[Part III: Geometric-Analytic Aspects] The Ricci Flow: Techniques and Applications [3] 9780821839461, 0821839462, 2007275659

Table of contents :
Contents
Preface
Contents of Part III of Volume Two
Notation and Symbols
Chapter 17. Entropy, $\mu$-invariant, and Finite Time Singularities
Chapter 18. Geometric Tools and Point Picking Methods
Chapter 19. Geometric Properties of $\kappa$-Solutions
Chapter 20. Compactness of the Space of $\kappa$-Solutions
Chapter 21. Perelman's Pseudolocality Theorem
Chapter 22. Tools Used in Proof of Pseudolocality
Chapter 23. Heat Kernel for Static Metrics
Chapter 24. Heat Kernel for Evolving Metrics
Chapter 25. Estimates of the Heat Equation for Evolving Metrics
Chapter 26. Bounds for the Heat Kernel for Evolving Metrics
Appendix G. Elementary Aspects of Metric Geometry
Appendix H. Convex Functions on Riemannian Manifolds
Appendix I. Asymptotic Cones and Sharafutdinov Retraction
Appendix J. Solutions to Selected Exercises
Bibliography
Index

Citation preview

Mathematical Surveys and Monographs Volume 163

The Ricci Flow: Techniques and Applications Part III: Geometric-Analytic Aspects Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni

American Mathematical Society

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http://dx.doi.org/10.1090/surv/163

The Ricci Flow: Techniques and Applications Part III: Geometric-Analytic Aspects

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Mathematical Surveys and Monographs Volume 163

The Ricci Flow: Techniques and Applications Part III: Geometric-Analytic Aspects 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

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EDITORIAL COMMITTEE Jerry L. Bona Ralph L. Cohen, Chair

Michael G. Eastwood J. T. Stafford

Benjamin Sudakov 2010 Mathematics Subject Classification. Primary 53C44, 53C25, 58J35, 35K55, 35K05, 35K08, 35K10, 53C21.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-163

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 a chapter 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. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 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

15 14 13 12 11 10

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Contents Preface What Part III is about Acknowledgments

ix ix x

Contents of Part III of Volume Two

xiii

Notation and Symbols

xvii

Chapter 17. Entropy, µ-invariant, and Finite Time Singularities 1. Compact finite time singularity models are shrinkers 2. Behavior of µ (g, τ ) for τ small 3. Existence of a minimizer for the entropy 4. 1- and 2-loop variation formulas related to RG flow 5. Notes and commentary

1 1 15 23 31 36

Chapter 18. Geometric Tools and Point Picking Methods 1. Estimates for changing distances 2. Spatial point picking methods 3. Space-time point picking with restrictions 4. Necks in manifolds with positive sectional curvature 5. Localized no local collapsing theorem 6. Notes and commentary

39 40 49 57 62 68 76

Chapter 19. Geometric Properties of κ-Solutions 1. Singularity models and κ-solutions 2. The κ-noncollapsed condition 3. Perelman’s κ-solution on the n-sphere 4. Equivalence of 2- and 3-dimensional κ-solutions with and without Harnack 5. Existence of an asymptotic shrinker 6. The κ-gap theorem for 3-dimensional κ-solutions 7. Notes and commentary

79 80 85 93 104 106 116 120

Chapter 20. Compactness of the Space of κ-Solutions 1. ASCR and AVR of κ-solutions 2. Almost κ-solutions 3. The compactness of κ-solutions 4. Derivative estimates and some conjectures

123 124 129 136 149

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vi

CONTENTS

5. Notes and commentary Chapter 21. Perelman’s Pseudolocality Theorem 1. Statement and interpretation of pseudolocality 2. Setting up the proof by contradiction and point picking 3. Local entropies are nontrivial near bad points 4. Contradicting the almost Euclidean logarithmic Sobolev inequality 5. Notes and commentary

154 157 158 166 171 178 179

Chapter 22. Tools Used in Proof of Pseudolocality 183 1. A point picking method 183 2. Heat kernels under Cheeger–Gromov 191
limits 197 3. Upper bound for the local entropy B v dµ 4. Logarithmic Sobolev inequality via the isoperimetric inequality 206 5. Notes and commentary 211 Chapter 23. Heat Kernel for Static Metrics 1. Construction of the parametrix for the heat kernel on a Riemannian manifold 2. Existence of the heat kernel on a closed Riemannian manifold via parametrix 3. Differentiating a convolution with the parametrix 4. Asymptotics of the heat kernel for a static metric 5. Supplementary material: Elementary tools 6. Notes and commentary

215 216 228 238 251 259 263

Chapter 24. Heat Kernel for Evolving Metrics 1. Heat kernel for a time-dependent metric 2. Existence of the heat kernel for a time-dependent metric 3. Aspects of the asymptotics of the heat kernel for a time-dependent metric 4. Characterizing Ricci flow by the asymptotics of the heat kernel 5. Heat kernel on noncompact manifolds 6. Notes and commentary

265 266 271

Chapter 25. Estimates of the Heat Equation for Evolving Metrics 1. Mean value inequality for solutions of heat-type equations with respect to evolving metrics 2. Li–Yau differential Harnack estimate for positive solutions of heat-type equations with respect to evolving metrics 3. Notes and commentary

305

Chapter 26. Bounds for the Heat Kernel for Evolving Metrics 1. Heat kernel for an evolving metric 2. Upper and lower bounds of the heat kernel for an evolving metric

333 333

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278 285 290 303

305 317 331

345

CONTENTS

3. Heat balls and the space-time mean value property 4. Distance-like functions on complete noncompact manifolds 5. Notes and commentary

vii

363 377 386

Appendix G. Elementary Aspects of Metric Geometry 1. Metric spaces and length spaces 2. Aleksandrov spaces with curvature bounded from below 3. Notes and commentary

387 388 401 412

Appendix H. Convex Functions on Riemannian Manifolds 1. Elementary aspects of convex analysis on Euclidean space 2. Connected locally convex subsets in Riemannian manifolds 3. Generalized gradients of convex functions on Riemannian manifolds 4. Integral curves to gradients of concave functions 5. Notes and commentary

413 413 417

Appendix I. Asymptotic Cones and Sharafutdinov Retraction 1. Sharafutdinov retraction theorem 2. The existence of asymptotic cones 3. A monotonicity property of nonnegatively curved manifolds within the injectivity radius 4. Critical point theory and properties of distance spheres 5. Approximate Busemann–Feller theorem 6. Equivalence classes of rays and points at infinity 7. Notes and commentary

457 457 465 468 472 482 487 495

Appendix J.

497

Solutions to Selected Exercises

433 442 456

Bibliography

503

Index

513

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Preface I didn’t have time to write you a short letter, so I wrote you a long one instead. – Samuel Clemens

What Part III is about I’m taking the time for a number of things That weren’t important yesterday. – From “Fixing a Hole” by The Beatles

This is Part III (a.k.a. ∆Rijk
), a sequel to Part I ([40]; a.k.a. Rijk
) ∂ Rijk
) of this volume (Volume Two) on techniques and Part II ([41]; a.k.a. ∂t and applications of the Ricci flow (we shall refer to Volume One ([42]; a.k.a. gij ) as Volume One). In Part I we discussed various geometric topics in Ricci flow such as Ricci solitons, an introduction to the K¨ahler–Ricci flow, Hamilton’s Cheeger– Gromov-type compactness theorem, Perelman’s energy and entropy monotonicity, the foundations of Perelman’s reduced distance function, the reduced volume, applications to the analysis of ancient solutions, and a primer on 3-manifold topology. In Part II we discussed mostly analytic topics in Ricci flow including weak and strong maximum principles for scalar heat-type equations and systems on compact and noncompact manifolds, B¨ohm and Wilking’s classification of closed manifolds with 2-positive curvature operator, Shi’s local derivative estimates, Hamilton’s matrix estimate, and Perelman’s estimate for fundamental solutions of the adjoint heat equation. Here, in Part III, we discuss mostly geometric-analytic topics in Ricci flow. In particular, we discuss properties of Perelman’s entropy functional, point picking methods, aspects of Perelman’s theory of κ-solutions including the κ-gap theorem, compactness theorem, and derivative estimates, Perelman’s pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in previous volumes, we have endeavored, as much as possible, to make the chapters independent of each other. ix Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

x

PREFACE

In Part IV we shall discuss some topics originally slated for Part III such as Hamilton’s classification of nonsingular solutions, the linearized Ricci flow, stability of the Ricci flow, the space-time formulation of the Ricci flow, and Type II singularities from the numerical perspective. Caveat: Many of the chapter numbers of references in Part II to Part III have changed and some of the referred chapters are in Part IV. Acknowledgments Now that your rose is in bloom, a light hits the gloom on the grey. – From “Kiss from a Rose” by Seal

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, Kate Okikiolu, Anton Petrunin, Justin Roberts, Xiaochun Rong, Peter Scott, Peter Topping, Bing Wang, 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. We would like to thank the editors of the Mathematical Surveys and Monographs series. We would like to thank Cristin Zanella 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. During the preparation of this volume, Bennett Chow was partially supported by NSF grants DMS-9971891, DMS-020392, DMS-0354540, and DMS-0505507. David Glickenstein was partially supported by NSF grant DMS 0748283. Christine Guenther was partially supported by the Thomas J. and Joyce Holce Professorship in Science. Jim Isenberg was partially supported by NSF grants PHY-0354659 and PHY-0652903. Dan Knopf was partially supported by NSF grants DMS-0511184, DMS-0505920, and DMS0545984. Peng Lu was partially supported by NSF grant DMS-0405255. Peng Lu was also partially supported by NSF funds for his visits to the UC San Diego mathematics department. Feng Luo was partially supported by NSF grant DMS-0103843. Lei Ni was partially supported by NSF grants DMS-0354540 and DMS-0504792. Bennett Chow and Lei Ni were partially supported by NSF FRG grant DMS-0354540 (joint with Gang Tian). We would like to thank the National Science Foundation, especially the Division of Mathematical Sciences and the Geometric Analysis subdivision. In particular, we would like to thank Christopher Stark, Helena Noronha, Alex

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ACKNOWLEDGMENTS

xi

Freire, and Zhongmin Shen. We are grateful for Drs. Noronha’s and Stark’s encouragement in the early stages of this book project. Bennett Chow would like to thank East China Normal University, the Mathematical Sciences Research Institute in Berkeley, Universit´e de CergyPontoise, and his home institution, University of California at San Diego, for providing a wonderful environment for writing this book. Ben would like to thank his parents, daughters, and friends for their help and encouragement. Ben is indebted to his parents, Yutze and Wanlin, for all of the nurturing, support, and encouragement they have given. Ben is especially grateful to Peng Lu for his persevering collaboration on this project. 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. 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, as well as his parents, Frank and Mary Ann, and his wife, Stephanie. Peng Lu would like to take this opportunity to thank all the people who helped him over the years. 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. He also thanks 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.

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xii

PREFACE

Bennett Chow, UC San Diego and East China Normal University 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] December 11, 2009

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Contents of Part III of Volume Two Well, you know, we’re doing what we can. – From “Revolution” by The Beatles

Chapter 17. Perelman’s entropy W leads to the µ-invariant. We discuss qualitative properties of the µ-invariant such as lower and upper bounds and we give a proof of the fact that limτ →0+ µ (g, τ ) = 0. We also discuss applications of the µ-invariant monotonicity formula. This includes the recent classification by Z.-L. Zhang of compact finite time singularity models as shrinking gradient Ricci solitons. We revisit the proof of the existence of a smooth minimizer for W, providing more details than in Part I, and we also show that when the isometry group acts transitively, the minimizer is not unique for sufficiently small τ . Related to renormalization group considerations, some low-loop calculations are presented. Chapter 18. We discuss some tools used in the study of the Ricci flow including the changing distances estimate for solutions of Ricci flow, point picking methods, rough monotonicity of the size of necks in complete noncompact manifolds with positive sectional curvature, and a local form of the weakened no local collapsing theorem. Chapter 19. With the goal of understanding compactness in higher dimensions, we introduce the notion of ‘κ-solution with Harnack’, which is a variant of Perelman’s notion of κ-solution. In dimensions 2 and 3 we show that κ-solutions with Harnack must have bounded curvature. We also discuss the construction of Perelman’s rotationally symmetric ancient solution on S n , the result that κ-solutions with Harnack must have bounded curvature, the existence of an asymptotic shrinker in a κ-solution (correcting a gap (no pun intended) in Part I), and the κ-gap theorem. Chapter 20. We show that noncompact κ-solutions have asymptotic scalar curvature ratio ASCR = ∞ and asymptotic volume ratio AVR = 0; the latter result does not require the κ-noncollapsed at all scales assumption. We show that solutions which are almost ancient and have bounded nonnegative curvature operator are collapsed at large scales and we obtain a curvature estimate in noncollapsed balls. We prove that the collection of κ-solutions with Harnack is compact modulo scaling. In dimension 3 this is equivalent to Perelman’s compactness theorem and implies scaled derivative of curvature estimates. xiii Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

xiv

CONTENTS OF PART III OF VOLUME TWO

Chapter 21. We discuss Perelman’s pseudolocality theorem. Assuming an initial ball with scalar curvature bounded from below and which is almost Euclidean isoperimetrically, we obtain a curvature estimate in a smaller ball; this estimate gets worse as time approaches the initial time. One may consider this as sort of a pseudolocalization of the curvature doubling time estimate. One of the ideas in the proof is that one can localize the entropy monotonicity formula by multiplying the integrand by a suitable time-dependent cutoff function. In the setting of a proof by contradiction, a main idea is to use point picking methods to locate an infinite sequence of ‘good’ high curvature points and to study a local entropy in their neighborhoods via Perelman’s Harnack-type estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow. Chapter 22. We discuss tools used in the proof of the pseudolocality theorem such as the point picking ‘Claims 1 and 2’, convergence of heat kernels under Cheeger–Gromov convergence, a uniform negative upper bound for the local entropies centered at the well-chosen bad points at time zero, and a sharp form of the logarithmic Sobolev inequality related to the isoperimetric inequality. Chapter 23. We discuss existence and asymptotics for heat kernels with respect to static metrics. We follow the parametrix method of Levi and its Riemannian adaptation by Minakshisundaram and Pleijel. Starting with a good approximation to the heat kernel, we prove the existence of the heat kernel by establishing the convergence of the ‘convolution series’. With this construction we compute some low-order asymptotics for the heat kernel. Chapter 24. We adapt the methods of the previous chapter to study the existence and asymptotics for heat kernels with respect to evolving metrics. We consider aspects of the adjoint heat kernel for evolving metrics related to §9.6 of Perelman’s paper [152]. We also discuss the existence of Dirichlet heat kernels on compact manifolds with boundary and heat kernels on noncompact manifolds with respect to evolving metrics. Chapter 25. We discuss estimates for solutions to the heat equation with respect to evolving metrics including the parabolic mean value property for solutions to heat equations and the Li–Yau differential Harnack estimate for positive solutions to heat equations. Chapter 26. Applying the estimates of the previous chapter, we discuss estimates for heat kernels with respect to evolving metrics including upper and lower bounds and the space-time mean value property. We also discuss the existence of distance-type functions on complete noncompact Riemannian manifolds with bounded gradient and Laplacian. Appendix G. With Perelman’s work, the space-time of a solution of the Ricci flow is given a quasi-length space structure. This geometric structure is foundational in the understanding of singularity formation under the

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CONTENTS OF PART III OF VOLUME TWO

xv

Ricci flow. We discuss notions of (quasi-)metric and (quasi-)length spaces, Gromov–Hausdorff convergence, and Aleksandrov spaces. Appendix H. We discuss convex analysis on Euclidean spaces and on locally convex subsets in Riemannian manifolds. Appendix I. We discuss the points at infinity for nonnegatively curved manifolds, the Sharafutdinov retraction theorem, and some consequences. Appendix J. We provide solutions to some of the exercises in the book.

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Notation and Symbols Confusion never stops, closing walls and ticking clocks. – From “Clocks” by Coldplay

The following is a list of some of the notation and symbols which we use in this book. × multiplication, when a formula does not fit on one line ∇ covariant derivative ‘set gradient’ of the distance function to a point ∇
r
heat operator ∗ adjoint heat operator
 defined to be equal to · dot product or multiplication ∇∇f Hessian of f
dual vector field to the 1-form α α ale asymptotically locally Euclidean Area area of a surface or volume of a hypersurface ASCR asymptotic scalar curvature ratio AVR asymptotic volume ratio B (p, r) ball of radius r centered at p bounded curvature bounded sectional curvature (for time-dependent metrics, the bound may depend on time) tangent cone at V of a convex set J ⊂ Rk CV J const constant d+ d− d+ d− dt , dt , dt , dt

d dGH dµ dµE dσ or dA ∆, ∆L , ∆d

various Dini time derivatives distance Gromov–Hausdorff distance volume form Euclidean volume form volume form on boundary or hypersurface Laplacian, Lichnerowicz Laplacian, Hodge Laplacian xvii

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xviii

NOTATION AND SYMBOLS

diam div En Er (x, t) exp F Γkij g (X, Y ) = X, Y  g (t) g∞ or g∞ (t) h or II H HV J for V ∈ ∂J Hess f I id int inj Isom IVP J λ L lhs log L  L L Cut L exp LI L JV L (v, X) µ (M, gˆ) Met MVP

diameter divergence Rn with the flat Euclidean metric heat ball of radius r based at (x, t) exponential map Perelman’s energy functional Christoffel symbols metric or inner product time-dependent metric, e.g., solution of the Ricci flow limit Riemannian metric or solution of Ricci flow second fundamental form mean curvature set of closed half-spaces H containing J ⊂ Rk with V ∈ ∂H Hessian of f (same as ∇∇f ) a time interval for the Ricci flow identity interior injectivity radius group of isometries of a Riemannian manifold initial-value problem Jacobian of the exponential map λ-invariant length left-hand side natural logarithm 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 µ-invariant static Riemannian manifold space of Riemannian metrics on a manifold mean value property

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NOTATION AND SYMBOLS

Mn,κ MHarn n,κ ν nωn ωn ode Pijk pco pde Rijk Rjk Rjk RF RG flow rhs R, Rc, Rm Rm# Rn SO (n, R) SV J for V ∈ ∂J sect Sn supp τ (t) Tx M Tx∗ M tr or trace V˜

xix

collection of n-dimensional κ-solutions n-dimensional κ-solutions with Harnack ν-invariant or unit outward normal volume of the unit Euclidean (n − 1)-sphere volume of the unit Euclidean n-ball ordinary differential equation the symmetric 3-tensor ∇i Rjk − ∇j Rik positive curvature operator partial differential equation
Rm gm (opposite of Hamilton’s convention)
m i ijk
i i Rijk = i, g Rijk (components of Ricci) a symmetric 2-tensor (Rjk = Rjk is a special case) Ricci flow renormalization group flow right-hand side scalar, Ricci, and Riemann curvature tensors the quadratic Rm # Rm n-dimensional Euclidean space real orthogonal group set of support functions of J ⊂ Rk at V sectional curvature unit radius n-dimensional sphere support of a function function satisfying dτ dt = −1 tangent space of M at x cotangent space of M at x trace reduced volume

Vˆ∞ V Vol W Wlin W k,p

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

k,p Wloc W W⊥ WMP

space of functions locally in W k,p tangential component of the vector W normal component of the vector W weak maximum principle

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http://dx.doi.org/10.1090/surv/163/01

CHAPTER 17

Entropy, µ-invariant, and Finite Time Singularities I’ll tip my hat to the new constitution. – From “Won’t Get Fooled Again” by The Who

Monotonicity formulas may be used to understand the qualitative behavior of solutions of the Ricci flow. As an example, in this chapter we consider the µ-invariant monotonicity formula and its applications to singularity analysis. In §1 we discuss lower and upper bounds for the µ-invariant. As an application, there is a lower bound for the volume of solutions g (t) of the Ricci flow with nonpositive λ-invariant. This implies that the corresponding finite time singularity models are noncompact in this case. As a further application, we discuss the classification of compact finite time singularity models as shrinking gradient Ricci solitons with no assumption on the sign of the λ-invariant. In §2 we prove the fact that limτ →0+ µ (g, τ ) = 0. This result was stated as Lemma 6.33(ii) in Part I but was not proved there. As an application we show that, for a closed Riemannian manifold on which the isometry group acts transitively, the minimizer for W is not unique for sufficiently small τ . In §3 we revisit the proof of the existence of a smooth minimizer for W while completing some additional details not discussed earlier in this book series. One may hope to extend Perelman’s energy and entropy monotonicity formulas. In §4 we discuss formulas relating Perelman’s energy F , the linear trace Harnack quadratic,
Hamilton’s matrix quadratic, the 2-tensor Rik
m Rjk
m , and the functional M |Rm|2 e−f dµ. Throughout this chapter we assume that Mn is a closed manifold unless otherwise indicated. 1. Compact finite time singularity models are shrinkers In this section and the next we discuss properties of the µ-invariant of a metric g at a scale τ > 0. This invariant is the infimum of Perelman’s entropy functional W (g, f, τ ), under a constraint, considered in Chapter 6 of Part I. In this section we present the results and proofs of Z.-L. Zhang on bounds for the µ-invariant and their geometric application to the classification of compact finite time singularity models. 1 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

2

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

1.1. Perelman’s entropy and its associated invariants. In this subsection we recall some basic facts regarding Perelman’s energy and entropy functionals, including the logarithmic Sobolev inequality, which shall be used in this chapter. 1.1.1. Energy, entropy, and their minimizers. Let g be a C ∞ Riemannian metric on a closed manifold Mn , let f : M → R be a C ∞ function, and let τ ∈ (0, ∞). Perelman’s energy functional is defined by (see (5.1) in Part I)     (R + ∆f ) e−f dµ. R + |∇f |2 e−f dµ = (17.1) F (g, f )  M

We may rewrite F as (17.2) where v  (17.3)

M



F (g, f ) =

M

  Rv 2 + 4 |∇v|2 dµ  G (g, v) ,

e−f /2 .

The associated λ-invariant is (see (5.45) in Part I)    ∞ −f e dµ = 1 λ (g)  inf F (g, f ) : f ∈ C (M), M    2 v dµ = 1 . = inf G(g, v) : M

C∞

minimizer f0 of F (g, f ) subject to the constraint There exists a unique
−f dµ = 1. Moreover, f satisfies the Euler–Lagrange equation (see e 0 M Lemma 5.23 in Part I) 2∆f0 − |∇f0 |2 + R = λ (g) .

(17.4)

Perelman’s entropy functional W is given by (see (6.1) in Part I)      τ R + |∇f |2 + f − n u dµ, (17.5) W(g, f, τ )  M

where u  (4πτ )−n/2 e−f  w2 .

(17.6) We may rewrite W as  W (g, f, τ ) = (17.7)

   τ Rw2 + 4 |∇w|2 dµ



M − log w2 + n2 log(4πτ ) + n w2  K (g, w, τ ) . 

Recall that the associated µ-invariant is defined by (see (6.49) in Part I)    ∞ (17.8) u dµ = 1 µ(g, τ )  inf W(g, f, τ ) : f ∈ C (M) , M    2 (17.9) w dµ = 1 . = inf K (g, w, τ ) : M

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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

The ν-invariant is (17.10)

3

ν(g)  inf µ(g, τ ) : τ ∈ R+ .

1.1.2. Monotonicity of Perelman’s entropy. Under the coupled Ricci flow system ∂ gij = −2Rij , (17.11a) ∂t n ∂f (17.11b) = −∆f + |∇f |2 − R + , ∂t 2τ dτ (17.11c) = −1, dt we have 2     1 d W(g(t), f (t), τ (t)) = 2τ Rc +∇∇f − g  u dµ ≥ 0. (17.12) dt 2τ M Indeed, let

    v  τ R + 2∆f − |∇f |2 + f − n u,
which satisfies W (g, f, τ ) = M v dµ. We have (see Lemma 6.8 in Part I)    1 2 ∗  (17.13)
v = −2τ Rc +∇∇f − g  u, 2τ

∂ − ∆ + R is the adjoint heat operator. This implies (17.12) where
∗ = − ∂t since     ∂ dW = − R v dµ = −
∗ v dµ. dt ∂t M M We remark that formula (17.13) is central to the proof of Perelman’s differential Harnack estimate (see Chapter 16 in Part II).  The functional W (g, · , τ ) is bounded from below under the constraint M

u dµ = 1 and there exists a smooth minimizer fτ , which satisfies the

equation (17.14)

  τ 2∆fτ − |∇fτ |2 + R + fτ − n = µ (g, τ )

(see Proposition 17.24 below). In terms of wτ = (4πτ )−n/4 e−fτ /2 , this is (17.15)  n log(4πτ ) + n wτ = µ (g, τ ) wτ . τ (−4∆wτ + Rwτ ) − wτ log wτ2 − 2 1.1.3. Logarithmic Sobolev inequality. The logarithmic Sobolev inequality is intimately tied to Perelman’s entropy functional due to their related forms. In our discussion of this, we shall assume n ≥ 3; we leave it to the reader to verify that these results carry over to the case n = 2 with only minor adjustments. The following is given as Lemma 6.36 in Part I.

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4

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Lemma 17.1 (Logarithmic Sobolev inequality). Let (Mn , g) be a closed Riemannian manifold.
For any a > 0, there exists a constant C (a, g) such that if ϕ > 0 satisfies M ϕ2 dµg = 1, then   ϕ2 log ϕdµg ≤ a |∇ϕ|2g dµg + C (a, g) , (17.16) M

M

where1 C (a, g) = a Vol (g)−2/n +

(17.17)

n2 . 4ae2 Cs (M, g)

Here Cs (M, g) denotes the L2 Sobolev constant, which we define to be the best (largest) positive constant such that (see Lemma 2 in [114]) (17.18)   n−2   n 2n 2 2 −n n−2 |∇ϕ|g dµg ≥ Cs (M, g) ϕ dµg − Vol (g) ϕ2 dµg M

M

M

C∞

for any function ϕ on M. We have the following elementary properties. Lemma 17.2 (Sobolev constants under scaling the metric). Let (Mn , g) be a closed Riemannian manifold. (i) The L2 Sobolev constant has the property that for any λ > 0,

Cs M, λ2 g = Cs (M, g) . (ii) The logarithmic Sobolev constant has the property that for any a > 0 and λ ≥ 1,

C a, λ2 g ≤ C (a, g) . Proof. (i) Let g˜ = λ2 g and ϕ˜ = λ−n/2 ϕ. Then dµg˜ = λn dµg , ϕ˜2 dµg˜ = g , and   2 ˜  |∇ϕ|2g dµg ∇ϕ˜ dµg˜ = λ−2

ϕ2 dµ 

M

g˜

M

−2

≥λ



Cs (M, g) 

= Cs (M, g)

ϕ M

ϕ˜ M

2n n−2

2n n−2

 n−2 n

dµg  n−2 n

dµg˜

−2

−λ

2 −n



Vol (g)

− Vol (˜ g )− n 2

M

 M

ϕ2 dµg

ϕ˜2 dµg˜.

