in H ˜ inside an embedded 3-ball in H. Define ˜ A  F (HA ) ∪ % NE , H E where the union is taken over those topological ends E for which TE,A is not con˜ A is ˜ For TE,A satisfying (b), we have NE ∩ F (HA ) = TE,A . Thus H tained in ∂ H. a compact 3-manifold with ˜A ⊂ H ˜ H

and

˜ A ⊂ ∂ H. ˜ ∂H

¯ by assumption, has at least as many cusp ends as H, if either the Since H, ¯ is strictly greater than the number of cusp ends of H or number of cusp ends of H ˜ then there exists a there exists a boundary component TE,A not contained in ∂ H, ¯ ˜ cusp end in H which is not truncated; i.e., H is noncompact. This contradicts the following properties:5 ˜ A ⊂ H, ˜ H

˜ A ⊂ ∂ H, ˜ ∂H

and

˜ A is compact. H

¯ is equal to the number of cusp We conclude that the number of cusp ends of H ¯ ends of H, that no TE,A bounds a solid torus in H or lies inside an embedded 3-ball ¯ and that in H, (34.67)

˜ A ) = int(H) ˜ ∼ ¯ H∼ = int (F (HA )) = int(H = H,

˜ and F (∂HA ) = ∂ H.) ˜ Corolwhere ∼ = denotes diffeomorphic. (In fact, F (HA ) = H lary 31.50 now implies Step 1. ¯ are isometric, we may henceforth assume that (H, h) is ¯ h) Since (H, h) and (H, ¯ ¯ equal to (H, h). Step 2. Extending isometries. For any isometric embedding Fˆ : HA → H, there exists an isometry I : H → H extending Fˆ . 5 In general, if N 3 is a noncompact manifold with boundary ∂N and if M3 ⊂ N is a submanifold with ∂M ⊂ ∂N , then M is noncompact.

4. APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES

277

Proof of Step 2. We first show the following. Claim. For each topological end E, TE,A = Fˆ ({rE,A } × TE ) is contained in the ε3 /2-thin part H(0,ε3 /2] of (H, h), where ε3 is the 3-dimensional Margulis constant. ˜ we have {rE,A } × TE ⊂ H[0,ε /2) . So, given Proof of the claim. Since A ≤ A, 3 y ∈ TE,A , there exists a piecewise smooth geodesic loop γ based at x  Fˆ −1 (y) ˜ we have that rE,A ≥ with [γ] ∈ π1 (H, x) − {1} and L (γ) ≤ ε3 /2. From A ≤ A/2, rE,A˜ + 12 ln 2 and hence γ ⊂ H − HA˜ . Therefore there exists a loop β ⊂ {rE,A } × TE which is homotopic to γ in H. Since {rE,A } × TE is incompressible in H, we obtain [β] ∈ π1 ({rE,A } × TE , x) − {1} . Moreover, since Fˆ is an isometry, Fˆ ◦ γ is a piecewise smooth geodesic loop based at y with L(Fˆ ◦ γ) = L (γ) ≤ ε3 /2. That [Fˆ ◦ γ] = 1 ∈ π1 (H, y) follows from [Fˆ ◦β] = 1 ∈ π1 (TE,A , y) and from the fact (from Step 1) that TE,A is incompressible  in H. Hence y ∈ H(0,ε3 /2] . By the claim, Fˆ maps truncated cusp ends in HA to truncated cusp ends in H. Using the fact that the hyperbolic metric h in a cusp end is a standard warped product, we may show that Fˆ may be extended to a global H. In  isometry I : H → particular, any cusp end is a warped product of the form [0, ∞) × T 2 , gcusp , where gcusp = dr 2 + e−2r gflat and (T , gflat ) is a flat 2-torus. The slices {r} × T foliating the cusp are intrinsically flat with constant second fundamental form. Since Fˆ is an isometry on HA , the hypersurfaces Fˆ ({r} × T ) share the same properties as {r} × T . From the uniqueness of such foliations, we conclude that the Fˆ ({r} × T ) are also standard slices in one of the cusp ends. In particular, for each end E there is an end E such that Fˆ : ([0, rE,A ] × TE , gcusp ) → ([a, b] × TE2 , g¯cusp ) is a product map of the form Fˆ (r, y) = ( (r) , ϕ (y)) , where  is linear and ϕ : (TE , gflat ) → (TE, g¯flat ) is a homothety. Now it is clear how to extend Fˆ to an isometry I on each cusp. This completes the proof of Step 2. Step 3. Existence of an isometry I near F . Suppose that Theorem 34.22 is false. Then there exist  ∈ N, ε > 0, a sequence ki ∈ N with ki → ∞, and embeddings Fi : HA → H such that (34.68)

Fi∗ h − hC ki (HA )
0, there exists i (η) ∈ N such that (34.70)

Fi (∂HA ) ⊂ Nη (∂HA )  {x ∈ H : d(x, ∂HA ) < η}

for all i ≥ i (η). Since Fi is an embedding, we may also conclude that Fi (HA ) ⊂ Nη (HA ) for i sufficiently large. Since ki → ∞, by (34.68) there exists a subsequence such that {Fi } converges in C ∞ to a smooth map F∞ : HA → H with F∞ (HA ) ⊂ HA . We have that F∞ is a local isometry since ∗ F∞ h − hC k (HA ,h) = 0

for each k.

To see that F∞ |int(HA ) is injective and hence an embedding, suppose that F∞ (x) = F∞ (y) for some x, y ∈ int(HA ) with x = y. Then, by the IFT, we have for i sufficiently large that there exists yi near y such that Fi (x) = Fi (yi ) and x = yi ; this contradicts Fi being an embedding. By (34.70), the sequence { Fi |∂HA } converges to a diffeomorphism F∞ |∂HA : ∂HA → ∂HA . In conclusion, F∞ : HA → H is an isometric embedding. By Step 2, we may extend F∞ to an isometry I∞ : H → H. Hence, by (34.69) 0 = dC  (HA ,h) (F∞ , I∞ ) ≥ ε > 0, which is a contradiction. The theorem is proved.



5. Notes and commentary §2. For Yano’s formula (34.18), see p. 57 of Yano and Bochner [443] and formula (2.5) of Smith [387]). §3. Regarding the regularity theory for (34.44), used to obtain Lemma 34.15, one may also consult Chapter 7 of J. L. Lions [206]. The scaling argument for elliptic operators in Simon [381], in addition to proving the Schauder estimate (34.47), establishes regularity for (34.44). As Simon says, his method . . . has completely general applicability to boundary value problems, without special consideration of the nature of the boundary conditions and boundary semi-norms beyond the natural assumptions that the corresponding constant coefficient problem in a halfspace is hypoelliptic and that the boundary semi-norms are nondegenerate H¨ older semi-norms which when applied to the boundary operators have the same κ-order as the H¨ older semi-norm being estimated. §4. Theorem 34.22 is Theorem 8.3 (“general rigidity”) in [143].

CHAPTER 35

Stability of Ricci Flow Look out kid. It’s something you did. God knows when, but you’re doin’ it again. – From “Subterranean Homesick Blues” by Bob Dylan

In the study of evolution equations, the concept of stability arises naturally and is easy to describe heuristically. Consider a nonlinear second-order evolution equation (35.1a) (35.1b)

∂ u (x, t) = A(x, t, u, Du, D2 u), ∂t u (x, 0) = u0 (x) ,

for a family of maps u (·, t) : Mn → N m between two manifolds. If (35.1) is well¯ is a fixed point, it is natural to ask whether solutions that start posed and u0 = u sufficiently near u ¯ in an appropriate topology exist for all time and converge to u ¯. More generally, if the solution u(t) starting at some u0 converges to some u∞ , one can ask whether solutions which start sufficiently close to u0 also converge to u∞ or perhaps converge to some other point in a specified set (near u∞ ). If the structure of (35.1) is sufficiently transparent, these questions can sometimes be answered by identifying and obtaining a priori estimates for key (geometric) quantities. However, another more general approach is often useful, especially if such estimates are impractical or otherwise unavailable. In this approach, answering the questions above involves two steps. In the first step, one computes the linearization DA(¯ u) of the differential operator on the right-hand side of (35.1a) at the fixed point u ¯ and analyzes its spectrum. If, say, the linearization has pure point spectrum, then its eigenvalues with negative real part together with their associated eigenfunctions indicate perturbations of the fixed point that are formally stable, while those eigenvalues with positive real part indicate unstable perturbations. If the linearization DA(¯ u) has eigenvalues with the real part bounded above by some ε < 0, then we say that the equation corresponding to A is linearly stable at u ¯ (see Definition 35.1). In the second step, one proves dynamic (i.e., asymptotic) stability by arguing that the linearization DA(¯ u) actually does determine the asymptotic behavior of solutions that belong to a sufficiently small neighborhood of u ¯ in a well-chosen function space. One effective set-up for doing this is to construct a qualitative geometric theory for the dynamics by regarding the pde as an evolution ode posed in an infinite-dimensional setting, essentially by using semigroup theory. Other approaches can be effective as well; indeed, it is sometimes possible to determine stability without explicit recourse to the linearized operator. But in any case, this second step can become markedly more difficult if zero is an eigenvalue of the linearization. If this happens, at least if the null eigendirection is integrable, 279

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35. STABILITY OF RICCI FLOW

one expects to encounter a center manifold on which the behavior of solutions is determined not by the linearization, but by higher-order terms in A. In a broad sense, any convergence theorem that states that any solution evolved from initial data in some open set converges to a unique fixed point in that set, such as the results proved for the Ricci flow by Hamilton [135], [136], Huisken [152], Margerin [217], [218], Nishikawa [294], Ye [449], B¨ohm and Wilking [30], Chen [65], and Brendle and Schoen [35] (we survey all of these in §4 of this chapter), implies a statement of stability. In this chapter however, we are primarily concerned with techniques for determining stability that start with the known existence of a stationary solution. This chapter then serves two purposes. The first is to give a detailed look at one approach to stability, namely that of linearizing the equation and then applying semigroup and maximal regularity theory to determine dynamical stability. The second purpose is to survey related stability results that are obtained using different methods. The level of detail varies correspondingly. In the first (main) part of this chapter—§1 and §2—we focus on methods for determining stability via linearization. In general, these methods have three useful features: (I) they are essentially pde techniques, hence generalize readily to a wide variety of geometric evolution equations; (II) in contrast to purely geometric techniques, which generally yield stability theorems only modulo diffeomorphisms, the methods we consider here prove convergence in normed spaces that allow the control of diffeomorphisms (which is useful for some applications); and (III) for quasilinear pde, these methods can provide a regularity boost—for example, yieldolder norm for initial data that are close only in a ing convergence in a C 2+α H¨ C 1+β -norm. In the second part of the chapter—§3—we survey a few of the dynamic stability results that have been obtained using linearization and the techniques described in §1 and §2 . As noted above, in the final part of the chapter—§4—we discuss some (though not all) dynamic stability results that have been obtained using different notions of stability, such as variational stability. The idea of variational stability is that if an evolution equation can be formulated as the gradient flow of a specified functional F , then the stability of the flow in the neighborhood of a fixed point u ¯ of that flow can be studied by examining the behavior of F near u ¯. Specifically, if u ¯ is a local minimum of F , one expects the flow to be stable at u ¯. If, on the other hand, u ¯ is either a local maximum or a saddle point, then one expects instability. One way of determining the local behavior of F is to use an infinite-dimensional version of the second derivative test from calculus. The analysis then focuses on the study of the u) of F at u ¯. second derivative D2 F (¯ 1. Linear stability of Ricci flow Definition 35.1. Let (35.1) be a parabolic pde system with a fixed point u ¯. Let Σ ⊆ C denote the spectrum of the (elliptic) linear operator Du¯ A, and let σ denote the projection of Σ onto the real axis. We say that u ¯ (or, equivalently, the pde at u ¯) is • strictly linearly stable if σ ⊂ (−∞, 0), • linearly stable if σ ⊂ (−∞, 0], • linearly unstable if σ ∩ (0, ∞) = ∅.

1. LINEAR STABILITY OF RICCI FLOW

281

In this section, we want to study the linear stability of the Ricci flow. However, as we have seen in the discussion of short-time existence for the Ricci flow in u) is not quite elliptic. Therefore, Chapter 3 of Volume One, the linearization D2 F (¯ we determine the stability of the Ricci flow indirectly, by analyzing the spectrum of the linearization of the equivalent Ricci–DeTurck flow. (The use of the Ricci– DeTurck flow to study Ricci flow is in many ways similar to the practice of imposing the Lorentz gauge on the Yang–Mills equations, the wave gauge on the Einstein equations, or the Bianchi gauge in the elliptic context.1 ) 1.1. Spaces and operators. To begin, let us fix some notation. The reader may reference this section as needed. Given a closed, connected smooth manifold Mn , we denote by S 2 (Mn ) the 2 (Mn ) the subset of positivebundle of symmetric (2, 0)-tensors over Mn , and by S+ definite tensors. For convenience, we write   2 2 S 2  C ∞ S 2 (Mn ) and S+  C ∞ S+ (Mn ) . We denote by Λp  Λp (T ∗ M) the bundle of p-forms on Mn and by Ωp  C ∞ (Λp ) the space of differential p-forms. 2 . Given g with volume form A smooth Riemannian metric g is an element of S+ 2 1 dμ, we let δ = δg : S → Ω denote the (“divergence”) map2 δ : h → δh = −g ij ∇i hjk dxk , whose formal adjoint under the L2 inner product,  (·, ·)   · , ·  dμ, ∗

is the (“Killing”) map δ =

δg∗

Mn 2

: Ω → S given by 1

1 1 L  g = (∇i ωj + ∇j ωi ) dxi ⊗ dxj . 2 ω 2 Here L is the Lie derivative, and ω  is the vector field metrically isomorphic to ω. We also denote by δ = δg the map Ωp → Ωp−1 formally adjoint to d : Ωp → Ωp+1 ; the meaning should be clear from the context. 2 ⊂ It is well known that S 2 with the C ∞ -topology is a Frech´et space and that S+ 2 S is an open convex cone. There is a natural right action of the group D (Mn ) 2 , given by (h, ϕ) → ϕ∗ h. For purposes of smooth diffeomorphisms of Mn on S+ of studying geometrically distinct metrics on Mn , we identify diffeomorphically 2 equivalent metrics and regard S+ as a union of orbits Og under diffeomorphism. 2 Ebin’s Slice Theorem [97] shows that, viewed in this way, S+ has many properties of an infinite-dimensional manifold. The theorem states that for any metric g, there is ∗ a map χ : U → D (Mn ) of a neighborhood U of g in Og such that (χ (ϕ∗ g)) g = ϕ∗ g 2 containing g such that the for all ϕ∗ g ∈ U, and there is a submanifold Γ of S+ ∗ 2 ∗ ∗ map U × Γ → S+ given by (ϕ g, γ) → (χ (ϕ g)) γ is a diffeomorphism onto a 2 . neighborhood of g in S+ δ ∗ : ω →

1 Note, however, that in the Yang–Mills and Einstein cases, imposing the Lorentz or the wave gauge involves setting restrictions on the fields and their derivatives without changing the field equations themselves, while the Ricci–DeTurck flow involves an evolution equation that is different from Ricci flow itself. 2 Note that δ = − div.

282

35. STABILITY OF RICCI FLOW

We need only the infinitesimal version of the slice theorem, which gives a useful 2 2 . For each g ∈ S+ , let δg and δg∗ be the maps defined above. decomposition of Tg S+  If W is a smooth vector field, it is clear that h, δg∗ W = (δg h, W ), hence that 2 ker δg ⊥ im δg∗ . In fact, these spaces span Tg S+ . (See [97] and [25].) One therefore obtains the orthogonal decomposition (35.2)

Tg S2+ = Hg ⊕ Vg ,

where

and Vg  im δg∗ . Hg  ker δg Because Tg Og = im δg∗ = Vg , this notation is meant to suggest “horizontal” and “vertical” subspaces. 1.2. Linearization of Ricci–DeTurck flow, revisited. We work with the linearized Ricci–DeTurck flow with a fixed background metric gˆ. We use the notation (35.3)

Agˆ (g)  −2 Rc (g) − Pgˆ (g),

where Pgˆ (g)ij  −(∇i Wj + ∇j Wi ), to make explicit the dependence on the background metric gˆ. Here W is the tensor field    −1 1 k pq g g ∇p gˆq − ∇ gˆpq . (35.4) Wj = gˆ jk 2 2 For smooth initial data g0 ∈ S+ , the Ricci–DeTurck flow is then

(35.5a) (35.5b)

∂ g = Agˆ (g) , ∂t g(0) = g0 .

We observe that if g0 = g¯ is a fixed point of the Ricci flow, then setting gˆ = g¯ one g ) = 0, so that g¯ is also a fixed point of the Ricci–DeTurck flow. Note that has Pg¯ (¯ this is not generally the case for other choices of gˆ. We have shown in Chapter 3 of Volume One that the unique solution of the Ricci–DeTurck flow with initial data g0 is equivalent, modulo diffeomorphisms, to the unique solution of the Ricci flow with the same initial data. Hence, if we show that the Ricci–DeTurck flow initial value problem (35.5) is linearly stable for certain metrics, then linear stability for the Ricci flow initial value problem for these same metrics readily follows. The parabolic system (35.5a) is quasilinear, which is important in our implementation of maximal regularity theory. We can see this explicitly as a consequence of the following result, which can be easily proved using formulas from Chapter 3 of Volume One. Exercise 35.2. Show that the operator Agˆ (g) is given in local coordinates by ∂2 gk ∂xp ∂xq  k ∂ gk + c x, gˆ, ∂ˆ g , ∂ 2 gˆ ij gk . + b (x, gˆ, ∂ˆ g , g)kp ij p ∂x The functions a (x, ·, ·), b (x, ·, ·, ·), and c (x, ·, ·) depend smoothly on x ∈ Mn and are analytic functions of their remaining arguments. (35.6)

Agˆ (g)ij = a (x, gˆ, g)kpq ij

1. LINEAR STABILITY OF RICCI FLOW

283

If gˆ = g¯ for a fixed point g¯ of Ricci flow, then the Lichnerowicz Laplacian ΔL is the natural linearization of Ricci–DeTurck flow: Proposition 35.3. The choice gˆ = g¯ yields g ) = ΔL h, (Dh Ag¯ ) (¯

(35.7) where

(ΔL h)ij  Δhij + 2Rkij hk − Rik hkj − Rjk hki . The operator ΔL is self-adjoint and strictly elliptic. Furthermore, if Mn is compact, then it is self-adjoint with respect to the L2 inner product (·, ·)  Mn ·, · dμ. The only stationary solutions of Ricci flow are Ricci-flat metrics, and the only stationary solutions of normalized Ricci flow are Einstein metrics. However, Ricci flow has a wealth of “generalized fixed points”, namely Ricci solitons. It is often useful to study the stability of these solutions using a modified flow. Shrinking solitons are converted into stationary solutions as follows. (We use  this calculation below in Subsections 1.6 and 3.2.) If Mn , g(t) is any solution of Ricci flow that becomes singular at some T < ∞ and if X is any smooth vector 1 log(T − t) and a field on Mn , we may introduce a dilated time variable τ := − 2λ rescaled family of metrics γ(τ ) defined by  g(t) := 2λ(T − t) ϕ∗t γ(τ (t)) , where ϕt denotes the 1-parameter family of diffeomorphisms generated by the vector fields (2λ(T − t))−1 X, and λ > 0. Then one calculates ∂ ∂ g(t) = (2λ(T − t)φ∗t γ(τ (t))) ∂t ∂t

    ∂γ ∂τ ∂ |s=0 (φ∗t+s (γ(τ (t)))) = −2λφ∗t (γ(τ (t)) + 2λ(T − t)) φ∗t + ∂τ ∂t ∂s   ∂γ = φ∗t −2λγ(τ (t)) + + LX γ(τ (t)) . ∂τ

Because ∂ g = −2 Rc(g) = φ∗t (−2 Rc(γ)) , ∂t we see that γ evolves by dilated Ricci flow, ∂ γ = −2Rc[γ] − LX γ + 2λγ. ∂τ  Clearly, any suitably normalized shrinking soliton Mn , γ, X becomes a stationary solution of the dilated Ricci flow constructed in this manner.  Exercise 35.4. Given a solution Mn , g(t) of Ricci flow that emerges from a “big bang” at some time T0 > −∞, and a smooth vector field X, show that replacing T − t by t − T0 in the construction above leads to a solution gˆ(τ ) of ∂ gˆ = −2Rc[ˆ g ] − LX gˆ − 2λˆ g, ∂τ 1 where here τ := 2λ log(t − T0 ). In this way, any suitably normalized expanding soliton becomes a stationary solution of a modified flow. Hint: See §3.1 of [129].

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1.3. Flat geometries. Proposition 35.5. If (Mn , g) is flat, then ΔL is negative semidefinite on S 2 with its kernel consisting exactly of parallel (2, 0)-tensors; hence, the dimension of the kernel is at most n (n + 1) /2. In particular, any flat metric is linearly stable for the Ricci–DeTurck flow. Proof. This follows from the fact that ΔL ≡ Δ on a flat manifold and from the nonpositivity of the eigenvalues of Δ.  1.4. K3 complex surfaces. Definition 35.6. A K3 surface is a closed connected smooth complex surface with vanishing first Chern class and with no global holomorphic 1-form. A K3 surface is a 4-dimensional real manifold. Kodaira [166] observed that each K3 surface is diffeomorphic to a unique simply-connected orientable manifold, the quartic hypersurface ⎧ ⎫ 3 ⎨ ⎬  zj4 = 0 ⊂ CP3 . X 4 = [z0 : z1 : z2 : z3 ] ∈ CP3 : ⎩ ⎭ j=0

Siu proved [384] that every K3 admits some K¨ ahler metric. Furthermore, Yau’s proof [447] of the Calabi conjecture showed that each K¨ahler class of a K3 surface contains a unique Ricci-flat K¨ahler metric. We have the following result (see [128] for an exposition).  Proposition 35.7. Any K¨ ahler–Einstein K3 surface M4 , g is linearly stable for the Ricci–DeTurck flow. That is, its Lichnerowicz Laplacian is negative semidefinite. The kernel is ε (g) ⊕ G, where ε (g) is the product of the 3-dimensional space of parallel self-dual 2-forms and the 19-dimensional space of harmonic anti-self-dual 2-forms. 1.5. Manifolds that admit nonzero parallel spinors. Work [88] of X. Dai, X. Wang, and G. Wei together with earlier results [430] of M. Wang allow a substantial generalization of Proposition 35.7: Theorem 35.8. If a compact Riemannian manifold (Mn , g) is covered by a spin manifold that admits nonzero parallel spinors, then ΔL has nonpositive spectrum. Remark 35.9. This result generalizes Proposition 35.7 because any manifold satisfying the hypotheses of Theorem 35.8 is necessarily Ricci flat. The authors of [88] prove Theorem 35.8 by exhibiting the Lichnerowicz Laplacian (with opposite sign convention) as the square of a twisted Dirac operator. The idea is to view h as the section of a vector bundle with a differential operator such that the Lichnerowicz Laplacian is the square of that operator. They achieve this for spin manifolds with parallel spinors using the Dirac operator: Let (Mn , g) be a compact spin manifold with spinor bundle S → Mn . If σ is a parallel nonzero spinor, then (Mn , g) is necessarily Ricci flat. Define a linear map Φ : S 2 → S ⊗ T ∗ Mn by Φ(h) = hij ei · σ ⊗ ej ,

1. LINEAR STABILITY OF RICCI FLOW

285

where ei is an orthonormal coframe. Then the Lichnerowicz Laplacian satisfies the formula D∗ D(Φ(h)) = Φ(−ΔL h), where D is the Dirac operator. In light of this result it is interesting to ask which compact Ricci flat manifolds admit nonzero parallel spinors. As a consequence of the Cheeger–Gromoll splitting theorem, it suffices to consider simply-connected, compact, Ricci-flat manifolds. One may also assume the manifold is irreducible.3 Then one has the following: Proposition 35.10 ([88]). A simply-connected, irreducible, compact, and Ricciflat manifold (Mn , g) either has holonomy SO(n) or admits a nonzero parallel spinor. Remark 35.11. All known examples of compact Ricci-flat manifolds have special holonomy and so admit nonzero parallel spinors. Consequently, there are no known compact Ricci-flat manifolds that are linearly unstable. In another paper, Dai, Wang, and Wei prove a related result: Theorem 35.12 ([89]). If a compact Einstein manifold (Mn , g) with nonpositive scalar curvature admits a nonzero parallel spinc spinor, then the linear operator A : hij → ΔL hij + Rik hkj + Rjk hik has nonpositive spectrum. In particular, A has nonpositive spectrum on any compact K¨ ahler–Einstein manifold with nonpositive scalar curvature. Note that a simply-connected manifold admits a nonzero parallel spinc spinor if and only if it is the product of a K¨ ahler manifold and a manifold that admits a nonzero parallel spinor. 1.6. Nontrivial Ricci solitons. As seen above in Subsection 1.2, every nontrivial (i.e. non-Einstein) Ricci soliton is a stationary solution of a dilated Ricci flow ∂ g = −2 Rc +2λg + LX g. ∂t It is natural to consider the linear stability of such stationary solutions. The cylinder soliton is linearly unstable, but the unstable perturbations are merely “coordinate instabilities” rather than geometrically meaningful ones. For example, it follows from the maximum principle that a slightly smaller cylinder vanishes under Ricci flow quicker than a slightly larger one does. Nonetheless, both remain round cylinders under Ricci flow. (See [14] for more details.) We discuss this phenomenon further in Subsection 4.1 below. A more interesting class of examples is the noncompact expanding homogeneous solitons, which in some cases model immortal solutions of Ricci flow that collapse with bounded curvature (see, e.g., [209]). An algebraic soliton is defined to be a left-invariant metric g on a homogeneous space G n that satisfies the equation Rc = λI +D, where Rc is the Ricci endomorphism of g, λ ∈ R, and D is a derivation 3 Here “irreducible” means “not locally a product space” in the sense of the de Rham decomposition of T ∗ Mn .