From this we can easily deduce that

(17.19) Cs M, λ2 g = Cs (M, g) , i.e., the L2 Sobolev constant is invariant under scaling the metric. 1 In (17.17) we correct formula (6.67) in Part I, where the n2 /4 factor originally e . appeared in the denominator; after (6.66) it should read cn = 2n

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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

5

(ii) Hence



−2/n + C a, λ2 g = a Vol λ2 g

n2 4ae2 Cs (M, λ2 g)

≤ C (a, g) if λ ≥ 1.




1.2. Lower and upper bounds for the µ-invariant. In this subsection we prove lower and upper bounds for the µ-invariant in terms of τ and certain geometric invariants of (Mn , g). 1.2.1. Upper bounds for µ. Taking f = c to be constant in (17.5), we obtain the following elementary upper bound for µ: n (17.20) µ(g, τ ) ≤ τ Ravg + log Vol (g) − log(4πτ ) − n, 2 where Ravg is the average scalar curvature of g. On the other hand, we may choose f in terms of the minimizer of F . The following is inequality (2) in Lemma 2.1 of [197]. Lemma 17.3 (Upper bound for µ in terms of λ, Vol, τ , and n). For any closed Riemannian manifold (Mn , g) and τ > 0 n 1 µ(g, τ ) ≤ τ λ (g) + Vol (g) − log (4πτ ) − n. e 2 

w2 dµ = 1, by (17.7) and log w2 w2 ≥ − 1e , Proof. For any w with

(17.21)

M

we have W (g, f, τ ) = τ

  M

2

Rw + 4 |∇w| 2

≤ τ G (g, w) +



 dµ −



n log w2 w2 dµ − log(4πτ ) − n 2 M

n 1 Vol (g) − log(4πτ ) − n, e 2

where G is defined by (17.2). Choosing w0  (4πτ )−n/4 e−f0 /2 to be the minimizer of the functional G (g, w), we conclude that µ(g, τ ) ≤ W (g, f0 , τ ) ≤ τ λ (g) +

n 1 Vol (g) − log(4πτ ) − n. e 2


When λ (g) ≤ 0, one may apply the scaling property of µ to obtain the following, which is Corollary 2.3 of [197]. Corollary 17.4 (Upper bound for µ when λ ≤ 0). If λ (g) ≤ 0, then (17.22)

µ (g, τ ) ≤ log Vol (g) −

1 n log (4πτ ) − n + . 2 e

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6

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Proof. By the scaling invariance of µ (see property (iii) on p. 236 of Part I) and (17.21), we have for any c ∈ (0, ∞), n 1 µ(g, τ ) = µ(cg, cτ ) ≤ cτ λ (cg) + Vol (cg) − log (4πcτ ) − n. e 2 Taking c = Vol (g)−2/n , we obtain 1 n µ(g, τ ) ≤ cτ λ (cg) + − log (4πτ ) + log Vol (g) − n e 2 −1 and (17.22) follows from λ (cg) = c λ (g) ≤ 0.




As a consequence of Corollary 17.4 we have that if λ (g) ≤ 0, then limτ →∞ µ (g, τ ) = −∞. This improves Exercise 6.32 in Part I. 1.2.2. Lower bounds for µ. Now we consider lower bounds for µ using the logarithmic Sobolev inequality; we have the following. Lemma 17.5 (Lower bound for µ). Let (Mn , g) be a closed Riemannian manifold and let τ > 0. We have n (17.23) µ (g, τ ) ≥ τ Rmin (g) − 2C (2τ, g) − log(4πτ ) − n, 2 where Rmin (g)  minx∈M R (x) and the constant C (2τ, g) is given by (17.17).
Proof. By Lemma 17.1, for any w ≥ 0 with M w2 dµg = 1 we have   w2 log w dµ ≤ 2τ |∇w|2 dµ + C (2τ, g) . M

M

Substituting this into (17.7), we obtain    n τ Rw2 + 4τ |∇w|2 dµ − log(4πτ ) − n K (g, w, τ ) = 2 M  w2 log w dµ −2 M

≥ τ Rmin (g) − 2C (2τ, g) −

 since M

n log(4πτ ) − n 2

τ Rw2 dµ ≥ τ Rmin (g). Taking the infimum over w, we obtain the


desired inequality (17.23).

Let Cs (g) denote the L2 Sobolev constant: the smallest number such that (17.24)    1/2 2 2 ≤ Cs (g) ϕ W 1,2 (g)  Cs (g) |∇ϕ| + ϕ dµ

ϕ n−2 2n L

for all ϕ ∈

(g)

M

C ∞ (M).

By (17.18), we have   2 1 max Vol (g)− n , 1 . Cs (g)2 ≤ Cs (M, g)

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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

7

The following is inequality (3) in Lemma 2.1 of [197] (compare with (6.63) in Part I); again we assume n ≥ 3 for simplicity. Lemma 17.6 (Lower bound for the µ-invariant). If τ ≥ n8 , then  n n n λ (g)− log(4πτ )−n−n log Cs (g)+ Rmin (g) . (17.25) µ (g, τ ) ≥ τ − 8 2 8 Remark 17.7. (1) Note that estimate (17.25) is not true for sufficiently small τ . One reason is because the limit as τ → 0+ of the rhs of (17.25) is equal to +∞, contradicting (17.47) below. (2) We see from (17.25) that if λ (g) > 0, then limτ →∞ µ (g, τ ) = ∞; see also Lemma 6.30 in Part I. 2 2
log w w dµ on the rhs of (17.7). Proof. We consider the term − M  w2 dµ = 1, Jensen’s inequality says that if φ : R → R For any w such that M

is convex and f ∈ L1 w4/n g , then    2 2 (φ ◦ f ) w dµ ≥ φ f w dµ . M

M

4 n−2

and φ (u) = − log u, we have In particular, taking f = w    4  2

n−2 2 w log w dµ = − log w n−2 w2 dµ − 2 M M   2n n−2 n−2 log w dµ ≥− 2 M = −n log w n−2 2n L (g)   ≥ −n log Cs (g) w W 1,2 (g)    n 2 (17.26) |∇w| dµ , = −n log Cs (g) − log 1 + 2 M where we used the L2 Sobolev inequality (17.24). Since log (1 + x) ≤ x for x ≥ 0, we have   2

n 2 w log w dµ ≥ −n log Cs (g) − |∇w|2 dµ. (17.27) − 2 M M Combining this with (17.7), we have n W (g, f, τ ) ≥ τ G (g, w) − log(4πτ ) − n − n log Cs (g) 2  n |∇w|2 dµ. − 2 M

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17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

On the other hand,   2 4 |∇w| dµ ≤ M

M

  4 |∇w|2 + (R − Rmin (g)) w2 dµ

= G (g, w) − Rmin (g) ,

so that

 n n n G (g, w) − log(4πτ ) − n − n log Cs (g) + Rmin (g) . W (g, f, τ ) ≥ τ − 8 2 8 Choosing fτ to be the constrained minimizer of W (g, · , τ ), we have  n n n G (g, wτ ) − log(4πτ ) − n − n log Cs (g) + Rmin (g) , µ (g, τ ) ≥ τ − 8 2 8 n −n/4 −f /2 τ e . If τ ≥ 8 , then we obtain (17.25). This where wτ  (4πτ ) completes the proof of the lemma.
1.3. Volume lower bound for Ricci flow solutions with λ ≤ 0. As a geometric application of the upper and lower bounds for µ, we have the following, which is Lemma 3.1 in [197]. Lemma 17.8 (Lower bound for the volume of a solution when λ ≤ 0). If (Mn , g (t)), t ∈ [0, T ), is a solution to the Ricci flow on a closed manifold with λ (g (t)) ≤ 0 for all t ∈ [0, T ), then there exists c1 , c2 ∈ (0, ∞) depending only on g (0) such that (17.28)

Vol (g (t)) ≥ c1 e−c2 t

for all t ∈ [0, T ). Proof. By taking g = g (0) and τ = n8 + t in (17.25) and by Perelman’s µ-invariant monotonicity formula (see Lemma 6.26 in Part I), we have    n n ≥ µ g (0) , + t µ g (t) , 8 8  n  n +t ≥ tλ (g (0)) − log 4π 2 8 n − n − n log Cs (g (0)) + Rmin (g (0)) . 8 On the other hand, since λ (g (t)) ≤ 0, by (17.22) we have  πn   n n ≤ log Vol (g (t)) − log − n + 1. µ g (t) , 8 2 2 Hence  n  n + t + c, log Vol (g (t)) ≥ tλ (g (0)) − log 4π 2 8 where  πn  n n − 1. c  −n log Cs (g (0)) + Rmin (g (0)) + log 8 2 2 We conclude that −n/2  n +t etλ(g(0)) . Vol (g (t)) ≥ ec 4π 8 The lemma follows easily.


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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

9

Remark 17.9. For an elementary upper bound of the volume of a solution when λ > 0, see the notes and commentary at the end of this chapter. Recall that a complete (finite time) singularity model (Mn∞ , g∞ (t)), t ∈ (−∞, 0], is obtained from taking the limit of rescalings of a finite time singular solution of the Ricci flow (Mn , g (t)), t ∈ [0, T ), on a closed manifold (see §1 in Chapter 19 in this volume and Chapter 8 of [45]). The volume lower bound has the following consequence for singularity models associated to solutions with λ ≤ 0; this is Corollary 3.2 in [197]. Lemma 17.10 (Finite time singularity models are noncompact when λ ≤ 0). If (Mn , g (t)), t ∈ [0, T ), where T < ∞, is a singular solution to the Ricci flow on a closed manifold with λ (g (t)) ≤ 0 for all t ∈ [0, T ), then any corresponding singularity model is noncompact. Proof. This follows since by (17.28) there exists c > 0 such that Vol (g (t)) ≥ c for all t ∈ [0, T ), whereas for any blow-up sequence the dilation factors tend to infinity. suppose that ti T and  More explicitly,  pi ∈ M are such that Ki  Rm g(ti ) (pi ) → ∞ and suppose that the sequence (Mn , gi (t), pi ), where   t , gi (t)  Ki · g ti + Ki converges to a complete ancient solution (Mn∞ , g∞ (t), p∞ ) to the Ricci flow in the sense of C ∞ Cheeger–Gromov convergence. Then n/2

lim Vol (gi (0)) = lim Ki

i→∞

i→∞

Vol (g (ti ))

=∞ since Vol (g (ti )) ≥ c > 0, independent of i. Now assume M∞ is compact. Then lim Vol (gi (0)) = Vol (g∞ (0)) < ∞,

i→∞

which is a contradiction.




Remark 17.11 (Noncompact singularity models have infinite volume). For any finite time noncompact singularity model (Mn∞ , g∞ (t)) with bounded curvature we have Vol (g∞ (t)) = ∞. This is because for each t there exists κ > 0 such that Volg∞ (t) Bg∞ (t) (x, 1) ≥ κ for all x ∈ M∞ (by Perelman’s no local collapsing theorem). Lemma 17.10 implies Corollary 17.12 (Compact singularity model implies λ > 0). Given a finite time singular solution (Mn , g (t)), t ∈ [0, T ), of the Ricci flow, if some associated singularity model is compact, then λ (g (t0 )) > 0 for some t0 ∈ [0, T ); by the λ-monotonicity formula (see Lemma 5.25 in Part I ) we then have λ (g (t)) > 0 for all t ∈ [t0 , T ).

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17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

On the other hand, it is possible for a finite time singular solution on a closed manifold that λ (g (t)) > 0 for all t ∈ [0, T ) and that all singularity models are noncompact. Such an example on S n is given by Angenent and one of the authors [7], where a finite time neckpinch (singularity model is S n−1 × R) is exhibited for a class of rotationally symmetric solutions with R (g (t)) > 0 (which implies λ (g (t)) > 0). 1.4. Classification of compact finite time singularity models. In this subsection we discuss the following application of bounds for the µ-invariant to the classification of compact finite time singularity models as shrinking gradient Ricci solitons by Z.-L. Zhang (see Theorem 1.1 in [197]). In the a priori special case of singularity models of Type I singular solutions, ˇ sum [169]. this result was proved earlier by Seˇ Theorem 17.13 (Compact finite time singularity models are shrinkers). If (Mn∞ , g∞ (t)), t ∈ (−∞, 0], is a finite time singularity model, where M∞ is a closed manifold, then g∞ (t) is a shrinking gradient Ricci soliton. Proof. Step 1. limt→T ν (g (t)) exists. By assumption, there exists a singular solution to the Ricci flow on a closed manifold (Mn , g (t)), t ∈ [0, T ), where T < ∞, and there exists a sequence (xi , ti ) with ti → T such that

n → (Mn∞ , g∞ (t)) (17.29) M , Qi g ti + Q−1 i t in the sense of C ∞ Cheeger–Gromov convergence for t ∈ (−∞, 0] and for Qi  |Rm| (xi , ti ) → ∞. By Corollary 17.12, we may assume by translating time that λ (g (t)) > 0 for all t ≥ 0. By (17.30)

−∞ < ν (g (t)) < 0

(see Remark 17.7(2) above and (17.47) and (17.42) below) and the monotonicity of ν (g (t)) (see Lemma 6.35(1) in Part I), we have that (17.31)

νT  lim ν (g (t)) ∈ (−∞, 0] t→T

exists. Step 2. λ (g∞ (t)) > 0. Since M∞ is compact, M∞ is diffeomorphic to M and there exist diffeomorphisms ϕi : M∞ → M such that

→ g∞ (t) ϕ∗i Qi g ti + Q−1 i t converges pointwise in C k on M∞ × [−k, 0] for each k ∈ N. Hence, by Lemma 5.24 in Part I,



t λ (g∞ (t)) = lim λ ϕ∗i Qi g ti + Q−1 i i→∞

−1 = lim Q−1 i λ g ti + Qi t i→∞

≥ 0.

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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

11

We claim that (17.32)

λ (g∞ (t)) > 0

for all t ∈ (−∞, 0]. To prove this, suppose by contradiction that λ (g∞ (t )) = 0 for some  t ∈ (−∞, 0]. Then by the λ-monotonicity formula, λ (g∞ (t)) ≡ 0 for all t ∈ (−∞, t ]. This implies that g∞ (t) is a steady gradient Ricci soliton for t ∈ (−∞, t ] (see Lemma 5.28 in Part I). Since M∞ is compact, g∞ (t) is Ricci flat for t ∈ (−∞, 0] (see Proposition 1.13 in Part I; we also use uniqueness to extend to the whole time interval). This contradicts the claim that any compact finite time singularity model cannot be Ricci flat. To see this claim, recall that (see Theorem 6.74 in Part I) any finite time singularity model is κ-noncollapsed at all scales for some κ > 0 in the sense that if Bg∞ (t) (x, r), r ∈ (0, ∞), is a metric ball such that Rg∞ (t) ≤ r−2

for all y ∈ Bg∞ (t) (x, r),

then Volg∞ (t) B(x, r) ≥ κrn . Since Rcg∞ (t) ≡ 0 on M∞ , this implies Volg∞ (t) (M∞ ) ≥ κrn for all r ∈ (0, ∞) and t ∈ (−∞, 0], which is a contradiction since the lhs is finite. This completes the proof of (17.32). Step 3. g∞ (t) is a shrinker. Now by (17.29) we have (we justify the first equality in Lemma 17.14 below)



t ν (g∞ (t)) = lim ν ϕ∗i Qi g ti + Q−1 i i→∞

= lim ν g ti + Q−1 i t i→∞

= lim ν (g (t)) t→T

= νT for all t ∈ (−∞, 0], where we used the fact that ν(cg) = ν(g) for any c ∈ (0, ∞). Since ν (g∞ (t)) is identically a constant and λ (g∞ (t)) > 0, by the equality case of the ν-monotonicity formula (see Lemma 6.35(2) in Part
I), we conclude that g∞ (t) is a shrinking gradient Ricci soliton. The following fact is used in the proof above. C∞

Lemma 17.14. If Mn∞ is a closed manifold and gi → g∞ pointwise in on M∞ , where λ (g∞ ) > 0, then

(17.33)

ν (g∞ ) = lim ν (gi ) . i→∞

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17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Proof. Step 1. ν (gi ) is bounded above by a negative constant. By (17.42) below there exists τ¯ > 0 such that µ (g∞ , τ¯) < 0. On the other hand, by Lemma 17.15 below, lim µ (gi , τ¯) = µ (g∞ , τ¯) ,

i→∞

so that

1 ν (gi ) ≤ µ (gi , τ¯) ≤ µ (g∞ , τ¯) < 0 2 for i sufficiently large. Thus (17.34)

ν (gi ) ≤ −ε0

for all i ∈ N ∪ {∞} and some ε0 > 0. Step 2. Properties of µ (gi , τ ). (i) By Lemma 17.15 again, we have for any C > 1, µ (gi , τ ) → µ (g∞ , τ )   uniformly with respect to τ ∈ C −1 , C . (ii) Since λ (gi ) → λ (g∞ ) > 0 and Cs (gi ) and R min (gi ) are uniformly bounded, Lemma 17.6 implies that there exists C1 ∈ n8 , ∞ such that (17.35)

(17.36)

µ (gi , τ ) ≥ 0

for all τ ≥ C1 and i ∈ N ∪ {∞}. (iii) By the proof of Proposition 17.20 below, we have that for any sequence τi → 0 there exists a subsequence such that lim µ (gi , τi ) = 0.

i→∞

This implies that for any ε > 0 there exists τ (ε) > 0 such that (17.37)

µ (gi , τ ) ≥ −ε

for all i ∈ N and τ ∈ (0, τ (ε)]. Step 3. Completion of the proof. Equation (17.33) now follows from combining (17.34), (17.35), (17.36), and (17.37).
Finally, we give the proof of Lemma 17.15 (Continuous dependence of µ (g, τ ) on g). For any n ≥ 2, C < ∞, and ε > 0, there exists δ > 0 such that if Mn is a closed manifold and if g and g˜ are Riemannian metrics such that g ) ≤ C, (1) Ravg (˜ (2) Vol (˜ g ) ≤ C, (3) |Rg − Rg˜| ≤ δ, (4) |g − g˜|g˜ ≤ δ, then µ (g, τ ) − µ (˜ g, τ ) ≤ ε  −1  for τ ∈ C , C .

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1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS

13

 
Proof. Given τ ∈ C −1 , C , let w ˜ with M w ˜ 2 dµg˜ = 1 be a minimizer of the entropy K (˜ g , · , τ ) in (17.7). Note that by (17.20) and assumptions (1) and (2), µ (˜ g , τ ) = K (˜ g , w, ˜ τ) g ) + log Vol (˜ g) + ≤ τ Ravg (˜ ≤ const (n, C)





n log C 2

for τ ∈ C −1 , C . Define c ∈ R+ so that  (cw) ˜ 2 dµg = 1 ; M

we may make c arbitrarily close to 1 by choosing δ sufficiently small. We have µ (g, τ ) ≤ K (g, cw, ˜ τ)         ˜ 2 + 4 |∇ (cw)| ˜ 2g − log (cw) τ Rg (cw) ˜ 2 (cw) ˜ 2 dµg = M

n log(4πτ ) − n 2 

w ˜ 2 c2 Rg dµg − Rg˜dµg˜ ≤ µ (˜ g, τ ) + τ   M  ˜ 2g dµg − |∇w| ˜ 2g˜ dµg˜ c2 |∇w| + 4τ   M 2 2

2 2 2 w ˜ log w ˜ w ˜ 2 dµg dµg˜ − c dµg − c log c + −

M

M

since K (˜ g , w, ˜ τ ) = µ (˜ g , τ ). Thus for any ε > 0, by taking δ sufficiently small in assumptions (3) and (4) and by making c sufficiently close enough to 1, we obtain µ (g, τ ) − µ (˜ g , τ ) ≤ ε. Here we used the fact that the logarithmic Sobolev inequality implies that   2

|∇w| ˜ 2g˜ dµg˜ and w ˜ 2 log w ˜ dµg˜ M

M

are bounded by µ (˜ g , τ ) + const (n, C), which in turn is uniformly bounded (see the proof of Lemma 6.24 in Part I or (17.58) below).
In dimension 3 any shrinking gradient Ricci soliton on a closed 3-manifold is a constant positive sectional curvature solution (see Theorem 9.79 in [45] for example; note that compact quotients of S 2 ×R cannot be κ-noncollapsed at scales), so that we have the following. Corollary 17.16 (Singularity models on closed 3-manifolds are round). If M3∞ , g∞ (t) , t ∈ (−∞, 0], is a finite time singularity model on a closed 3-manifold, then g∞ (t) is a shrinking spherical space form.

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14

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

ˇ sum [169] considers immortal solutions g (t) to the ‘Ricci flow with Seˇ cosmological constant 1’ ∂ g = −2 Rc +g ∂t

(17.38) with

|Rm| ≤ C

(17.39)

diam ≤ C

and



on M × [0, ∞), where C < ∞. Making the change of time variable t tˆ 



− ln 1 − tˆ , i.e., tˆ(t)  1−e−t , and rescaling the solution by defining g tˆ 



1 − tˆ g t tˆ , we have ∂  g = −2Rc ∂ tˆ on M × [0, 1). The conditions (17.39) correspond to the Type I condition   C    ˆ

Rm t ≤ 1 − tˆ and the diameter estimate
≤C diam

(17.40) This implies (17.41)

 1 − tˆ.

     ˆ

max Rm  t →∞ M

as tˆ → 1.

For if the (Ricci) curvature were uniformly bounded, then the diameter could not tend to zero as tˆ → 1.2 Moreover, we then also have   1    ˆ

max Rm  t ≥ M 8 1 − tˆ (see Lemma 8.19 in Volume One and Lemma 8.7 in [45]). In turn, if one does a Type I rescaling by defining



1 g tˆi + 1 − tˆi t˜ g˜i t˜ = 1 − tˆi  

   ˜

 for some tˆi → 1, then one obtains uniform bounds for Rm i  t and diami t˜ for t˜ ≤ 0. Problem 17.17. Show that if (Mn , g (t)), t ∈ [0, T ), is a finite time singular solution forming a singularity model on a closed manifold (which then must be diffeomorphic to M), then g (t) is Type I. 2

This implies





ˆ  max
Rm
ti → ∞ M

for some tˆi → 1,

from which (17.41) follows.

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2. BEHAVIOR OF µ (g, τ ) FOR τ SMALL

15

One could conceivably have the strange situation of a Type IIa singular solution which forms a Type I singularity model which is a compact shrinking gradient Ricci soliton. Certainly, one can have a Type IIa singular solution which forms a Type I singularity model which is a noncompact shrinking gradient Ricci soliton as evidenced by a degenerate neckpinch which includes the shrinking round cylinder as one of its singularity models. Mini-Problem 17.18 (Compact factors of singularity models are shrinkers). Show that if the universal cover of a (finite time) singularity model splits as (N m , h (t)) × Rn−m , where N is compact, then (N m , h (t)) is a shrinking gradient Ricci soliton. 2. Behavior of µ (g, τ ) for τ small In this section we present a detailed discussion of the limiting behavior of the µ-invariant as τ tends to 0. As a consequence, we shall show that for a closed Riemannian manifold on which the isometry group acts transitively, the minimizer fτ of W (g, · , τ ) is not unique for τ sufficiently small. 2.1. Behavior of µ (g, τ ) for τ small. In the next lemma and proposition we give a belated proof of Lemma 6.33(i), (ii) in Part I regarding the behavior of µ (g, τ ) for τ sufficiently small (this is a result of Perelman; see §3.1 of [152]). Lemma 17.19 (µ (g, τ ) is negative for τ small). If (Mn , g) is a closed Riemannian manifold, then there exists τ¯ > 0 such that (17.42)

µ (g, τ ) < 0

for all τ ∈ (0, τ¯).

Proof. Since M is closed, by the short time existence theorem, there is a τ¯ > 0 such that a (unique) solution g (t) to the Ricci flow with g (0) = g exists for t ∈ [0, τ¯]. Let τ (t)  τ¯ − t and x0 ∈ M and consider the corresponding fundamental solution u (x, t)  (4πτ (t))−n/2 e−f (x,t) ,

x ∈ M, t ∈ [0, τ¯),

to the adjoint heat equation ∂u = −∆g(t) u + Rg(t) u ∂t τ ) = 0). centered at (x0 , τ¯) (i.e., limt¯τ u (·, t) = δx0 ; note that τ (¯ As Perelman says in §3.1 of [152] and as we have seen in Chapter 16 of Part II (where we discussed Perelman’s differential Harnack estimate v ≤ 0), we have (17.43)

lim W (g (t) , f (t) , τ (t)) = 0.

t¯ τ

Hence, by the monotonicity of the entropy functional, (17.44) µ (g, τ¯) = µ (g, τ (0)) ≤ W (g (0) ,f (0) ,τ (0)) ≤ lim W (g (t) ,f (t) ,τ (t)) = 0. t¯ τ

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16

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

We now rule out µ (g, τ¯) = 0, from which the lemma follows. Suppose µ (g, τ¯) = 0. Since W is monotone, we then have W (g (t) , f (t) , τ (t)) = µ (g (t) , τ (t)) ≡ 0 for all t ∈ [0, τ¯). Hence, by (17.12) we have  g  (t) ≡ 0 Rc +∇∇f − 2τ for t ∈ [0, τ¯], so that g (t) is a shrinking gradient Ricci soliton with singular time t = τ¯. In particular, τ (t) max |Rm (g (t))| ≡ const M

for t ∈ [0, τ¯]. On the other hand, since g (¯ τ ) is a smooth metric (and recall that τ (¯ τ ) = 0), we conclude that |Rm (g (t))| ≡ 0 for t ∈ [0, τ¯]. We obtain a contradiction because there are no flat shrinking Ricci solitons on closed manifolds.
Recall that Gross’s Euclidean logarithmic Sobolev inequality (see Corollary 6.40 in Part I or Theorem 22.15 below) says that if f0 : Rn → R is a smooth function with  (2π)−n/2 e−f0 dµRn = 1, Rn

then





(17.45) Rn

 1 2 |∇f0 | + f0 − n (2π)−n/2 e−f0 dµRn ≥ 0, 2 2

0| for some x0 ∈ Rn . That is, for Euclidean with equality if f0 (x) = |x−x 2 space, the entropy is nonnegative and
the µ-invariant is zero. Note that if we let w0  (2π)−n/4 e−f0 /2 , then Rn w02 dµRn = 1 and we may rewrite (17.45) as     n n + log (2π) w02 dµRn ≥ 0. |∇w0 |2 − w02 log (w0 ) − (17.46) 2 4 Rn

Roughly speaking, since Riemannian manifolds are almost geometrically Euclidean on small scales, the Euclidean logarithmic Sobolev inequality implies that the entropy on small scales (τ small) is almost nonnegative and the corresponding µ-invariant is almost zero. The following is in §3.1 of ˇ sum, Tian, and Wang [170]). [152] (see also Proposition 3.2 in Seˇ Proposition 17.20 (µ (g, τ ) → 0 as τ → 0). If (Mn , g) is a closed Riemannian manifold, then (17.47)

lim µ (g, τ ) = 0.