286

35. STABILITY OF RICCI FLOW

of the associated Lie algebra g. (See Part I, Chapter 1, §6.) Every algebraic soliton on a simply-connected homogeneous space is an expanding Ricci soliton. (See [177] or [182].) However, to analyze the linear stability of expanding Ricci solitons, understood as stationary solutions of dilated Ricci flow, it is more effective to adopt this more algebraic perspective. By using this formulation, Jablonski, Petersen, and Williams have recently proved linear stability for a large class of expanding homogeneous solitons [157], greatly extending earlier work of three of the authors of this work [129]. Jablonski, Petersen, and Williams prove, for example, that all nilsolitons4 of dimensions n ≤ 6 as well as some families of solvsolitons are strictly linearly stable; we refer the reader to [157] for precise statements. 1.7. Linear stability of Perelman’s average energy. Cao, Hamilton, and Ilmanen [47] have proved a result that illustrates the interesting fact that the linear stability of Perelman’s average energy functional is equivalent to the linear stability of the Ricci flow itself. If (Mn , g) is a compact manifold and f : Mn → R is a smooth function, Perelman [312] defines the average energy functional  F(g, f ) = (|∇f |2 + R)e−f dμ Mn

and the diffeomorphism invariant quantity   ∞ n (35.8) λ(g) = inf F(g, f ) : f ∈ C (M ),

−f

e

 dμ = 1 .

Mn

As shown in Chapter 5 of Part I, the gradient flow of F is equivalent to the coupled modified Ricci flow ∂ g = −2(Rc +∇2 f ), ∂t   ∂ + Δ + R f = 0. ∂t Conversely, this flow is equivalent (up to diffeomorphism) to the system ∂ g = −2 Rc, ∂t   ∂ 2 + Δ + R f = |∇f | . ∂t Furthermore, the critical points of the gradient flow are precisely the compact steady gradient solitons. All such metrics are necessarily Einstein, that is, Ricci flat. Because of this correspondence, it is not surprising that the second variation of λ at a critical point is intimately related to the first variation of Ricci flow at a fixed point. Proposition 35.13 ([47]). Let (Mn , g) be a compact Ricci-flat manifold. Then the second variation of Perelman’s λ-invariant is given by  D2 λ(g) (h, h) = Lh, h dμ, 4 A soliton on a nilpotent group is called a nilsoliton, and a soliton on a solvable group is called a solvsoliton.

2. ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY

where

⎧ ⎨ Lh =

⎩

1 2 ΔL h,

h ∈ H,

0,

h ∈ V.

287

In [47], Cao, Hamilton, and Ilmanen also study the second variation of the infimum ν of Perelman’s entropy functional W. They show in particular that CPn is linearly stable (but not strictly so) with respect to ν, but that any positively curved K¨ ahler–Einstein manifold (Mn , g) with dim H (1,1) (M) ≥ 2 is unstable. Hall and Murphy [133] extend this result to shrinking K¨ahler–Ricci solitons, showing that any such manifold with dim H (1,1) (M) ≥ 2 is also unstable with respect to ν. A corollary of the Hall–Murphy result is that the Koiso–Cao soliton [168], [44] 2 2 on CP2 # CP and the Wang–Zhu soliton [431] on CP2 # (2CP ) are both linearly unstable. 1.8. Dynamic stability. Fix a compact manifold Mn and let M denote a normed space of Riemannian metrics on Mn . Suppose g¯ ∈ M is a fixed point of a geometric evolution equation  ∂ ∂t g = F (g), (35.9) g(0) = g0 . In the rest of this chapter, we are interested in dynamic stability at g¯, which we define in two forms: Definition 35.14. A stationary solution g¯ of (35.9) is • asymptotically stable if there is an open set U ⊆ M containing g¯ such that every solution of (35.9) with g0 ∈ U exists for all positive time and converges to g¯ as t → ∞, • stable if for every open set V ⊆ M containing g¯, there exists an open set U ⊆ V containing g¯ such that every solution of (35.9) with g0 ∈ U exists for all positive time and stays in V . We emphasize that passing from linear stability to one of these two forms of dynamic stability can be highly nontrivial. In §2, we present an approach that accomplishes this, particularly in the presence of a center manifold. In §3, we survey some successful applications of this and similar theories. Finally, in §4, we survey various dynamic stability results obtained via alternate approaches. 2. Analytic semigroups and maximal regularity theory Tools to pass from linear stability to dynamic stability using semigroups were first developed for semilinear autonomous pde (see, for example, [149] and [81]). These techniques have been extended to fully nonlinear autonomous equations in the work of Da Prato and Lunardi [92]. The case in which the linearization possesses a null eigenvalue is studied for quasilinear diffusion-reaction equations in Simonett [383]. Simonett’s method also provides the “regularity boost” noted in the introduction. To move from linear stability to dynamic stability, each of the nonlinear methods essentially involves an application of the implicit function theorem, in some

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35. STABILITY OF RICCI FLOW

form. For these stability techniques to be effective, one needs function spaces in which the linear Cauchy problem (35.10a)

u (t) = Au(t) + f (t) , 0 ≤ t ≤ T,

(35.10b)

u (0) = 0

has a unique solution for which u and Au possess the same regularity as f . Maximal regularity theory provides a way to do this, through the use of suitable interpolation spaces. Using some basic ideas from analytic semigroup theory, we now show how the interpolation spaces involved arise and how to use the theory to derive estimates that allow one to move from linear stability to stability. Parts of our exposition follow proofs in Lunardi [214], Clement and Heijmans [84], and [92], to which the reader is referred for further details. Throughout this section, E1 and E0 denote complex Banach spaces with E1 continuously embedded in E0 , and A : E1 → E0 denotes a densely defined linear operator. Remark 35.15. Given an operator Aˆ : E91 → E90 between real Banach spaces, one constructs an analytic semigroup using the complexified spaces Ek = {u + iv : u, v ∈ E9k } (k = 0, 1) and the complexified operator A : E1 → E0 defined by ˆ + iA(v). ˆ A(u + iv) = A(u) Below, we implicitly assume that everything in sight has been complexified in this way. Notice that the initial value problem (35.10) has the formal solution  t (35.11) u(t) = Sf (t)  e(t−s)A f (s)ds. 0

Semigroup theory, first introduced by Hille in 1948, provides a way to interpret etA . If this “variation of constants” formula is a solution of the initial value problem, then it can be used together with properties of etA to prove existence and uniqueness of solutions of (35.10), and to obtain the estimates that allow one to determine stability, instability, and center manifold behavior of solutions of (35.10). (An excellent reference for the calculus of semigroups and for interpolation and maximal regularity theory is found in Chapters 5 and 6 by Clement and Heijmans in [84]. For more detail on analytic semigroups, we refer the reader to [214].) To define etA , one uses the resolvent of A : E1 → E0 , given by R(λ, A)  (λj − A)−1 : E0 → E1 , where j : E1 ⊆ E0 is the inclusion map. For our purposes, we further wish to guarantee that t → etA is analytic and so restrict ourselves to a certain class of operators. We say that A is sectorial if the following two conditions are met: (1) There exists ω ∈ R such that (λj − A) : E1 → E0 is an isomorphism for all λ with Re λ ≥ ω. C f E0 . (2) There exists a constant C such that R(λ, A)f E1 ≤ |λ−ω| Remark 35.16. The term “sectorial” comes from the spectrum being contained in a wedge in a left (complex) half-plane. Remark 35.17. One can show that condition (2) is satisfied, for example, on the domain Rn for an elliptic operator A with bounded coefficients, via the Agmon– Douglis–Nirenberg inequalities. For linearized Ricci flow at a flat metric, A is the rough Laplacian, and so one can use standard Schauder estimates on a compact manifold (see, for example, Lemma 35.26 below).

2. ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY

One defines etA 

1 2πi

289

 etλ jR (λ, A) dλ, γ

where γ is a contour around the spectrum. If A is sectorial, then t → etA is a strongly continuous, analytic semigroup. That is, limt→0 ||etA x − x|| = 0 for all x ∈ E0 , the map t → etA is analytic, and e(t+s)A = etA esA . If f ∈ C([0, T ]; E1 ), it is not hard to show that formula (35.11) admits a rigorous interpretation. In this case, u(t) ∈ C([0, T ]; E1 ) ∩ C 1 ([0, T ]; E0 ) solves the initial value problem (35.10). However, without such a strong restriction on f , things may not work so smoothly. Assume that we only have f ∈ C([0, T ]; E0 ). In this case, we are only able to conclude that Sf defined by (35.11) satisfies Sf ∈ C θ ([0, T ]; E0 ) ∩ C([0, T ]; Eθ ), where Eθ is (for the chosen Banach spaces E1 ⊂ E0 and for any θ ∈ (0, 1)) the corresponding “interpolation space”, as defined below. One is not able to claim that u(t) = Sf is a solution of the initial value problem. Let us see why. We may assume that ω = sup{Re λ : λ ∈ σ(A)} < 0 without loss of generality since " otherwise we " may consider A − ωj. Then one has the operator bound "Ak etA "E ≤ C/tk , and 0 one calculates that " t  "   τ A  " " (τ +t−s)A (t−s)A e u(t) − u(t) = " " − e e f (s) ds " " E0 0 E0 " t  τ " " " =" Ae(σ+s)A f (t − s) dσ ds" " " 0

0

 t

E0

τ

dσds σ +s 0 0   1 ≤ C max ||f (s)||E0 τ log . s τ

≤ C max ||f (s)||E0 s

(See Lemma 6.9 of [84].) Note that in this calculation, the constant C depends upon the time limit T . This dependence is what  allows us to eliminate the dependence  on t of this quantity. So eτ A u (t) − u (t)E0 is only o(τ θ ) for θ < 1. One can show that u ∈ C θ ([0, T ], E0 ) similarly. Unfortunately, this result is optimal. One way to handle this is to assume more regularity on f . An alternative approach, which does not require enhanced regularity, is the maximal regularity method, developed by Da Prato and Grisvard in [91]. For this method, we need to carefully define interpolation spaces Eθ . For θ ∈ (0, 1), let hθ denote the little H¨ older space defined as follows: If A generates a strongly continuous semigroup, then for u ∈ E0 and u(t) = etA u, u(·) ∈ hθ Define

⇔

lim t−θ ||u(t) − u||E0 = 0. t↓0

 Eθ = Iθ (E0 , E1 )  u ∈ E0 : etA u ∈ hθ . (See Subsection 2.1 below for more comprehensive definitions of the little H¨older spaces.) Da Prato and Grisvard have proved the remarkable result that if f takes values in Eθ , then  t u(t) = e(t−s)A f (s)ds ∈ C 1 ([0, T ]; Eθ ) ∩ C([0, T ]; E1 ) 0

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is a unique solution of the initial value problem. (See Theorem 6.10 in [84] for a detailed proof.) So this says that if the spaces chosen are the interpolation spaces E1+θ and Eθ and if the sectorial operator A: E1+θ → Eθ extends to a sectorial operator A : E1 → E0 , then indeed the formal expression (35.11) is the unique solution of the initial value problem. We can use this expression, as defined in terms of these interpolation spaces, to derive estimates controlling the solutions of linear initial value problems and then proceed to prove stability for nonlinear evolution. We carry out this two-step procedure relying on the following two propositions: Proposition 35.18 (Estimates for linear initial value problems). Let u(t) be a solution of (35.10) with f ∈ C([0, T ]; Eθ ). If ω = sup{Re λ : λ ∈ σ(A)} < 0 and if η satisfies η + ω < 0, then       ≤ C sup eηt f (t) . (35.12) sup eηt u (t) + sup eηt u(t) t

Eθ

E1+θ

t

Eθ

t

Proof. Define E2 = {x ∈ E1 : Ax ∈ E1 } , E1+θ = Iθ (E2 , E1 ). If sup σ(A) < 0, then ||x||E1+θ = ||Ax||Eθ . If η = 0, then   ||u (t)||E1+θ = ||Au||Eθ = sup ξ 1−θ A2 eξA u(t)E0 ξ>0

   t   ξ+t−s ξ+t−s ≤ sup ξ 1−θ Ae 2 A Ae 2 A f (s) ds 0 ∞

 ≤ 21−θ sup ξ

0

E0

ξ 1−θ dσ ||f ||Eθ (ξ + σ)2−θ

≤ C ||f (t)||Eθ . The result follows since u is a solution of (35.10). For general η, the proof is similar, using f1 = eηt f and u1 = eηt u.  For nonlinear initial value problems, we apply the estimates of Proposition 35.18 to the linearization of the equation and then invoke an implicit function theorem to draw conclusions about the solutions of the nonlinear equation. We sketch a proof of asymptotic stability for nonlinear evolution of the following form. (See Theorem 2.2 in [92] and the proof of Theorem 9.1.2 in [214] for more details.) We work with (35.13)

u (t) = F (u(t)), t ≥ 0, u(0) = u0 ,

where F ∈ C 1 (E1 and E0 ), F ∈ C 1 (E1+θ , Eθ ) for some θ < 1, where F (0) = 0 (so 0 is a fixed point), and where for each u in an open ball B(0, R0 ) ⊂ E1+θ of radius R0 around 0, one has appropriate hypotheses on F  (u) : E1+θ → Eθ . (Assume, for example, that it generates an analytic semigroup and has bounded ¯ of the real parts of the eigenvalues of F  is negative, norm.) If the supremum λ then one can show that there exists an r > 0 so that if ||u0 ||E1+θ < r, then u ∈ Cη ([0, ∞); E1+θ ) ∩ Cη1 ([0, ∞); Eθ ) (where the Cη spaces are defined below), and it decays exponentially.

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291

¯ We choose η so that λ+η < 0, and we let Cη be the weighted space of functions f such that t → eηt f (t) is continuous and bounded. We define an operator A : B(0, R) ⊂ Cη1 ([0, ∞); Eθ ) ∩ Cη ([0, ∞); E1+θ ) → Cη ([0, ∞); Eθ ⊕ E1+θ ) by

A(u) = (u − F (u), u0 ), where R < R0 . It follows from maximal regularity theory (see for example the results of Da Prato and Grisvald [91]) that the linear initial value problem v  = F  (0)v, v(0) = v0 has a unique solution and that the linear operator A (0)v = (v  − F  (0)v, v0 ) is an isomorphism. For nonlinear equations, we can prove the following proposition. We assume that we can rewrite (35.13) in the form x = Ax + f (x(t)), x(0) = x0 , where A (1) (2) (3)

is linear, and that the following conditions hold: f ∈ C 1 (E1+θ , Eθ ). A is sectorial. f  (x) is locally Lipschitz continuous; i.e., there exists R0 such that sup

||f  (x) − f  (y)||L(E1 ,E0 ) ||x − y||E1

0 so that if ||u0 ||E1+θ ≤ r, then the initial value problem (35.13) (subject to conditions (1)–(3) enumerated above) has a unique solution u ∈ Cη1 ([0, ∞); Eθ ) ∩ Cη ([0, ∞); E1+θ ). Moreover,   =0 lim eηt u(t, u0 )E t→∞

1+θ

uniformly with respect to u0 . Proof. The claim is proved essentially by an implicit function theorem argument. We clarify the argument using fixed point theory. Consider a solution in the weighted space Y = Cη ([0, ∞); Eθ+1 ) ∩ Cη1 ([0, ∞); Eθ ), where η is chosen so that η + ω < 0, with ω = sup{Re λ : λ ∈ σ(A)} < 0. We look for a fixed point of the map Γ(x) = ξ, which maps x to the solution of the linear initial value problem ξ  (t) = Aξ + f (x(t)), ξ(0) = x0 . If u ∈ B(0, R0 ) ⊂ Y, then the assumptions on f imply that f (x(t)) ∈ Cη ([0, ∞); Eθ ). It then follows from the linear bound (35.12) that ||Γ(x)||Y ≤ C ||f (x)||Cη ([0,∞);Eθ ) ,

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35. STABILITY OF RICCI FLOW

so Γ maps into Y . Using the estimate  1 f  (σx(t) + (1 − σ)y(t)) dσ(x(t) − y(t)), f (x(t)) − f (y(t)) ≤ 0

we have ||Γ(x) − Γ(y)||Y ≤ CR ||x − y||Y . Hence ||Γ(x)||Y ≤ ||Γ(x) − Γ(0)||Y + ||Γ(0)||Y   1 ≤ ||x||Y + eA x0 Y . 2 Letting x be the fixed point, this implies that ||x||Y ≤ C ||x0 ||Eθ+1 , where we have used η + ω < 0. So for sufficiently small x0 , one has ||x ||Eθ + ||x||Eθ+1 ≤ Ce−ηt ||x0 ||Eθ+1 .



2.1. Little H¨ older spaces. We have seen how little H¨older spaces arise naturally in maximal regularity theory. They are the spaces that we use to prove stability of the Ricci–DeTurck flow at a flat metric. In this subsection we define them precisely for the space of symmetric (2, 0)-tensors and summarize some results from the theory that we find useful. The reader may reference this section as needed. Recall that if r ∈ N and ρ ∈ (0, 1), the H¨older space C r,ρ is the Banach space of all C r functions f : Rn → R for which the H¨older norm f r+ρ is finite. The subspace of C ∞ functions in C r,ρ is not dense, and so for our purposes we require a smaller space. We use the following formulation of the little H¨ older spaces: Definition 35.20. The little H¨ older space hr+ρ (Rn ) of functions is the ∞ closure of the subspace of C functions with respect to the ·r+ρ -norm. It is clear that hr+ρ is a Banach space. Furthermore, hr+ρ → hs+σ is a continuous and dense inclusion if s ≤ r and 0 < σ < ρ < 1. Corresponding statements hold for C r,ρ (Ω) if Ω ⊂ Rn is a bounded C ∞ domain. By fixing a smooth atlas, one can extend these definitions to functions defined on Mn and taking values in the bundle S2 (Mn ). (See [40].) Definition 35.21. The little H¨ older space hr+ρ of symmetric 2-tensors is the closure of C ∞ (Mn , S2 (Mn )) with respect to the H¨ older norm ·r+ρ . Note in particular that (35.14)

hr+ρ → hs+σ

remains a continuous and dense inclusion. An exact interpolation space Iθ of exponent θ ∈ (0, 1) takes any pair B1 ⊆ B0 of Banach spaces to a Banach space Iθ (B0 , B1 ) such that B1 ⊆ Iθ (B0 , B1 ) ⊆ B0 and such that T ∈ L (B0 , A0 ) ∩ L (B1 , A1 ) only if T ∈ L (Iθ (B0 , B1 ) , Iθ (A0 , A1 ))

2. ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY

293

and such that θ T L(Iθ (B0 ,B1 ),Iθ (A0 ,A1 )) ≤ T 1−θ L(B0 ,A0 ) T L(B1 ,A1 ) .

Various formulations of interpolation spaces are found in the literature. In studying the stability of the Ricci–DeTurck flow, we use the continuous interpolation spaces introduced in [91]. (As shown in [96], they are equivalent in norm to real interpolation spaces.) These continuous interpolation spaces, which one can verify are exact, are defined as follows: Definition 35.22. Let B0 and B1 be Banach spaces, and let θ ∈ (0, 1). The continuous interpolation space (B0 , B1 )θ is the set of all x ∈ B0 such that there exist sequences {yn } ⊆ B0 and {zn } ⊆ B1 with x = yn + zn , where    and zn B1 = o 2n(1−θ) yn B0 = o 2−nθ as n → ∞. The norm on (B0 , B1 )θ is equivalent to    inf sup 2nθ yn B0 , 2−n(1−θ) zn B1 , n≥1

where the infimum is taken over all such sequences (yn , zn ). Remark 35.23. This is compatible with the definition given in the introduction to this section. If A generates a strongly continuous semigroup and u ∈ E0 , then u ∈ Eθ if and only if etA u ∈ hθ . We need the following fact about continuous interpolation spaces: Lemma 35.24 ([419]). Let s ≤ r ∈ N, 0 < σ < ρ < 1, and 0 < θ < 1. If θ (r + ρ) + (1 − θ) (s + σ) is not an integer, then there is a Banach space isomorphism  s+σ r+ρ ∼ h(θr+(1−θ)s)+(θρ+(1−θ)σ) , ,h (35.15) h θ = and there exists C < ∞ such that for all η ∈ hr+ρ , (35.16)

θ η(hs+σ ,hr+ρ ) ≤ C η1−θ hs+σ ηhr+ρ . θ

2.2. The Ricci–DeTurck operator revisited. Let us now examine the Ricci–DeTurck operator (35.3) in the context of maximal regularity theory for nonlinear equations, as described in the introduction to §2 of this chapter. We fix a background metric gˆ and recall that Agˆ (g) is described locally by the expression (35.6). For fixed 0 < σ < ρ < 1, we define the following nested spaces of metrics, where the hr+ρ are defined in Definition 35.21:

(35.17)

E0 ∪ X0 ∪ E1 ∪ X1



h0+σ



h0+ρ



h2+σ

 h2+ρ .

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35. STABILITY OF RICCI FLOW

Notice that for θ  (ρ − σ) /2 ∈ (0, 1), it follows from (35.15) that (35.18)

X0 ∼ = (E0 , E1 )θ

and

X1 ∼ = (E0 , E1 )(1+θ) .

For fixed 0 < ε , 1 and 1/2 ≤ β < α < 1, we also define 0 1 g , ε)  g ∈ (X0 , X1 )β : g > εˆ g Gβ = Gβ (ˆ and Gα = Gα (ˆ g , ε)  Gβ ∩ (X0 , X1 )α , 2

where g > εˆ g means g (X, X) > ε for any vector X such that |X|gˆ = 1. Now for each g ∈ Gβ , expression (35.6) allows us to define a linear operator Agˆ,g on h2+σ = E1 by ∂2 γk ∂xp ∂xq  k kp ∂ + b (x, gˆ, ∂ˆ g , g)ij γk + c x, gˆ, ∂ˆ g , ∂ 2 gˆ ij γk . ∂xp

Agˆ,g (γ)ij = a (x, gˆ, g)kpq ij

For g ∈ Gβ , let us denote by A˜ (g) : E1 ⊆ E0 → E0  ˜ the unbounded linear operator Agˆ,g on E0 with dense domain D A(g) = E1 . Also we denote by A (g) : X1 ⊆ X0 → X0

 the unbounded linear operator Agˆ,g on X0 with dense domain D A(g) = X1 . By estimating appropriate H¨ older norms, one can prove that for each g ∈ Gα , the assignment γ → A (g) γ is a bounded linear map from X1 to X0 and that for each g ∈ Gβ , the assignment γ → A˜ (g) γ is a bounded linear map from E1 to E0 . We summarize these results as follows: Lemma 35.25. The functions g → A (g) and g → A˜ (g) define analytic maps Gα → L (X1 , X0 ) and Gβ → L (E1 , E0 ), respectively. Although for every g ∈ Gβ , A˜ (g) is bounded if regarded as an operator E1 → E0 , it is unbounded if regarded as an operator E0 → E0 , and it is in fact only defined  ˜ on a dense subspace D A (g) = E1 . Nonetheless, it has the desirable property of generating an analytic semigroup, which is bounded (and hence defined everywhere) as a map E0 → E0 . Lemma 35.26. For every g ∈ Gβ , A˜ (g) is the infinitesimal generator of a strongly continuous analytic semigroup on L (E0 ). Proof. For all g ∈ Gβ , the operator A˜ (g) is strongly elliptic. So its spectrum is discrete, having a limit point only at −∞. Hence there is λ0 > 0 such that λI − A˜ (g) is a topological linear isomorphism from E1 onto E0 so long as Re λ ≥ λ0 .

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295

In this case, a standard Schauder estimate applied to the operator λI − A˜ (g) yields C < ∞ such that " "  "  " " " " " ηE1  ηh2+σ ≤ C " λI − A˜ (g) η " 0+σ  C " λI − A˜ (g) η " h

E0

for every η ∈ E1 = D(A˜ (g)). Using Theorem 1.2.2 and Remark 1.2.1(a) of [6], we see that the result follows from this estimate.  2.3. A center manifold theorem. If A is a well-behaved quasilinear differential operator acting on appropriate function spaces and if its linearization DA(¯ u) at a fixed point u ¯ has an eigenvalue on the imaginary axis, then one expects the evolution of solutions starting near that fixed point to be characterized by the presence of exponentially attractive (or repulsive) center manifolds. Below, we state the Center Manifold Theorem, which makes this intuition precise and is an adaptation of results5 in [383]. However the statement of the theorem is rather technical, and so we first give a rough description of its ingredients and conclusions. The theorem follows from maximal regularity theory, and we therefore need to define appropriate interpolation spaces; its application also requires that the extension of the operator generate an analytic semigroup. We must work locally to apply the implicit function theorem, and to this end we use the space Gα . This is the content of the hypothesis of the theorem. The tangent space to the space of metrics at the fixed point g¯ must be decomposable into stable and (center) unstable subspaces, based on the signs of the real parts of the eigenvalues of the operator. We expect unstable behavior on the subspace of eigenvectors associated with positive eigenvalues, stable behavior on the subspace associated with negative eigenvalues, and center manifold behavior (possibly stable or unstable) if the eigenvalue has zero real part. The content of conclusion (1) of Theorem 35.27 below is that one can indeed decompose the space into these parts. Conclusion (2) of Theorem 35.27 says that there exists a (local) center manifold with locally invariant behavior. That is, there exists a small neighborhood such that if the solution is in that neighborhood, it can only move along the center manifold. For Ricci flow we can make an additional argument that once solutions land there they stay there, but that is not the case in general. From the variation of constants formula obtained via maximal regularity theory, we expect to obtain estimates of the solution in terms of its initial value, as well as its exponential behavior; this is the content of conclusion (3). We are ready for the statement of Simonett’s center manifold theorem (note that B (X , x, d) denotes the open ball of radius d centered at x in a metric space X ). For simplicity, we present a version of the theorem that does not include the optimal boost in regularity. Theorem 35.27 (Center Manifold Theorem). Let X1 → X0 be a continuous dense inclusion of Banach spaces, and let Xα and Xβ denote the continuous interpolation spaces corresponding to fixed 0 < β < α < 1. Let ∂ g = A (g) g (35.19) ∂t 5 See,

in particular, Theorems 4.1 and 5.8 and Remark 4.2 in [383].