τ →0+

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2. BEHAVIOR OF µ (g, τ ) FOR τ SMALL

17

Proof. Suppose that (17.47) is not true. Then there exist ε > 0 and a sequence {τi }i∈N with τi  0 such that µ (g, τi ) ≤ −ε. We assume that τi ≤ 12 . We shall derive a contradiction to (17.46). Consider the rescaled metrics 1 g (17.48) gi  2τi

(note that µ gi , 12 = µ (g, τi ) ≤ −ε). By Proposition 17.24 below, there exists a corresponding sequence {fi }i∈N of minimizers of   1 W (g, · , τi ) = W gi , · , 2 subject to the constraints   (4πτi )−n/2 e−fi dµg = (17.49) M

Then (17.50)



M

1 W (g, fi , τi ) = W gi , fi , 2

(2π)−n/2 e−fi dµgi = 1.

 = µ (g, τi ) .

Let {xi }i∈N be a sequence of points in M such that (17.51)

fi (xi ) = min fi (x) . x∈M

The pointed sequence of Riemannian manifolds {(Mn , gi , xi )}i∈N converges in the C ∞ Cheeger–Gromov sense to Euclidean n-space (Rn , gRn , 0). That is, there exists an exhaustion {Ui }i∈N of Rn by relatively compact open sets (Ui ⊂ Ui+1 ) and embeddings Φi : Ui → M such that Φi (0) = xi and (17.52)

g˜i  Φ∗i gi → gRn

uniformly in C ∞ on compact subsets of Rn .3 Consider the positive functions wi  (2π)−n/4 e−fi /2 , which by (17.15) satisfy

   n 1 1 log (2π) + n wi = µ gi , wi (17.53) −2∆gi wi + Rgi wi −2wi log wi − 2 2 2 with the constraint (17.49), i.e.,  wi2 dµgi = 1. (17.54) M

The contradiction to (17.45) is obtained via the following steps. 3 See the notes and commentary at the end of this chapter for explicit choices of Ui and Φi .

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18

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Step 1. For a subsequence, the functions w ˜i  wi ◦ Φi : Ui → R ˜∞ on Rn , for some converge in C 1,α on compact subsets to a C 1,α function w 1,2 n α ∈ (0, 1). Moreover, w ˜∞ ∈ W (R ) and  2 w ˜∞ dµRn ≤ 1. (17.55) Rn

Step 2. The limit function w ˜∞ is a weak solution to the elliptic equation   n ˜∞ w ˜∞ = − µ∞ + log (2π) + n + 2 log w ˜∞ . (17.56) 2∆Rn w 2 Step 3. The function w ˜∞ is positive (by Step 1, this implies the C 1,α 2 2 log w convergence of w ˜i log w ˜i to w ˜∞ ˜∞ ). ∞ Step 4. The function w ˜∞ is C and from the elliptic equation (17.56) that w ˜∞ satisfies, we obtain a contradiction to (17.45) since (17.55) holds. Now we prove Steps 1–4. From (17.50) and (17.7) we have for the minimizers wi of K gi , · , 12 that     1 1 = K gi , wi , µ gi , 2 2     1 2 + 4 |∇w |2 w R gi i i gi 2 (17.57) dµgi . =

2 n M − log wi + 2 log(2π) + n wi2 Proof of Step 1. By (17.57) and (17.54) we have    1 2 2 Rgi wi2 dµgi |∇wi |gi dµgi = 2 wi log wi dµgi − 2 M M M2   n 1 + log(2π) + n + µ gi , 2 2 (17.58) ≤ C1 , where C1 is independent of i; here we used the logarithmic Sobolev inequality, Lemma 17.2, and Rgi = 2τi Rg . Hence there exists C2 < ∞ such that (17.59)

wi W 1,2 (M,gi ) ≤ C2

for all i. By the L2 Sobolev inequality, i.e., (17.18), we then have4 (17.60)

wi

2n

L n−2 (M,gi )

≤ C3

when n ≥ 3 and we have wi Lp (M,gi ) ≤ C4 (p) for all p ∈ [1, ∞) when n = 2. Now consider w ˜i = wi ◦ Φi : Ui → R. From (17.19), the L2 Sobolev constant is independent of scaling and Vol (gi )−2/n ≤ Vol (g)−2/n . 4

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2. BEHAVIOR OF µ (g, τ ) FOR τ SMALL

19

Since g˜i = Φ∗i gi → gRn , by (17.116) below and (17.60), we have for any 2n (when n = 2, define compact domain Ω ⊂ Rn and any 2 < p < n−2 2n  ∞) n−2 (17.61)

˜i Lp (Ω,gRn ) ≤ C (Ω, p) < ∞,

w ˜i log w

where C (Ω, p) < ∞ is independent of i. Since (17.53) says (17.62)    n 1 1 log (2π) + n wi − µ gi , wi = 2wi log wi , −2∆gi wi + Rgi wi − 2 2 2 by the Lp estimate for solutions to second-order elliptic equations (see Theorem 9.11 in Gilbarg and Trudinger [71]) applied to w ˜i on Ω, we have (17.63)

w ˜i W 2,p (Ω,gRn ) ≤ C (Ω, p)

2n and all i. for any 2 < p < n−2 By the Sobolev inequality, we have 1 np

w ˜i p n 2 ≤ w ˜i 1, n−p ≤ C w ˜i W 2,p (Ω,gRn ) , W (Ω,gRn ) C L ( n−2 ) (Ω,gRn ) n n ≥ n−2 and where C = C (Ω, p) < ∞ is independent of i (note that n−p
2 dµ = 1). Thus, by applying (17.115) with δ > 0 arbitrarily small, w gi M i we have  Ω

|w ˜i log w ˜i |q dµRn ≤ C (Ω, q)

3  n , independent of i. From this and the standard elliptic for 2 ≤ q < 2 n−2 p L estimate for (17.62), we obtain the stronger (as compared to (17.63)) estimate

w ˜i W 2,q (Ω,gRn ) ≤ C (Ω, q) 3  n . By iterating the above argument (easy exercise), we for 2 ≤ q < 2 n−2 see that for any q ∈ (1, ∞) (17.64)

w ˜i W 2,q (Ω,gRn ) ≤ C (Ω, q) ,

where C (Ω, q) is independent of i.5 Because we have (17.64) with q > n, by the Sobolev inequality it follows that

w ˜i C 1,α (Ω,gRn ) ≤ C (Ω) for some α ∈ (0, 1) and where C (Ω) < ∞ is independent of i. By the Arzela– Ascoli theorem and a diagonalization argument, passing to a subsequence, we have that for some α ∈ (0, 1) there is a nonnegative function w ˜∞ in C 1,α (Rn , gRn ) such that ˜∞ w ˜i → w 5

Note that if w ˜i log w ˜i Lq (Ω,gRn ) ≤ C for some q ∈ (1, ∞), then w ˜i W 2,q (Ω,gRn ) ≤ C  .

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20

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

in C 1,α (Ω, gRn ) for all compact domains Ω ⊂ Rn . Note that now we have ˜i → w ˜∞ log w ˜∞ in C 0 (Ω, gRn ). w ˜i log w Moreover, (17.54) implies  w ˜i2 dµg˜i ≤ 1 Ω

for i large enough, so that

 Rn

Now by (17.58), we have

2 w ˜∞ dµRn ≤ 1.

 Rn

|∇w ˜∞ |2 dµRn ≤ C

for some C < ∞. In particular, w ˜∞ ∈ W 1,2 (Rn ). Proof of Step 2. First we show that, after passing to a subsequence, the limit   1 ≤ −ε (17.65) µ∞  lim µ gi , i→∞ 2 exists. By (17.23),   1 1 n ≥ Rmin (gi ) − 2C (1, gi ) − log(2π) − n µ gi , 2 2 2 n ≥ τi Rmin (g) − 2C (1, g) − log(2π) − n 2



is uniformly bounded from below since by Lemma 17.2 we have C 1,

1 2τi g



≤

C (1, g) because τi ≤ Now we integrate the equations (which follow from (17.53))    n 1 1 w ˜i + Rg˜i w log (2π) + n w ˜ i − 2w ˜i = −µ gi , ˜i − ˜i log w ˜i 2∆g˜i w 2 2 2 1 2.

in Ui against a compactly supported test function. Taking the limit of the integrations, we obtain (17.66)     n ˜∞ w ∇w ˜∞ , ∇ϕ dµRn = µ∞ + log (2π) + n + 2 log w ˜∞ ϕ dµRn 2 2 Rn Rn for all ϕ ∈ C0∞ (Rn ) since Rg˜i → 0. That is, w ˜∞ is a weak solution of (17.56). Proof of Step 3. By (17.88) below, we have    n 1 1 n 1 Rmin (gi ) − log(2π) − − µ gi , . (17.67) max wi ≥ exp M 4 4 2 2 2 Since τi ≤ 12 , we have (17.68)

Rmin (gi ) = 2τi Rmin (g) ≥ − |Rmin (g)|

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2. BEHAVIOR OF µ (g, τ ) FOR τ SMALL

21

for all i. Hence, by (17.67), (17.51), and Φi (0) = xi , (17.69)   1 n n . w ˜i (0) = wi (xi ) = max wi ≥ exp − |Rmin (g)| − log(2π) − M 4 4 2 Hence w ˜∞ (0) > 0.

(17.70)

By (17.70) and by the strong maximum principle for weak solutions (Lemma 17.26 below) applied to (17.56), we have w ˜∞ > 0

on Rn

1,α and w ˜∞ log w ˜∞ is contained in the local H¨ older space Cloc (Rn ). Proof of Step 4. Since w ˜∞ is a weak solution of (17.56), where the rhs 1,α n is contained in Cloc (R ), by the regularity theorem for weak solutions we have that w ˜∞ is a classical solution of (17.56). Now by Schauder theory, we have for any k ∈ N

w ˜∞ C k,α (Ω,gRn ) ≤ Ck (Ω)

˜∞ ∈ W 1,2 (Rn ) ∩ C ∞ (Rn ). for some Ck (Ω) < ∞. In particular, w Now we complete the proof of the proposition. For any R > 0 let ηR : Rn → [0, 1] be a radial cutoff function with  1 if 0 ≤ |x| ≤ R, ηR (x) = 0 if |x| ≥ R + 1, ∂ 2w ηR ≤ 0. Then by (17.66) with ϕ = ηR ˜∞ , we have and with −2 ≤ ∂r  2 

 w ˜∞ dµRn ∇w ˜∞ , ∇ ηR Rn    n µ∞ n + log (2π) + + log w ˜∞ (ηR w = ˜∞ )2 dµRn , 2 4 2 n R

so that

 0≤

Rn



− (17.71)

µ∞ = 2

|∇ (ηR w ˜∞ )|2 dµRn  n + log (ηR w ˜∞ ) (ηR w ˜∞ )2 dµRn 4 2    2 2 2 (ηR w ˜∞ ) dµRn + log ηR w dµRn , |∇ηR |2 − ηR ˜∞

n Rn



Rn

log (2π) +

Rn


2 dµ n ≤ 1. ˜∞ where the inequality is true by (17.46), which holds since Rn w R Taking R sufficiently large, the rhs of (17.71) is arbitrarily close to  µ∞ w ˜ 2 dµRn < 0, 2 Rn ∞
2 dµ n > 0. This is ˜∞ where this inequality is true because µ∞ < 0 and Rn w R a contradiction.


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22

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Problem 17.21. In view of Proposition 17.20, can one determine the more precise asymptotic behavior of µ (g, τ ) for τ near 0? 2.2. Possible nonuniqueness of minimizers of W for τ small. Proposition 17.20 has the following consequence for the nonuniqueness of certain minimizers for τ sufficiently small. This contrasts with the case of the energy functional F , for which the minimizer is unique. Lemma 17.22 (For small τ the minimizer is nonconstant and may not be unique). Let (Mn , g) be a closed Riemannian manifold. (1) Any minimizer fτ of W (g, · , τ ) cannot be a constant function for τ sufficiently small. (2) If the isometry group of (M, g) acts transitively, then for τ sufficiently small any minimizer fτ of W (g, · , τ ) is not unique. Proof. We prove both statements by contradiction. (1) Suppose that there exists a sequence τi → 0 such that for each i, there is a minimizer fτi of W (g, · , τi ) which is constant. Then by (17.14), i.e.,   τi 2∆fτi − |∇fτi |2 + Rg + fτi − n = µ (g, τi ) , we have that Rg is constant. Hence we have for all i,

(17.72)

µ (g, τi ) = τi Rg + fτi − n n = τi Rg − log (4πτi ) + log Vol (g) − n, 2

where the second equality follows from the constraint in (17.8).6 We obtain   n lim µ (g, τi ) = lim τi Rg − log (4πτi ) + log Vol (g) − n = ∞ i→∞ i→∞ 2 since τi → 0. This contradicts (17.47). (2) Let τ be sufficiently small so that part (1) holds for (M, g). Then suppose that the minimizer fτ of W (g, · , τ ) is unique. Since the isometry group of g acts transitively on M, for every x, y ∈ M there exists an isometry φ:M→M of the metric g with φ (x) = y. By the diffeomorphism invariance of the W-functional, we have that fτ ◦ φ is also a minimizer of W (g, · , τ ). Thus, by our uniqueness assumption, fτ ◦ φ = fτ , which implies fτ (x) = fτ (y). Since x and y are arbitrary, we conclude that fτ is constant, a contradiction to our assumption on τ .
6

When fτ is constant, the constraint − n2 log (4πτ ) + log Vol (g).

 M

(4πτ )−n/2 e−fτ dµ = 1 implies fτ =

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3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY

23

Now suppose that (Mn , g (τ ) , f (τ )), τ ∈ (0, ∞), is an expanding gradient soliton solution to the backward Ricci flow (i.e., g˜ (t)  g (−t) is a shrinking gradient Ricci soliton) satisfying 1 gij = 0. 2τ By the proof of Theorem 6.29 in Part I, f (τ ) is a minimizer for W(g (τ ) , · , τ ). Note that g (τ ) is isometric to τ g (1), so that Rij + ∇i ∇j f −

µ (g (τ ) , τ ) = µ (g (1) , 1) for all τ ∈ (0, ∞). In the special case of an Einstein solution, where 1 Rij − gij = 0, 2τ we have that a minimizer for W (g (τ ) , · , τ ) is a constant function.7 On the other hand, by Lemma 17.22(1), given g (τ ), for τ˜ sufficiently small, any minimizer fτ˜ of W (g (τ ) , · , τ˜) is not constant. Problem 17.23. It would be interesting to understand the behavior of minimizers in some special cases. (1) For τ > 0 what are the minimizers of W (gS n , · , τ ) on the unit nsphere? Are they always radial functions about some point in S n ? 1 1 and τ > 2(n−1) compare? (Note that How do the cases τ < 2(n−1) 1 RgS n = n (n − 1), so that in the case τ = 2(n−1) we have a constant minimizer.) (2) For which manifolds can one find minimizers with nice properties such as having a certain amount of symmetry? For example, one may consider the minimizers on complex projective space CP n . 3. Existence of a minimizer for the entropy In this section we discuss the proof of the existence of a minimizer for W which supplements the proof of Lemma 6.24 in Part I; we adopt here the notation used there. Included in our discussion is a proof of a strong maximum principle for weak solutions. We also consider a lower bound for the maximum value of the minimizer. 3.1. Proof of the existence of a minimizer for W. The following result is due to Rothaus and we follow his paper (see §1 of [161]). Proposition 17.24 (Existence of a smooth minimizer for W). For any metric g on a closed manifold Mn and for any τ > 0, there exists a smooth minimizer fτ of W (g, ·, τ ) which satisfies (17.14). 7

Note that the scale τ is related to the scalar curvature by τ =

n . 2Rg(τ )

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24

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Without loss of generality, we may assume τ = 1. Recall from (17.8) and (17.7) that    1,2 2 w dµ = 1 , µ (g, 1) = inf H (g, w) : w ∈ W (M, g) , M

where



H (g, w) 

 M

   n 4 |∇w|2 + R − log w2 − log (4π) − n w2 dµ 2

By Lemma 17.5, the functional H (g, ·) is bounded for w ∈ from below. Step 1. There exists a minimizer 0 ≤ w∞ ∈ W 1,2 of H. Let {wi }i∈N sequence of W 1,2 functions for the functional H (g, ·) with
be a minimizing 2 M wi dµ = 1 for all i ∈ N. We may assume that wi ≥ 0 for the following reason. If w ∈ W 1,2 , then |w| ∈ W 1,2 and W 1,2 (M, g).

|∇ |w|| ≤ |∇w| (see Corollary 2.1.8 of Ziemer [198]), so that H (g, |w|) ≤ H (g, w). Thus, if {wi } is a minimizing sequence in W 1,2 , then so is {|wi |}. Recall that we proved that there exists C < ∞ (independent of i) such that

wi W 1,2 (M,g) ≤ C

(17.73)

for all i ∈ N (we leave this as an exercise; see p. 238 in Part I or (17.58) above). By the Banach–Alaoglu theorem,8 there exists w∞ ∈ W 1,2 (M, g) and a subsequence such that wi converges to w∞ weakly in W 1,2 (M, g), i.e., for every v ∈ W 1,2 (M, g) lim wi , vW 1,2 (M,g) = w∞ , vW 1,2 (M,g) .

i→∞

As a standard consequence, we have (see also p. 205 in Part I)

w∞ W 1,2 (M,g) ≤ lim inf wi W 1,2 (M,g) .

(17.74)

i→∞

By (17.73) and the Rellich–Kondrachov compactness theorem, for every 2n ] we have that wi converges to w∞ in L n−2 −ε (M, g). In particuε ∈ (0, n+2 n−2
2 dµ = 1 and w lar, M w∞ ∞ ≥ 0. By this convergence and by Lemma 17.25 below, we have µ (g, 1) ≤ H (g, w∞ ) ≤ lim H (g, wi ) = µ (g, 1) . i→∞

We conclude that (17.75) 8

H (g, w∞ ) = µ (g, 1) .

See Theorems 3.15 and 3.17 in Rudin [164].

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3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY

25

Step 2. w∞ is a weak solution of (17.77). w∞ is a minimizer
Since 1,2 2 of H (g, w) subject to
the constraint M w dµ = 1, for any W 1,2 in W function φ : M → R such that M w∞ φ dµ = 0, we have (17.76)  d  H (g, w∞ + sφ) 0= ds s=0     d  2 + sφ)| 4 |∇ (w dµ = ∞ ds s=0 M     n d  2 log (4π) − n (w∞ + sφ)2 dµ + sφ) − R − log (w + ∞ ds s=0 M 2      2 n − log (4π) − n w∞ φ − w∞ φ dµ. 4∇w∞ , ∇φ + R − log w∞ =2 2 M That is, by definition w∞ is a weak solution to the following second-order elliptic equation  2 n log(4π) + n w∞ = µ(g, 1)w∞ . − (17.77) −4∆w∞ +Rw∞ −w∞ log w∞ 2 The constant µ(g, 1) is determined by (17.75) and by substituting φ = w∞ in (17.76). Step 3. w∞ is a positive C ∞ minimizer. Define  2 n log(4π) + n + µ(g, 1) w∞ . − (17.78) P (w∞ )  Rw∞ − w∞ log w∞ 2 2n

Since w∞ ∈ L n−2 , by (17.116) below we have P (w∞ ) ∈ Lp for any p ∈ 2n 2n ) (here n−2  ∞ if n = 2), so that w∞ ∈ W 2,p by the standard [1, n−2 p (interior) L estimate for weak solutions to second-order elliptic equations (see Theorem 9.11 in [71]). Bootstrapping, we obtain w∞ ∈ W 2,q for all q ∈ [1, ∞). By the Sobolev embedding theorem, this implies that w∞ ∈ C 1,α for some α ∈ (0, 1).
2 dµ = 1), Since w∞ ≥ 0 everywhere and w∞ > 0 somewhere (since M w∞ by the strong maximum principle for weak solutions of (17.77) (see Lemma 17.26 below), we have w∞ > 0 everywhere on M. Therefore w∞ log w∞ ∈ C 1,α , so that 4∆w∞ = P (w∞ ) ∈ C 1,α . By the regularity theory for weak solutions of the Poisson equation, w∞ is a classical solution and we may apply Schauder theory to conclude that w∞ ∈ C k,α for all k ∈ N. Hence w∞ is C ∞ , so that f1  − n2 log (4π) − 2 log w∞ is a C ∞ minimizer of K (g, · , 1). This completes the proof of Proposition 17.24.
To conclude this subsection, we prove the following result, which was used in the proof of Proposition 17.24 (see p. 112 of [161]).

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26

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

Lemma 17.25. Let (Mn , g) be a closed Riemannian manifold. The quan


tity M w2 log w2 dµ depends continuously on w with respect to the L2(1+δ) norm for any δ > 0. In particular, the dependence is continuous with respect to the W 1,2 -norm. Proof. To see this, suppose w1 , w2 ∈ L2(1+δ) (M), where δ > 0. At each x ∈ M we have



w22 log w22 − w12 log w12  |w2 |

d 2 u log u2 du = |w1 | du  |w2 |

2u 1 + log u2 du. = |w1 |

Applying the mean value theorem for integrals to this, we have





w22 log w22 − w12 log w12 = (|w2 | − |w1 |) · 2a 1 + log a2 , where a : M → [|w1 | , |w2 |] and a ∈ L2(1+δ) (M). Hence 



2 w2 log w22 − w12 log w12 dµ M

(17.79)

1/2 

 ≤

M 2

|w2 − w1 |2 dµ



2 4a2 1 + log a2 dµ

1/2

M

2

since (|w2 | − |w1 |) ≤ |w2 − w1 | . Note that   1 1 1+δ , a (17.80) |a log a| ≤ max e δe 1 δ a . Therefore since for a, δ > 0 we have a log a ≥ − 1e and log a ≤ δe


2 2 2 dµ in terms of w1 L2(1+δ) (M,g) and we can bound M 4a 1 + log a


w2 L2(1+δ) (M,g) . The lemma now follows from (17.79).

3.2. Strong maximum principle for weak solutions. We now give the proof of the strong maximum principle for weak solutions,9 which is used in the proofs of Proposition 17.20 and Proposition 17.24. The following proof is on pp. 114–116 of Rothaus [161] (we also follows his notation for the most part). Lemma 17.26 (Strong maximum principle for weak solutions). Suppose (Mn , g) is a complete Riemannian manifold. Let w∞ ≥ 0 be a C 1,α function which is a weak solution to (17.77). If w∞ (p) = 0 for some p ∈ M, then w∞ = 0 in a neighborhood of p. 9

On the other hand, Calabi [21] proved strong maximum principles for sub- and supersolutions in the support sense; see also Trudinger [181] and Theorem 2.4 of Andersson, Galloway, and Howard [5].

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3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY

27

Remark 17.27. Note that (17.77) is the same as (17.15) with τ = 1. By scaling, one sees that the lemma holds for solutions of (17.15) with arbitrary τ > 0. The proof is via a monotonicity formula in the radial direction emanating from p. Denote by S (p, r) the geodesic sphere of radius r centered at p. Let J (θ, r), where θ ∈ S n−1 (1) and r ∈ (0, inj (p)), be the Jacobian of the exponential map in spherical coordinates, so that





dµ expp (rθ) = J (θ, r) expp ∗ dσS n−1 (1) ∧ dr, where S n−1 (1) ⊂ Tp M is the unit sphere and dσS n−1 (1) is its volume (n − 1)J form. Note that lim n−1 = 1. r→0 r Now define F : (0, inj (p)) → R by

S(p,r) w∞ dσ S n−1 (1) w∞ (θ, r) J (θ, r) dσS n−1 (1)
= , F (r) =
S(p,r) dσ S n−1 (1) J (θ, r) dσS n−1 (1) where dσ denotes the induced volume (n − 1)-form on S (p, r).10 w∞ (p) = 0 and w∞ is continuous, we have

Since

lim F (r) = 0.

r→0+

We shall derive a differential inequality for F (r) to show that F (r) = 0 for r sufficiently small. Since w∞ ≥ 0, this implies w∞ = 0 in a neighborhood of p. If we set

∂w∞ (θ,r) ∂w∞ J (θ, r) dσS n−1 (1) ∂r S(p,r) ∂r dσ S n−1 (1)
= , G (r) =
S(p,r) dσ S n−1 (1) J (θ, r) dσS n−1 (1) then




F (r) =

S(p,r)







dσ

S(p,r) w∞ dσ







∂ S(p,r) w∞ ∂r




+

S(p,r) dσ


−

∂w∞ ∂r




S(p,r) dσ ∂ S(p,r) ∂r log J dσ 2

S(p,r) dσ



S(p,r) w∞

∂ ∂r

= G (r) +

∂ log J − ∂r 10




log J −


∂ S(p,r)
∂r

log J dσ

S(p,r)

dσ

 dσ

S(p,r) dσ




Since

log J dσ

∂ S(p,r) ∂r




log J dσ

S(p,r) dσ

= O (r) ,

That is, F (r) is the average value of w∞ on S (p, r).

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.

28

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

there exists a constant C < ∞ such that F  (r) ≤ G (r) + CrF (r) .