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35. STABILITY OF RICCI FLOW

be an autonomous quasilinear parabolic equation posed for t ≥ 0 such that A (·) ∈ C k (Gβ , L (X1 , X0 )) for some positive integer k and some open subset Gβ ⊆ Xβ . Assume that there exists a pair E0 ⊇ E1 of Banach spaces and extensions A˜ (·) of A (·) to domains D(A˜ (·)) that are dense in E0 such that the following conditions hold for each g ∈ Gα  Gβ ∩ Xα : • A˜ (g) ∈ L (E1 , E0 ) generates a strongly continuous analytic semigroup on L (E0 ); • X0 ∼ = (E0 , D(A˜ (g)))θ and X1 ∼ = (E0 , D(A˜ (g)))(1+θ) for some θ ∈ (0, 1); • A (g) agrees with the restriction of A˜ (g) to the dense subset D (A) ⊆ X0 ; • E1 → Xβ → E0 is a continuous and dense inclusion with the property that there are C > 0 and δ ∈ (0, 1) such that for all η ∈ E1 , one has δ ηXβ ≤ C η1−δ E0 ηE1 .

Let g¯ ∈ Gα be a fixed point of (35.19). Suppose that the spectrum Σ of the linearized operator Dg¯ A admits the decomposition Σ = Σs ∪Σc , where Σs ⊂ {z : Re z < 0} and where Σc ⊂ {z : Re z ≥ 0} consists of finitely many eigenvalues of finite multiplicity. Suppose further that Σc ∩ iR = ∅. Then the following hold: (1) If S (λ) denotes the eigenspace corresponding to λ ∈ Σc , then Xα ads c c mits = the decomposition Xα = Xα ⊕ Xα for all α ∈ [0, 1], where Xα ≡ λ∈Σc S (λ). (2) For each r ∈ N, there exists dr > 0 such that for all d ∈ (0, dr ], there g ) = 0 and is a bounded C r map ψ[r,d] : B (X1c , g¯, d) → X1s with ψ[r,d] (¯ s ¯ Dg¯ ψ = 0. The image of ψ[r,d] lies in the closed ball B (X1 , g¯, d), and its graph is a C r manifold  Mcloc  γ, ψ[r,d] (γ) : γ ∈ B (X1c , g¯, d) ⊂ X1 satisfying Tg¯ Mcloc ∼ = X1c . One says Mcloc is a local center manifold if Σc ⊂ iR and a local center unstable manifold otherwise. Mcloc is locally invariant for solutions of (35.19) as long as they remain in B (X1c , g¯, d) × B (X1s , 0, d). (3) For all α ∈ (0, 1), there are constants Cα > 0 independent of g¯ and ¯ one has constants ω > 0 and d¯ ∈ (0, d0 ] such that for each d ∈ (0, d], " " " " s "π g (t) − ψ[r,d] (π c g (t))" ≤ Cα e−ωt "π s g (0) − ψ[r,d] (π c g (0))" 1−α X1 Xα t for all solutions g (t) with g (0) ∈ B (Xα , g¯, d) and all times t ≥ 0 such that the solution g (t) remains in B (Xα , g¯, d). Here π s and π c denote the projections onto Xαs ∼ = (X1s , X0s )α and Xαc , respectively. The reader is strongly cautioned that local center (unstable) manifolds are not in general unique. For instance, it may be the case that dr → 0 as r → ∞. Furthermore, the Center Manifold Theorem does not in general tell us anything about the dynamics within a local center (unstable) manifold. In particular, it is not true in general that center manifolds consist of stationary solutions. 3. Dynamic stability results obtained using linearization 3.1. Stability of Ricci flow at a flat metric. We are now ready to illustrate the method described above by establishing the dynamic stability of Ricci flow at a flat compact manifold (Mn , g¯). We shall find

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION

297

that g¯ belongs to a unique exponentially attractive C ∞ center manifold consisting exactly of flat metrics. Furthermore, any solution starting sufficiently near g¯ converges to a flat metric in this center manifold. We begin with the observation that stability of the Ricci flow follows from the stability of the Ricci–DeTurck flow with background metric gˆ = g¯: ∂ g = Ag¯ g = −2 Rc (g) − Pg¯ (g), (35.20a) ∂t (35.20b) g (0) = g0 . Lemma 35.28. Let W (t) be a vector field on a compact Riemannian manifold (Mn , g (t)), where 0 ≤ t < ∞, and suppose there exist 0 < c ≤ C < ∞ such that sup |W (x, t)|g(t) ≤ Ce−ct .

x∈M

Then the diffeomorphisms ϕt generated by W converge exponentially to a fixed diffeomorphism ϕ∞ of Mn . Proof. Given x ∈ M, let γ : [0, ∞) → Mn be an integral curve for W starting at x. Then γ satisfies γ  (t) = W (γ (t) , t) , γ (0) = x, where we make the standard identification γ  ≡ γ∗ (d/dt). The length Lt [γ] of the integral curve is nondecreasing and bounded above because  t  t C C 1 − e−ct ≤ . Lt [γ] = |W (x, τ )|g(τ ) dτ ≤ C e−cτ dτ = c c 0 0 This proves that Lt [γ] converges as t → ∞. To see that the convergence is exponential, it suffices to note that  ∞  ∞ C |W (x, τ )|g(τ ) dτ ≤ C e−cτ dτ = e−ct . c t t n Since γ (t) = ϕt (x) and since x ∈ M is arbitrary, the result follows.  Corollary 35.29. Let g¯ be a flat metric on a compact manifold Mn . Suppose 2 there is a neighborhood O of g¯ in S+ with respect to the ·2+ρ H¨ older norm such that for every g0 ∈ O, the unique solution g (t) of the Ricci–DeTurck flow (35.20) converges exponentially fast to a flat metric g∞ . Then the unique solution gˆ (t)  ϕ∗t g of the Ricci flow with the same initial data gˆ (0) = g0 ∈ O converges exponentially fast to a flat metric gˆ∞ . Proof. It is clear that gˆ∞ must be flat if it exists, so all we need to do is show that gˆ (t) converges. But because g∞ and g¯ are both flat, one has ∇[g∞ ] g¯ = ∇[¯g] g¯ ≡ 0. Then since g (t) → g∞ exponentially fast, it follows that the infinitesimal generators W (t) of the DeTurck diffeomorphisms vanish exponentially, where    −1 1 i ij k pq g g W  g g¯ ∇p g¯q − ∇ g¯pq . jk 2 (Here ∇ denotes covariant differentiation with respect to the Levi-Civita connection of g (t).) Hence it follows from the lemma that the solution gˆ (t) converges  exponentially fast to some limit gˆ∞ .

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35. STABILITY OF RICCI FLOW

Combining this corollary together with Theorem 35.27, we obtain the following stability result for flat metrics under Ricci flow: Theorem 35.30 (Asymptotic stability at Rm = 0). Let (Mn , g¯) be a flat compact Riemannian manifold. For fixed ρ ∈ (0, 1), let X denote the closure of 2 with respect to the ·2+ρ H¨ older norm. Then: S 2 ⊃ S+ 2 ∼ (1) Tg¯ S = X admits the decomposition +

2 Tg¯ S+ = X s ⊕ X c,

where X c is the n (n + 1) /2-dimensional space of (2, 0)-tensors parallel with respect to the Levi-Civita connection of g¯. (2) There exists d0 > 0 such that for all d ∈ (0, d0 ], there is a bounded C ∞ map ψ : B (X c , g¯, d) → X s such that ψ (¯ g ) = 0 and Dg¯ ψ = 0. The image of ψ lies in the closed ball ¯ (X s , g¯, d), and its graph B Mcloc  {(γ, ψ (γ)) : γ ∈ B (X c , g¯, d)} satisfies Tg¯ Mcloc ∼ = X c . In other words, Mcloc is a unique C ∞ local center manifold of dimension n (n + 1) /2 that consists entirely of flat metrics. (3) There are constants C > 0, ω > 0, and d∗ ∈ (0, d0 ] such that for each d ∈ (0, d∗ ], one has π s gˆ (t) − ψ (π c gˆ (t))2+ρ ≤ Ce−ωt π s gˆ (0) − ψ (π c gˆ (0))2+ρ for all solutions gˆ (t) of the Ricci flow with gˆ (0) ∈ B (X , g¯, d) and all times t ≥ 0. Here π s and π c denote the projections onto X s and X c , respectively. In particular, any solution gˆ (t) of Ricci flow with initial data sufficiently near g¯ converges exponentially to a flat metric near g¯. Proof. By passing to a finite cover, we may assume that Mn is a torus. We shall apply the Center Manifold Theorem to the Ricci–DeTurck flow (35.20) and then use Corollary 35.29 to form a conclusion about the Ricci flow itself. Recall that any flat metric is a stationary solution of the Ricci–DeTurck flow. g ) is the rough Laplacian. This In the previous section, we proved that DAg¯ (¯ operator is negative semidefinite on S 2 with its kernel consisting exactly of the parallel (2, 0)-tensors, hence of dimension at most n (n + 1) /2. For the choices we have made above for X0 , X1 , E0 , and E1 (i.e., X0 = h0+ρ , X1 = h2+ρ , E0 = h0+σ , and E1 = h2+σ , for a fixed choice of σ and ρ satisfying 0 < σ < ρ < 1, with the little H¨older spaces defined in Definition 35.21) the results of Lemmas 35.25 and 35.26 allow us to apply the Center Manifold Theorem to the operator Ag¯ . It follows that local C r center manifolds r Mcloc exist and that the Ricci–DeTurck flow of any metric starting near g exponentially approaches r Mcloc . We now claim that the spaces r Mcloc are independent of r and consist entirely of flat metrics. To prove this, we observe that any flat metric g˜ sufficiently near g¯ belongs to r Mcloc for all r ∈ N: If not, then statement (3) of the Center Manifold Theorem would imply that g˜ converges exponentially to r Mcloc , contradicting the fact that g˜ is a stationary solution. The space of flat metrics on the torus is a convex n (n + 1) /2-dimensional set. Since each r Mcloc is at most n (n + 1) /2-dimensional, it follows that r Mcloc consists exactly of the flat metrics for all r ∈ N.

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION

299

Thus far in this argument, we have shown that there exists a neighborhood B (X , g¯, δ) such that the Ricci–DeTurck flow g (t) starting at any g0 ∈ B (X , g¯, δ) becomes flat exponentially fast in the ·2+ρ -norm for as long as g (t) ∈ B (X , g¯, δ). Since |Rij |, |∇j g¯k − ∂j g¯k |, and |∇i ∇j g¯k − ∂i ∂j g¯k | all decay exponentially fast if g (t) ∈ B (X , g¯, δ), there are C = C (δ) < ∞ and ω = ω (δ) > 0 such that   ∂   g  = |−2 Ric −Pg¯ (g)| ≤ Ce−ωt  ∂t  for as long as g (t) remains in B (X , g¯, δ). If we now choose 0 < ε < δ small enough so that C (ε) /ω (δ) < δ − ε, it then follows that for all solutions g (t) with g0 ∈ B (X , g¯, ε), we can estimate |g (t) − g¯| ≤ |g (t) − g0 | + |g0 − g¯| < δ − ε + ε = δ independently of t ≥ 0. Bounds on the derivatives of g are shown similarly. It follows that g¯ (t) remains in B (X , g, δ) for all time and hence converges to a unique flat metric. By Corollary 35.29, the same is true of the unique solution of the Ricci flow with the same initial data.  3.2. Dynamic stability in the presence of negative curvature. The situation regarding negatively curved metrics is considerably more delicate than the positively curved case. For example, for every n ≥ 4 and ε > 0, Gromov and Thurston [126] prove that there exists a closed n-manifold that admits a metric with pinched sectional curvatures −1 − ε < K ≤ −1 but no metric with K ≡ −1. Moreover, Farrell and Ontaneda [108] show that if n ≥ 10 and Mn admits a metric with negative sectional curvatures, then the space of negatively curved metrics on Mn has infinitely many path components. (Also see [107].) So it may be impossible to deform a given negatively curved metric continuously to an Einstein metric of negative sectional curvature. However, it is possible to establish both linear and asymptotic stability for a modified flow at Einstein manifolds with negative scalar curvature, if one assumes closeness to the hyperbolic metric in an appropriate norm, and not merely a curvature-pinching condition. (See Subsection 4.4 below for related work of Ye obtained using a very different method.) We now discuss this approach more precisely. Suppose (Mn , gˆ(t)) is a solution of unnormalized Ricci flow that exists for −∞ < T0 < t < T1 ≤ ∞. Given any constant r > 0, we define a scaling factor σ(t) = 2r(t − T0 ) and a rescaled time variable 1 log(t − T0 ), 2r noting that dτ /dt = 1/σ(t). Defining a 1-parameter family of metrics g˜(τ ) on Mn via gˆ(t) = σ(t)˜ g (τ (t)), 2rτ −1 2rτ i.e., g˜(τ ) = σ(T0 + e ) gˆ(T0 + e ), we see that the family g˜(τ ) is equivalent to gˆ(t) modulo rescaling of space and time. Moreover, g˜(τ ) evolves by the dilated Ricci flow ∂ g˜ = −2 Rc −2r˜ g. (35.21) ∂τ τ (t) =

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If we now specify a compact Einstein manifold (Mn , g) with Rc = −rg for some r > 0, then g is a fixed point of the rescaled Ricci flow (35.21). After applying the usual DeTurck trick, we verify that the linearization of this flow at the fixed point g is given by the formally self-adjoint operator L that sends a smooth symmetric (2, 0)-tensor h to ΔL h − 2rh, where ΔL is the Lichnerowicz Laplacian. In coordinates, one has (Lh)ij = (Δhij + 2Ripqj hpq − Rik hkj − Rjk hik ) − 2rhij = Δhij + 2Ripqj hpq , where (as usual) all indices are raised using g and covariant differentiation is carried out using the Levi-Civita connection of g. Define a smooth, compactly supported function Q(h) by Q(h) = Rijk hi hjk . Integration by parts gives  2 (35.22) (Lh, h) = − ∇h + 2 Q(h) dμ. Mn

Define a (3, 0)-tensor T (h) by Tijk = ∇k hij − ∇i hjk . Then Koiso’s Bochner-type formula [167]  1 {S(h), h − Rik hjk hij } dμ ∇h2 = δh2 + T (h)2 + 2 n M shows that

1 ∇h = T 2 + δh2 + r h2 + 2 This allows us to write (35.22) as



2

(35.23)

1 2 2 2 (Lh, h) = − T  − δh − r h + 2

Q(h) dμ. Mn

 Q(h) dμ. Mn

One can refine this further by using the irreducible decomposition of the Riemann curvature tensor Rm. Recall that for any n-dimensional Riemannian manifold, one has ◦ 1 1 R(g  g) + (Rc  g) + W, Rm = 2n(n − 1) n−2 ◦

where  denotes the Kulkarni–Nomizu product of symmetric tensors, Rc denotes the trace-free Ricci tensor, and W is the Weyl tensor. On the Einstein manifold (Mn , g), this reduces to r (g  g) + W, Rm = − 2(n − 1) which one may write in local coordinates as r (gi gjk − gik gj ) + Wijk . Rijk = − n−1 If p, q are (2, 0)-tensor fields, define W (p, q) = Wijk pi q jk . Then one can write Q(h) as Q(h) =

r {|h|2 − (trg h)2 } + W (h, h) n−1

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION

and hence estimate (35.23) by r n−2 1 trg¯ h2 − r h2 + (Lh, h) = − T 2 − δh2 − 2 n−1 n−1  n−2 2 r h + ≤− (35.24) W (h, h) dμ. n−1 Mn

301

 W (h, h) dμ Mn

Now suppose that n > 2 and that (Mn , g¯) has constant negative sectional curvature. Then the Weyl curvature W of g¯ vanishes, and g¯ is a stationary solution of the dilated Ricci flow (35.21). Calculation (35.24) shows that g¯ is strictly linearly stable. (The null eigenvalue that appears for n = 2 reflects the presence of Teichm¨ uller space as a center manifold.) For n ≥ 3, Mostow rigidity ensures that g¯ is unique up to scaling, hence that the choice of normalization in (35.21) forces a unique fixed point. Once we have (35.24) and the consequent linear stability of g¯, it follows for n > 2 that the argument presented in Proposition 35.19 implies asymptotic stability of the modified Ricci flow (35.21), modulo DeTurck diffeomorphisms, for all initial data in a 2 + α little H¨older neighborhood of a hyperbolic metric g¯, where 0 < α < 1 (see Theorem 9.1.2 or Theorem 9.1.5 in [214]).6 Because all nearby solutions converge exponentially fast to the unique fixed point of dilated Ricci flow (there is no center manifold present), it is easy to verify that the DeTurck diffeomorphisms converge exponentially fast as well; hence, every solution g˜(τ ) of the modified Ricci flow itself converges exponentially fast in τ (see Lemma 35.28 and Corollary 35.29). Moreover, the relation gˆ(t) = σ(t)˜ g (τ (t)) shows that the solution gˆ(t) of unnormalized Ricci flow obtained from g˜(τ ) exists for all time and its sectional curvatures become spatially homogeneous. Thus we have proved the following: Theorem 35.31. Let (Mn , g¯) be a compact Riemannian manifold with n > 2 and constant sectional curvature K < 0. Then there exists δ > 0 such that for older ball Bδ2+α (¯ g ) around g¯, the unique solution all initial data g0 in a little H¨ g(t) of normalized Ricci flow satisfying g(0) = g0 exists for all time and converges exponentially fast to a constant curvature metric (which takes the form λ¯ g for some λ > 0 since all constant negative curvature metrics are proportional to each other). Remark 35.32. By refining the techniques of the theorem, it is possible to show that for any initial data in a 1 + β little H¨older neighborhood of g¯, Ricci flow converges to g¯ in the 2 + α little H¨ older topology, where β ∈ (α, 1) depends on α. This result takes optimal advantage of the parabolic smoothing properties of the quasilinear nature of Ricci flow. See [162] (by one of the authors) for details. Remark 35.33. Farrell and Ontaneda [107] prove that for every n > 10 and ε > 0, there exists (Mn , g¯) such that Mn is compact, g¯ is hyperbolic, and there exists a Riemannian metric h on Mn whose sectional curvatures lie in [−1 − ε, −1 + ε] but such that the Ricci flow starting at h cannot converge smoothly to g¯. Their result is compatible with Theorem 35.31 (and its refinement in the remark above) 6 Maximal regularity theory is not necessary for this result, but it simplifies the proof and in general allows weaker regularity assumptions on the (nonlinear) right-hand side. We refer the reader to Chapters 8 and 9 of [214] for details.

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because the hypotheses of the theorem require not merely curvature pinching, but closeness in a C 2+α - (respectively, C 1+β -) topology. In the noncompact case, there has been considerable recent progress. Wu [438] has studied the stability of complex hyperbolic space of two complex dimensions and higher. To do so, he constructs weighted little H¨older spaces adapted to complete noncompact manifolds. Using this technology, Williams and Wu [436] prove dynamic stability under compact perturbations of the homogeneous solitons whose linear stability was established by Jablonski, Petersen, and Williams (see Subsection 1.6 above). 3.3. Stability for positively curved metrics. Motivated by the results above, it is natural to ask whether similar techniques might provide an alternative proof of asymptotic stability for metrics of constant positive sectional curvature. Consider normalized Ricci–DeTurck flow 2¯ ∂ g = −2 Rc + Rg + LW g (35.25) ∂t n   ¯ =( on a compact manifold Mn , where R R dμ)/( M dμ) and W is the vector M field (35.4) that we encountered in Subsection 1.2, with background metric gˆ = g (0). An Einstein metric g with Rc = Kg for any constant K is a stationary solution of (35.25). The linearization of (35.25) at the Einstein metric g is the operator   1 ¯ , (35.26) A : h → ΔL h + 2K h − Hg n   ¯ =( where H = trg h and H H dμ)/( M dμ). M Now let us make the stronger assumption that g is a metric of constant sectional curvature k > 0, so that K = (n − 1)k. By passing to a finite cover if necessary, we may further assume that Mn is the round sphere S n embedded in Rn+1 . Then one can make the following observations, whose proofs may be found in [162]: • A is linearly stable (but not strictly) if n = 2. Its null eigenspace is then the (n + 2)-dimensional space {cg : c ∈ R} ∪ {xj g : 1 ≤ j ≤ n + 1}, where the (x1 , . . . , xn+1 ) are the standard coordinate functions of Rn+1 . • A is linearly unstable for all n ≥ 3. The sole unstable eigenvalue is (n − 2)k with (n + 1)-dimensional eigenspace {xj g : 1 ≤ j ≤ n + 1}. The 1-dimensional null eigenspace is {cg : c ∈ R}. Recall that the coordinate functions act as infinitesimal conformal diffeomorphisms (M¨ obius transformations). If h = f g for some f ∈ {x1 , . . . , xn+1 }, then 1 h = − Lgrad f (g). 2k This shows that the linear instability of the operator A defined in (35.26) lies only in the direction of diffeomorphisms; hence (like the linear instability of the cylinder soliton discussed above) it is “ungeometric”. Round spheres are in fact dynamically stable, as we discuss in Subsection 4.1 below. The unstable eigenspace here is a consequence of the standard choice of DeTurck diffeomorphisms used to impose parabolicity, rather than a feature of Ricci flow itself. Indeed, Hamilton has observed [138] that those DeTurck diffeomorphisms solve a type of harmonic map

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION

303

flow with respect to a domain metric evolving by Ricci flow and a fixed target metric (in this case, the stationary solution of Ricci flow about which one is linearizing). It is well known [100] that the identity map of round spheres id : S n → S n is a weakly stable harmonic map for n = 2 and an unstable harmonic map for all n ≥ 3. 3.4. More dynamic stability results obtained by linearization. In [163], Young and one of the authors apply maximal regularity theory to the fully nonlinear cross-curvature flow, which has been developed as an alternate to the Ricci flow in the hope that it would give more information about negatively curved 3-manifolds. Given a manifold (M3 , g) with strictly negative sectional curvature, the flow is defined to be (35.27)

∂g = −2X, ∂t

where Xij = 12 P μν Riuvj and Pab = Rab − 12 Rgab is the Einstein tensor. The authors prove that hyperbolic metrics are stable stationary solutions of a normalized crosscurvature flow (as usual, obtained by rescaling in space and time to obtain a fixed point). They also prove stability of constant curvature hyperbolic manifolds under Ricci flow. The analysis uses maximal regularity theory and an asymptotic stability theorem for fully nonlinear systems [92]. In [130], another of the authors together with Todd Oliynyk applies the ideas of maximal regularity to the fully nonlinear second-order renormalization group flow α ∂ g = −2 Rc − Rm 2 . ∂t 2 (Here, α is a fixed positive constant.) In the flat case, the techniques from Ricci flow carry over in a straightforward way. For the hyperbolic case, one wishes to rescale the equation such that the metric is a fixed point. In this equation, however, the nonlinear term scales like Rm2 (cg) = 1c Rm2 (g); consequently after rescaling, the equation is no longer autonomous. Nevertheless, working instead with a related system, one does obtain a stability result for the rescaled equation, for a sufficiently small values of the parameter α. In [364], Sesum proves that for Ricci flat metrics, linear stability together with an integrability condition implies dynamical stability. The integrability condition essentially says that infinitesimal Ricci-flat deformations have a smooth manifold structure near the metric about which one linearizes. As described below in Subsection 4.2, Haslhofer and M¨ uller later showed that the integrability condition is a special case of being a maximizer of Perelman’s λ functional [147]. Sesum’s results have been used by Dai, Wang, and Wei to prove that K¨ahler–Einstein metrics of nonpositive scalar curvature are dynamically stable [89]. If an immortal solution of Ricci flow fails to converge, the failure typically manifests itself in collapse, with the injectivity radius going to zero while the curvature remains bounded. Using the machinery of ´etale groupoids, Lott [209] has proven that in many cases, properly rescaled solutions converge to solutions on bundles π G → M −→ B, with G a nilpotent Lie group and B n compact and orientable. The structure of these bundles reflects the dimension-reduction of the original solution, with the fibers G yielding geometric information about its collapsed directions. One of the authors has proven that if G is abelian (which is always true for 3-dimensional

304

35. STABILITY OF RICCI FLOW

solutions relevant to geometrization), stationary solutions of this type are stable attractors: solutions originating from initial data in a 1+α little H¨older neighborhood of a stationary solution converge to that solution exponentially fast in a 2+β H¨older norm [162]. 4. Dynamic stability results obtained by other methods 4.1. Dynamic stability in the presence of positive curvature. In this subsection, we briefly review various convergence theorems for Ricci flow obtained by methods other than linearization. For those cases in which a unique limit exists, each convergence theorem implies a stability theorem—though all are strictly stronger since none assumes existence of a stationary solution. Remark 35.34. We have inserted the hypothesis of simple connectivity in Theorems 35.35–35.37 and 35.39–35.40 below to ensure uniqueness (up to isometry) of the resulting limits. Hamilton’s seminal result for Ricci flow in dimension n = 3 implies a stability theorem because {g : Rc(g) > 0} is an open neighborhood of a metric of constant positive sectional curvature. Theorem 35.35 ([135]). If (M3 , g) is a closed simply-connected Riemannian manifold of positive Ricci curvature, then the volume normalized Ricci flow with initial data g converges to a unique metric of constant positive curvature. Of course, Hamilton’s theorem yields a much stronger result since one does not know a priori that M3 admits a constant sectional curvature metric, i.e., that M3 is a space form. Hamilton also proves the following in dimension n = 4. Theorem 35.36 ([136]). If (M4 , g) is a closed simply-connected Riemannian manifold with positive curvature operator, then the volume normalized Ricci flow with initial data g converges to a unique metric of constant positive curvature. In particular, M4 is diffeomorphic to the standard 4-sphere if it is orientable and to RP4 otherwise. Instead of assuming positivity of a curvature quantity, one may also impose a pinching condition. Huisken’s pinching theorem for dimensions n ≥ 4 implies stability of metrics of constant positive sectional curvature in those dimensions. To state it, we recall the orthogonal decomposition of the Riemann curvature tensor Rm = U + V + W into irreducible components (35.28)

U=

R (g  g) , 2n (n − 1)

V =

◦ 1 (Rc  g), n−2

W = Weyl tensor,

◦

where Rc denotes the trace-free Ricci tensor and  denotes the Kulkarni–Nomizu product of symmetric tensors: (h  k)(X1 , X2 , X3 , X4 )  h(X1 , X4 )k(X2 , X3 ) + h(X2 , X3 )k(X1 , X4 ) − h(X1 , X3 )k(X2 , X4 ) − h(X2 , X4 )k(X1 , X3 ).