(17.81)

Now we proceed to estimate G (r) from above. Let ϕ ∈ C0∞ (B) be a radial function (i.e., a function of d (·, p)), where B = B (p, inj (p)). We have  inj(p)   ∂w∞  ϕ (r) dσ ∇w∞ · ∇ϕ dµ = dr 0 B S(p,r) ∂r  inj(p) ϕ (r) A (r) G (r) dr, = 0

where

 dσ

A (r) = S(p,r)

is the (n − 1)-dimensional volume of S (p, r). Since w∞ is a weak solution to (17.77),  inj(p) (17.82) ϕ (r) A (r) G (r) dr 0  1 w∞ log (w∞ ) ϕ dµ = 2 B    n 1 −R + log(4π) + n + µ(g, 1) w∞ ϕ dµ. + 4 B 2 Let 1 L (r) = 2A (r)

(17.83) and (17.84)

K (r) =

1 4A (r)



 S(p,r)

w∞ log w∞ dσ

  n R − log(4π) − n − µ(g, 1) w∞ dσ, 2 S(p,r)

so that (17.82) implies that G (r) satisfies  inj(p)

 ϕ (r) G (r) − ϕ (r) L (r) + ϕ (r) K (r) A (r) dr = 0 0

for all radial functions ϕ ∈ C0∞ (B). This implies (Rothaus says, ‘By the usual one-dimensional regularity result, ...’) d (G (r) A (r)) = (K (r) − L (r)) A (r) . dr Now from definition (17.84) we have  C w∞ dσ K (r) ≤ A (r) S(p,r)

(17.85)

= CF (r) ,

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3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY

29

  where C = 14 maxB(p,inj(p)) R − n2 log(4π) − n − µ(g, 1) . Moreover, Jensen’s ¯ inequality says that if ϕ : [0, ∞) → R is convex, then     1 1 ϕ ◦ w∞ dσ ≥ ϕ w∞ dσ = ϕ (F (r)) . A (r) S(p,r) A (r) S(p,r) Applying Jensen’s inequality with ϕ (x) = x log x to (17.83), we have 2L (r) ≥ F (r) log F (r) . Applying these two inequalities to (17.85), we obtain   1 d (G (r) A (r)) ≤ CF (r) − F (r) log F (r) A (r) . (17.86) dr 2 Since w∞ ≥ 0, w∞ (p) = 0, and w∞ ∈ C 1,α , there exists a sequence ri → 0+ such that lim G (ri ) A (ri ) = 0. i→∞

Thus integrating (17.86) on the interval [ri , r] and taking i → ∞, we have   r 1 1 CF (s) − F (s) log F (s) A (s) ds. G (r) ≤ A (r) 0 2 Substituting this into (17.81) yields   r 1 1  CF (s) − F (s) log F (s) A (s) ds. F (r) ≤ CrF (r) + A (r) 0 2 Since limr→0+ F (r) = 0, we obtain   r  t  t dr 1 rF (r) dr + CF (s) − F (s) log F (s) A (s) ds F (t) ≤ C 2 0 0 A (r) 0 for t ∈ (0, inj (p)). Now assume that t0 ≤ inj (p) is small enough so that for t ∈ (0, t0 ], (1) C1 tn−1 ≤ A (t) ≤ C2 tn−1 , where C1 > 0 and C2 < ∞, (2) 0 ≤ F (t) ≤ 1. Then for t ∈ (0, t0 ],  t  rF (r) dr − F (t) ≤ C 0



t

+C 0

dr rn−1

t 0



dr rn−1



r

 sn−1 F (s) log F (s) ds

0

r

sn−1 F (s) ds 0

for some C < ∞. Now for a ∈ (0, t0 ] there exists b = b (a) < ∞ (where lima→0+ b (a) = 0) such that F (s) ≤ b

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30

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

for s ∈ (0, a]. If t ∈ (0, a], then    t  t  r 1 dr n−1 br dr + s ds +b F (t) ≤ C n−1 e 0 0 r 0   1 t2 e +b =C b+ n 2 since −x log x ≤ 1e . Choosing a small enough, we have F (t) ≤ t for t ∈ (0, a]. In general, if we have F (t) ≤ tk on (0, a] for some k ≥ 1, then   t  t  r dr k+1 n−1+k r dr − s k log s ds F (t) ≤ C n−1 0 0 r 0  r  t dr sn−1+k ds. +C n−1 r 0 0 Applying −x log x ≤ 1e again and integrating, we obtain  k+2   t k+1   r k t dr r t n−2+k + dr s ds + F (t) ≤ C n−1 k+2 e 0 r 0 0 n+k   k+2 k tk+1 tk+2 t (17.87) + + =C k + 2 e (k + 1) (n + k − 1) (k + 2) (n + k) provided t ∈ (0, min {a, 1/e}], where C is independent of k. Now (17.87) implies that there exists a0 ∈ (0, min {a, 1/e}) independent of k ≥ 1 such that11 F (t) ≤ tk+1/2 for t ∈ (0, a0 ]. By induction, we have F (t) ≤ t
for t ∈ (0, a0 ] and all  ≥ 1. This implies F (t) = 0 for t ∈ (0, a0 ] and Lemma 17.26 is proved.
3.3. The maximum value of a minimizer.
Under the constraint M w2 dµg = 1, let wτ be a minimizer of the functional K (g, w, τ ) defined by (17.7). The maximum value of wτ is related to an upper bound for the µ-invariant as follows (this is used in the proof of Proposition 17.20). 11

Indeed, we just need a small enough so that   2 √ k t2 t t + + ≤ t C k+2 e (k + 1) (n + k − 1) (k + 2) (n + k)

for t ∈ (0, a] and k ≥ 1.

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4. 1- AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW

31

Lemma 17.28 (Lower bound for the maximum value of a minimizer). On a closed manifold we have   n 1 τ n Rmin (g) − log(4πτ ) − − µ (g, τ ) . (17.88) max wτ ≥ exp M 2 4 2 2 Proof. By (17.15), a minimizer wτ satisfies  n log(4πτ ) + n wτ = µ (g, τ ) wτ . τ (−4∆wτ + Rwτ ) − wτ log wτ2 − 2 At a point xτ ∈ M where wτ attains its maximum, we have (∆wτ ) (xτ ) ≤ 0, so that  n log(4πτ ) + n wτ ≤ µ (g, τ ) wτ . τ Rwτ − 2wτ log wτ − 2 Hence   n n 1 τ R − log(4πτ ) − − µ (g, τ ) max wτ = wτ (xτ ) ≥ exp M 2 4 2 2


and the lemma follows. 4. 1- and 2-loop variation formulas related to RG flow

In this section we discuss formulas related to Perelman’s energy functional and its variation.12 One may wish that some of these formulas are related to ‘renormalization group flow’ (RG flow) in physics; the ‘loop’ terminology is from there. However the point of view we take is simply to calculate first variation formulas for certain Riemannian geometric invariants and to look for structure in these formulas. We leave the calculations as exercises for the reader. 4.1. Some 1-loop formulas. Let (Mn , g) be a closed Riemannian manifold and let f be a function on M. Recall from (17.1) that Perelman’s energy functional is    R + 2∆f − |∇f |2 e−f dµ (17.89) F1 (g, f ) = M

(we add the subscript 1 to F with the hope that this is the first in an infinite sequence of functionals). The integrand in (17.89) appears in a contracted second Bianchi-type identity (see §1.3 of [152]):   1   (17.90) div (Rc +∇∇f ) e−f = e−f ∇ R + 2∆f − |∇f |2 . 2 Let v be a symmetric 2-tensor on M and let X be a vector field on M. The linear trace Harnack quadratic is given by (17.91) 12

L (v, X)  div (div v) + v, Rc − 2 div v, X + v (X, X)

We would like to thank Shengli Kong for helpful discussions.

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32

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

(see Theorem A.57 in Part I for the corresponding linear trace Harnack estimate). When X = ∇f , we may rewrite this as (17.92) (17.93)

L (v, ∇f ) = (div −ι∇f ) ◦ (div −ι∇f ) v + Rc +∇∇f, v   = ef (div ◦ div + Rc +∇∇f ) e−f v .

Observe also that ∂R − 2 ∇R, X + 2 Rc (X, X) ∂t is Hamilton’s trace Harnack quadratic (see (15.17) in Part II). Let V = g ij vij .Two of the above quantities are related by the following (see also Lemma 6.82 in Part I) L (2 Rc, X) =

Lemma 17.29 (Variation of Perelman’s −f modified scalar curvature). If ∂ ∂ = v and ∂s f = V (so that ∂s e dµ = 0), then  ∂  R + 2∆f − |∇f |2 = L (v, ∇f ) − 2 v, Rc +∇∇f  . (17.94) ∂s Integrating the above formula by parts, we obtain (see §1.1 of [152]; compare with Exercise 6.16 in Part I) ∂ ∂s g

Lemma 17.30 (Perelman’s first variation formula for F ). If ∂ f ∂s = V , then  ∂ F1 (g, f ) = − (17.95) v, Rc +∇∇f  e−f dµ ∂s M (17.96) L (v, ∇f ) e−f dµ. =−

∂ ∂s g

= v and

M

4.2. Some 2-loop formulas. Let (∆L h)ij  ∆hij + 2Rkij
hk
− Rik hkj − Rjk hki be the Lichnerowicz Laplacian, which acts on symmetric 2-tensors. Given a function f on M, define the symmetric 2-tensor H∇f , which is a form of Hamilton’s matrix Harnack quadratic, by (compare with the expression in (15.11) of Part II) (17.97)



 1 2 H∇f (X, Y )  ∆L Rc − ∇∇R + Rc (X, Y ) 2 + P (X, ∇f, Y ) + P (Y, ∇f, X) + Rm (∇f, X, Y, ∇f ) ,

where P (X, Y, Z)  (∇X Rc) (Y, Z) − (∇Y Rc) (X, Z) for tangent vectors X, Y, Z (see Chapter 15 in Part II for the proof of Hamilton’s matrix Harnack estimate). This may be rewritten as   (17.98) H∇f = ef (div ◦ div + Rc +∇∇f )1,4 e−f Rm ,

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4. 1- AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW

33

where the subscripts 1, 4 denote the components on which the operator is acting. Note that if Rc +∇∇f ≡ 0, then H∇f ≡ 0. Recall that under ∂ (17.99a) g = −2 Rc, ∂t ∂f = −∆f − R + |∇f |2 , (17.99b) ∂t we have (see Proposition 6.95 in Part I) (17.100)   ∂ + ∆L − 2∇f · ∇ (Rc +∇∇f ) = 2H∇f − (Rc +∇∇f )  (Rc −∇∇f ) , ∂t where, for (1, 1)-tensors Aji and Bij , we define (A  B)ji  Aki Bkj + Bik Ajk . Define P ∗ (X, Y, Z)  P (Z, Y, X) , so that (see (15.50) in Part II) div (Rm) = P ∗ , where div (Rm) = trace1,2 g (∇ Rm) and where the superscripts 1, 2 indicate that the first two components are traced. Under the Ricci flow, the Riemann curvature (3, 1)-tensor Rm evolves by (see (6.2) in Volume One) ∂ Rm = − [∇, ∇] Rc +d∇ P ∗ , ∂t where (d∇ P ∗ ) (X, Y, Z, W ) = (∇X P ∗ ) (Y, Z, W ) − (∇Y P ∗ ) (X, Z, W ). Define the symmetric 2-tensor  Rm (·, ei , ej , ek ) Rm (·, ei , ej , ek ) , (17.102) α (17.101)

i,j,k

where {e
} is an orthonormal frame, i.e., αij = Rik
p Rjk
p . Then α satisfies the contracted second Bianchi-type identity   1 2 (17.103) div α − |Rm| g = Rm (·, ei , ej , ek ) P ∗ (ei , ej , ek ) . 4 Now we discuss some calculations due to two of the authors [44]. By (17.101), under the Ricci flow 1∂ |Rm|2 = Rc, α + Rm, ∇P ∗  4 ∂t (17.104) = Rc, α + div (Rm (·, ei , ej , ek ) P ∗ (ei , ej , ek )) − |P |2 . We then compute that under (17.99)    1 ∗
|Rm|2 e−f = e−f −L (α, ∇f ) + |P ∗ − ι∇f Rm|2 , (17.105) 4

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34

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

where
∗  −

∂ −∆+R ∂t

and where (ι∇f Rm) (X, Y, Z) = Rm (∇f, X, Y, Z). ∂ ∂ g = v and ∂s f = V2 , then If ∂s  1 d |Rm|2 e−f dµ (17.106) 4 ds      −f 1 = v · −H∇f − α + Rm(1,4) , Rc +∇∇f e dµ, 2 where the subscript (1, 4) denotes the components on which
the inner prodd |Rm|2 e−f dµ = uct is acting. As a special case, if Rc +∇∇f ≡ 0, then 14 ds
− 12 v, α e−f dµ. We summarize the above formulas. For a = 1, 2, define the 2-tensors β (a) , the functions γ (a) , and the functionals Fa (g, f ) by (17.107a)

β (1) = −2 (Rc +∇∇f ) ,

(17.107b)

β (2) = −α, γ (1) = −R − ∆f, 1 γ (2) = − |Rm|2 , 2  R + |∇f |2 e−f dµ, F1 (g, f ) = M  1 |Rm|2 e−f dµ. F2 (g, f ) = 4 M

(17.107c) (17.107d) (17.107e) (17.107f)

Given a functional G (g, f ), let δ(β (a) ,γ (a) ) G (g, f ) denote under

∂ ∂s g (s)

=

β (a)

and

∂ ∂s f

(s) =

γ (a) .

d ds G

(g (s) , f (s))

We have



(17.108)

 1 |Rm|2 e−f dµ 4   1 = − div e−f ∇ |Rm|2 dµ + L (α, ∇f ) e−f dµ 4 − |P ∗ − ι∇f Rm|2 e−f dµ

δ(β (1) ,γ (1) )

and (17.109) δ(β (2) ,γ (2) )



  R + 2∆f − |∇f |2 e−f dµ = −L (α, ∇f ) e−f dµ + 2 α, Rc +∇∇f  e−f dµ,

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4. 1- AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW

35

so that (17.110)



    1 2 −f δ(β (1) ,γ (1) ) − |Rm| e dµ + δ(β (2) ,γ (2) ) R + 2∆f − |∇f |2 e−f dµ 4   1 −f e ∇ |Rm|2 dµ + 2 α, Rc +∇∇f  e−f dµ = − div 4 − |P ∗ − ι∇f Rm|2 e−f dµ.

We obtain (17.111)

δ(β (1) ,γ (1) ) F2 (g, f ) + δ(β (2) ,γ (2) ) F1 (g, f )   |P ∗ − ι∇f Rm|2 e−f dµ + 2 α, Rc +∇∇f  e−f dµ. =− M

M

Note that (17.112)



−δ(β (1) ,γ (1) ) F2 (g, f ) + δ(β (2) ,γ (2) ) F1 (g, f ) = (0)

Remark 17.31. If one defines βij − M f e−f dµ, then


(17.113)

M

|P ∗ − ι∇f Rm|2 e−f dµ ≥ 0.

= gij , γ (0) =

n 2,

and F0 (g, f ) =

δ(β (0) ,γ (0) ) F1 (g, f ) + δ(β (1) ,γ (1) ) F0 (g, f ) = 0.

Problem 17.32. Do any of the above formulas fit into an infinite sequence of formulas? One would like to obtain a monotonicity formula extending Perelman’s entropy monotonicity formula. In particular, one may wish to consider an expression of the form (17.114)

  δ(β (1) ,γ (1) ) F1 (g, f ) + λ δ(β (1) ,γ (1) ) F2 (g, f ) + δ(β (2) ,γ (2) ) F1 (g, f )   + λ2 δ(β (1) ,γ (1) ) F3 (g, f ) + δ(β (2) ,γ (2) ) F2 (g, f ) + δ(β (3) ,γ (3) ) F1 (g, f ) +··· ,

where β (k) , γ (k) , Fk are suitably defined for k ∈ N and λ ∈ R+ . However it is not clear whether or not one should introduce new fields in addition to g and f . Some related calculations are in Oliynyk, Suneeta, and Woolgar [145]; see also Zamolodchikov [194] and Tseytlin [182].13 In the physics literature there are various 3- and 4-loop calculations. 13

We would like to thank C. Vafa for discussions at the California Institute of Technology during January and February of 2003. The first author would also like to thank A. Tseytlin for discussions at Ohio State University during May of 2003.

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36

17. ENTROPY, µ-INVARIANT, AND FINITE TIME SINGULARITIES

5. Notes and commentary The monotonicity formulas which are applied in this chapter were discovered partly based on the notion of self-similarity. On a more philosophical note, we venture to ask the following general question. Problem 17.33. In Ricci flow some of the guiding ideas and principles, among others, used to study the geometry of solutions are (1) analogies with the heat equation and its smoothing aspects, (2) applications of the maximum principle and monotonicity, (3) self-similar solutions, i.e., Ricci solitons, used to find quantities to estimate, (4) natural space-time quantities such as the reduced distance, (5) point picking and determining location and scale. Can one discover new guiding principles and ideas? (A) One goal (espoused by Hamilton) is to localize various formulas. (Perelman’s pseudolocality in effect does this for the curvature evolution under certain hypotheses.) Can one find new quantities on space-time (or some larger system) to help accomplish this? (B) An important problem (espoused by Perelman) is to formulate a notion of weak solution which enables the flow to continue past singularities. (C) Largely uncharted territory is the role of Riemannian Ricci flow in higher dimensions. Note that Lie algebra aspects of the evolution of the curvature operator, which began with Hamilton’s work on 4-manifolds, have been studied by B¨ohm and Wilking with applications to the classification of closed manifolds with 2-positive curvature operator. In addition, Hamilton and Perelman proved a number of results in arbitrary dimensions in their works. (D) Is it fruitful to study Ricci flow on spin manifolds or some other large class of manifolds? §1. In contrast to the lower bound given by Lemma 17.8 for the volume of a solution g (t) to Ricci flow with λ (g (t)) ≤ 0, we have the following. Upper bound for the volume of a solution with λ > 0. Suppose that a solution (Mn , g (t)) of the Ricci flow on a closed manifold and maximal time d λ (g (t)) ≥ n2 λ (g (t))2 (see Lemma interval [0, T ) has λ (g (0)) > 0. Since dt 5.25 in Part I), we have λ (g (t)) ≥

1 λ (g (0))−1 − n2 t

,

so that T ≤ n2 λ (g (0))−1 < ∞. Since d log Vol (g (t)) = −Ravg (g (t)) ≤ −λ (g (t)) , dt

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5. NOTES AND COMMENTARY

we also obtain

 Vol (g (t)) ≤

2 1 − λ (g (0)) t n

37

n/2 Vol (g (0)) .

§2. (1) The following is used in the proof of Proposition 17.20. From the formula  d  −δ x log x = x−δ−1 (1 − δ log x) dx for δ > 0, we see that the maximum of x−δ log x on (0, ∞) is 1δ e−1 which occurs at x = e1/δ . Hence 1 1 − ≤ w log w ≤ w1+δ . e δe Thus for any q > 0 and δ > 0,   1 1 q q w (log w) dµ ≤ wq(1+δ) dµ + q Vol (g) . (17.115) q (δe) M e M 2n Note that we have for any p < n−2   p p (17.116) ϕ (log ϕ) dµ ≤

−p  2n 2ne −e ϕ (n−2) dµ (n − 2) p M M 1 + p Vol (g) , e 1 δ 2n − 1 > 0. where we have used log x ≤ δe x with δ = (n−2)p Following the proof of Lemma 6.36 in Part I, we conclude that for a > 0   n4 2 2 (17.117) ϕ (log ϕ) dµg ≤ a |∇ϕ|2g dµg + 4 ae Cs (M, g) M M 1 + a Vol (g)−2/n + 2 Vol (g) . e (2) We can define Ui and Φi in (17.52) explicitly as follows. Let inj (g) √ → ∞ as i → ∞). denote the injectivity radius of g (note that inj (gi ) = inj(g) 2τi i n Choose orthonormal frames eα α=1 at xi with respect to gi and let ψi : (Rn , gRn ) → (Txi M, gi (xi ))

n be the linear isometry taking the standard basis in Rn to eiα α=1 . Let   inj (g)  ⊂ Rn Ui  B 0, √ 2τi and define Φi : Ui → M by Φi = expgxi ◦ψi , where expgxi denotes the exponential map of g at xi . Then (17.52) holds.

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http://dx.doi.org/10.1090/surv/163/02

CHAPTER 18

Geometric Tools and Point Picking Methods Funny how my memory slips while looking over manuscripts of unpublished rhyme. – From “Hazy Shade of Winter” by Simon and Garfunkel

In this chapter we primarily discuss some tools used extensively in the qualitative study of solutions of the Ricci flow. Applications of these tools appear in subsequent chapters and the impatient reader may wish to skip this chapter while referring back to it when necessary. A basic invariant reflecting the geometry of a Riemannian manifold is the distance function. We begin the chapter by discussing in §1 the changing distances estimate for solutions of Ricci flow. The results in this section are useful for a variety of applications: localizing monotonicity formulas, controlling geometric invariants depending on the distance function such as the diameter and the asymptotic scalar curvature ratio, and estimating space-time geometric invariants such as the reduced distance. To introduce the next topic, we recall the popular real estate saying “location, location, location.” This applies to Ricci flow in the form of point picking. Suppose that we are given a solution to the Ricci flow, such as a finite time singular solution to the Ricci flow on a closed manifold or a κsolution. The purpose of point picking is to obtain a good sequence of points and times (locations) so that the corresponding rescaled solutions based at these points and times admit a subsequence with a limiting solution having nice properties. We usually rescale so that the norms of the curvatures (or the scalar curvatures) of the rescaled solutions equal 1 at the basepoints. The reason for doing this is to obtain a nontrivial limit. One of Perelman’s innovations in [152] is the deep and clever use of point picking. These methods are useful for obtaining crucial curvature bounds via contradiction arguments and studying the geometry at infinity of κ-solutions (see Definition 19.7 in this volume).1 In §2 and §3 we discuss various point picking methods foundational to Perelman’s work on Ricci flow. Point picking methods are further discussed in Chapter 22 related to pseudolocality. 1

Results and tools which give strength to these arguments are the no local collapsing theorem, volume comparison, ASCR = ∞, AVR = 0, dimension reduction, classification of 2-dimensional ancient κ-solutions, and the strong maximum principle. 39 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

40

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

In §4 we discuss rough monotonicity of the size of necks in complete noncompact manifolds with positive sectional curvature. This will be useful in the study of κ-solutions. Finally, in §5 we discuss a local form of the weakened no local collapsing theorem presented in Chapter 8 of Part I. 1. Estimates for changing distances Now we turn to some geometric and intuitive aspects that we borrow from the study of manifolds with nonnegative sectional curvature and which influence the study of Ricci flow, including κ-solutions. By definition, κsolutions have nonnegative curvature operator, which implies nonnegative sectional curvature. Since compact 3-dimensional κ-solutions must have positive sectional curvature and hence are diffeomorphic to spherical space forms, we are most interested in noncompact κ-solutions. In relation to nonnegative sectional curvature, Greene wrote the following on p. 100 of [74]: ‘Nonnegativity, and especially positivity, of curvature tends to make long geodesics nonminimizing. Thus some tension arises between the necessary existence of rays and the curvature’s nonnegativity.’ Greene went on to write: ‘The fact that straight lines in Rn are, in this sense, just barely minimizing suggests that there should be quantitative estimates on just how much positivity of curvature could be possible without forcing a geodesic to be nonminimal.’ An important illustration of this idea is how much positivity of Ricci curvature a minimal geodesic γ can have. Namely, in Proposition 18.8 below
we shall obtain an upper bound for γ Rc (γ  (s) , γ  (s)) ds. An application to Ricci flow of this bound is given in Lemma 8.3(b) of Perelman [152], which discusses the change in the distance function under the Ricci flow (see Theorem 18.7(2) below). The precursors for this result are Theorems 17.2 and 17.4 of Hamilton [92]. See also Proposition 1.94 as well as Lemma 8.33 and Remark 8.34, all in [45]. 1.1. The time derivative of the distance function. First we recall the formula for the time derivative of the distance function using the lim inf of forward difference quotients. Our presentation follows the proof of Lemma 3.5 in Hamilton [89] (see also Lemma 10.29 in Part II). For a function f (x, t), let f (x, t + h) − f (x, t) ∂−f (x, t)  lim inf + ∂t h h→0

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1. ESTIMATES FOR CHANGING DISTANCES

41

denote the lim inf of forward difference quotients. We have a similar defini−f (x, t), the lim inf of backward difference quotients, with t + h tion for ∂∂t replaced by t − h. Let (Mn , g (t)), t ∈ [0, T ), be a smooth 1-parameter family of complete metrics. Given t0 ∈ [0, T ), let Z (t0 ) denote the set of all unit speed minimal geodesics joining x0 to x, with respect to g (t0 ). Note that Z (t0 ), with the subspace topology induced by the compact-open topology on the space of continuous paths, is compact and nonempty. The evolution of the distance function is given by the following. −

Lemma 18.1 (Time derivative ∂∂t of the distance function). For t0 ∈ [0, T ), in the sense of the lim inf of forward difference quotients     

∂g 1 ∂ −   (t0 ) η (s) , η (s) ds . d (x, x0 ) = min ∂t t=t0 g(t) η∈Z(t0 ) 2 η ∂t Proof. For t0 ∈ [0, T ) and each η ∈ Z (t0 ) (see Lemma 3.11 in Volume One)   

1 ∂g d   (t L (η) = ) η (s) , η (s) ds, (18.1) 0 g(t) dt t=t0 2 η ∂t where Lg(t) (η) is the length of the path η with respect to g (t) and where s denotes arc length with respect to g (t0 ). For each η ∈ Z (t0 )   d  ∂ −  d (x, x0 ) ≤  L (η) , ∂t t=t0 g(t) dt t=t0 g(t) where we have used dg(t) (x, x0 ) ≤ Lg(t) (η) for t ≥ t0 and we have also used dg(t0 ) (x, x0 ) = Lg(t0 ) (η). Therefore     

∂g 1 ∂ −   (t0 ) η (s) , η (s) ds . d (x, x0 ) ≤ min ∂t t=t0 g(t) η∈Z(t0 ) 2 η ∂t Now we prove the reverse inequality. Suppose for i ∈ N that hi → 0+ and ηi is a unit speed minimal geodesic joining x0 to x, with respect to g (t0 + hi ). Passing to a subsequence, we have ηi → η∞ , where η∞ is a unit speed minimal geodesic joining x0 to x, with respect to g (t0 ). Then Lg(t0 +hi ) (ηi ) − Lg(t0 ) (ηi ) dg(t0 +hi ) (x, x0 ) − dg(t0 ) (x, x0 ) ≥ hi hi  d  =  L (ηi ) , dt t=t0 +ki g(t) where 0 < ki < hi , by the mean value theorem. Taking the limit as i → ∞ and since hi → 0+ is a subsequence of an arbitrary positive sequence tending

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42

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

to 0, we have   d  ∂ −  d (x, x0 ) ≥  L (η∞ ) ∂t t=t0 g(t) dt t=t0 g(t)    

∂g 1  (t0 ) η (s) , η (s) ds . ≥ min η∈Z(t0 ) 2 η ∂t


This completes the proof of the lemma.

In the special case of the Ricci flow we immediately obtain the following. Corollary 18.2 (Time derivative of distance under Ricci flow). Let (Mn , g (t)), t ∈ [0, T ), be a complete solution to the Ricci flow. For t0 ∈ [0, T ), in the sense of the lim inf of forward difference quotients,    

 ∂ −   d (x, x0 ) = min − Rcg(t0 ) η (s) , η (s) ds . (18.2) ∂t t=t0 g(t) η∈Z(t0 ) η Mini-Problem 18.3. Is it true that under the Ricci flow,  

∂ −  d (x, x ) = − Rcg(t0 ) γ  (s) , γ  (s) ds 0 g(t)  ∂t t=t0

γ

for any γ ∈ Z (t0 )? Although the above discussion is sufficient for our needs, we recall the lower bound for the time derivative of the distance function using the lim inf of backward difference quotients, in view of the simplicity of its derivation. Lemma 18.4 (Time derivative ∂∂t− of the distance function). If (Mn , g (t)), t ∈ [0, T ), is a smooth 1-parameter family of complete metrics, then for t0 ∈ [0, T ) we have    1 ∂g ∂−    max (t0 ) (η (s), η (s))ds dg(t) (x, x0 ) ≥ (18.3) ∂t  2 η∈Z(t0 ) ∂t η

t=t0

in the sense of the lim inf of backward difference quotients. Proof. Let η : [0, s0 ] → M be any unit speed minimal geodesic joining x0 to x, with respect to g (t0 ). We have Lg(t) (η) ≥ dg(t) (x, x0 ), Lg(t0 ) (η) = dg(t0 ) (x, x0 ) for t ∈ [0, T ). Therefore we have   d  ∂−  d (x, x ) ≥ L (η) . 0 ∂t t=t0 g(t) dt t=t0 g(t) Taking the maximum over minimal η, we have   d  ∂−  dg(t) (x, x0 ) ≥ max ∂t  η∈Z(t0 ) dt  t=t0

Lg(t) (η) , t=t0

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1. ESTIMATES FOR CHANGING DISTANCES

43

where, as above, Z (t) is the compact set of all unit speed minimal geodesics joining x0 to x, with respect to the metric g (t), for t ∈ [0, T ]. The lemma now follows from (18.1).
1.2. Heat operator acting on the distance function. We are interested in the heat operator acting on the distance function. In view of the fact that the distance function is only Lipschitz continuous, recall the following. Definition 18.5 (Laplacian upper bound in the barrier sense). For a function ϕ continuous in a neighborhood of a point x, we say that ∆ϕ (x) ≤ C in the barrier sense if for any ε > 0 there exists a C 2 function ψ defined in a neighborhood of x such that ψ (x) = ϕ (x), ψ ≥ ϕ in a neighborhood of x, and ∆ψ (x) ≤ C + ε. We shall estimate ∆g dg (x0 , x) from above. Lemma 18.6 (Upper bound for ∆d). Let (Mn , g) be a complete Riemannian manifold. Given any x0 ∈ M and x ∈ M − {x0 }, we have  s0 

2

 (n − 1) ζ  (s) − ζ 2 Rc γ  (s), γ  (s) ds (18.4) ∆g dg (x0 , x) ≤ 0

for any unit speed minimal geodesic γ : [0, s0 ] → M joining x0 to x and any continuous piecewise C ∞ function ζ : [0, s0 ] → [0, 1] satisfying ζ (0) = 0 and ζ (s0 ) = 1. Proof. First we shall construct an ‘upper barrier’ for the distance function dg ( · , x0 ) in a neighborhood of x ∈ M − {x0 }. Let    1  ε = min min inj g (γ (s)) , dg (x0 , x) > 0. 2 s∈[0,s0 ] We extend γ to an n-parameter family of paths {γV¯ }V ∈B(ε) as follows (see below for notation), where B (ε) is the ball of radius ε centered at the origin in Tx M and where γ¯0 = γ. Given V ∈ Tx M, define V (s) ∈ Tγ(s) M,

for s ∈ [0, s0 ] ,

to be the parallel translation of V = V (s0 ) along γ, with respect to the metric g. We may define the continuous piecewise smooth vector field V¯ along γ by (18.5) V¯ (s) = ζ (s) V (s) ∈ Tγ(s) M, where ζ : [0, s0 ] → [0, 1] is a continuous piecewise C ∞ function satisfying ζ (0) = 0 and ζ (s0 ) = 1. Note that V¯ (0) = 0 and V¯ (s0 ) = V . Then there exists an n-parameter family of piecewise C ∞ paths {γV¯ }V ∈B(ε) so that (1) γ¯0 (s) = γ (s) for s ∈ [0, s0 ], (2) γV¯ (0) = x0 ,

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44

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(3) γV¯ (s0 ) = expx (V ), and  ∂  γ (s) = V¯ (s) for V ∈ B (ε). (4) ∂r r=0 rV For example, we may define

γV¯ (s) = expγ(s) V¯ (s) for s ∈ [0, s0 ] and V ∈ B (ε). We have Lg (γV¯ ) ≥ dg (expx (V ) , x0 )

for V ∈ B (ε) ,

Lg (γ¯0 ) = dg (x, x0 ) . This implies that the C ∞ function f : B (x, ε) → R+ defined by   f (y) = Lg γ −1 expx (y)

is an upper barrier for dg ( · , x0 ), that is, f (y) ≥ dg (y, x0 )

for y ∈ Bg (x, ε) ,

f (x) = dg (x, x0 ) . Thus, in the barrier sense of Definition 18.5, we have ∆g dg (x0 , x) ≤ ∆g f (x).

(18.6)

Now we turn to bounding ∆g f (x) from above. Let {E1 , . . . , En−1 , γ  (s0 )} be an orthonormal basis of Tx M, with respect to g. Then

E1 (s) , . . . , En−1 (s) , γ  (s) is an orthonormal basis of Tγ(s) M for s ∈ [0, s0 ]. Recall that the second variation in the direction V¯ of the length functional Lg , which holds for families of continuous piecewise smooth paths, is given by  ∂ 2  2 L g (γrV ) δV¯ L g (γ)  ∂r2 r=0  s0    

 ∇γ  V¯ (s)2 − R γ  (s), V¯ (s) V¯ (s), γ  (s) ds = 0 s0 

   2 (18.7) ζ (s) |V (s)|2 − ζ 2 R γ  (s), V (s) V (s) , γ  (s) ds = 0

for any V ∈ B (ε) (regarding the vanishing of the ‘endpoint’ term in the second variation formula, recall that γrV (s0 ) = expx (rV ) is a constant speed geodesic passing through x); in the above formula, R ( · , · ) · denotes the Riemann curvature (3, 1)-tensor.

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1. ESTIMATES FOR CHANGING DISTANCES

45

We compute ∆g f (x) =

n−1 



∇∇f (Ei , Ei ) + ∇∇f γ  (s0 ) , γ  (s0 )

i=1 n−1 

 

∂ 2  ∂ 2  f (expx (rEi )) + f expx rγ  (s0 ) =   2 2 ∂r r=0 ∂r r=0 i=1   n−1  ∂2  ∂ 2   L g (γrEi ) + (s0 + r) = ∂r2 r=0 ∂r2 r=0 i=1   s0  n−1 

  2 2   R γ (s), Ei (s) Ei (s), γ (s) ds, (n − 1) ζ (s) − ζ = 0

i=1

where we used (18.7) to obtain the last equality. The lemma now follows from (18.6) and n−1 



  R γ (s), Ei (s) Ei (s), γ  (s) = Rc γ  (s), γ  (s) .

i=1


1.3. Lower bounds for the time derivative of distance. We now prove the main result of this section, which is Lemma 8.3 in ∂ may denote the lim inf of either forward or [152]. In what follows, ∂t backward difference quotients. Theorem 18.7 (Estimate for changing distances). Let (Mn , g(t)), t ∈ [0, T ), be a complete solution to the Ricci flow. (1) (Heat-type inequality for distance function) Let (x0 , t0 ) ∈ M × [0, T ). If Rc(y, t0 ) ≤ (n − 1)K

(18.8)

for all y ∈ Bg(t0 ) (x0 , r0 ),

where K ≥ 0 and r 0 > 0, then for all x ∈ M − Bg(t0 ) (x0 , r0 ) the distance function is a supersolution to the heat-type equation       1 2 ∂ − ∆g(t) dg(t) (x, x0 ) Kr0 + ≥ −(n − 1) . ∂t 3 r0 t=t0

This inequality is understood in the barrier sense.2 (2) (Changing distances under Ricci flow) Let t0 ∈ [0, T ) and let x0 , x1 ∈ M be two points such that Rc(x, t0 ) ≤ (n − 1)K

for all x ∈ Bg(t0 ) (x0 , r0 ) ∪ Bg(t0 ) (x1 , r0 )

for some K ≥ 0 and r 0 > 0. 2 For differential inequalities in the barrier sense for the reduced distance function, see Chapter 7 in Part I.

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46

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(a) If dg(t0 ) (x0 , x1 ) ≥ 2r0 , then the time derivative of the distance function has the lower bound    2 1 ∂  Kr d (x , x ) ≥ −2(n − 1) + . 0 1 0 ∂t t=t0 g(t) 3 r0

(18.9)

(b) If dg(t0 ) (x0 , x1 ) < 2r0 , then  ∂  d (x0 , x1 ) ≥ −2(n − 1)Kr0 . ∂t t=t0 g(t)

(18.10)

Clearly, in either case we have    1 ∂  d (x0 , x1 ) ≥ −2(n − 1) Kr0 + . (18.11) ∂t t=t0 g(t) r0 Proof. (1) We first observe that as a consequence of Corollary 18.2, Lemma 18.4, and the fact that Z (t0 ) is compact, there exists γ ∈ Z (t0 ) such that  

∂  d (x, x ) ≥ − Rcg(t0 ) γ  (s) , γ  (s) ds. (18.12) 0 g(t)  ∂t t=t0 γ Since s0  dg(t0 ) (x, x0 ) > r0 by assumption, we may choose  s r0 if 0 ≤ s ≤ r0 , ζ (s) = 1 if r0 < s ≤ s0 in (18.4), so that ∆g(t0 ) dg(t0 ) (x0 , x) ≤

 0

−

 n − 1 s2   − 2 Rcg(t0 ) (γ (s), γ (s)) ds r02 r0

r0





s0 r0

Rcg(t0 ) (γ  (s), γ  (s))ds.

We simplify this as ∆g(t0 ) dg(t0 ) (x0 , x)  s0 Rcg(t0 ) (γ  (s), γ  (s))ds ≤− 0     r0  s2 n−1 Rcg(t0 ) (γ  (s), γ  (s)) 1 − 2 + ds + r0 r02 0    s0 2 1   Kr0 + Rcg(t0 ) (γ (s), γ (s))ds + (n − 1) ≤− 3 r0 0 since Rcg(t0 ) ≤ (n − 1)K along γ|[0,r0 ] ⊂ Bg(t0 ) (x0 , r0 ). Therefore, in the barrier sense, we have    s0 2 1   Kr0 + Rcg(t0 ) (γ (s), γ (s))ds + (n − 1) . ∆g(t0 ) dg(t0 ) (x0 , x) ≤ − 3 r0 0

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1. ESTIMATES FOR CHANGING DISTANCES

47

Combining this with (18.12), we obtain       2 1 ∂  . − ∆g(t) dg(t) (x, x0 ) Kr0 + ≥ −(n − 1) ∂t 3 r0 t=t0 This completes the proof of part (1). (2)(a) Let γ : [0, s0 ] → M joining x0 to x be as in part (1), where now we write x = x1 , and let s1 ∈ [r0 , s0 − r0 ] be such that / Bg(t0 ) (x0 , r0 ) ∪ Bg(t0 ) (x1 , r0 ) γ(s1 ) ∈ (here we used dg(t0 ) (x0 , x1 ) ≥ 2r0 ). Then the function h : M → [s0 , ∞) defined by h(x) = dg(t0 ) (x0 , x) + dg(t0 ) (x1 , x) attains its minimum s0 at γ(s1 ) and hence

(18.13) ∆g(t0 ) h (γ(s1 )) ≥ 0 in the barrier sense. Arguing as in the proof of part (1), if we define ζ : [0, s1 ] → [0, 1] by  s r0 if 0 ≤ s ≤ r0 , ζ (s) = 1 if r0 < s ≤ s1 , then we obtain (in the barrier sense) ∆g(t0 ) dg(t0 ) (x0 , γ(s1 ))    s1 2 1   Kr0 + Rc(γ (s), γ (s))ds + (n − 1) . ≤− 3 r0 0 On the other hand, if we define ζ : [s1 , s0 ] → [0, 1] by  1 if s1 ≤ s ≤ s0 − r0 , ζ (s) = s0 −s if s0 − r0 < s ≤ s0 , r0 then we obtain (in the barrier sense) ∆g(t0 ) dg(t0 ) (x1 , γ(s1 ))    s0 1 2 Kr0 + Rc(γ  (s), γ  (s))ds + (n − 1) . ≤− 3 r0 s1 From (18.12), we have

(18.14)

0 ≤ ∆g(t0 ) dg(t0 ) (x0 , γ(s1 )) + ∆g(t0 ) dg(t0 ) (x1 , γ(s1 ))    s0 2 1   Kr0 + Rc(γ (s), γ (s))ds + 2(n − 1) ≤− 3 r0 0    2 1 ∂ Kr0 + d (x0 , x1 ) + 2(n − 1) . ≤ ∂t t=t0 g(t) 3 r0

Part (2)(a) follows.

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(2)(b) When dg(t0 ) (x0 , x1 ) < 2r0 , the minimal geodesic γ (s) in (18.12) joining x0 to x1 is contained in Bg(t0 ) (x0 , r0 ) ∪ Bg(t0 ) (x1 , r0 ). By (18.12) we have  

∂  d (x, x ) ≥ − Rcg(t0 ) γ  (s) , γ  (s) ds 0 g(t)  ∂t t=t0 γ ≥ −(n − 1)K L g(t0 ) (γ) ≥ −2(n − 1)Kr0 .
1.4. Estimating Rc along a stable geodesic. From (18.7) or equation (18.14) in the proof of Theorem 18.7 just above, we can easily derive the following consequence. Recall that a geodesic is stable if its second variation of arc length is nonnegative. Proposition 18.8 (Estimate for Rc along a stable geodesic). If (Mn , g) is a Riemannian manifold and if γ : [0, L] → M is a stable (e.g., minimal ) unit speed geodesic with Rc ≤ (n − 1) K

in B (γ (0) , r) ∪ B (γ (L) , r) ,

where K > 0, r > 0, and L ≥ 2r, then    

1 2  Kr + . Rc γ (s) , γ (s) ds ≤ 2 (n − 1) 3 r γ √ In particular, taking r = 1/ K, we have that if   √  √  Rc ≤ (n − 1) K in B γ (0) , 1/ K ∪ B γ (L) , 1/ K , √ where K > 0 and L ≥ 2/ K, then  √

10 (n − 1) K. Rc γ  (s) , γ  (s) ds ≤ 3 γ Let (Mn , g (τ )) be a solution to the backward Ricci flow. Since   − 

∂ dg(τ ) (x, y) ≤ Rc β  (s) , β  (s) ds ∂τ β for any minimal geodesics β joining x to y (with respect to g (τ )), we conclude by Proposition 18.8 that if Rc ≤ (n − 1) K in B (x, r) ∪ B (y, r), where dg(τ ) (x, y) ≥ 2r, then     − 1 2 ∂ d Kr + . (x, y) ≤ 2 (n − 1) (18.15) ∂τ g(τ ) 3 r On the other hand, under the same assumptions as above except that dg(τ ) (x, y) < 2r, we have   − ∂ d (x, y) ≤ 2 (n − 1) Kr (18.16) ∂τ g(τ )

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2. SPATIAL POINT PICKING METHODS

since


β

49

Rc (β  (s) , β  (s)) ds ≤ (n − 1) K L (β).

Remark 18.9. The above results, which imply bounds for the time derivative of the distance function under the Ricci flow, are useful in the study of ancient solutions. For discussions of this use, see for example Lemma 8.33, Lemma 9.51, and Proposition 9.81, all in [45].

2. Spatial point picking methods In this section we discuss spatial point picking methods primarily due to Perelman. Earlier, basic examples of the use of point picking to obtain estimates occurred in Hamilton’s work on singularity analysis (see especially [92] and [94]) and Schoen’s work [166] and [167] on constant scalar curvature metrics in a conformal class (related to the Yamabe problem). Let ASCR denote the asymptotic scalar curvature ratio (see p. 472 in Part I or definition (19.8) below). In this section we consider point picking in the following scenarios: (1) a Riemannian manifold with sup R < ∞ and ASCR = ∞; (2) a Riemannian manifold with sup R = ∞ (which implies ASCR = ∞); (3) a sequence of Riemannian manifolds where the change in R is unbounded within some finite distance; (4) dimension reduction, where one seeks a limit which splits off a line. 2.1. Point picking when sup R < ∞ and ASCR = ∞. The asymptotic scalar curvature ratio is useful in studying the geometry at infinity of noncompact ancient solutions.3 One reason for why this is true is the following result (Lemma 22.2 of Hamilton’s [92]; see also Theorem 8.44 in [45]).4 Theorem 18.10 (Point picking on complete noncompact manifolds with ASCR = ∞). If (Mn , g, O) is a complete noncompact pointed Riemannian manifold with5 sup R < ∞ M

and

ASCR (g) = ∞,

Roughly speaking, ASCR = ∞ says that the curvature decays slower than quadratically. In particular, a manifold asymptotic to a cone has ASCR < ∞. 4 In the statement of Theorem 8.44 in [45] the assumption that supM R < ∞ was inadvertently omitted. 5 In this theorem, as well as the point picking results below, we do not need to assume that (Mn , g) has Rc ≥ 0; note that, in definition (19.8) of ASCR, we only need to assume that (Mn , g) is a complete noncompact Riemannian manifold. 3

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

then there exist a sequence of points {xi }∞ i=1 in M with d (xi , O) → ∞ and sequences εi → 0 and ri > 0 such that the balls B (xi , ri ) are disjoint and (18.17)

sup R ≤ (1 + εi ) R (xi ) , B(xi ,ri )

(18.18)

R (xi ) ri2 → ∞,

(18.19)

d (xi , O) /ri → ∞.

Remark 18.11 (For the above sequence, rescaled metrics have bounded R on large balls). In the above theorem, let gi  R (xi ) g. Then Rgi (xi ) = 1 and (18.20)

sup

Rgi ≤ 1 + εi ,

Bgi (xi ,˜ ri )

where r˜i  R1/2 (xi ) ri → ∞ as i → ∞. In particular, the rescaled metric gi is such that its scalar curvature Rgi at the center xi is close to its supremum over the ball Bgi (xi , r˜i ), where the radius tends to infinity. Note also that d (xi , O) →∞ ri as i → ∞, so that, even after rescaling, the centers xi of the balls are far from the basepoint O. dgi (xi , O) = R1/2 (xi ) ri

In §1 of Chapter 20 we shall prove that any complete noncompact κsolution with Harnack, as given by Definition 19.27, has ASCR = ∞ when n ≥ 3. 2.2. Point picking when sup R = ∞. The following result is a variant, for complete manifolds with supM R = ∞, of Theorem 18.10. This result enables us to perform dimension reduction (see subsection 2.4 of this section) in certain cases where the curvature is unbounded. Generally, one assumes unbounded curvature for the sole purpose of obtaining a contradiction to prove that the curvature is bounded. The point picking method here is a bit different than in the bounded curvature case (Theorem 18.10); the idea is to choose a sequence of points where the curvatures tend to infinity and then to adjust the choices of points locally. Proposition 18.12 (Point picking on noncompact manifolds with unbounded curvature, version I). Suppose that (Mn , g, O) is a complete noncompact pointed Riemannian manifold with sup R = ∞. M

Then there exist a sequence of points {xi }∞ i=1 and a sequence ri ∈ (0, 1] such that (1) supB(xi ,ri ) R ≤ 4R (xi ), (2) R (xi ) ri2 → ∞ (which implies that R (xi ) → ∞),

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2. SPATIAL POINT PICKING METHODS

51

(3) d (xi , O) → ∞, (4) the balls B (xi , ri ) are disjoint. Proof. (1) Since supM R = ∞, there exists a sequence of points {yi } such that R (yi ) → ∞ as i → ∞. Let xi ∈ B (yi , 2) be a point such that (18.21)

R (xi ) · d2 (xi , ∂B (yi , 2)) =

max R (x) · d2 (x, ∂B (yi , 2)) .

x∈B(yi ,2)

Let ri  12 d (xi , ∂B (yi , 2)); clearly ri ∈ (0, 1]. B (yi , 2) we have d (x, ∂B (yi , 2)) ≥ ri , so that

For any x ∈ B (xi , ri ) ⊂

R (x) ri2 ≤ R (x) · d2 (x, ∂B (yi , 2)) ≤ R (xi ) (2ri )2 . That is, R (x) ≤ 4R (xi ) (2) By (18.21) we have

for all x ∈ B (xi , ri ) .

4R (yi ) = R (yi ) · d2 (yi , ∂B (yi , 2)) ≤ R (xi ) (2ri )2 . Hence R (xi ) ri2 ≥ R (yi ) → ∞. (3) Since R (yi ) → ∞ implies that d (yi , O) → ∞ as i → ∞, by the triangle inequality we have d (xi , O) ≥ d (yi , O) − 2 → ∞ as i → ∞. (4) Since ri ≤ 1 and d (xi , O) → ∞, by passing to a subsequence, we
may assume that the balls B (xi , ri ) are disjoint. A modification of the above result yields the following improvement. Proposition 18.13 (Point picking on noncompact manifolds with unbounded curvature, version II). Let {εi }∞ i=1 be a bounded sequence of positive 6 n real numbers. If (M , g, O) is a complete noncompact pointed Riemannian manifold with supM R = ∞, then there exist sequences {xi }∞ i=1 and ri ∈ (0, 1] such that (1) supB(xi ,ri ) R ≤ (1 + εi ) R (xi ), (2) R (xi ) ri2 → ∞, (3) d (xi , O) → ∞, (4) the balls B (xi , ri ) are disjoint. Remark 18.14. Remark 18.11 equally applies to this proposition. Proof. (1) Since supM R = ∞, we may choose a sequence of points yi ∈ M such that (18.22)

R (yi ) ε2i → ∞.

(Since {εi } is bounded, we also have R (yi ) → ∞.) Define xi ∈ B (yi , 1) so that (18.23) 6

R (xi ) d2 (xi , ∂B (yi , 1)) =

max R (x) d2 (x, ∂B (yi , 1)) .

x∈B(yi ,1)

In applications one often chooses {εi } so that εi → 0 as i → ∞.

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

Define r˜i  d (xi , ∂B (yi , 1)) and take ε˜i ∈ (0, 1) to be chosen below. For any x ∈ B (xi , ε˜i r˜i ) we have d (x, ∂B (yi , 1)) ≥ (1 − ε˜i ) r˜i so that R (x) (1 − ε˜i )2 r˜i2 ≤ R (x) d2 (x, ∂B (yi , 1)) ≤ R (xi ) r˜i2 . Let ri  ε˜i r˜i ≤ 1 (i.e., ε˜i =

√

εi √ ), 1+εi (1+ 1+εi )

and

1 (1 − ε˜i )2

 1 + εi

so that

R (x) ≤ (1 + εi ) R (xi )

for all x ∈ B (xi , ri ) .

(2) By (18.23) we have R (yi ) ≤ R (xi ) r˜i2 , that is, ε2i R (yi )

2 → ∞ √ (1 + εi ) 1 + 1 + εi using (18.22) and the fact that {εi } is bounded. (3) Since R (yi ) → ∞, we have d (xi , O) ≥ d (yi , O) − 1 → ∞ as i → ∞.
(4) This follows from (3) and ri ≤ 1 by passing to a subsequence. R (xi ) ri2 ≥ ε˜2i R (yi ) =

2.3. Point picking when the change in R is unbounded. The next result will be used in the proof of Perelman’s compactness theorem for κ-solutions with Harnack (see Theorem 20.9 below). The idea of applying this result is to suppose that for a sequence of pointed solutions, a uniform curvature bound at the basepoints does not imply a uniform curvature bound in balls of fixed radii centered at these basepoints, i.e., one does not have ‘bounded curvatures at bounded distances’. One then wants to derive a contradiction. Lemma 18.15 (Point picking for sequences of manifolds where the change in R is unbounded). If (Nkn , hk , Ok ) is a sequence of complete7 pointed Riemannian manifolds with Rhk (Ok ) = 1 and if there exists a constant D > 0 such that sup

Rhk (w) → ∞

w∈Bhk (Ok ,D)

as k → ∞, then there exist wk ∈ Bhk (Ok , D + 1) and sk ∈ (0, D + 1) such that (i) Rhk (wk ) s2k → ∞ as k → ∞ and (ii) for all w ∈ Bhk (wk , sk ) Rhk (w) ≤ 2Rhk (wk ) . 7

¯h (Ok , D + 1) to be compact. We just need the balls B k

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2. SPATIAL POINT PICKING METHODS

53

Remark 18.16. Thus, relative to the curvature at the center, we have uniformly bounded curvature on balls of larger and larger radii by (i) and (ii). This conclusion is essentially the same as in Theorem 18.10 and Proposition 18.13. Proof. (i) Define Fk (w)  Rhk (w) (D + 1 − dhk (w, Ok ))2 for w ∈ Bhk (Ok , D + 1). Note that Fk = 0 on ∂Bhk (Ok , D + 1) and Fk (w) ≥

sup Bhk (Ok ,D+1)

sup Bhk (Ok ,D)

Rhk (w) → ∞

as k → ∞. We choose wk ∈ Bhk (Ok , D + 1) to be a point which satisfies Fk (wk ) =  Let sk  1 −

√1 2

sup

Fk (w) .

Bhk (Ok ,D+1)



(D + 1 − dhk (wk , Ok )). Then   3 √ − 2 Fk (wk ) → ∞ Rhk (wk ) s2k = 2

as k → ∞. This proves (i). (ii) For any w ∈ Bhk (wk , sk ) ⊂ Bhk (Ok , D + 1) we have by the triangle inequality sk ≤ D + 1 − dhk (w, Ok ) + dhk (wk , w) 1 − √12 ≤ D + 1 − dhk (w, Ok ) + sk , so that √

sk ≤ D + 1 − dhk (w, Ok ) . 2−1

It follows that Rhk (w) √

s2k

2 2

2 ≤ Fk (w) ≤ Fk (wk ) = Rhk (wk ) sk √

2 2−1 2−1

and hence Rhk (w) ≤ 2Rhk (wk ) for all w ∈ Bhk (wk , sk ). This proves the lemma.
2.4. Dimension reduction for ancient solutions. Dimension reduction is useful in the study of finite time singularity models. Recall that such solutions are ancient and, when their dimension is 3, have nonnegative curvature operator. In this subsection we first give a criterion for when an ancient solution on a manifold of a certain dimension reduces effectively to a solution on a manifold of one less dimension. As a consequence, we prove a dimension reduction result for κ-solutions with ASCR = ∞. In the last part of this subsection we mention Hamilton’s ‘curvature bumps’ theorem.

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

2.4.1. Point picking criteria for the existence of a line in the limit. The following result of Perelman in §11.4 and §11.7 of [152] relies on the Toponogov comparison theorem to construct a line in the limit; see also Theorem 8.46, especially the argument on p. 319, of [45]. Theorem 18.17 (Dimension reduction — splitting a line, version I). Suppose that (Mn , g (t) , O), t ∈ (−∞, ω), ω > 0, is a pointed complete noncompact ancient solution with bounded Rm ≥ 0 such that there exist sequences xi ∈ M, ri → ∞, and Ai → ∞ for which dg(0) (O, xi ) ≥ Ai ri and (18.24)

R(y, 0) ≤ ri−2

for all y ∈ Bg(0) (xi , Ai ri ).

Let gi (t)  ri−2 g(ri2 t). If inj(xi , g(0)) ≥ ι0 ri for some ι0 > 0 independent of i, then there exists a subsequence of solutions {(Mn , gi (t) , xi )} converging in the C ∞ pointed Cheeger–Gromov sense to a complete limit solution (Mn∞ , g∞ (t) , x∞ ), t ≤ 0, which is the product of an (n − 1)-dimensional ancient solution with bounded Rm ≥ 0 and a line. Remark 18.18. (1) Roughly speaking, the hypotheses of the theorem say that, in a relative sense, we have a bound on |Rm| in large balls centered at points far from the origin which satisfy an injectivity radius estimate. (2) Since ri → ∞, the limit in the theorem is a blow-down limit. Since in Theorem 18.17 we do not know if the limit solution is flat, it is useful to consider the following variant. Theorem 18.19 (Dimension reduction — splitting a line, version II). Suppose that a complete noncompact ancient solution (Mn , g (t)), t ∈ (−∞, ω), ω > 0, with bounded Rm ≥ 0 and O ∈ M are such that there exist C < ∞ and sequences xi ∈ M and ri ∈ (0, ∞) such that d2g(0) (O, xi )R (xi , 0) → ∞, ri2 R (xi , 0) → ∞, and (18.25)

R(y, 0) ≤ CR (xi , 0)

for all y ∈ Bg(0) (xi , ri ).