4. DYNAMIC STABILITY RESULTS OBTAINED BY OTHER METHODS

305

Theorem 35.37 ([152]). Let (Mn , g) be a closed simply-connected Riemannian manifold of positive scalar curvature. There exists δn depending only on n ≥ 4 such that if the irreducible components of the curvature tensor satisfy 2δn |V |2 + |W |2 < δn |U |2 ≡ R2 n (n − 1) pointwise, then the volume normalized Ricci flow with initial data g converges to a unique metric of constant positive curvature. Also see Margerin [217] and Nishikawa [294]. Remark 35.38. Note that the hypotheses above only need pointwise pinching, in sharp contrast to the global pinching assumptions necessary for classical sphere theorems in comparison geometry; e.g., Kmax (Mn , g) < 4Kmin (Mn , g). See the survey [2]. Margerin subsequently improved δ4 to its optimal value: Theorem 35.39 ([218]). Let (M4 , g) be a closed simply-connected Riemannian manifold of positive scalar curvature. If the irreducible components of the curvature tensor satisfy 1 2 2 2 |V | + |W | < |U | ≡ R2 6 pointwise, then the volume normalized Ricci flow with initial data g converges to a unique metric of constant positive curvature. In this case, M4 is diffeomorphic to the standard 4-sphere if it is orientable and to RP4 otherwise. The equality |V |2 + |W |2 = |U |2 implies that M4 is isometric either to CP2 with its Fubini– Study metric or else to a quotient of R×S 3 . Recall that the Riemannian curvature operator may be regarded as a selfadjoint map Rm : Λ2 Tx Mn → Λ2 Tx Mn at each x ∈ Mn . One says that Rm is 2-positive if the sum of its two smallest eigenvalues is positive. B¨ohm and Wilking prove the following remarkable convergence result in all dimensions n ≥ 5. Theorem 35.40 ([30]). If (Mn , g) is a closed simply-connected Riemannian manifold with 2-positive curvature operator, then the volume normalized Ricci flow with initial data g converges to a unique metric of constant positive curvature. The work of Brendle and Schoen [35] on the general n-dimensional 14 -pinching theorem (along with the earlier work by Chen [65] on the 4-dimensional case) provides another noteworthy stability (in fact, convergence) result for positively curved space forms. 4.2. Dynamic stability of Ricci-flat metrics. Haslhofer and M¨ uller have recently established a strong stability result for Ricci flow at Ricci flat metrics by exploiting Perelman’s λ-functional (35.8) and a L

ojasiewicz–Simon inequality: Theorem 35.41 ([147]). Let (Mn , gˆ) be a compact Ricci-flat manifold. If gˆ is a local maximizer of λ, then for every C k,α -neightborhood U of gˆ (k ≥ 2), there exists a C k,α -neighborhood V ⊂ U such that Ricci flow starting at any metric in V exists for all times and converges (modulo diffeomorphisms) to a Ricci-flat metric in U.

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The same authors also prove an instability result that demonstrates that the local maximizer hypothesis of the above theorem is natural. (As noted above, Sesum’s integrability condition is a special case of this.) Theorem 35.42. Let (Mn , gˆ) be a compact Ricci-flat manifold. If ˆ(g) is not a local maximizer of λ, then there exists a nontrivial ancient Ricci flow {g(t)}t∈(−∞,0] that converges (modulo diffeomorphisms) to gˆ for t → −∞. 4.3. Dynamic stability of noncompact flat space. Lang-Fang Wu has shown that under certain conditions on the initial data, Ricci flow originating at a complete metric on R2 exists for all time and approaches a soliton solution [440]. An alternate approach has been used by Schn¨ urer, Schulze, and Simon, who consider C0 -small perturbations of Euclidean space [346]. They show that if the space is asymptotically Euclidean, then solutions of a coupled Ricciharmonic heat flow exist for all time and converge uniformly to the flat Euclidean metric. Some of their results have been improved by Koch and Lamm [165], who prove the existence of global analytic solutions of Ricci–DeTurck flow on Euclidean space for bounded initial metrics that are L∞ close to the flat Euclidean metric. 4.4. Dynamic stability of negatively curved spaces. Ye has studied asymptotic stability of Ricci flow [449] for certain geometries. He considers compact manifolds (Mn , g) with n ≥ 3 on which the self-adjoint elliptic operator   1 L : hij → ΔL hij + Rjk + Rgjk hik n has strictly negative spectrum. This occurs, for example, at a metric of negative sectional curvature whose Ricci eigenvalues are sufficiently pinched, or at a metric of negative scalar curvature whose sectional curvatures are sufficiently pinched. Ye’s approach requires certain hypotheses on diameter, volume, and the L2 -norm of the (traceless) Ricci tensor—but does not require a priori knowledge of the existence of a hyperbolic metric. In dimensions n ≥ 6, Li and Yin prove stability of the constant curvature metric on Hn under small perturbations that decay sufficiently quickly at spatial infinity [186]. (Smallness here is measured by a technical condition related to Ye’s work.) Schn¨ urer, Schulze, and Simon prove a similar result for coupled Ricci-harmonic map flow, assuming perturbations that are bounded in L2 and small in C 0 [347]. More generally, Bamler establishes stability for noncompact symmetric spaces evolving by Ricci flow [18]; this strengthens the Schn¨ urer, Schulze, and Simon result for Hn by imposing weaker assumptions on the perturbation. Bamler also proves that finitevolume noncompact hyperbolic manifolds of dimensions n ≥ 3 are stable under a rescaled Ricci flow, without imposing any decay on the perturbation [17].

CHAPTER 36

Type II Singularities and Degenerate Neckpinches One of the key features of Hamilton’s scenario for the Ricci flow of 3-dimensional geometries is the role of neckpinch singularities. The claim is that if a 3-dimensional Ricci flow solution develops a local singularity, it should generally take the form of a constricting neckpinch (see §3 of [138] or Subsection 2.3 of Chapter 8 of [77]). Roughly speaking, a neckpinch is characterized by the local topology S 2 × I (for an interval I), by the curvature achieving a local maximum at the center of the interval, and by the curvature in the S 2 direction dominating that in the I direction. If the neckpinch is sufficiently constricted, in the sense that the S 2 radii are sufficiently small, with correspondingly large curvature, then under Ricci flow the neckpinch must further constrict and become singular in finite time. In the process of proving the Poincar´e and geometrization conjectures, Perelman’s work [312], [313] confirms Hamilton’s general conjectures regarding this central role of neckpinches in the formation of singularities in 3-dimensional Ricci flow. Although Hamilton’s and especially Perelman’s works say a tremendous amount about neckpinch singularities, one can further study the detailed nature of these singularities and their asymptotic form. At least for rotationally symmetric geometries, work by Angenent together with one of the authors (see [13] or Appendix A in Part I) analyzes the details of the asymptotic behavior of neckpinch singularities. Their work also shows, as expected, that (rotationally symmetric) neckpinch singularities are Type I (in the sense that | Rm |(T −t), for T the time of singularity formation, is uniformly bounded). In other work by two of the authors and Sesum [155], support is found for the conjecture that the asymptotic behavior of nonrotationally symmetric neckpinch singularities in Ricci flow is accurately modeled by that of rotationally symmetric neckpinches. Like the neckpinch singularities, the Ricci flow singularities which develop from the extinction of a round 3-sphere, from the collapse of an S 3 whose geometry has positive Ricci curvature, and from the extinction of a round product S 2 × S 1 are all Type I. How might Type II singularities develop? One possibility which has been proposed by Hamilton is the following: Consider a family of 3-dimensional geometries (say, on S 3 ), each with a neckpinch centered at the equator and with the degree of pinching parametrized (continuously) by λ ∈ (0, ∞). For very small values of λ, the pinching is tightly constricted and the Ricci flow becomes singular in finite time, while for larger values of λ the constricting is very loose, the scalar curvature is positive, and the (normalized) Ricci flow converges to the round sphere. Since a neckpinch singularity forms for small λ and does not for large λ, what happens for intermediate values? Is there a transitional “threshold” or “critical” value λc for

307

308

36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES

the parameter, and what is the asymptotic behavior of the Ricci flow starting at such a critical geometry? Rigorous studies by Angenent and Vel´ azquez along with intuitive argument (see [15]) of analogous mean curvature flows have suggested that the Ricci flows of critical geometries might develop Type IIa singularities (finite-time singularity, | Rm |(T − t) unbounded, singularity model an eternal solution) and further that these singularities would be marked by the concentration of curvature at the poles rather than at the neckpinch, with the flow at the pole modeled by the (steady) Bryant soliton [37]. Such behavior has been labeled a degenerate neckpinch singularity. Strong support for this conjecture has been obtained from two programs of research. The first, that of Gu and Zhu [127], relies on a number of results from Perelman’s work [312], [313] to show that indeed there do exist (rotationally symmetric) Ricci flow solutions which develop degenerate neckpinch singularities. They prove that such solutions exist on S n+1 for all n ≥ 3; however they provide no details regarding the nature of these singularities apart from showing that they are Type IIa and that the curvature does concentrate at one or both of the poles, where the singularity is modeled by a Bryant soliton. The second program of research, carried out by two of the authors and their collaborators, is focussed more on the detailed nature of the flow geometry in the neighborhood of degenerate neckpinch singularities which form in Ricci flow. This program, which we survey in this chapter, has been carried out in three stages. The first stage involves numerical simulations of the Ricci flow of rotationally symmetric critical1 geometries. This work, done by Garfinkle and one of the authors [116], [117] very strongly and clearly indicates that for a wide variety of families of initial geometries with parametrized degrees of neckpinching, the Ricci flow of the critical initial geometry in that family forms a degenerate neckpinch singularity. Restricted for numerical convenience to S n+1 , these simulations provide models of the behavior of the geometry near the developing singularity. We describe this work in §1, where we present some of the general considerations which arise in using numerical simulations as a tool for studying Ricci flow, along with the details of this particular numerical study. The second stage of this program is a formal matched asymptotics study of the formation of degenerate neckpinch singularities under Ricci flow. As we discuss in §2, this work (done by Angenent and two of the authors [11]) involves the construction of a class of very detailed approximate solutions for the flow in a neighborhood of one of the poles, with all of these approximations (done on S n+1 , with rotational symmetry) exhibiting degenerate neckpinch behavior. A formal study of this nature does not prove that Ricci flow solutions which form degenerate neckpinch singularities and have the prescribed near-the-pole behavior do exist, but it does strongly suggest that this is the case. The proof that such solutions do indeed exist is detailed in [12] and is discussed here in §3. The key idea is to use a modified barrier-type argument to show that for every approximate solution constructed using formal matched asymptotics, there is at least one solution of Ricci flow which, near the poles, converges to that

1 Here we use the word “critical” in the sense described above for a λ-parametrized family of initial geometries.

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309

approximate solution. In addition to proving that there are (rotationally symmetric) solutions with degenerate neckpinches, these results show that there is a wide variety of rates of curvature blow-up—all Type IIa—to be found in such solutions. We recall that finite-time singularities may be categorized in the following way: (I) Type I singularity: supM ×[0,T ) (T − t) |Rm| < ∞, (IIa) Type IIa singularity: supM ×[0,T ) (T − t) |Rm| = ∞. Similarly, we may categorize infinite-time singularities as follows: (III) Type III singularity: supM ×[0,∞) t |Rm| < ∞, (IIb) Type IIb singularity: supM ×[0,∞) t |Rm| = ∞. Often we think of singularities forming forward in time. On the other hand, for ancient solutions we have the following: (I-ancient) Type I ancient solution: supM ×(−∞,0] |t| |Rm| < ∞, (II-ancient) Type II ancient solution: supM ×(−∞,0] |t| |Rm| = ∞. To this we may also add the following categorization for singularities forming backward in time at time 0: (IV) Type IV singularity: supM ×(0,T ] t |Rm| < ∞, (IIc) Type IIc singularity: supM ×(0,T ] t |Rm| = ∞. We say that a singularity is Type II if it is either Type IIa, IIb, or IIc. Problem 36.1. Given a reasonable definition of “generic”, are Type II singularities nongeneric? One may ask this question separately for singularities of Type IIa, IIb, or IIc. 1. Numerical simulation of solutions with degenerate neckpinches 1.1. The role of numerical simulations. For nonlinear partial differential equations which model physical systems—e.g., Navier–Stokes and Einstein’s equations—numerical simulation is a familiar and increasingly important tool for studying the behavior of solutions of the pde. The motivation for such study has historically come primarily from the intrinsic physical interest of the particular solutions being simulated—e.g., ocean flow solutions of Navier–Stokes and black hole collision solutions of Einstein’s equations. Recent experience has shown that, in addition to its effectiveness in probing the behavior of particular solutions of some intrinsic interest (physical or otherwise), numerical simulation can be very useful as a tool for exploring the mathematical behavior of large classes of solutions. It has been used to seek support or to seek counterexamples to conjectures regarding the generic behavior of solutions—e.g., the cosmic censorship and the Belinskii–Khalatnikov–Lifschitz conjectures for cosmological solutions of Einstein’s equations [26]. It can also play a major role in uncovering new, often unexpected, phenomena in families of solutions. Here, we cite the numerical discovery by Choptuik [66] of critical behavior in sets of gravitational collapse solutions of the Einstein equations. Ricci flow does not appear to model any physical systems (apart from its tie-in with the renormalization group from quantum field theory). There are, however, a number of outstanding unproven conjectures regarding the behavior of Ricci flow solutions. As we discuss here, numerical simulations have proven to be very useful in exploring one such conjecture, and they should be useful in many others. In

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addition, with so much of the behavior of Ricci flow not yet understood, such exploration is likely to uncover new and surprising phenomena in Ricci flow. While there has been much discussion of possible numerical simulation of Ricci flow solutions (it has been listed as a topic for discussion at many Ricci flow workshops during the past several years), to date, very few such simulations have been carried out in a serious way. One of the very few questions which has been explored via numerical simulation is that of the Ricci flow of critical geometries (in the sense defined above) and whether these flows produce Type II singularities and the conjectured characteristic degenerate neckpinch behavior. This work [116], [117], which was largely carried out before the Gu and Zhu proof appeared, strongly indicates that (rotationally symmetric) critical geometries do produce Type II singularities, with degenerate neckpinch behavior. Apart from the role of this numerical study in extending our understanding of the geometries whose Ricci flows develop Type II singularities and in providing the details of the formation of these singularities, this work provides a useful example of what numerical simulation can and cannot do in exploring mathematical conjectures. We note especially that these numerical simulations have proven to be very useful in guiding subsequent analytical work detailing the formation of degenerate neckpinch singularities, as described below. One feature of many of the conjectures concerning Ricci flow which has to some extent hindered their exploration by numerical simulations is that in most cases the evolving geometries under study live on manifolds with nontrivial topology. This is an issue because the vast majority of numerical techniques have been developed for use on manifolds which are subsets of Rn . We note, however, recent work [205] which has developed numerical techniques for use on manifolds with nontrivial topology. 1.2. Setting up and carrying out the numerical simulation. We now focus on the work done in [116] and [117], which uses numerical simulation to study Type II rotationally symmetric degenerate neckpinches. 1.2.1. Initial data and evolution equations for rotationally symmetric critical neckpinch solutions. Numerical explorations of the behavior of solutions of pdes involve the explicit numerical construction of (approximations to) particular solutions to the pde system. Hence the first step in such an exploration is to identify and parametrize the solutions to be considered and to write out the pde system in terms of the functions appearing in those solutions. We describe now how this is done for the critical neckpinch study. While it would be very useful to explore what happens for geometries which are not rotationally symmetric, the first stage of the critical neckpinch studies focuses on geometries with this strong assumption of isometry. Such metrics on S 3 may be written in the form  (36.1) g = e2X e−2W dψ 2 + e2W sin2 ψ[dθ 2 + sin2 θdφ2 ] , where (ψ, θ, φ) gives the standard angular coordinates on the three sphere and where spherical symmetry holds as long as the functions X and W are functions of ψ only. To also ensure regularity of the geometry at the poles (which are marked by the coordinate values ψ = 0 and ψ = π), one requires that the quantity S  sinW2 ψ approach a constant at ψ = 0 and ψ = π.

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Imposing a restriction like rotational symmetry on the geometries under study is useful only if the restriction is preserved by the Ricci flow equations. As noted in §8 of Chapter 4 in [77], Ricci flow does indeed preserve such Lie group generated isometries. Hence we may proceed to calculate the Ricci flow equations and evaluate their evolution for the problem at hand working solely in terms of the functions X(ψ, t) and W (ψ, t), or, equivalently, X(ψ, t) and S(ψ, t). The Ricci flow system becomes a 1 + 1 initial value boundary value problem for these two functions on (0, π) × R+ . We note that, in reducing the system in this way, the rotational symmetry assumption allows us to avoid confronting the problem of doing numerical analyses on a nontrivial manifold. As discussed in Chapter 3 of Volume One, the Ricci flow pde system itself is not parabolic, but one can prove that its initial value problem is well-posed by modifying the system via the addition of a Lie derivative term to obtain a Ricci– DeTurck flow pde system, which is parabolic, and then using the well-posedness of the Ricci–DeTurck flow initial value problem together with the fact that Ricci flow solutions and Ricci–DeTurck flow solutions are related by well-defined diffeomorphisms. One might guess that while the Ricci–DeTurck flow is useful as a tool for proving the well-posedness of the Ricci flow, in numerical simulations one can work directly with the Ricci flow itself. Experience shows that this is not the case and that the Ricci–DeTurck flow is just as important for numerical simulations. In particular, one finds in carrying out the numerical simulation for the (rotationally symmetric) critical neckpinch study that the strictly parabolic Ricci–DeTurck flow system is numerically stable, while the weakly parabolic Ricci flow system is not. In retrospect, this makes sense since strict parabolicity is useful in proving wellposedness because it allows one to use energy estimates to prove convergence of successive approximations, and of course this same sort of convergence is crucial to the success of a numerical simulation. Since the Ricci–DeTurck flow is needed for the critical neckpinch study, we now display the evolution equations for X(ψ, t) and S(ψ, t) corresponding to such a flow2 , also including a volume normalizing term:  ∂X 1  2 2(W −X) =e (X ) + (W  ) 2 + 3X  W  X  + 2 cot ψX  − 2 + ∂t 2   1 rˆ −4W  + 1 + 2 cot ψW (36.2) + (1 − e ) + , 3 2sin2 ψ  ∂S 3  2(W −X) 1 − 4W − e−4W =e S  + 6 cot ψS  − 8S − ∂t 2sin4 ψ  1 − e−4W  1 − 2 cot ψX  + 2 sin ψ cos ψS  + 4cos2 ψS + 2 sin ψ 

 2  X 6X  1  2  (sin ψS + 2 cos ψS) . (36.3) + (sin ψS + 2 cos ψS) + − 2 sin ψ sin ψ

2 As described in Chapter 3 of Volume One, there is not a unique Ricci–DeTurck flow; for each choice of a background connection, there is a different version of it. All of these are equally useful for proving well-posedness of the Ricci flow. It may, however, turn out that different versions of Ricci–DeTurck flow have different numerical stability properties, and therefore different versions may be useful for different numerical simulations.

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Here we note that the average scalar curvature rˆ is given by (36.4)  π   2 2 rˆ = dψeX+3W e−4W − 1 − 4 sin ψ cos ψW  + sin2 ψ[3 + (X  + W  ) ] , N 0 where the normalization constant N is given by  π (36.5) N≡ dψ e3X+W sin2 ψ. 0

With the evolution equations in hand, the next task is to choose a 1-parameter family of initial data geometries which correspond to the variable neckpinches of interest. To do this, one may set W = X, and one may then choose X so that 4e4X sin2 ψ = sin2 2ψ for cos2 ψ ≥ 1/2 and 4e4X sin2 ψ = sin2 2ψ + 4λcos2 2ψ for cos2 ψ ≤ 1/2. Here λ, which is a constant taking values between 0 and +∞, serves as the parameter, and we note that for λ = 0 this data represents two round 3spheres joined at the poles, while for λ sufficiently large, the central cusp smooths out and the geometry more closely approximates that of a single round sphere. 1.2.2. Numerical considerations. At least three main approaches are commonly used for carrying out the explicit numerical simulation of solutions of partial differential equation systems on manifolds. Most traditional is partial differencing; also widely used are spectral methods and finite-element techniques. Each of these approaches has advantages and disadvantages, each has strong advocates, and all three have been used in numerical relativity, with partial differencing being the predominant choice in that field. For the case being described here—rotationally symmetric solutions of the Ricci flow equations on S 3 —the spatial domain reduces to an interval of the real line, and consequently partial differencing is very simple to implement, is not very expensive in computer time, and is expected to be sufficiently accurate. Hence that is the approach which has been used. We now outline how it works for this application. One starts by dividing the spatial coordinate range (0, π) of ψ into N −2 pieces, so that one has Δψ = π/(N − 2). One then chooses N grid points, including a pair which run outside the coordinate range. Hence the first spatial grid point is at ψ = −Δψ/2, while the last is at ψ = π + Δψ/2. Now in terms of these grid points, a function  of the form F (ψ, t0 ) for fixed time t0 is replaced by a set of N numbers Fi = F (i − 32 )Δψ, t0 where 1 ≤ i ≤ N , and spatial derivatives are replaced by centered finite differences in the following way: (36.6) (36.7)

∂F Fi+1 − Fi−1 → , ∂ψ 2Δψ ∂2F Fi+1 + Fi−1 − 2Fi → . 2 2 ∂ψ (Δψ)

For the time  dependence of these functions, one chooses a fixed time step Δt and replaces F (i − 32 )Δψ, nΔt by the numbers Fin . Now, for an evolution equation of the form ∂F ∂t (ψ, t) = G(ψ, t) one numerically evolves using the approximation (36.8)

Fin+1 = Fin + ΔtGni .

This evolution is implemented for all values of i except 1 and N . Note that these two “ghost zones” are not part of the manifold since ψ is not in the range 0 ≤ ψ ≤ π.

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At the ghost zones one uses smoothness of the metric, which implies that X  and S  vanish at the poles. This condition may be implemented as X1 = X2 and XN = XN −1 (and correspondingly for S). 1.2.3. Numerical results. To study neckpinching and the related critical behavior, the numerical evolution outlined above was carried out and reported in [116] for a wide range of the λparametrized neckpinch initial data, as is made explicit in Subsection 2.2 of that work. The behavior observed in these numerical simulations matches that which had been speculated. In particular, the following is seen. For the “large value” 0.2 of the pinching parameter λ (subcritical flow), one finds that the geometry evolves towards the round sphere geometry. This is seen in the graphs in Figure 36.1 and Figure 36.2, which show X approaching a constant and S approaching 0. One also sees this asymptotic behavior by tracking the

Figure 36.1. X for subcritical Ricci flow.

Figure 36.2. S for subcritical Ricci flow.

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Figure 36.3. RS 2 for subcritical Ricci flow.

Figure 36.4. R⊥ for subcritical Ricci flow. evolution of the Ricci curvature eigenvalues3 RS 2 and R⊥ which are found to graph out as shown in the Figures 36.3 and 36.4. For the “small value” 0.1 of λ corresponding to tight pinching (supercritical flow), we find that in the neighborhood of the equator, RS 2 grows without bound as t increases, while R⊥ stays bounded. As well, away from the equator, both are bounded; see Figures 36.5 and 36.6. This signals the formation of an S 2 neckpinch singularity at the equator. The “critical value” for λ is found by numerical experiment. That is, one examines the behavior of the flow for values of λ between 0.1 and 0.2 and finds neckpinching singularity behavior for all λ < 1.639, while the geometry flows to a round sphere for λ > 1.639. For the flow starting with λ = 0.1639, the behavior is markedly different. There, as seen in Figures 36.7 and 36.8, R⊥ gets small everywhere except at the poles, while RS 2 slowly grows everywhere except at the 3 These

(36.9)

(36.10)

are calculated to be



R⊥ = −2e2(W −X) −1 + X  + W  + (X  + 3W  ) cot ψ + 2(X  + W  )W  ,  1 − e−4W RS 2 = −e2(W −X) −2 + + X  + W  + (3X  + 5W  ) cot ψ sin2 ψ  + (X  + W  )(X  + 3W  )

(where we note that W, W  and W  can be expressed in terms of S, S  and S  ).

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Figure 36.5. RS 2 for supercritical Ricci flow.

Figure 36.6. R⊥ for supercritical Ricci flow. poles. At the poles, both curvatures get very large. In a sense, the geometry approaches that of a 3-dimensional javelin (i.e., a long thin cylinder with pointed ends), with S 2 cross sections. This is the conjectured behavior for a degenerate neckpinch. As noted above, it has been further conjectured that the behavior of the flow at the poles for critical initial data is locally modeled by the Bryant steady soliton solution. More specifically, the idea is that if tk is a sequence of times approaching the singular time T and if each snapshot of the flow g(tk ) is dilated in a neighborhood of the pole in such a way that the curvature is normalized to unity at the pole, then the dilated metrics gˆ(tk ) locally (about the pole) approach the geometry of the Bryant steady soliton. In the numerical work discussed in [117], this conjecture is tested by examining the Ricci flow of a sequence of subcritical initial metrics (i.e., a sequence of λi parametrized geometries with λi approaching 0.1639 from above). For each of these flows one seeks the time ti at which the maximal curvature is achieved on the pole, one dilation-normalizes each gλi (ti ) as described above, and then one compares

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Figure 36.7. RS 2 for critical Ricci flow.

Figure 36.8. R⊥ for critical Ricci flow. with the geometry of the Bryant steady soliton. As shown in Figures 36.9 and 36.10 (note that the solid line is the critical solution at the time of maximum curvature and the dashed line is the Bryant steady soliton), for λi very close to the critical value, the match is striking. 1.3. Future numerical work and conclusions. The numerical work described below in Subsection 2.2 of this chapter is far from complete. It is important that a much wider variety of parametrized families of initial data—all having the feature that for one extreme of the parameter there is clearly a neckpinch singularity forming during the flow while for the other extreme the flow approaches the round sphere geometry—be used as starting points for the Ricci flow evolution. Included should be families which are not rotationally symmetric. Is there always a unique transition value for the parameter, dividing pinching from converging flows? Does one always find the same sort of very special behavior—formation of a javelin-type geometry, with the curvature concentrating

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317

Figure 36.9. RS 2 for critical Ricci flow compared to Bryant steady soliton.