Let gi (t)  R (xi , 0) g(R (xi , 0)−1 t). If inj(xi , g(0)) ≥ ι0 R (xi , 0)−1/2 for some ι0 > 0 independent of i, then there exists a subsequence of rescaled solutions {(Mn , gi (t) , xi )} converging in the C ∞ pointed Cheeger–Gromov sense to a complete limit solution (Mn∞ , g∞ (t) , x∞ ) which is the product of an (n − 1)-dimensional nonflat solution with bounded Rm ≥ 0 and a line.

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2. SPATIAL POINT PICKING METHODS

55

Remark 18.20. In particular, the hypotheses of the theorem imply that ASCR (g (0)) = ∞. Note also that the curvature bounds in the balls in (18.25) are comparable to the curvatures at the centers (this guarantees a nonflat limit). Proof of Theorem 18.19. Since Rm ≥ 0 is bounded, by (18.25) and the trace Harnack estimate, i.e., ∂R ∂t ≥ 0, we have R(y, t) ≤ CR (xi , 0)

for y ∈ Bg(0) (xi , ri ) and t ∈ (−∞, 0].

By assumption we also have an injectivity radius estimate so that Hamilton’s Cheeger–Gromov-type compactness theorem implies that there exists a subsequence of {(Mn , gi (t) , xi )} which converges to a limit solution (Mn∞ , g∞ (t) , x∞ ). Since ri2 R (xi , 0) → ∞, the metrics g∞ (t) are complete with uniformly bounded curvature, Rg∞ (x∞ , 0) = 1, and Rmg∞ ≥ 0. We shall now show that this limit solution is the product of an (n − 1)dimensional solution (which must be nonflat and have bounded Rm ≥ 0) with a line. Consider the metric g (0) at time zero. By passing to a subsequence, we may assume that dg(0) (xi , xi+1 ) ≥ dg(0) (O, xi )

  and that a sequence of unit speed minimal geodesics γi : 0, dg(0) (O, xi ) → M joining O to xi converges to a ray emanating from O.8 Let αi denote a minimal geodesic joining xi to xi+1 . Since γi converges to a ray, we have that the angle between γi and γi+1 at O tends to zero, i.e., xi Oxi+1 → 0. Since the sectional curvatures of g (0) are nonnegative, by the triangle version of the Toponogov comparison theorem (see Theorem G.33(1) and definition (G.17) in Appendix G), the Euclidean comparison angle tends to zero, i.e., ˜ i Oxi+1 → 0. x Since dg(0) (xi , xi+1 ) ≥ dg(0) (O, xi ), this implies that the other Euclidean ˜ i xi+1 tends to π. comparison angle Ox Now for any ρ ∈ (0, ∞) and for i large enough (depending on ρ), there exist points pi ∈ γi and qi ∈ αi such that dg(0) (pi , xi ) = dg(0) (qi , xi ) = ρR (xi , 0)−1/2 since dg(0) (xi , xi+1 )R (xi , 0)1/2 ≥ dg(0) (O, xi )R (xi , 0)1/2 → ∞. There exists a subsequence such that −γi and αi converge to rays −γ∞ and α∞ (respectively) emanating from x∞ in the limit manifold M∞ . The Toponogov 8

For the definition of ray, see Definition I.1 in Appendix I.

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

monotonicity principle, i.e., Lemma G.38,9 then implies that the Euclidean comparison angle ˜ i xi qi → π, p which in turn implies that di (pi , qi ) → di (pi , xi ) + di (qi , xi ) = 2ρ, where di denotes the distance with respect to the metric gi (0), so that d∞ (p∞ , q∞ ) = d∞ (p∞ , x∞ ) + d∞ (q∞ , x∞ ) = 2ρ in M∞ , where p∞ and q∞ are the limits of pi and qi . Since ρ > 0 is arbitrary, this implies that (M∞ , g∞ (0)) contains a line passing through x∞ , namely the concatenated path −γ∞ ∪ α∞ . It follows, as in the proof of Theorem 18.17, that (Mn∞ , g∞ (t)) splits as the product of an (n − 1)-dimensional nonflat solution with bounded Rm ≥ 0 and a line.
Nonexample. Consider the rotationally symmetric expanding soliton on Rn , n ≥ 3, with positive sectional curvature; we discussed its construction in §5 of Chapter 1 in Part I. The sectional curvatures of these solutions decay quadratically (Proposition 1.43 in Part I) and the warping function grows linearly (Proposition 1.42 in Part I). In this case one does not have dimension reduction; indeed a blow-down limit about a sequence of points tending to spatial infinity will limit to a rotationally symmetric cone. 2.4.2. Existence of a line in a limit of a κ-solution with ASCR = ∞. By Theorem 18.10 and Theorem 18.19 we have the following. Corollary 18.21 (Dimension reduction for ancient κ-solutions). If (Mn , g (t)), t ∈ (−∞, ω), where ω > 0, is a noncompact κ-solution with ASCR(t) = ∞, then (M, g (t)) dimension reduces to the product of an (n − 1)-dimensional κ-solution with bounded Rm ≥ 0 and a line. Remark 18.22. As we shall see below, by Theorem 20.1, a noncompact κ-solution with Harnack must have ASCR(t) = ∞ for all t ∈ (−∞, ω). 2.4.3. Alternate method for dimension reduction — curvature bumps. Another way to obtain dimension reduction is via the following result of Hamilton (see §21 and §22 of [92]), which also relies on the Toponogov comparison theorem (see also Theorem 8.51 in [45]). Theorem 18.23 (Finite number of curvature bumps in manifolds with sect ≥ 0). Given any ε > 0 and n ≥ 2, there exists λ < ∞ depending only on ε and n such that if (Mn , g) is a complete Riemannian manifold with 9 Let (Mn , g) be a complete Riemannian manifold with nonnegative sectional curvature. Given p ∈ M and two minimal geodesics α(s) and β(t) such that α(0) = β(0) = p, the Toponogov monotonicity principle says that the Euclidean comparison angle θ(s, t)
 (s) pβ (t) is nonincreasing in both s and t. α

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3. SPACE-TIME POINT PICKING WITH RESTRICTIONS

57

nonnegative sectional curvature and O ∈ M, then there are at most a finite number of disjoint balls B (p, r) with ε in B (p, r) sect (g) ≥ 2 r (called curvature ε-bump) and d (p, O) ≥ λr (called λ-remote). 3. Space-time point picking with restrictions Besides spatial location, as considered in the previous section, in Ricci flow we also have temporal location. In this section we discuss space-time point picking techniques which we shall apply in §2 of Chapter 20 when discussing almost κ-solutions. Let (N n , h (t)), t ∈ [t0 , 0], be a (not necessarily complete) solution to the Ricci flow with nonnegative Ricci curvature. Let B and C be positive constants and define

(18.26) NB,C  (y, t) ∈ N × (t0 , 0] : R (y, t) > C + B(t − t0 )−1 . We call NB,C the set of ‘large curvature points’. Let p ∈ N and suppose the metric ball Bh(0) (p, 1) is compactly contained in N . Assume there exists (y1 , t1 ) ∈ Bh(t1 ) (p, 14 ) × (t0 , 0] which satisfies R(y1 , t1 ) > C + B(t1 − t0 )−1 ,

(18.27) i.e., (y1 , t1 ) ∈ NB,C . Let σ ∈ (0, 1) and (18.28)



A1 (σ)  C + B(t1 − t0 )−1 · min



1 σ (t1 − t0 ) , 2 2304

 .

Lemma 18.24 (Existence of a large curvature point with local control). Let (N n , h (t)) with Rc ≥ 0, p ∈ N , (y1 , t1 ), and A1 (σ) be as above. If A1 (σ) > 1, then the following property 1 holds. For any A ∈ (1, A1 (σ)] there exist t∗ ∈ (t0 , t1 ] and y∗ ∈ Bh(t∗ ) p, 3 such that t∗ − t0 ≥ (1 − σ) (t1 − t0 ), R (y∗ , t∗ ) > C + B(t∗ − t0 )−1 , and (18.29)

R (y, t) ≤ 2R(y∗ , t∗ )

for those points (y, t) ∈ NB,C which satisfy (18.30a) (18.30b)

t ∈ (t∗ − AR−1 (y∗ , t∗ ), t∗ ],   y ∈ Bh(t) p, dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ) .

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

Proof. We shall construct by recursion a finite sequence of points {(yi , ti )} which ends with the desired point (y∗ , t∗ ). Step 1. Construction of a finite sequence of points. Start with (y1 , t1 ) and suppose that we have constructed points (yi , ti ) ∈ NB,C for 1 ≤ i ≤ k, where k ≥ 1, which satisfy for 1 ≤ i ≤ k − 1, (1) ti+1 ∈ (ti − AR−1 (yi , ti ), ti ], (2) (18.31)

dh(ti+1 ) (yi+1 , p) ≤ dh(ti ) (yi , p) + A1/2 R−1/2 (yi , ti ) ,

(3) R (yi+1 , ti+1 ) > 2R (yi , ti ) .

(18.32)

If the curvature bound (18.29) holds for (y∗ , t∗ ) = (yk , tk ) and all (y, t) ∈ NB,C satisfying (18.30), then we stop the construction of the sequence. Otherwise, we may choose (yk+1 , tk+1 ) so that (i) (yk+1 , tk+1 ) ∈ NB,C , (ii) (18.33)

tk+1 ∈ (tk − AR−1 (yk , tk ), tk ],

(iii) (18.34)

dh(tk+1 ) (yk+1 , p) ≤ dh(tk ) (yk , p) + A1/2 R−1/2 (yk , tk ) ,

(iv) (18.35)

R (yk+1 , tk+1 ) > 2R (yk , tk ) .

Clearly (yk+1 , tk+1 ) satisfies (1), (2), and (3). In this way we obtain a sequence {(yi , ti )}αi=1 , where α ∈ [1, ∞]. We now proceed to prove that this sequence must be finite, i.e., α < ∞. By (18.32) and (18.27), we have

(18.36) R (yk , tk ) > 2k−1 R (y1 , t1 ) > 2k−1 C + B(t1 − t0 )−1 for k < α + 1.10 In particular, if α = ∞, then limk→∞ R (yk , tk ) = ∞. Using (18.27), (18.31), and (18.36), we estimate dh(tk ) (yk , p) ≤ dh(tk−1 ) (yk−1 , p) + A1/2 R−1/2 (yk−1 , tk−1 ) ≤ dh(t1 ) (y1 , p) + (18.37) 10


2R(y∗ , t∗ ).

(18.44) Then s satisfies

s − t0 ≥ (t∗ − t0 ) − DR−1 (y∗ , t∗ ) D ≥ (t∗ − t0 ) − C + B (t∗ − t0 )−1 t∗ − t0 , ≥ 2 where we have used assumption (iv) and, in the last inequality, D ≤ C(t∗ −t2 0 )+B . Hence, using (iv) again, we have R (w, s) > 2R(y∗ , t∗ )   > 2 C + B (t∗ − t0 )−1 ≥ C + B (s − t0 )−1 . Thus (w, s) ∈ NB,C . It follows from D ≤

A1/2 100(n−1)

< A that

s ∈ (t∗ − AR−1 (y∗ , t∗ ), t∗ ]. If we can show the claim that   (18.45) w ∈ Bh(s) p, dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ) , then we can apply Lemma 18.24 to get R (w, s) ≤ 2R(y∗ , t∗ ), which is the desired contradiction to (18.44). Now we estimate the distance dh(s) (w, p). Let   (t$ , t∗ ] ⊂ t∗ − DR−1 (y∗ , t∗ ), t∗

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3. SPACE-TIME POINT PICKING WITH RESTRICTIONS

61

be the maximal time interval such that for any t ∈ (t$ , t∗ ], 1 (18.46) dh(t) (w, p) < dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ), 2 where either Case 1. t$ = t∗ − DR−1 (y∗ , t∗ ) or Case 2. t$ > t∗ −DR−1 (y∗ , t∗ ) and equality in (18.46) holds when t = t$ . (Clearly there is such a t$ .)   If Case 1 is true, then since by (18.43), s ∈ t∗ − DR−1 (y∗ , t∗ ), t∗ , we have   w ∈ Bh(s) p, dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ) and we obtain claim (18.45). If Case 2 is true, then we shall obtain a contradiction. Since at time t∗ we have dh(t∗ ) (w, p) ≤ dh(t∗ ) (w, y∗ ) + dh(t∗ ) (y∗ , p) 1 < dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ), 10 there exists a smallest time t# ∈ (t$ , t∗ ) such that 1 (18.47) dh(t) (w, p) > dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ) for t ∈ [t$ , t# ), 5 1 (18.48) dh(t# ) (w, p) = dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ ). 5 For any t ∈ [t$ , t# ], to estimate the change of the distance dh(t) (w, p) using the changing distances inequality (Theorem 18.7(2)), we consider the balls Bh(t) (w, r0 ) and Bh(t) (p, r0 ), for some 1 1/2 −1/2 A R (y∗ , t∗ ) 10 to be chosen later. By (18.47), we have 2r0 ≤ dh(t) (w, p) and by (18.46) and (18.49) we have (18.50)   1/2 −1/2 (y∗ , t∗ ) . Bh(t) (w, r0 ) ∪ Bh(t) (p, r0 ) ⊂ Bh(t) p, dh(t∗ ) (y∗ , p) + A R (18.49)

r0 ≤

Now we verify the curvature assumption for z ∈ Bh(t) (w, r0 )∪Bh(t) (p, r0 ) in Theorem 18.7(2). If (z, t) ∈ / NB,C , then, using R(y∗ , t∗ ) > C + B(t∗ − t0 )−1 , we have R (z, t) ≤ C + B (t − t0 )−1 ≤ C + B (t$ − t0 )−1

−1 ≤ C + B t∗ − DR−1 (y∗ , t∗ ) − t0 ≤ C + 2B (t∗ − t0 )−1 ≤ 2R(y∗ , t∗ ). On the other hand, if (z, t) ∈ NB,C , then by (18.50) and Lemma 18.24 we get R (z, t) ≤ 2R(y∗ , t∗ ).

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

We can now apply Theorem 18.7(2) to get that for any t ∈ [t$ , t# ],   2 1 ∂ d · 2R(y∗ , t∗ )r0 + (w, p) ≥ −2 (n − 1) , ∂t h(t) 3 r0 where on the rhs we used the fact that Rc ≤ R ≤ 2R(y∗ , t∗ ) (since the Ricci curvature is nonnegative). Using 0 ≤ t# − t$ ≤ DR−1 (y∗ , t∗ ), we may integrate this inequality to obtain dh(t# ) (w, p) − dh(t$ ) (w, p)   2 1 ≥ −2 (n − 1) · 2R(y∗ , t∗ )r0 + DR−1 (y∗ , t∗ ); 3 r0 it then follows from (18.48) that 1 dh(t$ ) (w, p) ≤ dh(t∗ ) (y∗ , p) + A1/2 R−1/2 (y∗ , t∗ )   5 R−1 (y∗ , t∗ ) 4 r0 + D. + 2 (n − 1) 3 r0 Take r0 =

1 −1/2 (y∗ , t∗ ). 10 R

We obtain

13 1/2 −1/2 A R (y∗ , t∗ ). 30 However, this is a contradiction to the definition of t$ . This both rules out Case 2 and completes the proof of Proposition 18.25.
dh(t$ ) (w, p) ≤ dh(t∗ ) (y∗ , p) +

4. Necks in manifolds with positive sectional curvature Any singularity model in dimension 3 is either a shrinking spherical space 2 form S 3 /Γ, a shrinking

round cylinder S × R, its quotient under the an2 tipodal map S × R /Z2 , or noncompact with positive sectional curvature (see Corollary 17.16). In the latter case, the underlying manifold is diffeomorphic to R3 . In this section we consider ε-necks in complete noncompact manifolds with positive sectional curvature. One may think of an ε-neck as a good region as compared to a good location. The main result of this section is Proposition 18.33 below, which shall be used in the first proof of Proposition 19.44 and which is a consequence of a result of Sharafutdinov [171]; techniques used in this section include Busemann functions and the Toponogov comparison theorem. 4.1. ε-necks.  First we give the definition of an ε-neck. Let cn  (n − 1) (n − 2) and let gS n−1 , n ≥ 3, be the standard metric on the unit sphere S n−1 (so that the scalar curvature is equal to (n − 1) (n − 2) = c2n ). Definition 18.26 (Embedded ε-neck). Let (N n , h) be a complete Riemannian manifold. Given ε > 0 and p ∈ N with R (p) > 0, let r  cn R (p)−1/2 . A geodesic ball B(p, ε−1 r)  N in (N n , h) is called an embedded ε-neck, if, after scaling h by r−2 , the metric on B(p, ε−1 r) is ε-close

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4. NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE

63

−1 in the C ε +1 -topology12 to a piece of the cylinder metric gS n−1 + du2 on S n−1 × R. More precisely, there exists an embedding

ϕ : B(p, ε−1 r) → S n−1 × R with ϕ (p) ∈ S n−1 × {0} such that on B(p, ε−1 r)   ∗

ϕ gS n−1 + du2 − r−2 h ε−1 +1 < ε. C (r −2 h)

We call ϕ−1 S n−1 × {0} the center sphere of the embedded ε-neck. Note that if ϕ is an embedded ε-neck, for ε sufficiently small, we have the embedding   ψ  ϕ−1 : S n−1 × −ε−1 + 4, ε−1 − 4 → N. Later we shall abuse terminology by saying that ψ also is an ε-neck. Remark 18.27. Note that if ϕ is an embedded ε-neck, where ε ∈ (0, ε (n)], then ϕ is an embedded ε (n)-neck. As an aside, we also note that given any (ε0 , k0 , L0 ), provided ε > 0 is small enough depending on (ε0 , k0 , L0 ), an embedded ε-neck as in Definition 18.26 will contain an embedded (ε0 , k0 , L0 )-neck as defined in §3 of Hamilton’s [94]. The following, Theorem 1.1 on p. 88 in §7 of [94] (see also Lemma 9 in [43] by two of the authors), is what Hamilton calls ‘a replacement for the [smooth] Schoenflies conjecture’. Lemma 18.28 (Center spheres of ε (n)-necks bound smooth balls). There exists ε (n) > 0 such that if (N n , h) is a complete noncompact Riemannian manifold with positive sectional curvature and if   ψ : S n−1 × −ε(n)−1 + 1, ε(n)−1 − 1 → N ⊂ N

is an embedded ε (n)-neck, then the center sphere ψ S n−1 × {0} bounds a smooth n-ball in N . By the Gromoll–Meyer theorem (see [77]), N n is diffeomorphic to Rn . Hence, if n = 3, then Lemma 18.28 follows from a classical result. In particular, since a center sphere ψ S 2 × {0} is an embedded smooth 2sphere, by a result of Alexander [4], it bounds a smooth 3-ball in N .13 However, when n = 4 the corresponding result is unknown. The smooth 4-dimensional Schoenflies conjecture says the following. Conjecture 18.29 (Schoenflies). Every smooth embedded 3-sphere in R4 bounds a smooth 4-ball.

Here ε−1 denotes the least integer greater than or equal to ε−1 . 13 Note that in dimension 3 the topological, piecewise-linear, and differentiable categories are the same. 12

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

4.2. The diameter of a level set of a Busemann function in an ε-neck. The following lemma will be used to prove Proposition 18.33. Given a subset S ⊂ M, let diam (S)  supx,y∈S d (x, y). Lemma 18.30 (Bounds for the diameter of a level set of a Busemann function intersecting the center sphere of a neck). For any n ≥ 3 there exists ε(n) > 0 which has the following property. Suppose (N n , h) is a complete noncompact Riemannian manifold with positive sectional curvature and suppose   ψ : S n−1 × −ε(n)−1 + 1, ε(n)−1 − 1 → N ⊂ N is an embedded ε(n)-neck. Let γ : [0, ∞) → N be a (unit speed ) ray and let bγ : N → R be the Busemann function defined by γ.14 Let q0  ψ (z0 , 0) ∈ ψ S n−1 × {0} be a point on the center sphere S and let b0  bγ (q0 ). Then 15 the level set b−1 γ (b0 ) has the property

(18.51) where cn 





11 −1/2 πcn R (q0 )−1/2 ≤ diam b−1 , γ (b0 ) ≤ 11πcn R (q0 ) 12 (n − 1) (n − 2).

Proof. Let ε (n) ∈ (0, 1), to be chosen later sufficiently small. By Lemma 18.28, the manifold minus the center sphere of the neck, i.e., N − ψ S n−1 × {0} , is disconnected and consists of two connected components: one component, which we call Ninn , is diffeomorphic to Rn and the other component, which we call Nout , is diffeomorphic to Rn − 0 . Without loss of generality we may assume

∂inn N  ψ S n−1 × −ε(n)−1 + 1 ⊂ Ninn and



∂out N  ψ S n−1 × ε(n)−1 − 1 ⊂ Nout . Note that Ninn ∪ N ⊂ N is a closed smooth n-ball. Since γ (s) is a ray, there exists s0 ∈ (0, ∞) such that for s ≥ s0 we have γ (s) ∈ N − (Ninn ∪ N) .



} Hence for s ≥ s0 , any minimal geodesic joining any point p ∈ ψ S n−1 × {u 0

with u0 ≤ ε(n)−1 − 2 to γ (s) must intersect each slice ψ S n−1 × {u} for u ∈ u0 , ε(n)−1 − 1 . 14

That is, bγ (x)
lim (s − d (γ (s) , x)) s→∞

=

sup (s − d (γ (s) , x)) . s∈[0,∞)

15

  Keep in mind that diam S n−1 (r) = πr ≡ πcn R−1/2 .

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4. NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE

65

Let A be a constant to be chosen later, and ε (n) will be chosen so that it satisfies ε(n)−1 − 1 > A. First we prove that the Busemann function satisfies

bγ (q) < b0 for any q ∈ Ninn − ψ S n−1 × (−A, 0) provided A ≥ 85 π. For s sufficiently large, any unit speed minimal geodesic βs (t) joining βs (0) = q and βs (ts ) = γ (s), where ts  d (q, γ (s)), must intersect ψ S n−1 × {u} for each u ∈ −A, ε(n)−1 −

1 . In particular there exist t0 ∈ [0, ts ] such that βs (t0 ) ∈ ψ S n−1 × {0} . By the triangle inequality we have s − d (γ (s) , q) = s − d (γ (s) , βs (t0 )) − d (βs (t0 ) , q) ≤ s − d (γ (s) , q0 ) + d (βs (t0 ) , q0 ) − d (βs (t0 ) , q) ≤ bγ (q0 ) + d (βs (t0 ) , q0 ) − d (βs (t0 ) , q) .  (n − 1) (n − 2). If ε(n) is small Recall that we have earlier set cn  enough, since the geometry of the neck N is close to that of the standard cylinder, we have R (q0 ) > 0 and16 (18.52)

(18.53)

6 d (βs (t0 ) , q0 ) ≤ πcn R (q0 )−1/2 5

and (18.54)

5 d (βs (t0 ) , q) > Acn R (q0 )−1/2 . 6

Hence for A ≥ 85 π, by (18.52), (18.53), and (18.54), we have s − d (γ (s) , q) ≤ bγ (q0 ) −

2 πcn R (q0 )−1/2 . 15



Thus for q ∈ Ninn − ψ S n−1 × (−A, 0) , where A ≥ 85 π, (18.55)

bγ (q) = lim (s − d (γ (s) , q)) ≤ bγ (q0 ) − s→∞

Second we prove that bγ (q) > b0

2 πcn R (q0 )−1/2 < b0 . 15



for any q ∈ Nout − ψ S n−1 × (0, A)

provided A ≥ 5π. For s sufficiently large, any unit speed minimal n−1 geodesic

× {A} at ηs (t) joining ηs (0) = q0 and ηs (ts ) = γ (s) intersects ψ S ηs (t1 ). Let ζ (t) be a unit speed minimal geodesic joining ζ (0) = q0 and ζ (tq ) = q. The minimal geodesic ζ intersects ψ S n−1 × {A} at ζ (t2 ). Consider the triangle ∆ηs (t1 ) q0 ζ (t2 ); since the geometry of the neck N is 16

The points βs (t0 ) and q0 lie on the same center sphere. For the standard (round) cylinder S n−1 (r) × R of radius r we have R ≡ c2n r−2 ; note that diam S n−1 (r) = πr = πcn R−1/2 .

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66

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

close to that of the standard cylinder, we have that, provided A ≥ 5π,17 the Euclidean comparison angle satisfies ˜ s (t1 ) q0 ζ (t2 ) ≤ π . η 6 By applying the Toponogov comparison theorem (hinge version) to the triangle ∆q0 γ (s) q and the law of cosines, we have d2 (q, γ (s)) ≤ d2 (q0 , γ (s)) + d2 (q0 , q)

  ˜ s (t1 ) q0 ζ (t2 ) − 2d (q0 , γ (s)) d (q0 , q) cos η

(18.56)

≤ d2 (q0 , γ (s)) + d2 (q0 , q) − d (q0 , γ (s)) d (q0 , q) .

If s is large enough,18 then (18.57)

1 d (q0 , γ (s)) ≥ d (q0 , q) + 2

!

3 d2 (q, γ (s)) − d2 (q0 , q) 4

1 ≥ d (q, γ (s)) + d (q0 , q) , 4 where the second inequality holds provided d (q, γ (s)) ≥

13 8 d (q0 , q).

Hence

1 s − d (γ (s) , q) ≥ s − d (q0 , γ (s)) + d (q0 , q) . 4

By taking the limit s → ∞, we have for q ∈ Nout − ψ S n−1 × (0, A) , where A ≥ 5π, (18.58)

1 bγ (q) ≥ bγ (q0 ) + d (q0 , q) > b0 . 4

Third we prove the second inequality in (18.51), i.e., diam b−1 γ (b0 ) ≤ 11πcn R (q0 )−1/2 . Now choose A = 5π. Note that for any q ∈ b−1 γ (b0 ), by

19 n−1 × [−5π, 5π] . That is, (18.55) and (18.58), we have q ∈ ψ S

n−1 × [−5π, 5π] . b−1 γ (b0 ) ⊂ ψ S In fact we can make the angle ηs (t1 ) q0 ζ (t2 ) as small as we like by choosing A sufficiently large and ε (n) sufficiently small. In the standard unit cylinder S n−1 × R, if we have points a ∈ S n−1 × {0} and b, c ∈ S n−1 × {A}, where A ≥ 5π, then   π 1 < . bac ≤ 2 tan−1 10 12 17

18 19

If d (q0 , γ (s)) ≥ 12 d (q0 , q), then the first inequality in (18.57) is equivalent to (18.56). If       q ∈ Ninn − ψ S n−1 × (−5π, 0) ∪ Nout − ψ S n−1 × (0, 5π)   = N − ψ S n−1 × [−5π, 5π] ,

then bγ (q) = b0 .

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4. NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE

67

If ε(n) > 0 is small enough, since the geometry of the neck N is close to that of the standard cylinder, then we have20

n−1

diam b−1 × [−5π, 5π] γ (b0 ) ≤ diam ψ S

11 cn R (q0 )−1/2 diam S n−1 × [−5π, 5π] ≤√ 101 " 11 −1/2 cn R (q0 ) π 2 + (10π)2 ≤√ 101 ≤ 11πcn R (q0 )−1/2 .

Finally we prove the first inequality in (18.51), i.e., diam b−1 γ (b0 ) ≥ −1/2 11 . Note that for fixed z ∈ S n−1 , the function bγ (ψ (z, u)) is 12 πcn R (q0 ) continuous in u. Inequalities (18.55) and (18.58) imply that for any z ∈ S n−1 we have bγ (ψ (z, 5π)) > b0 and bγ (ψ (z, −5π)) < b0 . Hence there exists u (z) ∈ [−5π, 5π] such that bγ (ψ (z, u (z))) = b0 . If we choose z ∈ S n−1 to be the antipodal point of z0 (recall ψ (z0 , 0) = q0 ), then since the geometry of the neck N is close to that of the standard cylinder, it is clear that

diam b−1 γ (b0 ) ≥ d (ψ (z, u (z)) , ψ (z0 , 0)) 11 ≥ cn R (q0 )−1/2 dS n−1 ×[−5π,5π] ((z, u (z)) , (z0 , 0)) 12 11 ≥ πcn R (q0 )−1/2 . 12
The existence of a ‘sufficiently good’ neck implies the following. Lemma 18.31 (Manifold containing an ε(n)-neck has bγ attaining its minimum). There exists ε (n) > 0 such that if (N n , h) is a complete noncompact Riemannian manifold with positive sectional curvature containing an embedded ε(n)-neck, then for any ray γ the corresponding Busemann function bγ is bounded from below and attains its minimum. Proof. By the proof of Lemma 18.30, if ε (n) > 0 is sufficiently small and if N ⊂ N is an embedded ε(n)-neck, then Ninn ∪ N is compact and for x ∈ N − Ninn ∪ N , we have bγ (x) > b0 . Hence inf N bγ = minNinn ∪N bγ is
attained at some point in Ninn ∪ N ⊂ N . Remark 18.32. Without the existence of a sufficiently good neck, it is easy to come up with a counterexample to bγ being bounded from below. See Exercise I.14 in Appendix I. 20

In particular we choose ε(n) > 0 small enough so that the second inequality holds > 1. simply because √11 101

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

4.3. Bounding radii of farther ε-necks by radii of closer ε-necks. We now estimate the scalar curvature of ε-necks farther from an origin (i.e., a fixed point) by the scalar curvature of ε-necks closer to the origin. Note √ that the radius of an ideal neck, i.e., an exact cylinder, is equal to cn / R. Proposition 18.33 (In manifolds with sect > 0 the radii of farther ε(n)necks are almost larger). There exists ε(n) > 0 depending only on n ≥ 3 which has the following property. Suppose (N n , h) is a complete noncompact Riemannian manifold with positive sectional curvature. Let γ : [0, ∞) → N be a (unit speed ) ray and let bγ : N → R be the Busemann function defined by γ. If for i = 1, 2, Ni are two disjoint embedded ε(n)-necks with yi on the center sphere Si of the neck Ni , then the scalar curvatures of yi satisfy (18.59)

R (y2 ) ≤ 144R (y1 )

21 provided d (p∗ , y1 ) ≤ d (p∗ , y2 ), where p∗ is any point in b−1 γ (minx∈N bγ (x)).

Proof of Proposition 18.33. Let bi  bγ (yi ) for i = 1, 2. By our assumptions on the necks and by (18.55) and (18.58), we have b2 > b1 . Note that bγ is a convex function (see Proposition B.54 in Volume One). Therefore we may apply Sharafutdinov’s theorem (see Theorem 3 in [171] or the expository Theorem I.24 in Appendix I) to obtain

−1

diam b−1 γ (b1 ) ≤ diam bγ (b2 ) . Hence by Lemma 18.30 we have 11 πcn R (y1 )−1/2 ≤ 11πcn R (y2 )−1/2 , 12 that is, R (y2 ) ≤ (12)2 R (y1 ) .
5. Localized no local collapsing theorem The aim of this section is to prove a localized version of the no local collapsing theorem given by Theorem 6.74 and Theorem 8.26 in Part I. We shall assume that the reader is familiar with the L-length L (γ), the reduced 1 ¯ L (q, τ ), and the reduced volume V˜ (τ ) distance  (q, τ ) = 2√1 τ L (q, τ ) = 4τ (see Chapter 7 in Part I or §5 in Chapter 19 in this volume for definitions and properties). 21

See Lemma 18.31 for why bγ attains its minimum.

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5. LOCALIZED NO LOCAL COLLAPSING THEOREM

69

5.1. Statement of the main theorem. Recall that, as compared to Definition 19.1 below, there is another version of κ-noncollapsing for solutions of the Ricci flow which uses a stronger assumption (see also Definition 8.23 in Part I). Definition 18.34 (Weakly κ-noncollapsed). Let (Mn , g (t)) be a complete solution to the Ricci flow with bounded curvature for each time t ∈ [0, T ). Given κ, ρ ∈ (0, ∞), we say that g(t) is weakly κ-noncollapsed at (x0 , t0 ) ∈ M × [0, T ) below  the scale  ρ if for any r ∈ (0, ρ] satisfying |Rm| ≤ r−2 in Bg(t0 ) (x0 , r) × t0 − r2 , t0 , we have Volg(t0 ) Bg(t0 ) (x0 , r) ≥ κrn . Remark 18.35. Since the function Ψ (r)  r2

max

¯g(t ) (x0 ,r)×[t0 −r 2 ,t0 ] B 0

  Rmg(t) (x)

is nondecreasing and since limr→0 Ψ (r) = 0, there exists a maximum radius r¯ ∈ (0, ∞] such that Ψ (r) ≤ 1 for r ∈ (0, r¯). We have that g(t) is weakly κ-noncollapsed at (x0 , t0 ) below the scale ρ if and only if Volg(t0 ) Bg(t0 ) (x0 , r) ≥ κrn

for all r ∈ (0, min {ρ, r¯}].

The main result of this section is the following, which is Theorem 8.2 in Perelman [152]. Theorem 18.36 (Localized no local collapsing). For any n ≥ 2 and A > 0 there exists κ(n, A) > 0 satisfying the following property. For any complete solution g(t), t ∈ [0, r02 ], of the Ricci flow on a manifold Mn with bounded curvature and any x0 ∈ M, if   |Rm| ≤ r0−2 in Bg(0) (x0 , r0 ) × 0, r02 and if Volg(0) Bg(0) (x0 , r0 ) ≥ A−1 r0n , then g(t) is weakly κ (n, A)-noncollapsed below the scale r0 at any point in Bg(r02 ) (x0 , Ar0 ) × r02 . n Remark 18.37. Given √ a nonflat complete solution (M , g (t)), t ∈ [0, T ], and x0 ∈ M, let r¯0 ≤ T be the maximum radius such that   Rmg(t) (x) ≤ 1. max r¯02 ¯g(0) (x0 ,¯ B r0 )×[0,¯ r02 ]

Define A (r0 ) 

r0n . Volg(0) Bg(0) (x0 , r0 )

In this respect the theorem says that for any r0 ∈ (0, r¯0 ] the solution g(t) is below the scale r0 at any point in the ball weakly κ (n, A (r0 ))-noncollapsed Bg(r02 ) (x0 , A (r0 ) r0 ) × r02 .

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

5.2. Proof of Theorem 18.36. The proof is similar to the proof of the ‘weakened no local collapsing theorem’ in §7.3 of [152] (see also Theorem 8.26 in Part I). We may assume that r0 = 1 by replacing g (t) by the solution r0−2 g r02 t . Let (Mn , g(t)), t ∈ [0, 1], be a complete solution to the Ricci flow with bounded curvature and suppose x0 ∈ M is such that in Bg(0) (x0 , 1) × [0, 1]

|Rm| ≤ 1 and Volg(0) Bg(0) (x0 , 1) ≥

A−1 .

Let

τ (t) = 1 − t

and

g¯ (τ ) = g (t) .

Then g¯ (τ ), τ ∈ [0, 1], is a solution to the backward Ricci flow. Suppose that x ∈ Bg(1) (x0 , A)

(18.60)

  and r ∈ (0, 1] are such that |Rm| ≤ r−2 in Bg(1) (x, r)× 1 − r2 , 1 . We define Volg(1) Bg(1) (x, r) . rn Note that, by Definition 19.50 below, g (t) is strongly (κ + ε)-collapsed at (x, 1) at scale 1 for any ε > 0. The theorem shall follow from bounding κ from below by a positive constant depending only on n and A. In the following, the reduced distance  and reduced volume V˜ are defined with respect to the solution g¯ (τ ) and the basepoint (x, 0). Let c1 (n) ∈ (0, 12 ] be the constant given in Theorem 8.24 in Part I (see also Theorem 19.51 below). (1) If κ1/n ≥ c1 (n), then there is nothing to prove since we have a lower bound for κ depending only on n. (2) On the other hand, if κ1/n < c1 (n), then by Theorem 8.24 in Part I the reduced volume has the upper bound   (18.61) V˜ κ1/n r2 ≤ c2 (n, κ) , κ = κ (x, r) 

where c2 (n, κ) 

exp

1

6 n (n − n/2

(4π)

1)

1 2

κ + ωn−1 (n − 2)

n−2 2

e

− n−2 2

 exp −



1 1

.

2κ 2n

Observe that (18.62)

lim c2 (n, κ) = 0.

κ→0+

We shall show that there exists a constant c3 (n, A) > 0 independent of κ such that (18.63) V˜ (1) ≥ c3 (n, A) . Then, by the monotonicity of the reduced volume (see Corollary 8.17 in Part I) and (18.61), we have   (18.64) c3 (n, A) ≤ V˜ (1) ≤ V˜ κ1/n r2 ≤ c2 (n, κ)

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5. LOCALIZED NO LOCAL COLLAPSING THEOREM

71

(here we used κ1/n r2 < c1 (n) ≤ 12 ). From this and (18.62), we conclude that there exists κ (n, A) > 0 such that κ ≥ κ (n, A). Theorem 18.36 is proved, except that we need to give a proof of (18.63). We shall need the following lemma, which will be proved in the next subsection, to estimate V˜ (1) from below. Lemma 18.38. Let (Mn , g(t)), t ∈ [0, 1], be a complete solution of the Ricci flow with bounded curvature. Suppose for some x0 ∈ M that we have (18.65)

|Rm| ≤ 1

in Bg(0) (x0 , 1) × [0, 1] .

Then (i) |W |g(t) ≤ en−1 |W |g(0) for any z ∈ Bg(0) (x0 , 1), t ∈ [0, 1], and W ∈ Tz M; (ii) Bg(t) (x0 , e1−n ) ⊂ Bg(0) (x0 , 1) for all t ∈ [0, 1]; (iii) there exists c4 (n, A) < ∞ such that the reduced distance, as defined above, satisfies min

¯ 1 (x0 ,e1−n ) y∈B g( )

(y, 12 ) ≤ c4 (n, A) .

2

Next we give an upper bound (depending on n and A) of  (q, 1) for q ∈ Bg¯(1) (x0 , 1) = Bg(0) (x0 , 1). By Lemma 18.38(iii), there exists x1 ∈ Bg¯( 1 ) (x0 , e1−n ) such that 2

1 (x1 , ) ≤ c4 (n, A) . 2 1 Let γ1 : [0,

2 ] → M be a minimal L-geodesic with (γ1 (τ ) , τ ) joining (x, 0) 1 to x1 , 2 , where x is as in (18.60). By Lemma 18.38(ii), for any q ∈ Bg¯(1) (x0 , 1) there exists a minimal (Riemannian) geodesic

1 γ2 : [ , 1] → Bg¯(1) (x0 , 1) 2 joining x1 to q, with respect to the metric g¯(1), which satisfies    γ (s) = 2dg¯(1) (x1 , q) ≤ 4. 2 g¯(1) By Lemma 18.38(i), we have    γ2 (s)2

g¯(s)

2  ≤ e2(n−1) γ2 (s)g¯(1) ≤ 16e2(n−1) .

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18. GEOMETRIC TOOLS AND POINT PICKING METHODS

We can estimate  (q, 1) by the reduced length of the concatenated path γ1 ∪ γ2 : [0, 1] → M:  (q, 1) ≤

1 2



1 2

0

1 + 2



 2  √  s R (γ1 (s) , s) + γ1 (s)g¯(s) ds   2  s R (γ2 (s) , s) + γ2 (s)g¯(s) ds

1√ 1 2

1 ≤ c4 (n, A) + 2



  s n (n − 1) + 16e2(n−1) ds

1√ 1 2

 c5 (n, A) . We have proved  (q, 1) ≤ c5 (n, A)

for any q ∈ Bg¯(1) (x0 , 1).

Now we can estimate the reduced volume V˜ (1) from below:    n  − −
(q,τ ) (4πτ ) 2 e dµg¯(τ ) (q) V˜ (1) ≥  Bg¯(1) (x0 ,1) τ =1  n ≥ (4π)− 2 e−c5 (n,A) dµg¯(1) Bg¯(1) (x0 ,1) −n 2

≥ (4π)

e−c5 (n,A) A−1 .

Now (18.63) is proved by taking c3 (n, A)  (4π)− 2 e−c5 (n,A) A−1 ; hence Theorem 18.36 is proved. n

5.3. Proof of Lemma 18.38. (i) This standard estimate holds since (18.65) implies      ∂  g (t) (W, W ) = 2 Rcg(t) (W, W ) ≤ 2 (n − 1) |W |2 g(t)   ∂t for any t ∈ [0, 1] and any tangent vector W on Bg(0) (x0 , 1) (see also (3.3) in Part I ). (ii) Let γ (s), 0 ≤ s ≤ s0 , be any unit speed geodesic, with respect to g (t), such that γ (0) = x0 and s0 < e1−n . It follows from (i) that Lg(0) (γ) ≤ en−1 s0 < 1. In particular Bg(t) (x0 , e1−n ) ⊂ Bg(0) (x0 , 1). (iii) We first define a smooth nondecreasing function φ : R → [1, ∞]

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5. LOCALIZED NO LOCAL COLLAPSING THEOREM

73

such that

⎧ 1 ⎪ ⎪

⎪ ⎨ 0 ≤ φ (u) ≤ 12 exp en+1 + n   φ(u) = 1 ⎪ exp ⎪ 1−n e −u ⎪ ⎩ ∞

if u ∈ (−∞, 12 e1−n ], if u ∈ ( 12 e1−n , 23 e1−n ), if u ∈ [ 23 e1−n , e1−n ), if u ∈ [e1−n , ∞).

For any A > 0 there exists C1 (n, A) < ∞ such that 2  2 φ (u) − φ (u) ≥ (2A + nen )φ (u) − C1 (n, A)φ(u) (18.66) φ(u)

for u ∈ −∞, e1−n , that is, 2 φ(u)

(18.67)

(φ (u))2 − φ (u) − (2A + nen )φ (u) φ(u)

≥ −C1 (n, A).

To see this, define C3 (n) 

2 φ(u)

min

(φ (u))2 − φ (u) − (2A + nen )φ (u) φ(u)

u∈[ 12 e1−n , 23 e1−n ]

< ∞.

On the other hand, for u ∈ [ 23 e1−n , e1−n ) we have 2 φ(u)

(φ (u))2 − φ (u) − (2A + nen )φ (u) φ(u)

(18.68)

=

1 (e1−n − u)4

−

2 (e1−n − u)3

−

2A + nen (e1−n − u)2

.

For u ∈ [ 23 e1−n , e1−n ) the rhs of (18.68) is bounded from below by some finite number −C2 (n, A). Therefore (18.67) holds for u ∈ −∞, e1−n with −C1 (n, A)  min {−C2 (n, A), C3 (n)} . The next step is to apply the weak maximum principle to the function h : M× [0, 1] → [−∞, ∞] defined by ¯ 1 − t) + 2n + 1), h(w, t)  φ(dg(t) (x0 , w) − (2t − 1)A) · (L(w, ¯ where L(w, τ ) = 4τ  (w, τ ). Claim. Define H (t)  min h (w, t) . w∈M

Then (1) H (1) = h (x, 1) = 2n + 1, where x is as in (18.60). (2) ¯ L(w, τ ) ≥ −2n for τ ≤

1 2

and w ∈ M.

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74

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(3) There exists y ∈ Bg¯( 1 ) (x0 , e1−n ) such that 2

H

1

2



= h y, 12 .

(4) At any point (y, t) where h(y, t) = H (t) with t ≥ 12 , we have  (18.69)

 ∂ − ∆ h(y, t) ≥ − (2n + C1 (n, A)) h(y, t). ∂t

Assuming the claim, we finish the proof of part (iii) of Lemma 18.38. It follows from part (4) of the claim and Lemma 3.5 in [89] (on differentiating a minimum function; see subsection 1.1 of this chapter) that d H (t) ≥ − (2n + C1 (n, A)) H (t) dt

for t ≥ 12 .

Integrating this on [ 12 , 1] while using H (1) = 2n + 1, we obtain H( 12 ) ≤ c6 (n, A) . This and part (3) of the claim imply that there exists y ∈ Bg¯( 1 ) (x0 , e1−n ) 2 such that

c6 (n, A) ≥ h y, 12 = φ(dg( 1 ) (x0 , y)) · (2(y, 12 ) + 2n + 1) 2

≥ 2(y, 12 ) + 2n + 1 whenever (y, 12 ) ≥ −n −

1 2

(since φ ≥ 1). Thus we have (iii).

Finally we give a proof of the claim. ¯ 0) = (1) This part follows from dg(1) (x, x0 ) ≤ A (by (18.60)) and L(w, which is justified by the equation after (7.94) in Part I. Indeed, we have

d2g¯(0) (w, x),

¯ 0) + 2n + 1) h(w, 1) = φ(dg(1) (x0 , w) − A) · (L(w, = φ(dg(1) (x0 , w) − A) · (d2g¯(0) (w, x) + 2n + 1) ≥ 2n + 1 with equality when w = x since φ (u) = 1 when u ≤ 0. (2) Let Rmin (t)  minz∈M R (z, t). From the evolution equation for the d 2 , so that Rmin ≥ n2 Rmin scalar curvature, we have dt Rg¯(τ ) (·) ≥ −

n . 2 (1 − τ )

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5. LOCALIZED NO LOCAL COLLAPSING THEOREM

75

Hence, for a minimal L-geodesic γ from (x, 0) to (w, τ ), we have  τ  2  √  √ ¯ s Rg¯ (γ (s) , s) + γ  (s)g¯(s) ds L(w, τ) = 2 τ   0 τ √ √ n ≥2 τ ds s − 2 (1 − s) 0 1 ≥ −2n if τ ≤ . 2 (3) This part follows directly from (2), which implies for w ∈ M 1 ¯ ) + 2n + 1 ≥ 1. L(w, 2 ¯=L ¯ (w, 1 − t). (4) Let u = dg(t) (x0 , w)−(2t−1)A, d = dg(t) (x0 , w), and L We compute   ¯

∂d ∂h  ¯ + 2n + 1 + φ(u) ∂ L , (18.70a) (w, t) = φ (u) − 2A L ∂t ∂t ∂t  ¯ ¯ + 2n + 1)φ (u)∇d + φ(u)∇L, (18.70b) ∇h(w, t) = (L



 ¯ + 2n + 1 φ (u) |∇d|2 ¯ + 2n + 1 φ (u)∆d + L ∆h(w, t) = L   ¯ + φ(u)∆L ¯ + 2 ∇φ(u), ∇L



 ¯ + 2n + 1 φ (u) ¯ + 2n + 1 φ (u)∆d + L (18.70c) = L   ¯ + φ(u)∆L. ¯ + 2 ∇φ(u), ∇L Hence       ∂ ∂h ¯ − 2 ∇φ(u), ∇L ¯ − ∆h (w, t) = φ(u) −∆ L ∂t ∂t    ∂d   ¯ (18.71) − ∆d − 2A . + (L + 2n + 1) −φ (u) + φ (u) ∂t Given t, at a minimum point y of h( · , t) we have ∇h = 0. Hence, by (18.70b), we have at (y, t) ¯ = −(L ¯ + 2n + 1)φ (u)∇d φ(u)∇L and hence



2   ¯ = −(L ¯ + 2n + 1) (φ (u)) . ∇φ(u), ∇L φ(u) It now follows from (18.71) that     ∂ ∂h ¯ − ∆h (y, t) = φ(u) −∆ L ∂t ∂t   2 (φ (u))2 −φ (u) + φ(u) ¯ + 2n + 1) (18.72) . + (L

 + ∂d − ∆d − 2A φ (u) ∂t

Note that for t ≥ 12 we have dg(t) (w, x0 ) ≥ 12 e1−n whenever φ (u) = 0 at (w, t). From Bg(t) (x0 , e1−n ) ⊂ Bg(0) (x0 , 1) for all t ∈ [0, 1] and from

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76

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

|Rc(z, t)| ≤ n − 1 for all z ∈ Bg(t) (x0 , 12 e1−n ), we can apply Theorem 18.7(1) to obtain   ∂ − ∆ dg(t) (w, x0 ) ≥ −(n − 1)en ≥ −nen (18.73) ∂t whenever (w, t) has φ (u) = 0. ¯ ≤ 2n. Applying (18.66) ¯ τ + ∆L Recall from Lemma 7.46 in Part I that L   and (18.73) to (18.72) presuming φ (u) = 0 (if φ (u) = 0, then this follows from (18.72) directly), we find that at any point (y, t) where both h(y, t) = H (t) and t ≥ 12  

∂ ¯ + 2n + 1 C1 (n, A)φ(u) − ∆ h(y, t) ≥ −2nφ(u) − L ∂t ≥ − (2n + C1 (n, A)) h(y, t). ¯ τ ) ≥ −2n for τ ≤ 1 to obtain the last inequality. Here we have used L(y, 2 This proves part (4) of the claim and hence both Lemma 18.38 and Theorem 18.36.
Remark 18.39. Regarding the standard issue of the nonsmoothness of h, see the remark at the end of §2 in Chapter 25. 6. Notes and commentary §2. There are several other point picking lemmas used in the Ricci flow besides the ones we give in this chapter. For example, see (1) the point picking results in §1 of Chapter 22, which are used in the proof of the pseudolocality theorem, (2) the point picking result in the proof of Theorem 12.1 in Perelman [152]. Now we raise a general question. Problem 18.40 (General point picking). Formulate Bourbaki-type results about point picking (in particular, show general results exhibiting the essences of point picking). It would be desirable for the results to include the results in this section as well as other point picking results in Ricci flow, in particular, the methods in Perelman [152] and Hamilton [92]. Perhaps such a result may include phrases such as the following for example. Let S be a metric space, let f : S → R be a function, and let φ : S → R be a reference function, etc., where examples of f are R and |Rm| and examples of φ are d (·, O)2 and T − t. In contrast to the nontrivial limits we get from the above methods of point picking, consider the following pathological case. Exercise 18.41 (Trivial limits). Show that for any sequence of pointed Riemannian manifolds {(Mni , gi , Oi )}i∈N , there exists a sequence Ki → ∞ such that (Mni , Ki gi , Oi ) converges to flat Euclidean space.

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6. NOTES AND COMMENTARY

77

Hint. Just choose Ki sufficiently large. Note that the manifolds need not even be complete. §4. In Appendix H we discuss the proof of Sharafutdinov’s theorem.

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http://dx.doi.org/10.1090/surv/163/03

CHAPTER 19

Geometric Properties of κ-Solutions Will the wind ever remember the names it has blown in the past? – From “The Wind Cries Mary” by Jimi Hendrix

In this chapter we discuss some basic properties of κ-solutions. Topics include the example of a nontrivial κ-solution on the n-sphere, a generalization of the notion of κ-solution, the existence of an asymptotic shrinker in a κ-solution, and the κ-gap theorem. In §1 we review Perelman’s no local collapsing theorem with application to the existence of singularity models and we introduce the notion of κ-solution. We also collect some examples which provide some intuition regarding ancient solutions. For example, we give an elementary discussion of the existence of ε-necks in the Bryant soliton. In §2 we modify Perelman’s notion of κ-solution by replacing the bounded curvature assumption in his definition by the a priori weaker requirement that the solution satisfies the trace Harnack estimate. Such solutions are called ‘κ-solutions with Harnack’. We also introduce two notions reflecting the geometry at infinity. In §3 we present the construction of Perelman’s κ-solution on the nsphere S n for n ≥ 3, which is rotationally symmetric, is invariant under a reflection, and has the Bryant soliton as a backward limit and the shrinking cylinder as another backward limit. Although Perelman’s solution is analogous to the 2-dimensional King–Rosenau solution (discussed in both Volume One and Part I),1 it is qualitatively different. In particular, Perelman’s solution is κ-noncollapsed at all scales for some κ > 0, whereas the King–Rosenau solution is not. It is possible (not proven) that Perelman’s solution is unique among Type II κ-solutions on S 3 . In §4 we first recall Hamilton’s classification of 2-dimensional κ-solutions; namely, the only such solutions are the round shrinking 2-sphere or its Z2 quotient. In dimensions 2 and 3 we show the result that κ-solutions with Harnack must have bounded curvature and hence are κ-solutions. In §5 we discuss additional details (supplementing the proof of Theorem 8.32 in Part I) regarding the proof of the existence of an asymptotic gradient shrinking soliton in a κ-solution. 1

Perelman’s solution and the King–Rosenau solution share the following characteristics. They are rotationally symmetric, invariant under reflection, shrink to a round point forward in time, and limit to either a cylinder or a steady soliton backward in time. 79 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

80

19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

In §6 we also concentrate on dimension 3 and prove the κ-gap theorem for 3-dimensional noncompact κ-solutions. This result of Perelman says that there exists a universal constant κ0 > 0 such that any 3-dimensional nonspherical space form κ-solution is actually a κ0 -solution. The proof of this theorem exploits the fact that for 3-dimensional noncompact κ-solutions, the asymptotic shrinking soliton is a round cylinder (or its Z2 -quotient). It also exploits the connection between a lower bound for the reduced volume and κ-noncollapsing. 1. Singularity models and κ-solutions In this section we briefly recall Perelman’s no local collapsing theorem, its application to the existence of finite time singularity models, and its motivation to study κ-solutions. We also discuss intuitive aspects of a special κ-solution, namely the Bryant soliton. 1.1. κ-noncollapsing and existence of singularity models. First we recall the following definition introduced by Perelman. Definition 19.1 (κ-noncollapsed Riemannian metric). Let κ > 0 and ρ ∈ (0, ∞] be two constants. A Riemannian metric gˆ on a manifold Mn is said to be κ-noncollapsed below the scale ρ if, for any geodesic ball B(x, r) with r < ρ satisfying | Rm(y)| ≤ r−2 for all y ∈ B(x, r), we have that the volume ratio is bounded from below: Vol B(x, r) ≥ κ. (19.1) rn If gˆ is κ-noncollapsed below the scale ∞, then gˆ is said to be κ-noncollapsed at all scales. A complete solution (Mn , g (t)), t ∈ I, where I ⊂ R is an interval, to the Ricci flow, is said to be κ-noncollapsed below the scale ρ if for every t ∈ I, g (t) is κ-noncollapsed below the scale ρ. A main relevance of Definition 19.1 to the Ricci flow is Perelman’s no local collapsing theorem (see Theorem 4.1 in [152]). Theorem 19.2 (Perelman’s no local collapsing). For any finite time solution of the Ricci flow (Mn , g (t)), t ∈ [0, T ), on a closed manifold and any finite positive scale ρ, there exists κ > 0 (where κ depends only on the initial metric g (0), T , and ρ) such that the solution is κ-noncollapsed below the scale ρ. The proof of this result, using the entropy monotonicity formula (17.12), is discussed in Chapter 6 of Part I. To obtain a local injectivity radius estimate, we recall the following local result in [33] relating volume, curvature, and injectivity radius (see also Lemma 4.5 of Petersen [155]). Proposition 19.3 (Cheeger, Gromov, and Taylor). For any constant κ > 0 and dimension n, there exists a constant ι0 > 0 depending only on κ

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1. SINGULARITY MODELS AND κ-SOLUTIONS

81

and n such that if (Mn , gˆ) is a complete Riemannian manifold, p ∈ M, and ≥ κ, then r ∈ (0, ∞) are such that |Rm| ≤ r−2 in B (p, r) and Vol B(p,r) rn inj (p) ≥ ι0 r. Combining the above two results with Hamilton’s Cheeger–Gromov-type compactness theorem, we have the following. Theorem 19.4 (Existence of finite time singularity models). Let g(t), t ∈ [0, T ), be a solution to the Ricci flow on a closed manifold Mn with T < ∞. Suppose that we have a sequence of times ti T , points pi ∈ M, radii ri ∈ (0, ∞), and a constant C < ∞ such that (19.2)

Ki  | Rm |(pi , ti ) → ∞,

(19.3)

| Rm | (x, t) ≤ CKi

(19.4)

Ki ri2

for all x ∈ Bg(ti ) (pi , ri ) and t < ti ,

→ ∞.