Figure 36.10. R⊥ for critical Ricci flow compared to Bryant steady soliton. at the poles and with the flow at the poles approaching the Bryant soliton model— for flows starting at the critical value? Does the flow always approach rotational symmetry for supercritical and subcritical flows as well as for the critical flows? Two of the authors, together with Garfinkle, have begun exploration of some of these issues for the analogous case of mean curvature flow. In [118], a variety of families of embeddings of S 1 × S 1 in S 1 × R2 flat space are considered, all with initially convex or near-convex cross sections and all with initial profiles consistent

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with the formation of a neckpinch singularity under mean curvature flow. Very consistently, it is found [118] that in a neighborhood of where the neckpinch occurs, the embedding becomes rounder and rounder, and the features of the developing neckpinch are essentially identical to those seen in a mean curvature neckpinch singularity which is initially and always rotationally symmetric. Experience suggests that similar behavior is likely to be seen in similar families of nonrotationally symmetric Ricci flow degenerate neckpinch singularities. No matter how emphatically numerical studies might suggest that solutions of Ricci flow have a certain behavior, such studies do not prove that this is indeed the case. On the other hand, numerical studies can serve as a strong guide toward how best to go about proving that this behavior is present. The next two sections outline a successful program for proceeding from detailed numerical simulations to a verification that degenerate neckpinch singularities with the numerically observed behavior indeed occur in 3-dimensional Ricci flow. 2. Matched asymptotic studies of degenerate neckpinches The goal of a formal matched asymptotics study of the conjectured behavior of a class of solutions of a given pde system is to produce a set of approximate solutions which exhibit that conjectured behavior. To do this, one first formulates a detailed ansatz which encapsulates most of the features of the conjectured behavior. Based on this ansatz, one simplifies the pde system (e.g., via asymptotic expansions and linearization) and then obtains solutions to the simplified system. One then checks these solutions for consistency with the ansatz, removing those which fail this consistency test. Those remaining are the formal (approximate) solutions. By construction, they should match the conjectured behavior. In most cases, including the study of degenerate neckpinch singularities in [11] which we describe here, the ansatz is best defined and applied by cutting the relevant domain into a set of regions and initially working in each region independently. The simplified pde systems are generally very different from one region to another; hence a crucial part of the consistency check is to match the approximate (regionally defined) solutions at the regional boundaries. We see this in Subsection 2.7 below. 2.1. Degenerate neckpinch ansatz. The full ansatz used in [11] for constructing formal approximate Ricci flow solutions with degenerate neckpinch singularities is best stated in stages. The first stage—the Primary Ansatz—can be stated without reference to a choice of coordinates or to the four local regions of study. We state this here. The remaining stages of the full ansatz depend on coordinate choices and pertain to the individual separate regions. These are stated below in Subsections 2.3, 2.4, 2.5, and 2.6, in the context of the regional analyses. The Primary Ansatz consists primarily of conditions which the initial geometries (Mn+1 , g0 ) of the Ricci flows of interest (Mn+1 , g(t)) must satisfy. One verifies that most of these conditions are preserved throughout the flow. The Primary Ansatz also includes a condition that the flow becomes singular in finite time. Ansatz 36.2 (Primary Ansatz). The manifold Mn+1 is the sphere S n+1 (for n ≥ 2), and the initial geometries (S n+1 , g0 ) are rotationally symmetric in the sense that they admit an SO(n + 1) isometry group. (This isometry group action defines a pair of poles in S n+1 and a foliation of S n+1 by S n sections orthogonal

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319

to the rotation axis (through the poles).) The initial geometries satisfy the following curvature conditions: (i) the Ricci curvature is positive in a neighborhood of each pole; (ii) the sectional curvature of each of the planes tangent to each of the S n sections is positive; (iii) the scalar curvature is everywhere positive. In a neighborhood of one of the poles (which we label the right pole), there is at least one bump-neck-bump sequence, consisting of a pair of local maxima of the function D(z) (the bumps) sandwiching a local minimum (the neck) for D(z). (Here D(z) specifies the diameter of the S n section a parameter distance z along the axis.) A singularity, marked by unbounded curvature, occurs in finite time T at the right pole, with no singularity occurring anywhere prior to T . We note that these conditions are chosen so that the initial geometries are close to those which, according to the numerical simulations, form degenerate neckpinch singularities. 2.2. Coordinates and Ricci flow equations. Following [11], we discuss the formal matched asymptotic studies using the (rotationally symmetric) metric representation (36.11) (36.12)

g(t) = φ2 (x, t)dx2 + ψ 2 (x, t)ground = ds2 + ψ 2 (s(x, t), t)ground.

Here ground is the round metric on the sphere S n , x ∈ (−1, 1) is a coordinate running between the poles PL and PR and labeling the S n cross sections, and x s(x, t) := 0 φ(ξ, t)dξ is the corresponding arclength coordinate. One readily shows that the Ricci flow equations for these metrics takes the form ∂ss ψ (36.13) φ, ∂t φ = n ψ (1 − (∂s ψ)2 ) (36.14) , ∂t ψ = ∂ss ψ − (n − 1) ψ where we note that ∂s ψ := φ1 ∂x ψ, so that [∂t , ∂s ] = −∂φt φ ∂s . Based on the Primary Ansatz and arguments from [13] (see also [11]) which indicate that the bump-neck-bump configuration is preserved until immediately before the singularity occurs at time T , we divide the formal analysis into four overlapping regions: the tip (a neighborhood of the right pole PL ), the intermediate region (a neighborhood of the bump closest to PL ), the parabolic region (a neighborhood of the neck), and the outer region (which extends from the far side of the neck outwards). We now discuss each of these regions in turn, describing the coordinate dilations, the auxiliary ans¨atze, the series expansions and pde linearization, and the approximate solutions characteristic of each region. We do this in an order—parabolic, intermediate, tip, outer—which best clarifies the analysis, though it is not the sequential geometric order of the regions. 2.3. The parabolic region. In the region surrounding the center of the neck, the geometry (suitably dilated) is approximately cylindrical. The quadratic deviation from cylindricality motivates the labeling of this portion of the geometry as the “parabolic region”. As noted, it follows from the Primary Ansatz condition R > 0 that the flow becomes singular at some finite time T < ∞. Hence, using x0 = 0 to label the central

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point of the developing neckpinch singularity and defining the arclength parameter s relative to this choice, we are led to rescale the time and space coordinates in the following way: τ (t) := − log(T − t),

(36.15)

s(x, t) σ(x, t) := √ = eτ /2 s(x, t). T −t Further, since maximum principle arguments show that  ψ(x, t) ≥ (n − 1)(T − t) (36.16)

everywhere in the parabolic region, we are led to rescale the evolving geometric quantity ψ(x, t) as follows: ψ(s, t) . U (σ, τ ) :=  2(n − 1)(T − t)

(36.17)

One readily calculates the evolution equation for U (σ, τ ) to be    σ 1 (∂σ U )2 1 + nI ∂σ U + (n − 1) + (36.18) ∂τ U = ∂σσ U − U− , 2 U 2 U σ U where I := 0 ∂σσ U dσ. Motivated by Perelman’s arguments that neckpinches in three dimensions become cylindrical and presuming that this behavior extends to higher dimensions, we make the first of the auxiliary (regional) ans¨atze: Ansatz 36.3 (Parabolic Ansatz 1). In the parabolic region, as the singularity time T is approached (i.e., as t → T , or, equivalently, as τ → ∞), the quantity U (σ, τ ) approaches unity uniformly. Note that we may use this ansatz to define the extent of the parabolic region, delineating it as that region in which the quantity V (σ, τ ) := U (σ, τ ) − 1 is sufficiently small. In essence, then, the content of Parabolic Ansatz 1 is that there indeed is a region in which this is the case. Within this region, it is useful to rewrite (36.18) as an evolution equation for V (σ, τ ); segregating terms which are linear and nonlinear in V , we have ∂τ V = LV + N (V ),

(36.19) where

σ ∂σ V + V 2 is the indicated linear operator and N (V ) includes the higher terms. As a consequence of Parabolic Ansatz 1, in that region the term N (V ) may be neglected. The operator L is a familiar one from studies of neckpinches in mean curvature flow [15] as well as from studies of singularity formation in solutions of equations of the form ∂t w = ∂xx w + up [121]. The operator is self-adjoint in the space LV := ∂σσ V −

(36.20)

L2 (R, e−

σ2 4

dσ) and has the pure point spectrum

k 2 with the associated eigenfunctions being the Hermite polynomials hk (σ). It follows from standard Bernoulli (separation ansatz) considerations that solutions of the

(36.21)

λk = 1 −

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321

 equation ∂τ V = LV can be written in series form k bk eλk τ hk (σ) for a sequence of constants bk . It is useful to recall that the leading term in hk (σ) is σ k and that for k even, hk (σ) involves only even powers of σ, while for k odd, it involves only odd powers. If we were considering an initial value boundary value problem for the equation ∂τ V = LV , we would use the boundary conditions and initial conditions to determine the constants bk . For the purposes here of modeling degenerate neckpinches, we add a further ansatz. Ansatz 36.4 (Parabolic Ansatz 2). For some positive integer k ≥ 3, one can write (36.22)

V (σ, τ ) = bk eλk τ σ k + O(eλk τ σ k−2 )

(σ → ∞),

with this term much smaller than one in the parabolic region. We note that in [11], this ansatz is stated more phenomenologically, in terms of certain aspects of the neckpinch singularity formation. We also note that since we have defined the parabolic region as that in which V (σ, τ ) is small, it follows 1 1 from (36.22) that this region is delineated by the condition |σ| , e( 2 − k )τ . 2.4. The intermediate region. As opposed to the parabolic region in which V is very small, the intermediate region is marked by V being of order 1. Since it follows from Parabolic Ansatz 2 and the definitions of the Hermite polynomials that in the parabolic region the leading order term in V is comparable to eλk τ σ k , we may demarcate the neighboring intermediate region as that within which eλk τ σ k is of order 1. To approximate the behavior of Ricci flow solutions in the intermediate region, it is useful to work with the spatial variable (36.23)

ρ := (eλk τ )1/k σ = e( k − 2 )τ σ 1

1

(of order 1 in the intermediate region) and the metric variable  (36.24) W (ρ, t) := U (σ, τ ) = 1 + V (σ, τ ) . One may then write equation (36.18) in the form 2 1 1 ρ ∂ρ W − W − W −1 = e−τ ∂t W − e( k −1)τ F(W, ∂ρ W, ∂ρρ W ) (36.25) k 2 2 where F is a functional of the indicated quantities. Assume we now impose Ansatz 36.5 (Intermediate Ansatz). ∂t W as well as F(W, ∂ρ W, ∂ρρ W ) is bounded in the intermediate region as τ → ∞. Then it follows that the right-hand side of equation (36.18) may be neglected, and consequently the behavior of the flow in the intermediate region may be modeled by solutions of the ode 1 ˜ 1 ˜ −1 ρ ˜ ∂ρ W − W − W = 0. (36.26) k 2 2 One readily determines that the general solution to (36.26) is ! ˜ (ρ, t) = 1 − (ρ/c)k , (36.27) W with the constant of integration c(t) generally a function of t.

322

36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES

2.5. The tip region. The numerical simulations discussed in §1 of this chapter, as well as the results in [127], indicate the Ricci flow for a critical initial geometry should asymptotically approach the configuration of a Bryant soliton. It takes a bit of work to incorporate this behavior into the approximate solutions of the formal matched asymptotic analysis. One starts by noting that since the conditions for regularity at the right pole require that ∂s ψ(s(1, t), t) = −1 for all t < T , it follows that ∂s ψ < 0 in a neighborhood of that pole. It then follows from the implicit function theorem that one may replace “s” by “ψ” as a local spatial (radial) coordinate, and one may write the metric in a neighborhood of the right pole in the form4 (36.28)

g = z(ψ, t)−1 dψ 2 + ψ 2 ground .

In this form, z is the function governing the configuration of the geometry of the tip region. One readily verifies that the evolution of z is governed by the flow equation (36.29)

∂t z = Fψ [z]   1 1 = 2 ψ 2 zzψψ − (ψzψ )2 + (n − 1 − z)ψzψ + 2(n − 1)(1 − z)z ; ψ 2

we note that the (nonlinear) differential operator Fψ [z] acting on z(ψ, t) depends explicitly on the coordinate ψ. It is important to make one further coordinate change in the tip region: We replace ψ by a new radial coordinate γ, which is related to ψ by a yet to be determined time-dependent expansion factor Γ(τ (t)); explicitly, γ(s, t) = Γ(τ (t))ψ(s, t). We then set Z(γ, t) := z(ψ, t) and calculate (from (36.29)) the evolution equation for Z: (36.30)

Γ−2 (Zt + Γ−1 Γτ γZγ ) = Fγ [Z];

here Fγ [·] is the operator defined in (36.29), but with ψ replaced by γ, wherever it appears. As discussed in [11], the solutions of the ode boundary value problem consisting of the ode Fγ [Z] = 0 together with the boundary conditions Z(0) = 1 and Z(∞) = 0 are closely related to the Bryant steady soliton metrics. More specifically, if one writes the Bryant steady soliton metric [37] in the form (36.31)

g = B(r)−1 dr 2 + r 2 ground ,

then the solutions of Fγ [Z] = 0 with these boundary conditions all take the form γ  (36.32) Z(γ) = B a for some positive scaling function a. We note that the function B(r) is not known in explicit analytic form; it can, however, be represented arbitrarily accurately numerically from the definition of the Bryant steady soliton, which essentially reduces to the ode boundary value problem under discussion here. 4 More precisely, as discussed in [11] one shows that there exists a negative definite function y(ψ, t) such that ∂s (s, t) = y(ψ(s, t), t) and then sets z := y 2 .

2. MATCHED ASYMPTOTIC STUDIES OF DEGENERATE NECKPINCHES

323

As determined above, the evolving geometry at the tip is modeled by solutions of (36.30). To relate these solutions to the Bryant soliton function B(r), one imposes Ansatz 36.6 (Tip Ansatz). In the tip region, as t approaches the singularity τ time, the scaling function Γ(τ ) satisfies the estimates Γ(τ ) . e 2 and∂τ Γ = o(Γ3 ), and the derivative of the geometric function Z satisfies the estimate ∂t Z = o(Γ2 ). Taken together, these assumptions imply that the left-hand side of equation (36.30) may be neglected, so Z may be presumed to be a solution of Fγ [Z] = 0. 2.6. The outer region. The simplest model for a degenerate neckpinch is reflection symmetric across its equator and has just one neck and two bumps. For the matched asymptotic analysis of such models, one has sequentially (from the right pole to the left pole) a tip region, an intermediate region, a parabolic region, an intermediate region, and a tip region. However, more generally, and for less symmetric neckpinch models, while one side of the parabolic region includes an intermediate region and a tip region, on the other side is an outer region, which we now describe. Generally, the outer region does not lead to another neckpinch. If the arc length coordinate s is chosen with its zero at the center of the parabolic region, the outer region is identified as the region where the rescaled coordinate τ σ = e 2 s approaches −∞. The geometry in this region is controlled by the quantity ψ(s, t), and without imposing any further ans¨ atze, one determines by matching with quantities in the adjacent parabolic region that   s 2  (36.33) ψ(s, t)2 ≈ 2(n − 1) (T − t) − , c where k is the integer (≥ 3) introduced in the Parabolic Ansatz 2 and where c is ˜. the constant appearing in the expression (36.27) for W It follows from this expression for ψ in this region that if k is even, then the full 1 diameter of the solution, from pole to pole, decays to zero at the rate |s| ≤ c(T −t) k . If rather k is odd, then the distance from the equator to the right pole (at which the singularity is forming) decays at this same rate. 2.7. Matching at regional overlaps. As discussed above, the regional analyses and ans¨ atze lead to regional approximate solutions with free parameters: In the parabolic region, expression (36.22) for V (σ, τ ) has the free parameter bk (along with the choice of the integer k); in ˜ (ρ, t) involves the free function the intermediate region, expression (36.27) for W of integration c(t); and in the tip region, expression (36.32) for Z(γ) has the free scaling factor a and also depends implicitly on the free expansion factor function Γ(τ ) which relates γ(s, t) and ψ(s, t). The regional solution in the outer region, as described here, has no free parameters. These free parameters are crucial for matching the regional approximate solutions across the regional overlaps. Only by restricting their relative values does one obtain a globally consistent approximate solution. As shown in [11], one finds the following: (i) the integer k ≥ 3 may be freely chosen; (ii) matching the parabolic and intermediate regions requires that c(t) be constant and that bk = − 12 c−k ; (iii) matching the tip and the intermediate regions requires that a = k(n−1) and that 2c

324

36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES 1

Γ(τ ) = e(1− k )τ . With these restrictions imposed, two free parameters are left: the choice of the integer k ≥ 3 and the choice of the constant c. For any choice of these parameters, one obtains a globally consistent approximate Ricci flow solution which models a degenerate neckpinch. As constructed, these approximate solutions become singular at time T at the right pole. One readily calculates their rate of curvature blow-up: C . (36.34) sup |Rm(x, t)| ∼ n+1 (T − t)2−2/k x∈S For any choice of k ≥ 3, the indicated blow-up rate corresponds to a Type II singularity. It is interesting to speculate whether this quantization of blow-up rates is a real feature of degenerate neckpinch singularities of Ricci flow or if it is rather just an artifact of the particular approximate models constructed in [11]. We note that in the study [439] of the asymptotics of degenerate neckpinch singularities on noncompact manifolds, this quantization of blow-up rates does not occur. 3. Ricci flow solutions with degenerate neckpinch singularities It is known from [127] that Ricci flow solutions which develop degenerate neckpinch singularities and have Type II curvature blow-up behavior exist. The numerical work and formal matched asymptotics work discussed above indicate what some of the detailed features of such solutions could be. These studies do not, however, guarantee that Ricci flow solutions with these features exist. In this section we describe work which shows that indeed they do exist. In particular, we show that for every one of the approximate formal solutions modeling degenerate neckpinch singularity formation (as discussed in §2 above), there exist rotationally symmetric Ricci flow solutions which asymptotically approach this formal solution and consequently share its asymptotic properties. The formation of a degenerate neckpinch singularity is not expected to be a stable property of Ricci flow solutions, so one does not expect to be able to use sub and super solution methods to prove that Ricci flow solutions approach each of the (n, k, c)-parametrized approximate formal solutions. However, as shown in [12], the closely related Wa˙zewski retraction method [433] does the job. We note its use to prove a similar result for mean curvature flow solutions [15]. The idea of the Wa˙zewski retraction method, as applied to the present problem, is the following: We consider the space M(S n ) of rotationally symmetric metrics5 on S n and represent the Ricci flow solutions of such metrics as trajectories in M(S n ) × R1 . A particular choice of one of the approximate solutions constructed above in §2 via formal matched asymptotics is also represented by a trajectory in M(S n ) × R1 . Reflecting the parametrization of these approximate solutions, we label them as gˆ{n,k,c} (t). For each choice of an approximate solution gˆ{n,k,c} (t) and for any positive , we construct a tubular neighborhood Ξ{n,k,c;} of that solution in M(S n ) × R1 which has the following features: (i) as t → T (the singularity time for gˆ{n,k,c} (t)), Ξ{n,k,c;} progressively narrows so that any trajectory contained within it approaches gˆ{n,k,c} (t) arbitrarily closely; (ii) the boundary of Ξ{n,k,c;} consists of three pieces (36.35)

+ o ∂Ξ{n,k,c;} = ∂Ξ− {n,k,c;} ∪ ∂Ξ{n,k,c;} ∪ ∂Ξ{n,k,c;} ,

5 We note that in practice, each point in M(S n ) can be specified by a pair of (metric) functions φ(s) and ψ(s).

3. DEGENERATE NECKPINCH SINGULARITIES

325

with no solution inside Ξ{n,k,c;} ever contacting ∂Ξ+ {n,k,c;} and with every solution − inside Ξ{n,k,c;} which contacts ∂Ξ{n,k,c;} necessarily exiting immediately. As a consequence of this property of ∂Ξ− {n,k,c;} , one readily determines that the exit times (from Ξ{n,k,c;} ) for solutions that do not reach ∂Ξo{n,k,c;} depend continuously on the choice of initial data for such solutions. Presuming that a tubular neighborhood Ξ{n,k,c;} with the properties just described can be constructed, the Wa˙zewski method argues as follows that a Ricci flow solution must asymptotically approach gˆ{n,k,c} (t): Fixing an initial time t0 < T (with t0 very close to T ), one considers a k-dimensional subspace of the set M(S n ) × {t0 } of all initial data sets for the Ricci flow solutions of interest. One then specifies a closed connected subset B k of the parameter space for this set of initial data sets, choosing it so that every initial data set g[α] (t0 ) corresponding to a parameter choice α ∈ ∂B k lies in the tubular neighborhood instant exit set k ∂Ξ− {n,k,c;} and so that the map λ : α ∈ ∂B → g[α] (t0 ) is not contractible. One also k chooses B so that the corresponding initial data sets g[β] (t0 ) for all β ∈ B k are far enough away from the neutral set ∂Ξo{n,k,c;} so that the Ricci flow solutions g[β] (t) which evolve from this initial data have the property that so long as they remain in Ξ{n,k,c;} , they stay bounded away from ∂Ξo{n,k,c;} . Consequently, these solutions either remain in Ξ{n,k,c;} or exit somewhere at ∂Ξ− {n,k,c;} . Continuing the Wa˙zewski argument, we now suppose that every solution g[β] (t) (with β ∈ B k ) leaves Ξ{n,k,c;} at some time tβ . As noted above, exits can only happen at ∂Ξ− {n,k,c;} , so it follows from this supposition that there is a continuous map − Λ → ∂Ξ{n,k,c;} . Moreover, since we know that initial data g[β] (t0 ) with β ∈ ∂B k must be contained in ∂Ξ− {n,k,c;} and consequently have tβ = t0 , we see that the map Λ must be an extension of the map λ. It then follows that λ must be contractible. This violates the construction of λ as a noncontractible map. We hence conclude that at least some of the Ricci flow trajectories evolving from initial data g[β] (t0 ) (with β ∈ B k ) must remain in Ξ{n,k,c;} for all t ∈ [t0 , T ) and must therefore asymptotically approach the formal approximate solution gˆ{n,k,c} (t). Consequently, there are Ricci flow solutions with the degenerate neckpinch singularity behavior of these formal models. It should be evident from this discussion that to implement the Wa˙zewski retraction method in proving this degenerate neckpinch result, there are two primary tasks that must be carried out. The most difficult of these tasks is verifying the existence of the tubular neighborhoods Ξ{n,k,c;} with their necessary properties. In this survey, we do not discuss the details of the construction of these tubular neighborhoods, nor the verification of their properties. We refer the reader to [12], where this is done. We do note that in the portion of Ξ{n,k,c;} corresponding to the tip region, the intermediate region, and the outer region, sub and super solution barriers can be constructed. These barriers prevent escape from that part of Ξ{n,k,c;} corresponding to those regions; hence the instant exit portion of the boundary of the tubular neighborhood does not occur in these regions. In the parabolic region, standard barriers do not exist, and exit from the corresponding portion of Ξ{n,k,c;} can occur. Consequently, the analysis of the portion of the tubular neighborhood corresponding to the parabolic region is in fact where the most intricate estimates are needed; we refer the reader to §6 of [12].

326

36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES

The other necessary task is finding the set B k and the corresponding sets of initial data g[β] (t0 ) with properties as described above. As shown in [12], this task is relatively straightforward, with a strong reliance on the Hermite polynomials and eigenvalues which arise in the formal matched asymptotic analysis. 4. Concluding remarks The work discussed here determines the asymptotic behavior of a range of rotationally symmetric Ricci flow solutions which develop degenerate neckpinch singularities. Should one expect to find the asymptotic behavior discussed here in generic solutions which develop this sort of singularity? Hamilton has conjectured that, in the space of all Ricci flow solutions on a closed manifold, degenerate neckpinches are nongeneric; see Problem 36.1 in this chapter and see p. 294 in [77]. On the other hand, do degenerate neckpinch singularities always develop from initial data sets lying at the threshold between sets leading to nondegenerate neckpinch singularities and sets which (for volume normalized Ricci flow) do not become singular? Compare with Problem 8.25 in [77]. The numerical work discussed in Subsection 1.1 indicates that at least among rotationally symmetric solutions, degenerate neckpinch singularities do generically develop from threshold data, and they generically show the behavior discussed here. However, such a result is far from proven. What about neckpinch singularities forming in the Ricci flow of nonrotationally symmetric geometries? A consequence of Perelman’s work is that all finite-time singularities on closed 3-manifolds are asymptotically rotationally symmetric after possibly passing to a finite cover. Perelman proved that 3-dimensional singularity formation is modeled by canonical neighborhoods. In particular, unless one is on a spherical space form, one can always extract a round cylinder limit or its Z2 quotient. In the case of 3-dimensional Type II singularities, a Bryant soliton limit always exists (a proof of this fact also uses Brendle’s classification of κ-noncollapsed 3-dimensional steady solitons). In all dimensions it has been conjectured that if the initial geometries are close to being rotationally symmetric or if they differ from rotationally symmetric geometries in relatively controlled ways, then the resulting Ricci flow solutions are likely to become increasingly round and then are likely to exhibit the same asymptotic behavior as rotationally symmetric solutions. This conjecture has been explored to a limited extent for nondegenerate neckpinch Ricci flow solutions [155] and more extensively for nondegenerate neckpinch formation in mean curvature flows [115], [114]. While these results are very limited and have not yet explored degenerate neckpinch formation in nonrotationally symmetric Ricci flow, they do all support the conjecture. This suggests that the study of the asymptotic behavior of rotationally symmetric Ricci flow solutions in which neckpinch singularities form could in fact be describing the asymptotic behavior of a much wider range of Ricci flow neckpinch solutions.

APPENDIX K

Implicit Function Theorem I can endure no more, I demand you remember who you are. – From “I Need a Doctor” by Dr. Dre featuring Eminem and Skylar Grey

In this appendix we collect results that are used in various places in the book. We discuss the implicit function theorem, H¨ older and Sobolev spaces, harmonic maps, and the spectrum of the Hodge–de Rham Laplacian acting on differential forms on the unit sphere S n . 1. The implicit function theorem In this section we recall some basic results in functional analysis related to the applications of the implicit function theorem (IFT) discussed in Chapters 33 and 34. All of the Banach and Hilbert spaces in this section are real. 1.1. The Lax–Milgram–Lions theorem. We used the following result to find weak solutions to the boundary value problem in Lemma 34.14. Theorem K.1 (Lax–Milgram–Lions). Suppose that A is a Hilbert space, B is a normed linear space, and f : A×B → R is a continuous bilinear functional. Then the following two statements are equivalent: (1) (Coercive) There exists a constant c > 0 such that inf

sup |f (a, b)| ≥ c.

b=1 a≤1

(2) (Existence of a weak inverse) For every continuous linear functional L : B → R there exists aL ∈ A such that f (aL , b) = L (b) for all b ∈ B. 2

Note that if A = B and if there exists c > 0 such that f (a, a) ≥ c aA , then f is coercive. 1.2. Contraction mapping theorem. The following contraction mapping theorem is useful in proving existence results. Lemma K.2 (Contraction maps have unique fixed points). Let (X, d) be a complete metric space. If f : X → X is such that there exists λ ∈ [0, 1) such that d (f (x) , f (y)) ≤ λd (x, y) for all x, y ∈ X, then there exists a unique point x∞ ∈ X for which f (x∞ ) = x∞ . 327

328

K. IMPLICIT FUNCTION THEOREM

Proof. The idea is to iterate the map f (starting at any point) and obtain a Cauchy sequence which converges to the fixed point. Let x0 ∈ X and for each i ≥ 0 define xi ∈ X inductively by xi+1  f (xi ). We have d (xi , xi+1 ) ≤ λd (xi−1 , xi ) ≤ · · · ≤ λi d (x0 , x1 ) for all i ∈ N and hence for any j > i, d (xi , xj ) ≤ d (xi , xi+1 ) + · · · + d (xj−1 , xj )  ≤ λi + · · · + λj−1 d (x0 , x1 ) ≤

λi d (x0 , x1 ) . 1−λ ∞

Since λ ∈ [0, 1), we have that {xi }i=0 is a Cauchy sequence. Since X is complete, the limit x∞  limi→∞ xi exists. Taking the limit as i → ∞ of xi+1 = f (xi ) and using the continuity of f , we have x∞ = f (x∞ ). The fixed point x∞ is unique since if x∞ is also a fixed point of f , then d (x∞ , x∞ ) = d (f (x∞ ) , f (x∞ )) ≤ λd (x∞ , x∞ ) , which implies that d (x∞ , x∞ ) = 0.



1.3. Inverse function theorem. We first discuss the differentiability of maps between Banach spaces. Let (X,  · X ) and (Y,  · Y ) be Banach spaces. Let Lk (X, Y ) denote the Banach space of k-multilinear continuous maps from X k  X × · · · × X to Y with the standard norm: A  sup{A (x1 , . . . , xk )Y : x1 X ≤ 1, . . . , xk X ≤ 1} ; A

we write L (X, Y ) = L1 (X, Y ). There are natural isomorphisms between L2 (X, Y ) and L(X, L(X, Y )), between L3 (X, Y ) and L(X, L(X, L(X, Y ))), etc. Let U ⊂ X be an open set. We say that a map f : U → Y is (Fr´ echet) differentiable at x ∈ U if there exists Df (x) ∈ L (X, Y ) such that lim

h→0

f (x + h) − f (x) − Df (x) (h)Y = 0. hX

We call Df (x) the (Fr´ echet) derivative of f at x. We say that f is C 1 if Df : U → L (X, Y ) is continuous. If Df is differentiable at every point in U , then we have D2 f : U → L2 (X, Y ) and we say that f is C 2 if D2 f is continuous. Continuing this way we may define the k-th derivative Dk f : U → Lk (X, Y ) of f and what it means for f to be C k . Next we state and prove the Banach space inverse function theorem. Theorem K.3. Let (X,  · X ) and (Y,  · Y ) be Banach spaces and let U ⊂ X be an open set. If f : U → Y is a C k map where k ≥ 1 and x0 ∈ U is such that Df (x0 ) : X → Y is a bijection, then there exists an open neighborhood W of x0 such that f |W : W → f (W ) is a bijection onto an open neighborhood of f (x0 ) and the inverse of f |W , i.e., ( f |W )−1 : f (W ) → W , is a C k map. Proof. We prove the case where k = 1; the cases where k ≥ 2 is left as an exercise. Since Df (x0 ) is a bijection, by the closed graph theorem we have

1. THE IMPLICIT FUNCTION THEOREM

329

that (Df (x0 ))−1 is a bounded linear operator. Without loss of generality we may assume, by replacing f by the map x → (Df (x0 ))

−1

(f (x + x0 ) − f (x0 )) ,

that X = Y , x0 = 0, f (0) = 0, and Df (x0 ) = idX . Define g : U → X by g (x)  x − f (x), which is the difference of f and its linearization at 0. We have Dg (0) = 0 and since g is C 1 , there exists δ > 0 such that (K.1)

Dg (x) ≤

1 2

¯ (0, δ)  {x ∈ X : x ≤ δ}. Note that this implies for x ∈ B ¯ (0, δ) that for x ∈ B (K.2)

Df (x) (v) ≥ v − Dg (x) (v) ≥

1 v 2

for all v ∈ X. By the fundamental theorem of calculus and (K.1), we have that for ¯ (0, δ), any x1 , x2 ∈ B " 1 " " " " g (x1 ) − g (x2 ) = " (K.3) Dg (tx + (1 − t) x ) (x − x ) dt 1 2 1 2 " " 0  1 ≤ x1 − x2  Dg (tx1 + (1 − t) x2 ) dt 0

1 ≤ x1 − x2  . 2 ¯ (0, δ/2) for x ∈ B ¯ (0, δ). Since g (0) = 0, we obtain that g (x) ∈ B Given y ∈ X, we define gy : U → X by (K.4)

gy (x)  y + x − f (x) = y + g (x) .

¯ In particular, f (x) = y if and  only if gy (x) = x. For y ∈ B (0, δ/2) we have ¯ ¯ by the triangle inequality, gy B (0, δ) ⊂ B (0, δ). Moreover, by (K.4) and (K.3), ¯ (0, δ/2) the map gy | ¯ ¯ ¯ for any y ∈ B B(0,δ) : B (0, δ) → B (0, δ) is a contraction mapping of a complete metric space. Therefore Lemma K.2 implies that there ¯ (0, δ) of gy | ¯ exists a unique fixed point x ∈ B B(0,δ) ; that is, we have an inverse map −1 ¯ ¯ f : B (0, δ/2) → B (0, δ). ¯ (0, δ), Next, we show that f −1 is Lipschitz continuous. For any x1 , x2 ∈ B f (x1 ) − f (x2 ) ≥ x1 − x2  − g (x1 ) − g (x2 ) 1 ≥ x1 − x2  . 2 ¯ (0, δ/2), we have Hence, given any y1 , y2 ∈ B " " −1 "f (y1 ) − f −1 (y2 )" ≤ 2 y1 − y2  . (K.5) Finally, we show that f −1 is differentiable. By (K.2), we have " " " −1 " sup "(Df (x)) " ≤ 2. ¯ x∈B(0,δ)

330

K. IMPLICIT FUNCTION THEOREM

Hence given y ∈ B (0, δ/2) and h ∈ X with 0 < h ≤ δ/2 − y, we have " "   −1 " −1 " (h)" "f (y + h) − f −1 (y) − Df f −1 (y) " "   ≤ 2 "Df f −1 (y) f −1 (y + h) − f −1 (y) − h" " " = o "f −1 (y + h) − f −1 (y)" = o (h) , where we also used (K.5). Therefore, f −1 is differentiable in B (0, δ/2). Since Df and f −1 are continuous, we have that    −1 D f −1 (y) = Df f −1 (y) is a continuous function of y; i.e., f −1 is C 1 .



A simple consequence of the Banach space inverse function theorem is the following Banach space implicit function theorem. Theorem K.4 (Banach space IFT). Let X , Y, and Z be Banach spaces, let U ⊂ X × Y be an open set, and let f : U → Z be a C k map, where k ≥ 1. Suppose (x0 , y0 ) ∈ U is such that the partial derivative (∂Y f )(x0 ,y0 ) : Y → Z, defined by f (x0 , y0 + sv) − f (x0 , y0 ) s→0 s for v ∈ Y, is a Banach space isomorphism. Then there exists ε > 0 and δ > 0 such that for every x ∈ B (x0 , ε) there exists a unique solution y  φ (x) ∈ B (y0 , δ) to the equation f (x, y) = f (x0 , y0 ) . The map φ : B (x0 , ε) → B (y0 , δ) is C k and its derivative Dφ : X → Y is given by −1  ◦ (∂X f )(x,φ(x)) , (Dφ)x = − (∂Y f )(x,φ(x)) (∂Y f )(x0 ,y0 ) (v) = lim

where ∂X f : X → Z is the partial derivative. 1.4. Banach manifolds. Just as a manifold is a topological space modeled on a Euclidean space (of a fixed dimension), a Banach manifold is a topological space modeled on (possibly different) Banach spaces. A C k Banach manifold, where k ∈ N ∪ {0, ∞}, is defined to be a set B with k a C Banach manifold atlas for B. Such an atlas is defined to be a collection of subsets {Ui }i∈I which cover B, together with bijections ϕi : Ui → ϕi (Ui ) ⊂ Xi , where (1) Xi is a Banach space and ϕi (Ui ) is an open subset, (2) ϕi (Ui ∩ Uj ) ⊂ Xi is open, and (3) the overlap map ϕj ◦ ϕ−1 : ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj ) i ∩ ∩ Xi Xj is C k for all i, j ∈ I.

1. THE IMPLICIT FUNCTION THEOREM

331

We call each (Ui , ϕi ) a Banach space coordinate chart. A Banach manifold atlas defines a topology T on B, with a subset S of B defined to be open if for every i ∈ I, ϕi (S ∩ Ui ) ⊂ Xi is open. Let Mn and N m be smooth closed manifolds and let C k (M, N ) denote the set of C k maps from M to N . In the following, we construct a Banach manifold structure on C k (M, N ). Let gN be a C ∞ Riemannian metric on N and define the open neighborhood of the zero-section (K.6)

O  {V ∈ T N : |V | < inj gN (π (V ))} ⊂ T N ,

where inj gN (q) denotes the injectivity radius of gN at q ∈ N . We note that for each q ∈ N the exponential map expgqN : O ∩ Tq N → N is a C ∞ diffeomorphism onto its image. Given f ∈ C k (M, N ), consider the set f ∗ O ⊂ f ∗ T N . The set Uf  C k (f ∗ O) of C k sections of f ∗ O → M is an open subset of the Banach space C k (f ∗ T N ). We define a Banach space chart (Uf , ϕf ) for C k (M, N ) containing (centered at) the point f ∈ C k (M, N ) by (K.7a)

ϕf : Uf → C k (M, N ) ,

(K.7b)

ϕf (σ) (x)  expgfN(x) (σ (x)) .

With the collection {(Uf , ϕf )}f ∈C k (M,N ) of Banach space local coordinate charts, we obtain a Banach manifold atlas for C k (M, N ). Now let Mn and N m be arbitrary smooth manifolds. k,α (M, N ) if ψ ◦ Definition K.5. We say that a map f : M → N is in Cloc k,α f ◦ ϕ−1 ∈ Cloc for any smooth chart (U, ϕ = {xi }) of M and any smooth chart (V, ψ = {y a }) of N , whenever this function is defined. k,α (M, N ) by Whenever the manifolds M and N are closed, we denote Cloc k,α ∗ C (M, N ). If f is smooth, then the map ϕf : C (f O) → C k,α (M, N ) defined by (K.7b) is a Banach space chart about f in C k,α (M, N ). Since the IFT (Theorem K.4) is a local result, by using Banach space charts, one easily obtains a version of the IFT for Banach manifolds. k,α

Now we will consider examples of Banach manifolds of maps between manifolds. Let Mn be a compact manifold with boundary ∂M. We define C k,α (M; ∂M) to k,α be the subset of maps F in Cloc (M, M) satisfying F (∂M) ⊂ ∂M. We construct a Banach space coordinate chart for C k,α (M; ∂M) about any smooth map f . For reasons we shall see below, we change the metric g to a uniformly equivalent metric g˜ where, in a collar of ∂M, the metric g˜ is a product metric with the property that ∂M is totally geodesic in (M, g˜). Let expx denote the exponential map with respect to g˜. Define the injectivity radius inj g˜ (x) of the manifold with boundary (M, g˜) at a point x ∈ M to be the supremum of r such that expx : exp−1 ˜(x) (0, r) → B (x, r) x (B (x, r)) ∩ Bg is a diffeomorphism, where B (x, r)  {y ∈ M : dg˜ (y, x) < r}. Define the injectivity radius of M to be inj(˜ g ) = inf x∈M inj g˜ (x). For example, for the upper half-space {x ∈ Rn : xn ≥ 0}, the injectivity radius of the Euclidean metric is equal to infinity.

332

K. IMPLICIT FUNCTION THEOREM

Let B be the Banach space of C k,α vector fields on M which when restricted to ∂M are tangent to ∂M. Define the open subset C  {V ∈ B : |V (x) | < inj(˜ g ) for each x ∈ M}. Given f ∈ C k,α (M; ∂M), consider the map ϕf : C → C k,α (M; ∂M) defined by ϕf (V ) = {fV : x → expgf˜ (x) (V (f (x)))}. Similarly to (K.6), we define  (K.8) O∂  V ∈ T M : |V | < inj g˜ (π (V )) , expg˜ (V ) ∈ M

⊂ T M.

Given f ∈ C ∞ (M; ∂M), consider the subset Uf of σ ∈ C k,α (f ∗ (O∂ )) satisfying σ (x) ∈ T (∂M) for x ∈ ∂M. Similarly to the previous subsection, define a Banach space chart (Uf , ϕf ) for C k,α (M; ∂M) by ϕf : Uf → C k,α (M; ∂M) , ϕf (σ) (x)  expgf˜ (x) (σ (x)) . Note that if x ∈ ∂M, then ϕf (σ) (x) ∈ ∂M since σ (x) ∈ Tf (x) (∂M) and since ∂M is totally geodesic in (M, g˜). 2. H¨ older spaces and Sobolev spaces on manifolds In this section we discuss H¨ older spaces and, briefly, Sobolev spaces on manifolds. We also include a discussion of jet bundles. 2.1. C k,α H¨ older spaces for tensors. Let (Mn , g) be a Riemannian manifold, which may be noncompact and may have boundary. Let π : E r → M be a C ∞ real vector bundle with metrics on its fibers and a compatible connection (such as a tensor bundle). For the tensor > bundle E  E ⊗  T ∗ M we have the covariant derivative ∇ : E → E+1 . Given an k extended integer k ∈ [0, ∞], let Cloc (M, E) be the set of sections φ of E|int(M) such that for each 0 ≤  < min {k + 1, ∞}, the -th covariant derivative ∇ φ extends continuously to M as a section of E (we shall use the same notation ∇ φ for this extension). We define the seminorms [φ]  supx∈M |∇ φ(x)| for  ≥ 0. Definition K.6. Given an integer k ∈ [0, ∞), the space C k (M, E) is the k (M, E) for which the norm Banach space of all sections φ ∈ Cloc φk 

k 

[φ]j

j=0

is finite. i n n Let ({Uβ }N β=1 , ψβ = {xβ }i=1 : Uβ → R ) be a covering of M by coordinate charts. Further assume that there are trivializations E r |Uβ → Uβ × Rr correspondk (M, E), its covariant ing to bases of local sections {s1β , . . . , srβ }. Given φ ∈ Cloc derivative may be expressed as  ∇ φ  φβj;i1 ···i sjβ ⊗ dxiβ1 ⊗ · · · ⊗ dxiβ ∈ Γ(Uβ , E ), 1≤j≤r, 1≤i1 ,...,i ≤n

¨ 2. HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS

333

where the {φβj;i1 ···i } are locally defined functions on Uβ . For any α ∈ (0, 1], we define the H¨older seminorm ! β β 2  k  j,i1 ,...,ik (φj;i1 ···i (p) − φj;i1 ···i (q)) , (K.9) ∇ φ α  sup sup α d (p, q) β p,q∈Uβ p=q

where d is the Riemannian distance with respect to g. This seminorm depends on the choices of local coordinates and trivializations. Definition K.7. Given an integer k ∈ [0, ∞) and α ∈ (0, 1], the H¨ older space k (M, E) for which the H¨ older C k,α (M, E) is the Banach space of all sections φ ∈ Cloc norm k      φk,α  φk + ∇k φ α = [φ]j + ∇k φ α j=0

is finite. Using parallel translation, we may alternatively define a H¨ older seminorm as follows. Given p, q ∈ M, let γp,q be a minimal geodesic from p to q in M. Let γp,q : (E )p → (E )q denote the parallel translation along γp,q defined by the connection ∇. Define the seminorm   γ (∇k φ(p)) − ∇k φ (q)  k  p,q (Ek )q (K.10) ∇ φ α  sup sup . α d (p, q) γ p,q∈M p,q p=q

When M is compact, we may restrict ourselves to finite coverings. Then, up to uniform equivalence, it is not hard to see that the definition of the H¨older norm φk,α is independent of the choice of connections, coordinate systems, and local     trivializations of E; we may also replace the seminorm ∇k φ α by ∇k φ α . Although Banach space norms used in this book are primarily H¨ older norms, on occasion we used Sobolev norms, such as in Subsection 3.6 of Chapter 34. Here we recall Definition K.8. Given an integer k ∈ [0, ∞) and p ∈ [1, ∞), for any φ ∈ k (M, E) we define its Sobolev W k,p -norm by Cloc 1/p k    j p   (K.11) φk,p  . ∇ φ (E )q dμ(q) j=0

M

k

We define the Sobolev space W k,p (M, E) to be the completion of C k (M, E) with respect to the W k,p -norm. In this subsection we have given just the bare definitions. To conclude, we remind the reader that for the use of H¨older spaces and Sobolev spaces in geometric analysis, compactness theorems, interpolation inequalities, and embedding theorems play crucial roles. 2.2. Jets of maps and distances between maps. In the proof of Proposition 33.16, on harmonic parametrizations of almost hyperbolic pieces, we used the notion of C k -distance between maps of manifolds. To define this, we need to consider jets of maps.

334

K. IMPLICIT FUNCTION THEOREM

  n Let (Mn , gM ) and (N m , gN ) be Riemannian manifolds. Let U, xi i=1 be a   m coordinate neighborhood of p ∈ M and let V, y  =1 be a coordinate neighborhood of q ∈ N . Given k ≥ 1, we say that pointed maps F, G : (U, p) → (V, q) have the same k-jet (at p) if and only if ∂ |α| F  ∂ |α| G (p) = (p) ∂xα ∂xα for all multi-indices α = (α1 , . . . , αn ) with 0 ≤ |α|  α1 + · · · + αn ≤ k and 1 ≤  ≤ m, where F   y  ◦ F and G  y  ◦ G. Having the same k-jet at a point p is an equivalence relation whose definition is independent of the choice of coordinate systems in neighborhoods of p and q. k (M, N ) to be the set of k-jet equivalence classes [F ]kp,q of maps F satDefine Jp,q isfying F (p) = q. We define the vector bundle of k-jets of maps J k (M, N ) → M × N as the union 

J k (M, N ) 

k Jp,q (M, N ) .

p∈M, q∈N k  = Spi M ⊗ Tq N , where S i M denotes the

k The fiber Jp,q (M, N ) is isomorphic to

i=1

vector bundle of symmetric covariant i-tensors. A coordinate dependent vector k  = k space isomorphism from Jp,q Spi M ⊗ Tq N is given by (M, N ) to i=1

[F ]p,q

⎞  ∂ |α| F  ∂ ⎝ −→ (p) dxα ⊗  ⎠ . α ∂x ∂y i=1 k ?

⎛

|α|=i

Now we consider a coordinate-free point of view. Recall that dF is a section of T ∗ M ⊗ F ∗ T N . Let (K.12)

∇ = ∇g,h : C ∞ (T ∗ M ⊗ F ∗ T N ) → C ∞ (T ∗ M ⊗ T ∗ M ⊗ F ∗ T N )

i  denote the covariant derivative induced by ∇g and F ∗ ∇h . Define ∇i = ∇g,h  i-th covariant derivative acting on sections of this bundle. ∇g,h ◦ · · · ◦ ∇g,h to be the > i+1 ∗ Tp M ⊗ TF (p) N . Two maps F ang G have the same For example, ∇i dF (p) ∈ k-jet at p if and only if they have the same value at p and the same higher covariant derivatives at p; i.e., F (p) = G (p) and ∇i dF (p) = ∇i dG > j(p)∗ for 0 ≤ i ≤ k − 1. Tp M ⊗ Tq N is given by Recall that the symmetrization of an element Ω ∈ Sym (Ω) (X1 , . . . , Xj ) =

1   Ω Xσ(1) , . . . , Xσ(j) , j! σ

where the summation is taken over all permutations σ. By standard commutator formulas, the tensor ∇i−1 dF (p) is uniquely determined by the curvature tensor and the symmetrizations of ∇j−1 dF (p) for all j ≤ i. For this reason there is a

¨ 2. HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS

335

natural bundle isomorphism defined by J : J k (M, N ) →

k ?  ∗ i p1 S M ⊗ p∗2 T N , i=1

J([F ]kp,q ) =

k ?

 Sym ∇i−1 dF (p) ,

i=1

where p1 : M × N → M and p2 : M × N → N denote projections. Each vector bundle p∗1 S i M ⊗ p∗2 T N over M × N has a natural inner product gi on each fiber induced by gM and gN and also has a natural connection (i) ∇ induced by the Riemannian connections ∇gM and ∇gN . We obtain a fiber metric and compatible connection on the vector bundle J k (M, N ) via pull-back by J. The connection defines parallel translation and horizontal spaces for J k (M, N ). So, by a general construction, we obtain a unique Riemannian metric gJ k on the total space J k (M, N ) with the following properties. The restriction of gJ k to the fibers is the fiber metric on J k (M, N ), the horizontal spaces are gJ k -orthogonal to the fibers, and the gJ k restricted to the horizontal spaces project isometrically to gM + gN on M × N . Given k ≥ 1, let dJ k be the Riemannian distance on J k (M, N ) associated to the metric gJ k . Given C k maps F, G : M → N , we may now define the C k (M, N ) distance between them as dC k (M,N ) (F, G)  sup dJ k ([F ]kp,F (p) , [G]kp,G(p) ).

(K.13)

p∈M

For k = 0, we may define dC 0 (M,N ) (F, G)  sup dgN (F (x) , G (x)) .

(K.14)

x∈M

The metric space (C k (M, N ) , dC k (M,N ) ) is complete for k ≥ 0. Following Hamilton’s presentation in §8 of [143] we may also consider jets of mappings as homomorphisms of jets of paths. Given a smooth manifold P, let I = (−1, 1) and define the vector bundle  k J kP = J0,q (I, P) → P. q∈P

The space J P is the quotient space of equivalence classes of paths α : I → P having = the same k-jet at 0. Note that J k P is naturally bundle isomorphic to k T P. Via this isomorphism, given a Riemannian metric on P and its associated Levi-Civita connection, we may define a Riemannian metric on the total space J k P. Hence we may define the Riemannian distance dJ k P between two k-jets. Given k ≥ 1 and a C k map F : M → N , we define the induced map k

J kF : J kM → J kN by k

k

J k F ([α]0,α(0) ) = [F ◦ α]0,F (α(0)) . We may also define the C k,α -distance between nearby hypersurfaces. Let (M , g) be a Riemannian manifold with sectional curvature bounded above by κ2 > 0. Let X n−1 ⊂ M be a closed smooth hypersurface. Let ν be a choice of smooth unit normal vector field to X and assume that −Λ I X ≤ IIX ≤ Λ I X , where n

336

K. IMPLICIT FUNCTION THEOREM

I and II denote the first and second fundamental forms, respectively. Let ι (Λ, κ) > 0 be as in Lemma I.42 in Part III. Then for any smooth function ϕ : X → R with |ϕ| < ι (Λ, κ), we have that the normal graph Yϕ  {expx (ϕ (x) νx ) : x ∈ X } is a smooth hypersurface. For such a hypersurface, define dC k,α (Yϕ , X ) = ϕC k,α (X ) . Definition K.9. If dC k,α (Yϕ , X ) ≤ ε, then we say that Yϕ is ε-close to X in C k,α -norm. 3. Harmonic maps and their linearization In this section we will derive several formulas related to harmonic maps and the harmonic map heat flow. Some of these formulas are used in Chapters 33 and 34. As throughout this book, we find it useful to compute and express formulas using local coordinates. In addition, we often indicate invariant ways to calculate the formulas that we present. 3.1. Harmonic maps. Let (Mn , g) and (N m , h) be Riemannian manifolds with or without boundary. Given a map f : M → N , its derivative (i.e., tangent map) df : T M → T N may be considered as a section of the vector bundle E  T ∗ M ⊗ f ∗ (T N ), where n f ∗ (T N ) → M is the pull-back vector bundle of T N by f . If (U, xi i=1 ) and m (V, {y α }α=1 ) are local coordinates on M and N , respectively, then df =

∂f α i ∂ dx ⊗ α , ∂xi ∂y

where f γ  y γ ◦ f and where the expression on the rhs is defined on U ∩ f −1 (V). On E we have the natural bundle metric g  h  g −1 ⊗ f ∗ h. Using this we define the energy density 2

|df |gh  g ij (hαβ ◦ f ) α

∂f α ∂f β . ∂xi ∂xj β

∂f ∂f ∂f ∂f Since (f ∗ h)ij = h( ∂x i , ∂xj ) = (hαβ ◦ f ) ∂xi ∂xj , an invariant way to write the energy density is |df |2gh = g ij (f ∗ h)ij = tr g (f ∗ h) .

Definition K.10 (Map energy). The map energy of the map f is defined by   2 |df |gh dμg = tr g (f ∗ h) dμg . (K.15) Eg,h (f )  M

M

We shall determine the critical points of Eg,h in the next subsection. First, we recall some basic facts about the pull-backs of connections, their tensor products, and the definition of harmonic maps. The pull-back by f of the Levi-Civita ∗ connection ∇h to the linear connection ∇f h on f ∗ (T N ) is uniquely defined by  ∗     ˜ ◦ f, C  f ∗ ∇h X C  ( ∇h f X C) (K.16) ∇f h X

∗

3. HARMONIC MAPS AND THEIR LINEARIZATION

337

where C  C˜ ◦ f , for C˜ ∈ C ∞ (T N ). In local coordinates, this is (∇fi

(K.17)

∗

h

C)γ =

∂ γ ∂f α γ C + (Γ (h)αβ ◦ f )C β , ∂xi ∂xi

γ

where the Γ (h)αβ denote the Christoffel symbols of h. −1

Let ∇g denote the connection on T ∗ M dual to ∇g . Let ∇gh  ∇g be the unique linear connection on E satisfying (K.18)

∇gh (A ⊗ B) = (∇g

−1

A) ⊗ B + A ⊗ ∇f

∗

h

−1

⊗ ∇f

∗

h

B

for A ∈ C ∞ (T ∗ M) , B ∈ C ∞ (f ∗ (T N )). This connection is compatible with the bundle metric g  h; that is, # $ # $ gh α, β + α, ∇ β X α, βgh = ∇gh X X gh

gh

∞

for any sections α, β ∈ C (E) and X ∈ T M. Now the map-Hessian ∇gh df is a section of the bundle T ∗ M ⊗S T ∗ M ⊗ ∗ f T N . The map-Laplacian of f is the trace with respect to g of ∇gh df ; i.e.,   (K.19) Δg,h f  trg ∇gh df ∈ C ∞ (f ∗ T N ) . In local coordinates,  γ ∂ ∇gh df = ∇gh df dxi ⊗ dxj ⊗ γ , ∂y ij where

  β γ  γ ∂f ∂ k ∇gh df = ∇gh dx ⊗ i ∂/∂x ∂xk ∂y β ij j γ ∂2f γ ∂f α ∂f β k ∂f γ − Γ (g) + (Γ (h) ◦ f ) ij αβ ∂xi ∂xj ∂xk ∂xi ∂xj α β ∂f ∂f = ∇gi ∇gj (f γ ) + (Γ (h)γαβ ◦ f ) i . ∂x ∂xj

=

(Note that the above displayed expression is symmetric in i and j.1 ) Hence, by tracing the Hessian of f with respect to g, we obtain the following. Lemma K.11 (Map-Laplacian in local coordinates). If f : M → N , then (K.20a) (K.20b)

∂f α ∂f β (Δg,h f )γ = Δg (f γ ) + g ij (Γ (h)γαβ ◦ f ) i ∂x ∂xj  2 γ  γ ∂ f ∂f ∂f α ∂f β k γ − Γ (g) + (Γ (h) ◦ f ) = g ij . ij αβ ∂xi ∂xj ∂xk ∂xi ∂xj

1 See also p. 5 of [99], where this symmetry is expressed invariantly. In particular, given vector fields X, Y ∈ C ∞ (M), we have ∗

∇fX

h

∗

f (df (Y )) − ∇Y

h

(df (X)) = df ([X, Y ]) .

Since df ([X, Y ]) = (df ) (∇X Y ) − (df ) (∇Y X), another way to write this, using the product rule, is 



(df ) (Y ) = ∇gh (df ) (X) . ∇gh X Y That is, the Hessian of f is symmetric.

338

K. IMPLICIT FUNCTION THEOREM

As a special case, if M = N and f is the identity map and if we choose the γ coordinates x and y to be the same, then ∂f = δkγ so that ∂xk (K.21)

k

k

k

(Δg,h id) = g ij (−Γ (g)ij + Γ (h)ij )

is the trace of the difference of the two Christoffel symbols. Note that although a connection is not a tensor, the difference of two connections is a tensor. One way to rewrite formula (K.21) is k

1 ij  −1 k  g ∇i hj + ∇gj hi − ∇g hij g h 2    −1 k 1 g = h div (h) − ∇ H , 2

(Δg,h id) =

where H  trg (h) = g ij hij . If (P n , k) and (N m , h) are Riemannian manifolds, f : P → N is a map and ϕ : M → P is a diffeomorphism, then (see for example (2.56) in [77])  (K.22) (Δk,h f ) (z) = (Δϕ∗ k,h (f ◦ ϕ)) ϕ−1 (z) . In particular, if f : M → N is a diffeomorphism, then we may rewrite its mapLaplacian as (K.23)

(Δg,h f ) (x) = (Δ(f −1 )∗ g,h idN ) (f (x)) .

We also have the formula (f −1 )∗ (Δg,h f ) = Δg,f ∗ h idM . Definition K.12 (Harmonic map). A map f : (Mn , g) → (N m , h) is called a harmonic map if (K.24)

Δg,h f = 0.

As we shall see below, harmonic maps are critical points of the map energy. If N = Rm , then a map is harmonic if and only if each of its components is a harmonic function. If M is 1-dimensional, then a harmonic map is the same as a constant speed geodesic. Example K.13. Let f be a smooth immersion of a differentiable manifold Mn  of f (M) into a Riemannian manifold (N m , h). Then the mean curvature vector H is equal to Δf ∗ (h),h f . In particular, an isometric immersion is a minimal immersion  = 0) if and only if it is a harmonic map. (i.e., H 3.2. First variation formula for the map energy. Let f : Mn → N m be a map between Riemannian manifolds (with or without boundary) and let {fs }s∈(−ε,ε) be a 1-parameter family of maps with f0 = f . Define  s , which is a section of the vector bundle f ∗ (T N ). the variation field V  ∂f ∂s  s=0   ∗ 2 ∂  We have2 ∂s |dfs |gh = 2 ∇f h V, df gh . We verify this equation in local s=0 2 ∗ ∗  Since dfs is a section of T M ⊗ fs (T N ) and this bundle depends on s, the expression dfs in and of itself does not make sense.

∂  ∂s s=0

3. HARMONIC MAPS AND THEIR LINEARIZATION

339

coordinates as follows:  ∂  (f ∗ h) (p) ∂s s=0 s ij    ∂  ∂fsα ∂fsβ = (p) (p) (hαβ ◦ fs ) (p) ∂s s=0 ∂xi ∂xj

 ∂ Vβ ◦f ∂ (V α ◦ f ) ∂f β ∂f α = (hαβ ◦ f ) (p) (p) (p) + (p) (p) ∂xi ∂xj ∂xi ∂xj    α ∂ ∂f ∂f β h (p) (p) ◦ f (p) + V γ (f (p)) αβ γ i ∂y ∂x ∂xj   β α f ∗h f ∗h α ∂f β ∂f (K.25) + (∇j V ) (hαβ ◦ f ) (p) . = (∇i V ) ∂xj ∂xi The last equality above may be seen from (K.17) and the following identity: ∂f γ ∂f β ∂f γ ∂f α α β hαβ Γ (h)γδ + hαβ Γ (h)γδ i j ∂x ∂x ∂xj ∂xj   1 ∂f γ ∂f β ∂hδβ ∂hγβ ∂hγδ = + − 2 ∂xi ∂xj ∂y γ ∂y δ ∂y β   γ α ∂hδα 1 ∂f ∂f ∂hγα ∂hγδ + + − 2 ∂xj ∂xi ∂y γ ∂y δ ∂y α =

∂f γ ∂f β ∂hγβ . ∂xi ∂xj ∂y δ

In invariant notation we write (K.25) as  # ∗ $ # $ ∗ ∂  (fs∗ h) = ∇f h V, df + df, ∇f h V ,  ∂s f ∗h f ∗h s=0

where the inner products contract the last components of the tensors. Tracing (K.25), we have   ∂  ∂  2 (K.26) |dfs |gh = g ij (fs∗ h)ij ∂s  ∂s  s=0

s=0

∗ ∂f β = 2g ij (∇fi h V )α j (hαβ ◦ f ) # ∗ $ ∂x f h = 2 ∇ V, df .

gh

In the following we shall assume that V has compact support in the interior of M. From (K.26) we may deduce the first variation formula for the map energy:      d  ∂  2 Eg,h (fs ) = |dfs |gh dμg ds s=0 ∂s  M  # s=0 $ ∗ ∇f h V, df =2 dμg gh M  #  $ V, tr g ∇gh df = −2 dμg f ∗h M  = −2 V, Δg,h f f ∗ h dμg . M

340

K. IMPLICIT FUNCTION THEOREM

To obtain the third equality above we used the divergence theorem and  # ∗ $ # $ + V, tr g ∇gh df , (K.27) div g (V, df f ∗ h ) = ∇f h V, df ∗ gh

f h

β

where (V, df f ∗ h )i = (hαβ ◦ f ) V α (df )i . Summarizing, we have the following. 

Lemma K.14 (First variation formula for the map energy). If then  # ∗ $ ∂  2 f h |df | = 2 ∇ V, df . (K.28) s gh ∂s s=0 gh

∂fs  ∂s  s=0

 V,

If V has compact support in the interior of M, then   d  (K.29) E (f ) = −2 V, Δg,h f f ∗ h dμg . g,h s ds s=0 M Hence, a map is a critical point of the map energy if and only if it is a harmonic map. Next we give an alternate, invariant, proof of (K.29). Define (K.30)

F : M × (−ε, ε) → N

by F (x, s)  fs (x). We have DF ∈ C ∞ (T ∗ (M × (−ε, ε)) ⊗ F ∗ T N ). Abusing notation, let T ∗ M also denote the pull-back by the projection M × (−ε, ε) → M of T ∗ M. Let dM F ∈ C ∞ (T ∗ M ⊗ F ∗ T N ) denote the spatial derivative of F , let ∂ ∂s denote the tangent vector in the positive (−ε, ε) direction, and let ∇ denote the natural connection on T ∗ M ⊗ F ∗ T N induced by g, h, and F . ˜ = X ∈ C ∞ ( T (M × (−ε, ε))| For any X ∈ Tp M, let X {p}×(−ε,ε) ). Since ∇∂/∂s X = 0 and ∇ (dF ) is symmetric, we have     ˜  (K.31) ∇∂/∂s (dM F )s=0 (X) = ∇∂/∂s dF X    s=0  ∂  = ∇X˜ dF  ∂s   s=0  ∗ ∂F  = ∇fX h . ∂s s=0 Using this, one computes    ∂  d  Eg,h (fs ) = 2 dM F, dM F gh dμg ds s=0 ∂s s=0 M    ∇∂/∂s (dM F )s=0 , df gh dμg =2   / M . ∂F f ∗h =2 dμg , ∇ , df ∂s M gh from which one deduces (K.29) by integration by parts, i.e., (K.27), and

∂F ∂s

=V.

3. HARMONIC MAPS AND THEIR LINEARIZATION

341

3.3. Second variation formula for the map energy. We adopt the notation of the previous subsection. In particular, let V  ∂  Although in local coordinates the scalar ∂s (Δg,h fs )γ is well des=0  (Δg,h fs ) is not well defined since Δg,h fs is a section fined, the expression ∂  ∂fs  . ∂s  s=0

∂s s=0

of fs∗ (T N ) which is a bundle that depends on s. With this understanding, by differentiating (K.20a) we see that the first variation of the map-Laplacian is given by  ∂V α ∂f β ∂  γ γ γ ij (K.32) (Δ f ) = Δ (V ) + g (Γ (h) ◦ f ) g,h s g αβ ∂s s=0 ∂xi ∂xj ∂f α ∂V β + g ij (Γ (h)γαβ ◦ f ) i ∂x ∂xj    α β ∂f ∂f ∂ + g ij . V δ δ Γ (h)γαβ ◦ f ∂y ∂xi ∂xj We wish to put this formula in a nicer form. Define   ∗ Δg,h V = tr g ∇gh ∇f h V . From now on we shall suppress the compositions with f in our formulas. We have (∇fj

∗

h

V )γ =

∂V γ ∂f β + Γ (h)γβα V α j ∂x ∂xj

and ∇fj (∇gh i

∗

h

 ∂V γ ∂f β γ α + Γ (h) V βα ∂xj ∂xj   ∂V γ ∂f β γ α − Γ (g)kij + Γ (h) V βα ∂xk ∂xk   ∂V δ ∂f α ∂f β γ δ ε + Γ (h)αδ + Γ (h)βε V ∂xi ∂xj ∂xj  = ∇gi ∇gj (V γ ) + ∇gi ∇gj f β Γ (h)γβα V α   α ∂ ∂f β ∂f δ ∂f β γ γ ∂V α + Γ (h) + Γ (h) V βα βα ∂xj ∂xi ∂y δ ∂xj ∂xi   ∂V δ ∂f α ∂f β γ δ + Γ (h)αδ + Γ (h)βε V ε . i j ∂x ∂x ∂xj

V )γ =

∂ ∂xi



Hence  γ γ (Δg,h V ) = Δg (V γ ) + Δg f β Γ (h)βα V α   β δ β α ∂ γ γ ∂V ij ∂f ∂f α ij ∂f +g Γ (h) + g Γ (h) V βα βα ∂xj ∂xi ∂y δ ∂xj ∂xi   α δ β ∂V ∂f ∂f + g ij i Γ (h)γαδ + Γ (h)δβε V ε . j ∂x ∂x ∂xj

342

K. IMPLICIT FUNCTION THEOREM

Combining this with (K.32), we have   ∂  (Δg,h fs )γ = (Δg,h V )γ − Δg f β Γ (h)γβα V α  ∂s s=0   α β ∂ ∂f ∂f ∂ γ γ + g ij V δ Γ (h) − Γ (h) αβ βδ ∂y δ ∂y α ∂xi ∂xj + g ij V δ (−Γ (h)γαε Γ (h)εβδ )

∂f α ∂f β . ∂xi ∂xj

We may further rewrite this and simplify as (K.33)    δ ε  β ∂  γ γ β ij ∂f ∂f f + Γ (h) (Δ f ) = (Δ V ) − Δ g Γ (h)γαβ V α g,h s g,h g εδ ∂s s=0 ∂xi ∂xj   α β ∂ ∂f ∂f ∂ γ γ + g ij V δ Γ (h) − Γ (h) αβ βδ ∂y δ ∂y α ∂xi ∂xj   ∂f α ∂f β + g ij V δ Γ (h)γδε Γ (h)εβα − Γ (h)γαε Γ (h)εβδ ∂xi ∂xj γ β γ α = (Δg,h V ) − (Δg,h f ) Γ (h)αβ V  γ ∂f α ∂f β + V δ RN δαβ g ij . ∂xi ∂xj Note that the terms forming the expression on the rhs of (K.33) are not the components of a tensor. Using (K.33) we compute for V independent of s that  ∂  (K.34) V, Δg,h fs f ∗ h s ∂s s=0   ∂  γ ((hαγ ◦ fs ) (V α ◦ fs ) (Δg,h fs ) ) = ∂s s=0     ∂ γ γ α ∂  δ = hαγ V (Δg,h fs ) + V hαγ V α (Δg,h f ) ∂s s=0 ∂y δ   γ β ∂ α + hαγ V V (Δg,h f ) ∂y β  γ ∂f α ∂f β = hαγ V α (Δg,h V )γ + hεγ V ε V δ RN δαβ g ij ∂xi ∂xj  ∂ β γ β − hαγ V α (Δg,h f ) Γ (h)δβ V δ + V δ hαβ V α (Δg,h f ) ∂y δ   ∂ + hαγ V β β V α (Δg,h f )γ . ∂y Since −hαγ Γ (h)γδβ

and

  ∂ ∂ 1 ∂ ∂ + δ hαβ = hβα − β hδα + α hδβ ∂y 2 ∂y δ ∂y ∂y γ = hβγ Γ (h)δα ∂ f ∗h δ α V α + Γ (h)α βδ V = (∇β V ) , β ∂y

3. HARMONIC MAPS AND THEIR LINEARIZATION

343

we may rewrite (K.34) as   ∂  ∂f α ∂f β γ (K.35) V, Δg,h fs f ∗ h = hαγ V α (Δg,h V ) + RN δαβε V δ i g ij j V ε  s ∂s s=0 ∂x ∂x ∗

γ

+ hαγ V β (∇fβ h V )α (Δg,h f ) .  s ∗ When V = Vs depends on s, we simply add the term  ∂V ∂s s=0 , Δg,h f f h to the rhs of (K.35). In conclusion, we have the following. s Lemma K.15 (Second variation formula for map energy). If ∂f ∂s  Vs and V0 = V , then   ∂  Vs , Δg,h fs f ∗ h = h (V, Δg,h V ) + tr g RN (V, df, df, V )  s ∂s s=0  . / ∂Vs  f ∗h , Δg,h f , + h(∇V V, Δg,h f ) + ∂s s=0 f ∗h

where  N  ∂f α ∂f β R (V, df, df, V )  RN δαβε V δ i g ij j V ε . ∂x ∂x Assume that V has compact support in the interior of M. Then differentiating (K.29) and integrating by parts yields       N d2   f ∗ h 2 Eg,h (fs ) = 2 − tr g R (V, df, df, V ) dμg ∇ V  ds2 s=0 gh M  . /

 ∂Vs  f ∗h −2 , Δg,h f h(∇V V, Δg,h f ) + dμg . ∂s s=0 M f ∗h tr g

In particular, if f is a harmonic map, then       N d2   f ∗ h 2 R (V, df, df, V ) (K.36) E (f ) = 2 V − tr dμg  ∇ g,h s g ds2 s=0 gh M  I (V, V ) . The rhs of (K.36) is called the index form. We say that a harmonic map f is weakly stable if I (V, V ) ≥ 0 for all V ∈ f ∗ (T N ) with compact support in the interior of M. If (N , h) has nonpositive sectional curvature, then     f ∗ h 2 I (V, V ) ≥ 2 dμg ≥ 0, ∇ V  M

gh

so that any harmonic map f mapping into a Riemannian manifold with nonpositive sectional curvature is weakly stable. Exercise K.16. Adopt the notation in (K.30). Show that   N ∇∂/∂s ΔM g,h F = Δg,h V + tr g R (V, df ) df .

344

K. IMPLICIT FUNCTION THEOREM

3.4. The harmonic map heat flow. In 1964, Eells and Sampson [100] proved the following existence theorem for harmonic maps. Essentially, this result began the study and application of nonlinear parabolic equations in differential geometry. It was a main motivation for Hamilton’s discovery of the Ricci flow. Theorem K.17 (Existence of harmonic maps into nonpositively curved targets). Let (Mn , g) and (N m , h) be closed Riemannian manifolds where (N , h) has nonpositive sectional curvature. If f0 : M → N is a smooth map, then there exists a harmonic map f∞ : M → N which is homotopic to f0 . The original proof 3 of Eells and Sampson uses the harmonic map heat flow ∂f = Δg,h f. (K.37) ∂t They proved that if (N , h) has nonpositive sectional curvature, then for any C ∞ map f0 : M → N , there exists a C ∞ solution f (t) : M → N to the harmonic map heat flow with f (0) = f0 and defined for all time t ∈ [0, ∞). Moreover, as t → ∞, the map f (t) converges in C ∞ to a C ∞ harmonic map f∞ . When the target has negative sectional curvature, Hartman [145] proved a uniqueness result. In this subsection we discuss a couple of energy-type estimates used in the proof of Theorem K.17; these estimates give indications that solutions to the harmonic map heat flow behave well. The interested reader may see [100] for a complete proof. First, we compute the evolution of the energy density under the harmonic map heat flow. By taking V = Δg,h f in (K.28) we have that # ∗ $ ∂ 2 |df |gh = 2 ∇f h (Δg,h f ) , df . ∂t gh To rewrite this as a heat-type equation we shall deduce a Bochner–Weitzenb¨ ocktype formula. We compute  2   , Δg |df |2gh = 2 Δg,h (df ) , df  + 2 ∇gh (df ) gh

which implies (K.38)

 2 ∂   2 2 |df |gh = Δg |df |gh − 2 ∇gh (df ) ∂t gh # ∗ $ f h +2 ∇ (Δg,h f ) − Δg,h (df ) , df

. gh

We claim that as sections of E = T ∗ M ⊗ f ∗ (T N ) we have   ∗ (K.39) ∇f h (Δg,h f ) − Δg,h (df ) = − RcM (df ) + tr2,3 g RN (df, df ) df , where the trace tr2,3 g is with respect to the second and third components of RN . Recall that for a 1-form ω on M, we have   ∇i ∇j ωk − ∇j ∇i ωk = − RM ijk ω , whereas for a vector field W on N we have

δ  ∇α ∇β W δ − ∇β ∇α W δ = RN αβγ W γ .

3 Schoen [348] later gave a variational proof of this theorem using a convexity property of the map energy functional which holds when (N , h) has nonpositive sectional curvature.

3. HARMONIC MAPS AND THEIR LINEARIZATION

345

In view of this, we compute that δ

δ

δ

δ

(∇i ∇j df )k − (∇j ∇k df )i = (∇i ∇j df )k − (∇j ∇i df )k  ∂f δ  N δ ∂f α ∂f β ∂f γ  = − RM ijk + R αβγ . ∂x ∂xi ∂xj ∂xk Tracing this by multiplying by g jk , we obtain 

∇fi

∗

h

δ (Δg,h f )

 ∂f δ  N δ ∂f α ∂f β ∂f γ  jk R αβγ − (Δg,h (df ))δi = − RcM i + g ∂x ∂xi ∂xj ∂xk

from which (K.39) follows. Define    m ∂f δ ∂f ε  M (df  df )  RcM (df ) , df gh = hδε RcM Rc ∂x ∂xm and  N    R (df  df  df  df )  tr2,3 g RN (df, df ) df , df gh  δ ∂f α ∂f β ∂f γ ∂f ε = g im hδε g jk RN αβγ . ∂xi ∂xj ∂xk ∂xm By (K.39) and (K.38), we have the following. Lemma K.18 (Evolution of the energy density). Under the harmonic map heat flow we have (K.40)

 2  ∂   |df |2gh = Δg |df |2gh − 2 ∇gh (df ) − 2 RcM (df  df ) ∂t gh  + 2 RN (df  df  df  df ) .

In particular, if the sectional curvatures of (N , h) are nonpositive and the Ricci curvatures of (M, g) are bounded below by −K ∈ R, then (K.41)

 2 ∂   2 2 2 + 2K |df |gh . |df |gh ≤ Δg |df |gh − 2 ∇gh (df ) ∂t gh

Assuming that M is closed, we may apply the maximum principle to (K.41) 2 and conclude that |df |gh (x, t) is bounded above by Ce2Kt on M × [0, ∞), where C = max |df |2gh (·, 0). Second, we shall show that the norm squared of the Laplacian of f is a subsolution to the heat equation. By taking V = Δg,h f in (K.33), we see that ∂ γ γ β γ α (Δg,h f ) = (Δg,h (Δg,h f )) − (Δg,h f ) Γ (h)αβ (Δg,h f ) ∂t  γ ∂f α ∂f β + (Δg,h f )δ RN δαβ g ij ∂xi ∂xj

346

K. IMPLICIT FUNCTION THEOREM

so that ∂ ∂ ∂ |Δg,h f |2 = 2hγε (Δg,h f )ε (Δg,h f )γ + (hγε ◦ f ) (Δg,h f )ε (Δg,h f )γ ∂t ∂t ∂t   ε γ β γ α = 2hγε (Δg,h f ) (Δg,h (Δg,h f )) − (Δg,h f ) Γ (h)αβ (Δg,h f )  γ ∂f α ∂f β + 2hγε (Δg,h f )ε (Δg,h f )δ RN δαβ g ij ∂xi ∂xj   ∂ + (Δg,h f )δ hγε (Δg,h f )ε (Δg,h f )γ ∂y δ  2 2 = Δ |Δg,h f | − 2 |∇ (Δg,h f )| + 2 RN (Δg,h f, df, df, Δg,h f ) , where  N γ ∂f α ∂f β ε δ R (Δg,h f, df, df, Δg,h f )  hγε (Δg,h f ) (Δg,h f ) RN δαβ g ij . ∂xi ∂xj In particular, if the sectional curvatures of (N , h) are nonpositive, then ∂ (K.42) |Δg,h f |2 ≤ Δ |Δg,h f |2 − 2 |∇ (Δg,h f )|2 ≤ Δ |Δg,h f |2 . ∂t We may apply the maximum principle to (K.42) and conclude that |Δg,h f | is uniformly bounded on M × [0, ∞). 3.5. Linearization of the harmonic map equation with normal boundary condition. In §3 of Chapter 34, we considered an existence problem for harmonic maps satisfying the normal boundary condition. In particular, adopting the notation in that section, let Mn be a manifold with boundary ∂M and let F be a diffeomorphism from (M, ∂M) to itself. Correspondingly, we defined in (34.31) the map   Φ (˜ g , F )  F −1 ∗ (Δg,˜g F ) , F −1 ∗ (F∗ (N )) . We now compute the second component of the linearization of the map Φ at (g, id), where id denotes the identity map of M. Note that, formally, the tangent space at id of the space of diffeomorphisms preserving ∂M comprises the vector fields V on M such that ( V |∂M )⊥ = 0. Let d  F  V ∈ C ∞ (T M). Fs be a 1-parameter family of maps with F0 = id and ds s=0 s  i k Given p ∈ M, choose coordinates x satisfying Γij (p) = 0. Then at p we have   −1 d  Fs ∗ (W ) = [V, W ] that by (K.32) and ds s=0     −1 k ∂ k d  k ij F (Δ F ) = Δ (V ) + g Γ V . g,g s g s ∗ ds s=0 ∂x ij We also have (Δg V )k = g ij ∇i ∇j V k     ∂ ∂ k ij k   k k  =g V + Γj V − Γij ∇ V + Γi ∇j V ∂xi ∂xj   ∂ k = Δg (V k ) + g ij Γ V ∂xi j 

and Rk

=g

ij

k Rij

=g

ij

 ∂ k ∂ k Γ − Γ . ∂x ij ∂xi j

4. SPECTRUM OF Δd ON p-FORMS ON S n

347

Hence, in invariant form, the variation formula for the first component of Φ(g, Fs ) is   −1 d  (K.43) Fs ∗ (Δg,g Fs ) = Δg V + Rc(V ). ds  s=0

Regarding the second component of Φ(g, Fs ), since   −1 Fs ∗ ((Fs )∗ (N )) = N − (Fs )∗ (N ) , N  Fs−1 ∗ N and

 d  (Fs )∗ (N ) = − [V, N ] , ds s=0   −1 d  F (N ) = [V, N ] ,  ds s=0 s ∗

we have that (K.44)

  −1 d  Fs ∗ (((Fs )∗ (N )) ) = [V, N ] , N  N − [V, N ]  ds s=0 = ([N, V ]) = (∇N V ) − II(V )

since (∇V N ) = II(V ). 4. Spectrum of Δd on p-forms on S n Let L be a linear operator acting on p-forms on a manifold. We say that λ is an eigenvalue of L with nonzero eigenform α if (L + λ) α = 0. Let (S n , gsph ) be the unit n-sphere (in Rn+1 ). In this section we recall some basic facts about the eigenvalues of the Hodge–de Rham Laplacian acting on functions and p-forms on S n . The case p = 1 is used in §2 of Chapter 34 to study harmonic maps of S n near the identity. The main reference for this section is the book by Berger, Gauduchon, and Mazet [27]. 4.1. Spectrum of the Laplacian acting on functions on S n . Recall that the Laplacian on Rn+1 may be written in polar coordinates as (K.45)

ΔRn+1 =

∂2 n ∂ 1 + + 2 ΔS n . 2 ∂r r ∂r r

We say that a function ϕ on Rn+1 is homogeneous of degree k if ϕ (x) = k x |x| ϕ( |x| ) for all x ∈ Rn+1 − {0}. In this case we have k ∂ϕ = ϕ ∂r r

and

∂2ϕ k (k − 1) = ϕ. ∂r 2 r2

Hence 1 1 k (k + n − 1) ϕ + 2 ΔS n ϕ. r2 r Therefore, if ϕk is the restriction to S n of a degree k homogeneous harmonic polynomial on Rn+1 , then by taking r = 1 we have ΔRn+1 ϕ =

(K.46)

ΔS n ϕk + k (k + n − 1) ϕk = 0 ;

in particular, ϕk is an eigenfunction on S n .

348

K. IMPLICIT FUNCTION THEOREM

Let Pk denote the space of degree k homogeneous polynomials on Rn+1 and let Hk denote the space of degree k homogeneous harmonic polynomials on Rn+1 . For k ∈ N ∪ {0} one has the orthogonal decomposition (K.47)

Pk+2 = Hk+2 ⊕ r 2 Pk .

Since P0 = H0 and P1 = H1 , applying induction to (K.47) yields the following. Proposition K.19 (Space of homogeneous (harmonic) polynomials). For all j ∈ N ∪ {0} we have the decompositions (K.48a)

P2j = H2j ⊕ r 2 H2j−2 ⊕ · · · ⊕ r 2j−2 H2 ⊕ r 2j H0 ,

(K.48b)

P2j+1 = H2j+1 ⊕ r 2 H2j−1 ⊕ · · · ⊕ r 2j−2 H3 ⊕ r 2j H1 ,

which are orthogonal with respect to the L2 inner product on S n for the functions restricted to S n . That is, for any f ∈ Hk and h ∈ H , where k = , we have  f |S n h|S n dμgsph = 0. Sn

Let ˜ k  { f | n : f ∈ Hk } . P˜ k  { f |S n : f ∈ Pk } and H S = % ˜ ˜ Note that P˜ k ⊂ P˜ k+2 . By the proposition, k∈N∪{0} Hk = k∈N∪{0} P k , where the lhs is%an orthogonal direct sum. On the other hand, by the = Stone–Weierstrass ˜ k forms a theorem, k∈N∪{0} P˜ k is dense in C ∞ (S n ). It follows that k∈N∪{0} H complete orthogonal system in L2 (S n ). As a consequence, we have Proposition K.20 (Eigenvalues of S n ). For k ∈ N ∪ {0}, the k-th eigenvalue λk of the Laplacian acting on functions on S n is given by λk = k (k + n − 1) ˜k. and the corresponding eigenspace is H (K.49)

4.2. Elements of Hodge theory on Riemannian manifolds. Let (Mn , g) be a closed Riemannian manifold. Let d : Ωp (M) → Ωp+1 (M) and δ : Ωp (M) → Ωp−1 (M) denote the exterior derivative and its formal adjoint, respectively, acting on differential forms on M, where Ωp (M)  C ∞ (Λp T ∗ M) is the space of C ∞ p-forms. Recall that the Hodge star operator (for p = 0, . . . , n) ∗ : Λp T ∗ M → Λn−p T ∗ M is defined to be the unique linear isomorphism such that γ, η dμg = γ ∧ ∗η for any γ, η ∈ Λp T ∗ M. The operator δ acting on Ωp (M) may be written as δα = (−1)np+n+1 ∗ d ∗ α. We say that a differential form α is closed if dα = 0; likewise, a differential form β is said to be co-closed if δβ = 0. Note that a differential form is closed if and only if its Hodge star is co-closed; i.e., dα = 0 if and only if δ (∗α) = 0. Likewise, δβ = 0 if and only if d (∗β) = 0. acting Let Δd = − (dδ + δd) denote the associated Hodge–de Rham Laplacian  on differential forms.4 With respect to the L2 -inner product η, ωL2  M η, ω dμ we have the following properties. The Hodge–de Rham Laplacian is self-adjoint: Δd α, βL2 = α, Δd βL2 4 This

for α, β ∈ Ωp (M)

is the opposite of the usual sign convention.

4. SPECTRUM OF Δd ON p-FORMS ON S n

349

and nonpositive: Δd α, αL2 = − dα, dαL2 − δα, δαL2 ≤ 0. We also have that (K.50)

Δd (dα) = d (Δd α) ,

Δd (∗α) = ∗ (Δd α) .

Hence, if Δd ϕ + λϕ = 0, then λ ≥ 0 and (K.51)

Δd (dϕ) + λ (dϕ) = 0,

Δd (∗ϕ) + λ (∗ϕ) = 0.

Let Hp (M) ⊂ Ωp (M) denote the space of harmonic p-forms. The Hodge decomposition theorem says the following. Theorem K.21. Let (Mn , g) be a closed Riemannian manifold. The space Ω (M) of C ∞ p-forms may be written as p

Ωp (M) = Δd (Ωp (M)) ⊕ H p = dδ (Ωp (M)) ⊕ δd (Ωp (M)) ⊕ H p (M)   = d Ωp−1 (M) ⊕ δ Ωp+1 (M) ⊕ H p (M).

(K.52) (K.53)

4.3. Spectrum of Hodge–de Rham Laplacian on p-forms on S n . The spectrum of the Hodge–de Rham Laplacian acting on p-forms on the nsphere S n of radius 1 has been computed by Gallot and Meyer, following an idea of Calabi. By the Hodge decomposition theorem and (K.50), the discussion is reduced to the cases of closed and co-closed forms. [p]

Theorem K.22 (Closed eigenforms). The k-th eigenvalue c λk (S n ) of the Hodge–de Rham Laplacian acting on closed p-forms on the unit n-sphere S n is given by c [p] λk

(K.54)

(S n ) = (k + p) (n + k − p + 1)

for k ∈ N ∪ {0}. The corresponding eigenspace is the space of degree k homogeneous harmonic p-forms on Rn+1 restricted to S n .5 Now the Hodge star operator maps the eigenspaces of closed (n − p)-forms to the eigenspaces of co-closed p-forms (see (K.51)). Hence we have the following: [p]

Theorem K.23 (Co-closed eigenforms). The k-th eigenvalue cc λk (S n ) of the Hodge–de Rham Laplacian acting on co-closed p-forms on the unit n-sphere S n is given by cc [p] λk

(K.55)

(S n ) = (k + n − p) (k + p + 1)

for k ∈ N ∪ {0}. Corollary K.24. The lowest eigenvalue of Δd acting on p-forms on S n is min {p (n − p + 1) , (p + 1) (n − p)}. We now proceed to prove Theorem K.22. Let ∇ denote the covariant derivative on S n and let D denote the covariant derivative on Rn+1 . Let ν denote the unit outward normal vector field to S n so that ν (x) = x for x ∈ S n . Recall that the 5 See

[p]

[113] for a formula for the multiplicity of the k-th eigenvalue c λk (S n ).

350

K. IMPLICIT FUNCTION THEOREM

second fundamental form of S n equals the first fundamental form of S n . Our sign conventions are such that for vector fields X and Y on S n , (K.56)

DX ν = II (X) = X

and (K.57)

DX Y = ∇X Y − II (X, Y ) ν = ∇X Y − X, Y  ν.

x Now extend ν to Rn+1 − {0} as ν (x) = |x| so that Dν ν = 0. Given a vector n n+1 to be homogeneous of degree 1. In particular, field X on S , extend X to R x ) for x ∈ Rn+1 − {0}. We then have Dν X = X and X (x) = |x| X( |x|

[X, ν] = DX ν − Dν X = X − X = 0. We have the following. Lemma K.25 (δ on S n in terms of δ on Rn+1 ). If η is a p-form defined on a neighborhood of S n in Rn+1 , then (δS n η|S n ) (X1 , . . . , Xp−1 ) − (δRn+1 η) (X1 , . . . , Xp−1 ) = (n − 2p + 2) η (ν, X1 , . . . , Xp−1 ) + ν (η (ν, X1 , . . . , Xp−1 )) , where the vector fields Xj on S n are extended to Rn+1 to be homogeneous of degree 1, so that Dν Xj = Xj . Proof. Recall that on a Riemannian manifold (Mn , g) we have that δ : Ω (M) → Ωp−1 (M) is given by p

(δη) (X1 , . . . , Xp−1 ) = −

n 

(∇ei η) (ei , X1 , . . . , Xp−1 )

i=1

for all vector fields X1 , . . . , Xp−1 , where {ei }ni=1 is a local orthonormal frame field. On S n we have by the product rule and by (K.57) that for each i, (Dei η) (ei , X1 , . . . , Xp−1 ) − (∇ei η|S n ) (ei , X1 , . . . , Xp−1 ) = η (∇ei ei − Dei ei , X1 , . . . , Xp−1 ) +

p−1 

η (ei , X1 , . . . , ∇ei Xj − Dei Xj , . . . , Xp−1 )

j=1

= η (ν, X1 , . . . , Xp−1 ) + ei , Xj 

p−1  j=1

η (ei , X1 , . . . , Xj−1 , ν, Xj+1 , . . . , Xp−1 ) .

4. SPECTRUM OF Δd ON p-FORMS ON S n

351

Hence (δS n η|S n ) (X1 , . . . , Xp−1 ) − (δRn+1 η) (X1 , . . . , Xp−1 ) =−

n 

(∇ei η|S n ) (ei , X1 , . . . , Xp−1 ) +

i=1

n 

(Dei η) (ei , X1 , . . . , Xp−1 )

i=1

+ (Dν η) (ν, X1 , . . . , Xp−1 ) = n · η (ν, X1 , . . . , Xp−1 ) +

p−1 

η (Xj , X1 , . . . , Xj−1 , ν, Xj+1 , . . . , Xp−1 )

j=1

+ (Dν η) (ν, X1 , . . . , Xp−1 ) = (n − p + 1) η (ν, X1 , . . . , Xp−1 ) + (Dν η) (ν, X1 , . . . , Xp−1 ) = (n − 2p + 2) η (ν, X1 , . . . , Xp−1 ) + ν (η (ν, X1 , . . . , Xp−1 )) where we used Dν ν = 0 and Dν Xj = Xj to obtain the last equality.



Note that if ω is a p-form defined on a neighborhood of S n in Rn+1 , then (dS n ω|S n ) (X1 , . . . , Xp+1 ) = (dRn+1 ω) (X1 , . . . , Xp+1 ) for vector fields Xj on S n . By a straightforward calculation using this and Lemma K.25, one obtains R

Lemma K.26 (Δd on S n in terms of Δd on Rn+1 ). If η is a closed p-form on , then  n  n+1 ΔSd η|S n − ΔR η (X1 , . . . , Xp ) d

n+1

= ν (ν (η (X1 , . . . , Xp ))) + (n − 2p + 2) ν (η (X1 , . . . , Xp )) , where the vector fields Xj on S n are extended to Rn+1 to be homogeneous of degree 1. Consequently, we have the following. Lemma K.27 (Closed eigenforms on S n ). If η is a harmonic p -form on Rn+1 which is homogeneous of degree k, then η|S n is a closed p-form on S n and  n  ΔSd η|S n (X1 , . . . , Xp ) = (k + p) (n − p + k + 1) η|S n (X1 , . . . , Xp ) . Proof. Since η is homogeneous of degree k and since we extend Xj to be homogeneous of degree 1, we have that η (X1 , . . . , Xp ) is homogeneous of degree n+1 η = 0 and Lemma K.26, we have k + p. Now by ΔR d   n ΔSd η|S n (X1 , . . . , Xp ) = (k + p) (k + p − 1) η|S n (X1 , . . . , Xp ) + (n − 2p + 2) (k + p) η|S n (X1 , . . . , Xp ) and the lemma follows (note that η|S n is closed since it is the restriction of the closed form η to a submanifold).  From this and a density result one may derive Theorem K.22 (see p. 283 of [27]).

352

K. IMPLICIT FUNCTION THEOREM

4.4. Bochner–Weitzenb¨ ock formula for p-forms. Recall that the Lichnerowicz theorem says that if Rc ≥ (n − 1)k > 0, then the lowest eigenvalue of the Laplacian acting on functions satisfies λ1 (Δ) ≥ nk, √ with equality if and only if M is the n-sphere of radius 1/ k. One may ask if this eigenvalue comparison theorem generalizes to Δd acting on differential forms. In regards to this, Gallot and Meyer [113] proved a Bochner–Weitzenb¨ ock formula which says that if α is a p-form on a Riemannian manifold (Mn , g), then Δd α, α = Δα, α + Rp (α, α) , where Rp is a certain expression which is linear in the Riemann curvature tensor. Furthermore, they showed that if the smallest eigenvalue λ1 (Rm) of the curvature operator Rm : Λ2 T ∗ M → Λ2 T ∗ M (which at each point is a self-adjoint linear map) is at least k, then Rp (α, α) ≥ p (n − p) k |α|2 . As a consequence one can prove the following, with Corollary K.24 representing the [p] model case with equality. Let λ1 (M, g) denote the first positive eigenvalue of Δd acting on p-forms. Theorem K.28 (Gallot and Meyer). If (Mn , g) is a closed Riemannian manifold with λ1 (Rm) ≥ k > 0, then [p]

λ1 (M, g) ≥ min {p (n − p + 1) , (p + 1) (n − p)} k. 5. Notes and commentary §1. For Theorem K.1, see Lions [206] and Lions and Magenes [207]. A general reference for the inverse function theorem is Lang [179]. §4. For the Hodge Decomposition Theorem, Theorem K.21, see Theorem 6.8 on p. 223, of Warner [432] for example. Our discuss of the spectrum of the Hodge–de Rham Laplacian on p-forms on S n follows Gallot and Meyer [113].

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Index

Bianchi gauge, 281 Bochner formula, 77 Bochner–Weitzenb¨ ock formula, 352 Bochner-type inequality, 53 boundary torus incompressibility of , 218, 238 incompressible, 182 Brouwer fixed point theorem, 175 Bryant soliton, 2, 44 volume growth, 45

adjoint heat operator, 153 almost hyperbolic piece persistence of , 227 persistent, 216 ancient solution 3-dimensional with pinched Ricci curvatures, 65 incomplete, 39 must have R ≥ 0, 39 on S 2 , 69 Type I, 45, 73, 161, 309 Type I with PCO, 63 Type II, 45, 165 with finite width, 53 asymptotic cone, 29, 42 of 3-dimensional κ-solution, 46 asymptotic limit, 226 asymptotic shrinker existence of, 45 of 3-dimensional Type II ancient solution, 165 asymptotic soliton backward, 163 forward, 163 asymptotic volume ratio, 17 existence, 20 of shrinker, 32 positive, 21 automorphism group inner, 192 outer, 192

canonical neighborhood theorem, 43 center manifold, 280 Center Manifold Theorem, 295 Cheeger–Gromov compactness theorem, 204 convergence, 204 theory, 201 Christoffel symbols on S 2 , 75 circular average, 59, 101 CMC boundary conditions, 216 CMC spheres in necks, 260 co-area formula, 17 Cohn-Vossen inequality, 55, 61, 102 collapse, 201 compactness theorem for the normalized Ricci flow, 204 complete vector field, 6 compressible surface, 179 torus, 180 concentration-compactness, 102 Condition H, 213 constant mean curvature, 215 contraction mapping principle, 327 coordinates on S 2 , 71 curvature scalar, 2 curve shortening flow, 85

backward limit Cheeger–Gromov, 106 cylinder, 90, 104 is not the plane, 98 of scalar curvature, 79 on S 2 , 75 backwards uniqueness, 87 Bakry–Emery volume comparison, 24 Bakry–Emery Ricci tensor, 3 Banach manifold, 330 atlas, 330 371

372

cusp hyperbolic, 186 maximal, 186 cut and paste argument, 195 cutoff function, 36 de Rham splitting theorem, 164 derivative estimate, 62, 78 local, 18, 55, 134 differential form closed, 348 co-closed, 348 Dimension Classic, xiii Dini derivative, 88 disk model, 171 ε-neck, 47, 91 ε-thick part, 188 ε-thin end, 188 part, 188 eigenvalues of Laplacian on S n , 348 Einstein manifold, 2 metric, 200 end topological, 186 end-complementary, 185 energy density, 336 Dirichlet-type, 80 of maps, 336 entropy Nash, 27 Perelman’s, 26 eternal solution, 209 Euclidean metric cone, 46 Euclidean space characterization, 9 expanding soliton, 2 fBochner formula, 23 Laplacian, 2 mean curvature, 23 Riccati equation, 23 Ricci tensor, 3 scalar curvature, 3 volume, 3 F -structure, 201 finite graph, 194 foliation, 193 Fr´ echet derivative, 328 differentiable, 328

INDEX

Gaussian soliton, 2 geometric decomposition, 201 geometric perimeter, 50 relative, 50 gradient Ricci soliton, 2 equality case, 8 expanding K¨ ahler, 15 lower bound for R, 3 graph manifold, 194 Grayson’s theorem, 85 GRS, 2 normalized, 2 Hamilton–Ivey estimate, 42 harmonic embedding continuing, 222 existence of , 220 harmonic map, 338 near the identity, 219, 260, 266 Harnack estimate trace, 39 Hausdorff measure, 50 heat operator adjoint, 153 Hessian, 2 Hodge star operator, 348 Hodge–de Rham Laplacian, 348 homotopy equivalent, 192 horoball, 177 horosphere, 177 hyperbolic cusp, 186 hyperbolic isometry, 175 hyperbolic limit, 210 stable, 216, 230 hyperbolic manifold finite-volume, 177 hyperbolic piece, 185 hyperbolic space, 171 disk model, 171 hyperboloid model, 172 upper half-space model, 171 hyperboloid model, 172 hyperplane at infinity, 171 immortal almost hyperbolic piece, 216 implicit function theorem, 330 incompressibility of boundary torus, 218, 238 incompressible boundary torus, 182 surface, 179 inner automorphism group, 192 inverse function theorem for Banach spaces, 328 irreducible 3-manifold, 178 isometry hyperbolic, 175

INDEX

parabolic, 175 isometry group of a hyperbolic manifold, 192 isoperimetric constant, 83 jet, 334 jets bundle of, 334 K3 complex surface, 284 κ-noncollapsed below the scale ρ, 40 κ-solution, 43 King–Rosenau solution, 73 Kulkarni–Nomizu product, 304 L-distance, 135 Laplacian Hodge–de Rham, 348 Lax–Milgram–Lions theorem, 327 leaf, 193 length spectrum, 190 level set, 9, 50 linear stability/instability, 280 local estimate for scalar curvature, 35 locally homogeneous, 200 logarithmic Sobolev constant, 26 long-time existence criterion, 42 Loop Theorem, 179 μ-invariant, 26 map energy, 336 map-Laplacian, 337 Margulis constant, 187 Margulis lemma, 189 algebraic version, 187 geometric consequence, 189 local consequence, 187 Margulis tube, 189, 190 maximal cusp end, 186 maximal regularity theory, 289 Meeks and Yau theorem, 239 Mercator projection, 71 minimal disk evolution of the area of, 244 M¨ obius transformation, 174 monotonicity of Rmin (t), 203 Mostow rigidity theorem, 192 ν-invariant, 64 nilpotent, 176 noncollapsed, 201 nonsingular solution, 200 Condition H, 213 normalized GRS, 2

373

normalized Ricci flow, 200 notation, xvii outer automorphism group, 192 parabolic isometry, 175 parabolic rescaling, 154 plaque, 193 point at infinity, 171 Polyakov’s energy, 80 porous medium equation, 72 positive curvature operator, 200 potential function, 2 bounds for, 9 Cao–Zhou estimate, 10 normalized, 2 sublevel set of, 18 pressure function, 72 backward limit, 81 estimates, 77 primitive element, 179 properly discontinuously, 177 pseudo-metric, 46 Q definition, 109 heat-type equation, 111 must vanish for ancient solution, 125 plane version, 118 plane version evolution, 124 vanishing characterization, 117 ray, 41, 46 reduced boundary, 50 reduced distance, 44 based at the singular time, 135, 141 of Type I solution, 146 reduced volume at the singular time, 145 relative Fisher information functional, 27 relative isoperimetric inequality, 50 RG flow, 303 Ricci flat, 45 Ricci flow normalized, 200 with cosmological constant, 204 Ricci soliton lower bound for R, 3 Ricci tensor, 2 Ricci–DeTurck flow, 282 scalar curvature, 2 bounded, 42 evolution of, 203 local estimate for, 35 lower bound, 15 nonnegative, 39 Schwarz–Ahlfors–Pick lemma, 57

374

sectional curvature nonnegative, 40 Seifert fibered manifold, 193 Seifert–Van Kampen theorem, 182 sequential collapse, 201 set of singular points, 158 shrinker asymptotic volume ratio, 32 must be gradient, 168 must be κ-noncollapsed, 168 volume growth of, 24 with nonnegative Ricci curvature, 29 shrinking soliton, 2 singular solution, 40 singularity Type A, 133 Type I, 309 Type IIa, 309 Type IIb, 309 Type IIc, 309 Type III, 309 Type IV, 309 singularity model, 40 3-dimensional, 43 bounded scalar curvature, 42 compact, 41 existence of, 40 linear volume growth, 41 volume growth, 45 Slice Theorem, 281 soliton Bryant, 2 Gaussian, 2 Ricci gradient, 2 sphere at infinity, 171 spherical space form, 207 stable hyperbolic limit, 216, 230 steady soliton, 2 stereographic projection, 71 sublevel set, 18 surface, 200 compressible, 179 incompressible, 179 sweep out, 219

INDEX

tensor symmetrization, 108 trace-free part, 109 thick-thin decomposition, 188 topological end, 186 set of, 186 totally umbillic, 187 trace operator, 268 theorem, 268 Trudinger–Moser-type inequality, 60 Type A solution, 134 local derivative estimates for, 134 Type I ancient solution, 45 Type I singularity, 309 admits shrinker as a singularity model, 157 reduced distance of, 146 Type II ancient solution, 45 Type IIa singularity, 309 Type IIb singularity, 309 Type IIc singularity, 309 Type III singularity, 309 Type IV singularity, 309 uniformization theorem, 177 unit vector normal, 50 upper half-space model, 171 vector field complete, 6, 166 virtually abelian, 187 virtually nilpotent, 187 volume comparison Bakry–Emery, 24 relative, 96 volume converges, 49 volume growth linear, 41 of shrinker, 24 weak solution, 82 width, 51

Selected Published Titles in This Series 206 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow: Techniques and Applications: Part IV: Long-Time Solutions and Related Topics, 2015 204 Victor M. Buchstaber and Taras E. Panov, Toric Topology, 2015 203 Donald Yau and Mark W. Johnson, A Foundation for PROPs, Algebras, and Modules, 2015 202 Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, Asymptotic Geometric Analysis, Part I, 2015 201 Christopher L. Douglas, John Francis, Andr´ e G. Henriques, and Michael A. Hill, Editors, Topological Modular Forms, 2014 200 Nikolai Nadirashvili, Vladimir Tkachev, and Serge Vl˘ adut ¸, Nonlinear Elliptic Equations and Nonassociative Algebras, 2014 199 Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of Free Noncommutative Function Theory, 2014 198 J¨ org Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, 2014 197 Richard Evan Schwartz, The Octagonal PETs, 2014 196 Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of Isotropic Convex Bodies, 2014 195 Ching-Li Chai, Brian Conrad, and Frans Oort, Complex Multiplication and Lifting Problems, 2014 194 Samuel Herrmann, Peter Imkeller, Ilya Pavlyukevich, and Dierk Peithmann, Stochastic Resonance, 2014 193 Robert Rumely, Capacity Theory with Local Rationality, 2013 192 Messoud Efendiev, Attractors for Degenerate Parabolic Type Equations, 2013 191 Gr´ egory Berhuy and Fr´ ed´ erique Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication, 2013 190 Aleksandr Pukhlikov, Birationally Rigid Varieties, 2013 189 Alberto Elduque and Mikhail Kochetov, Gradings on Simple Lie Algebras, 2013 188 David Lannes, The Water Waves Problem, 2013 187 Nassif Ghoussoub and Amir Moradifam, Functional Inequalities: New Perspectives and New Applications, 2013 186 Gregory Berkolaiko and Peter Kuchment, Introduction to Quantum Graphs, 2013 185 Patrick Iglesias-Zemmour, Diffeology, 2013 184 Frederick W. Gehring and Kari Hag, The Ubiquitous Quasidisk, 2012 183 Gershon Kresin and Vladimir Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, 2012 182 Neil A. Watson, Introduction to Heat Potential Theory, 2012 181 Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay Representations, 2012 180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, 2012 179 Stephen D. Smith, Subgroup Complexes, 2011 178 Helmut Brass and Knut Petras, Quadrature Theory, 2011 177 Alexei Myasnikov, Vladimir Shpilrain, and Alexander Ushakov, Non-commutative Cryptography and Complexity of Group-theoretic Problems, 2011 176 Peter E. Kloeden and Martin Rasmussen, Nonautonomous Dynamical Systems, 2011 175 Warwick de Launey and Dane Flannery, Algebraic Design Theory, 2011

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton’s program to prove Thurston’s geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton’s theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives. This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow.

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