Then there exists a subsequence of rescaled solutions   t gi (t)  Ki · g ti + Ki such that (Mn , gi (t), pi ) converges to a complete ancient solution (Mn∞ , g∞ (t), p∞ ) in the C ∞ Cheeger–Gromov sense. This limit solution has nonnegative scalar curvature and is κ-noncollapsed at all scales, where κ > 0 is a constant depending only on g (0) and T . Recall that the limit solution (Mn∞ , g∞ (t)) is called a finite time singularity model. Exercise 19.5. Prove Theorem 19.4. Hint: Compare with Corollary 3.18 and Corollary 3.29 in Part I. Note that if the solution g (t) is κ-noncollapsed below the scale ρ, then the solution 1/2 gi (t) is κ-noncollapsed below the scale Ki ρ, while Ki → ∞. Remark 19.6. There is a corresponding result for a complete solution on a noncompact manifold with bounded curvature provided the initial metric is κ-noncollapsed below some scale ρ; see Theorem 8.26 in Part I. Now recall the following important concept originally introduced by Perelman in §11.1 of [152] (see also Definition 8.31 in Part I). Definition 19.7 (κ-solution). Given a positive constant κ, a complete ancient solution (Mn , g˜(t)), t ∈ (−∞, 0], of the Ricci flow is called a κsolution if it satisfies the following. (i) For each t ∈ (−∞, 0] the metric g˜(t) is nonflat with nonnegative curvature operator and κ-noncollapsed at all scales.

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(ii) There is a constant C < ∞ such that the scalar curvature Rg˜ (x, t) ≤ C for all (x, t) ∈ M × (−∞, 0]. The relevance of κ-solutions to singularity analysis is as follows. A finite time singularity model satisfies all of the properties in the definition of a κ-solution except possibly the nonnegativity of the curvature operator. However in dimension n = 3, by applying the Hamilton–Ivey 3-dimensional curvature estimate to Theorem 19.4, finite time singularity models have nonnegative curvature operator. Corollary 19.8 (3-dimensional finite time singularity models). When n = 3, under the same hypotheses of Theorem 19.4, there exists a subsequence such that (M3 , gi (t), pi ) converges to a complete κ-solution (M3∞ , g∞ (t), p∞ ) in the C ∞ Cheeger–Gromov sense, where κ > 0 is a constant depending only on g (0) and T . Problem 19.9 (Rotationally symmetric ancient solutions). Does there exist a rotationally symmetric ancient solution with bounded nonnegative curvature operator in dimension at least 3 which is not κ-noncollapsed at all scales for any κ > 0? 1.2. Some examples of κ-solutions. For the first class of examples of κ-solutions, we have the products of spheres and Euclidean spaces and the quotients of these products. Example 19.10 (Cylinders and their quotients). Consider the compact

and noncompact (smooth manifold) quotients S n−k × Rk /Γ of the shrinking round cylinder with n − k ≥ 2 and k ≥ 0. Here, the largest κ > 0 where the ancient solution is κ-noncollapsed on all scales depends on the number |Γ| of elements of Γ and the action of Γ on Rk .2 In particular (n = 3 and k = 0) for smooth quotients S 3 /Γ, where Γ ⊂ SO (4), there does not exist a uniform lower bound for κ. (This is in general the case for odd-dimensional spherical space forms; on the other hand, the only even-dimensional spherical space forms are S n and RP n .) The above examples are also useful to keep in mind while considering the noncompactness and dimension assumptions on the manifold Mn in the hypotheses of the ‘κ-gap theorem’ of §6 of this chapter. Remark 19.11. The cigar soliton and the King–Rosenau solution (a.k.a. sausage model) are not κ-solutions because they violate the condition of κnoncollapsing at all scales. In fact, Hamilton proved in [92] that the round shrinking 2-sphere and its Z2 -quotient (i.e., the constant curvature RP 2 ) are the only 2-dimensional κ-solutions. See also §1 of Chapter 9 (especially Corollary 9.19) in [45]. 2 For example, if k = 0, then the largest κ is at least cn / |Γ|, where cn > 0 depends only on n.

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1. SINGULARITY MODELS AND κ-SOLUTIONS

83

As a higher-dimensional analogue of the 2-dimensional cigar soliton solution we have the following. Example 19.12 (Bryant solitons). The Bryant soliton (Mn , gBry ), where n ≥ 3 and Mn ∼ = Rn , is the unique (up to scaling) complete nonflat rotationally symmetric steady gradient Ricci soliton (see §4 of Chapter 1 in Part I). The 3-dimensional Bryant soliton is useful to keep in mind while developing intuition in the study of singularity formation in dimension 3. In particular, it occurs as a singularity model for degenerate neckpinch singularities3 (for a proof of this, see Gu and Zhu [84]; for some heuristic discussion, see also §6 of Chapter 2 in Volume One and §2.3 of Chapter 8 in [45]). Note that the sectional curvatures of the Bryant soliton decay inverse linearly and quadratically, depending on the directions of the 2-planes (see Exercise 19.13 below). On the other hand, the curvature of the cigar soliton decays exponentially. Consider the Bryant soliton at a fixed time, i.e., Mn ∼ = Rn with the rotationally symmetric metric gBry  dr2 + w (r)2 gS n−1 . Here w together with a radial potential function f satisfies the ode (ordinary differential equation) corresponding to the steady gradient Ricci soliton equation Rc(g) + ∇∇f = 0 (see (1.45) in Part I): w , ww f  = ww − (n − 1)(1 − (w )2 ). w We have (see p. 24 in Part I) 1 1/2 r ≤ w (r) ≤ Cr1/2 (19.5) 0 ≤ w (r) ≤ Cr−1/2 and C for some constant C ∈ (0, ∞) and all r ∈ (0, ∞). When considering ancient solutions, we may keep in mind the following property of the Bryant soliton. Existence of ε-necks in the Bryant soliton.4 Given r0 ∈ (0, ∞), consider the rescaled metric   w (r0 + w (r0 ) r˜) 2 r2 + gS n−1 , g0 = w (r0 )−2 gBry = d˜ w (r0 ) f  = n

where r˜  w (r0 )−1 (r − r0 ).5 We have       w (r0 + w (r0 ) r˜)  − 1 ≤   w (r0 )

r˜ 0

  w (r0 + w (r0 ) r) dr . 

Note also that given a rotationally symmetric Type IIa singular solution on S n , there is a corresponding rotationally symmetric steady gradient soliton singularity model, which must be the Bryant soliton. Numerical evidence of this was first given by Garfinkle and one of the authors [67], [68]. 4 See Definition 18.26. 5 The effect of the translation by −r0 given in the change of coordinates defining r˜ is the same as choosing the center of the neck to have distance r0 from O with respect to gBry . 3

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Now

19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

≤ C 2 r0−1 and    r˜       w (r0 + w (r0 ) r) dr ≤ C  

1 w(r0 )2

0

  (r0 + w (r0 ) r) dr 0  r˜ 2C  1/2   (r0 + w (r0 ) r)   = w (r0 )  0 2C |˜ r| . = 1/2 (r0 + w (r0 ) r˜)1/2 + r0 r˜

−1/2

Hence

    w (r0 + w (r0 ) r˜) 2C |˜ r|  − 1 ≤ (19.6)  1/2 w (r0 ) (r0 + w (r0 ) r˜)1/2 + r0   1/2 for any r0 ∈ (0, ∞) and r˜ ∈ −w (r0 )−1 r0 , ∞ . Since w (r0 ) ≤ Cr0 , this shows that (see the next exercise) for any ε > 0, there exists r (ε) < ∞ such that any point x ∈ M with d (x, O) > r (ε) is the center of a ball which is ε-close in C 0 to a round unit cylinder of length ε−1 . (That is, we choose ε−1 ≤ r˜  d (x, O)1/2 .) Exercise 19.13 (Estimates for curvatures of the n-dimensional Bryant soliton). Show the following. (1) The estimates for w and w in (19.5). (2) Estimates for the sectional curvatures: There exist constants C ∈ (1, ∞) depending only on the dimension n such that C −1 d (x, O)−1 ≤ ν1 ≤ Cd (x, O)−1 , C −1 d (x, O)−2 ≤ ν2 ≤ Cd (x, O)−2 , where ν1 and ν2 denote the sectional curvatures of the planes tangent to the spheres and the planes with one radial and one spherical direction, respectively. (3) Show that C −1 d (x, O)−1 ≤ |II (x)| ≤ Cd (x, O)−1 , where II (x) denotes the second fundamental form of the geodesic (n − 1)-sphere centered at O passing through x. Hint: See §4 of Chapter 1 in Part I. Also note that the second fundamental form of the level hypersurface f −1 (a) is given by II = ∇∇f |∇f | . To show that the C 0 ε-neck in (Mn ,Bry ) obtained above is actually a C k ε -neck (where ε tends to zero and k tends to infinity as ε tends to zero), we need to estimate the higher derivatives; we leave this as an exercise. Exercise 19.14. For the Bryant soliton (Mn , gBry ), show that for any ε > 0, there exists r¯ (ε) < ∞ such that any point x ∈ M with d (x, O) > r¯ (ε) is the center of a ball which is an embedded ε-neck.

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2. THE κ-NONCOLLAPSED CONDITION

85

Hint: Estimate the higher derivatives of w (r).

2. The κ-noncollapsed condition In this section we recall the definitions of asymptotic scalar curvature ratio and asymptotic volume ratio, which will be used in the next chapter to study the geometry of κ-solutions. We introduce the notion of κ-solution with Harnack and we discuss a characterization of the κ-noncollapsed at all scales condition for ancient solutions by the boundedness of the entropy. 2.1. Invariants reflecting the geometry at infinity. Perhaps the most interesting case of κ-solutions is when the underlying manifold is noncompact. In this case we are interested in the geometry at spatial infinity of the solution. 2.1.1. Asymptotic volume and scalar curvature ratios. Given a complete noncompact manifold (N n , h) with Rc ≥ 0 and p ∈ N , the asymptotic volume ratio is defined by (19.7)

Vol B(p, r) ∈ [0, 1] , r→∞ ωn rn

AVR(h)  lim

where ωn is the volume of the unit Euclidean n-ball. Recall that the Bishop–Gromov volume comparison theorem says that if (N n , h) is a complete Riemannian manifold with Rc ≥ 0 and p ∈ N , then Vol B(p,r) is a nonincreasing function of r for all r > 0. The fact in (19.7) rn that AVR(h) ≤ 1 follows from this. Given a complete noncompact manifold (N n , h) and p ∈ N , the asymptotic scalar curvature ratio is defined by (19.8)

ASCR (h)  lim sup R(x) · d(x, p)2 . d(x,p)→∞

It is easy to show that the limits in the definitions of AVR(h) and ASCR (h) are independent of the choice of p ∈ N . Remark 19.15. Note that if the scalar curvature of (N n , h) is unbounded, then ASCR (h) = ∞. Essentially, ASCR (h) < ∞ if and only if the scalar curvature has quadratic (or faster) decay from above. In particular and more specifically, if ASCR (h) < ∞, then for every ε > 0 there exists ρ < ∞ such that R(x) · d(x, p)2 < ASCR (h) + ε for all x ∈ M − B (p, ρ).

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

2.1.2. Examples. Now we consider some examples.

Example 19.16 (Cylinders have AVR = 0). If P n−1 , k is a closed Riemannian manifold with Rc ≥ 0, then the product P n−1 , k × R has AVR = 0. Indeed, if (x, s) ∈ P n−1 ×R, then B ((x, s) , r) ⊂ P ×[s − r, s + r], so that Vol B ((x, s) , r) ≤ 2r Vol (P, k) . Although the above example is simple, in the next chapter we shall find it useful in dimension 3. Example 19.17. For the Bryant soliton (Mn , gBry ) we have for any p∈M (1) limr→∞

Vol B(p,r) r (n+1)/2

∈ (0, ∞) and AVR = 0 and

(2) limd(x,p)→∞ R (x) · d (x, p) ∈ (0, ∞) and ASCR = ∞. Although in the next example the metric is incomplete near the vertex, it is instructive to consider its geometry at infinity. Example 19.18 (The geometry at infinity of a Riemannian cone). Let

n−1 , k be a closed Riemannian manifold with Rc ≥ 0 and consider the P Riemannian cone (CP, h), where

CP  [0, ∞) × P n−1 / ∼ and {0} × {x} ∼ {0} × {y} for all x, y ∈ P and h = dr2 + r2 k. We then have (19.9)

ASCR (h) < ∞

and AVR (h) =

Vol (P, k) > 0. nωn

Exercise 19.19 (The curvature of a Riemannian cone). To verify that ASCR (h) < ∞ in the above

example, compute the curvature of a Riemannian cone CP, dr2 + r2 k . Hint: Use (20.55) below. Let (Mn , g, O) be a pointed complete Riemannian manifold. The condiis related tion of the asymptotic cone6 being top dimensional

to ASCR (g) < ∞. In particular, suppose that the family Mn , ds−2 g , O s∈[1,∞) converges as s → ∞ in the pointed Gromov–Hausdorff distance to its asymptotic cone 2 suppose (AX, dAX , 0). Furthermore,

that AX − {0} has a C n-manifold2 n −2 structure such that M , s g, O s∈[1,∞) converges as s → ∞ in the C Cheeger–Gromov sense to a C 2 Riemannian metric g∞ on AX − {0}. Then 6

See subsection 1.3 of Appendix H.

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2. THE κ-NONCOLLAPSED CONDITION

87

by (19.9) there exists a constant C < ∞ such that R (x) d (x, O)2 ≤ C for all x ∈ M. That is, ASCR (g) < ∞. 2.1.3. A relation between AVR and ASCR. Let (N n , h) be a complete Riemannian manifold with Rm ≥ 0 and which is κ-noncollapsed at all scales. Suppose that there exists C ∈ [1, ∞) such that |Rm| (x) ≤ Cd (x, O)−2 for all x ∈ N − B (O, 1) . Then for all x ∈ N − B (O, 2) we have   1 −2 for all y ∈ B x, √ d (x, O) . |Rm| (y) ≤ 4Cd (x, O) 2 C n Since (N , h) is κ-noncollapsed at all scales, for such x we have   κ 1 Vol B x, √ d (x, O) ≥ n n/2 d (x, O)n . 2 C 2 C Therefore, for r ∈ [2, ∞), by taking any x with d (x, O) = r, we have       r κ 1 r ≥ Vol B x, √ ≥ n n/2 rn . Vol B O, 1 + √ 2 C 2 C 2 C κ > 0. In particular, we have shown that if Thus AVR (h) ≥ 2n C n/2 ωn n (N , h) is κ-noncollapsed at all scales and AVR (h) = 0, then we conclude ASCR (h) = ∞; compare with the related Theorem 20.1.

Remark 19.20. Although being κ-noncollapsed at all scales and having AVR = 0 are certainly not contradictory, there is some tension between these two properties. 2.1.4. AVR and ASCR under Ricci flow. Next we consider some basic properties of AVR and ASCR for solutions of Ricci flow. In Theorem 19.1 of [92] Hamilton proved the following (see also Theorem 8.32 in [45]). Theorem 19.21 (ASCR is independent of t). If (Mn , g (t)) is a complete ancient solution on a noncompact manifold with bounded nonnegative curvature operator, then ASCR (g (t)) is independent of t. In Theorems 18.2 and 18.3 of [92] Hamilton proved the following (see also Proposition 8.37 in [45]). Proposition 19.22 (AVR is independent of t). If (Mn , g (t)) is a complete solution with bounded nonnegative Ricci curvature and (19.10)

lim dt (x,O)→∞

|Rm (x, t)| = 0

for all t, then AVR (g (t)) is independent of t. Moreover, if (19.10) holds at some time, then it holds for all later times.

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

There are two classes of solutions to Ricci flow which have infinite asymptotic scalar curvature ratio. Theorem 19.23. (1) (Steady gradient Ricci soliton) If (Mn , g (t) , ∇f (t)), t ∈ (−∞, ∞), n ≥ 3, is a complete steady gradient Ricci soliton with sect (g (t)) > 0 and with the property that R attains its maximum at some point, then ASCR (g (t)) = ∞. (2) (Type I ancient solution) If (Mn , g(t)) is a complete noncompact Type I ancient solution with bounded positive curvature operator, then ASCR(g(t)) = ∞. For (1) see the original Theorem 20.2 in [92] or Theorem 9.44 in [45] and for (2) see Theorem 9.32 in [45]. Regarding backward limits of Type II ancient solutions,7 we have the following (see Proposition 9.29 in [45]). Theorem 19.24. If (Mn , g(t)) is a complete noncompact Type II ancient solution with Rm ≥ 0 and bounded positive sectional curvature, then there is a backwards limit which is a steady gradient Ricci soliton which attains its maximum scalar curvature. This result and Theorem 19.23(1) may suggest the following. Problem 19.25. If (Mn , g(t)) is a complete noncompact Type II ancient solution with bounded positive curvature operator, then is ASCR(g(t)) = ∞? Here we are not assuming that the solution is κ-noncollapsed at all scales. Problem 19.26. Let (Mn , g(t)) be a complete noncompact ancient solution and let (Mn∞ , g∞ (t)) be a backwards limit. Is ASCR (g (t)) = ∞ if and only if ASCR (g∞ (t)) = ∞? More ambitiously, one may ask if ASCR (g∞ (t)) = ASCR (g (t)). In this chapter and the next chapter we shall use the invariants AVR and ASCR to study κ-solutions. 2.2. κ-solutions with Harnack. We now discuss a variant on the notion of κ-solution, which we shall find useful in formulating Perelman’s compactness theorem in higher dimensions; see § 4 of this chapter and §3 of the next chapter.8 Definition 19.27 (κ-solution with Harnack). If the bounded curvature condition (ii) in Definition 19.7 is replaced by the requirement that g˜ (t) ∞.

7

By definition, Type II (backward in time) means that supM×(−∞,0] |t| |Rm| (x, t) =

8

Definition 19.27 was introduced by one of the authors [142]; see also p. 406 in Part

I.

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2. THE κ-NONCOLLAPSED CONDITION

89

satisfies the trace Harnack estimate ∂R + 2 ∇R, X + 2 Rc (X, X) ≥ 0 (19.11) ∂t for all vector fields X, then we say that g˜ (t) is a κ-solution with Harnack. By taking X = 0 in (19.11), we immediately see that, for a κ-solution with Harnack, the scalar curvature R is pointwise nondecreasing in time, ˜ (t) defined for t ≤ 0 does i.e., ∂R ∂t ≥ 0. Thus, if a κ-solution with Harnack g not have bounded curvature in space-time, then sup R (x, 0) = ∞. x∈M

In this case we clearly have ASCR (˜ g (0)) = ∞. Note that by Hamilton’s trace Harnack estimate, i.e., Corollary 15.4 in Part II, a κ-solution is a κ-solution with Harnack. The only possible distinction between these two notions is when M is noncompact and the curvatures of g˜ (t) are unbounded. Remark 19.28. A reason for why the notion of κ-solution with Harnack is useful is that under certain limits, the condition that the trace Harnack estimate holds is preserved while, a priori, the condition that the curvature is bounded may not be preserved. Perhaps one may improve Hamilton’s matrix Harnack estimate in the following direction. Problem 19.29 (Hamilton’s matrix Harnack estimate without a bound on Rm). In Hamilton’s matrix Harnack estimate, i.e., Theorem 15.1 in Part II, can one weaken the part of the hypothesis requiring the curvature to be bounded? In particular, is there a class of complete solutions (Mn , g˜ (t)), g (t))| = ∞ t ∈ [0, ω), to the Ricci flow with Rm (˜ g (t)) ≥ 0 and supM |Rm (˜ for each t ∈ (0, ω) for which Hamilton’s matrix Harnack estimate holds for solutions in this class? Perhaps one may assume that the curvatures satisfy a growth condition in terms of the distance function to a fixed point. A simpler question than Problem 19.29 may be whether Hamilton’s trace Harnack estimate holds for a class of solutions on surfaces with unbounded curvature. As far as we know, not much is known about complete solutions to the Ricci flow on noncompact surfaces with unbounded curvature. A related problem is that of ‘localizing’ the Harnack estimate. In the case of the heat equation on a fixed Riemannian manifold, this was accomplished by Li and Yau [121]. Remark 19.30 (Harnack estimate when Rm is bounded but not uniformly bounded). Note that if (Mn , g˜ (t)), t ∈ [0, ω), is a complete solution g (t))| < ∞ to the Ricci flow such that Rm (˜ g (t)) ≥ 0 and supM×(t0 ,ω) |Rm (˜ 9 for each t0 ∈ (0, ω), then Hamilton’s matrix Harnack estimate holds for 9

Note that here we have not assumed a uniform (in time) bound on the curvatures.

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

this solution. One may simply apply Hamilton’s matrix Harnack estimate on each interval [ε, ω), where ε ∈ (0, ω), and then take ε → 0+ . Remark 19.30 may also lead one to ponder the following. Problem 19.31 (Continuity of the supM |Rm| (t) function). Suppose that (Mn , g˜ (t)), t ∈ (α, ω), is a complete solution to the Ricci flow such g (t))| < ∞ for each t ∈ (α, ω). Is supM |Rm (˜ g (t))| a conthat supM |Rm (˜ tinuous function of t? A weaker question is to ask this under the additional assumption that Rm (˜ g (t)) ≥ 0.10 2.3. A characterization of the κ-noncollapsed condition for ancient solutions. In this subsection we shall discuss the following characterization of κsolutions in terms of the boundedness of the entropy of the fundamental solution to the adjoint heat equation. At the end of §11.1 in [152], Perelman wrote: We impose one more requirement on the solutions; namely, we fix some κ > 0 and require that gij (t) be κ-noncollapsed on all scales (the definitions 4.2 and 8.1 are essentially equivalent in this case). It is not hard to show that this requirement is equivalent to a uniform bound on the entropy S, defined as in 5.1 using an arbitrary fundamental solution to the conjugate heat equation. Let      τ R + |∇f |2 + f − n (4πτ )−n/2 e−f dµ (19.12) W(g, f, τ )  M

denote Perelman’s entropy functional and let ∂
∗  − − ∆ + R ∂t denote the adjoint heat operator. Recall from (17.12) that under the coupled system ∂ g = −2 Rc, ∂t
∗ u = 0, dτ = −1, dt where u  (4πτ )−n/2 e−f , we have (19.13)

d W(g (t) , f (t) , τ (t)) = 2τ dt



2    1 Rc +∇∇f − g  udµ ≥ 0.  2τ  M

Now we may interpret §11.1 of [152] as saying the following. 10 Another weaker question is to ask only whether for every [β, ψ] ⊂ (α, ω) we have supM×[β,ψ] |Rm| < ∞.

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2. THE κ-NONCOLLAPSED CONDITION

91

Claim 19.32 (Entropy and κ-noncollapsing for ancient solutions). Let t ∈ (−∞, 0], be a complete ancient solution to the Ricci flow g (t))| < ∞ for each t ∈ (−∞, 0]. such that Rm (˜ g (t)) ≥ 0 and supM |Rm (˜ Then there exists κ > 0 such that g˜ (t) is κ-noncollapsed at all scales if and only if there exists C < ∞ such that for any fundamental solution u : M × (−∞, t0 ) → R+ of the adjoint heat equation

(Mn , g˜ (t)),


∗ u = 0, where limt→(t0 )− u (t) = δp0 for some (p0 , t0 ) ∈ M × (−∞, 0], we have (19.14)

sup

|W (˜ g (t) , f (t) , t0 − t)| ≤ C,

t∈(−∞,t0 ]

where u (t) = (4π (t0 − t))−n/2 e−f (t) .11 Problem 19.33. Prove this claim.12 For comparison, consider the case of the heat equation on a static Riemannian manifold (Mn , gˆ).13 Let    −n/2 t |∇f |2 + f − n e−f dµ Wlin (f, t) = (4πt) M

denote the linear entropy functional. If (Mn , gˆ) is a closed Riemannian manifold with nonnegative Ricci curvature and u (x, t) = (4πt)−n/2 e−f (x,t) is a solution to the heat equation, then (see Theorem 0.1 in [139])  2     1 d ∇∇f − g  + Rc (∇f, ∇f ) u dµ ≤ 0. Wlin (f (t) , t) = −2t  dt 2t  M In Proposition 3.2 of [139], one of the authors proved the following. Theorem 19.34. Let (Mn , gˆ) be a complete noncompact Riemannian manifold with nonnegative Ricci curvature. Given any p0 ∈ M, let fp0 (x, t) be defined by H (x, p0 , t) = (4πt)−n/2 e−fp0 (x,t) being the heat kernel (minimal positive fundamental solution). Recall that Wlin (fp0 (t) , t) is well defined and finite (see Corollary 16.17 in Part II ). Then the following two properties are equivalent: (1) (ˆ g has maximum volume growth) There exists c > 0 such that Vol B (p, r) ≥c rn for all p ∈ M and r ∈ (0, ∞). (2) (Linear entropies of heat kernels are uniformly bounded) There exists C < ∞ such that for all p0 ∈ M |Wlin (fp0 (t) , t)| ≤ C

for all t ∈ (0, ∞) .

11

Needless to say, where C is independent of (p0 , t0 ). At the time of this writing we are not aware of a complete proof of this claim in the literature. 13 Some of this material was also discussed in Sections 1–3 of Chapter 16 in Part II. 12

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Proof. Let H (x, p0 , t) = (4πt)−n/2 e−f (x,t) be the heat kernel (we drop the subscript p0 on f ). One can show that   n |∇H|2 dµ − H ln H dµ − ln (4πt) − n. 0 ≥ Wlin (f (t) , t) = t H 2 M M Suppose property (1) holds. Then by Li and Yau’s upper bound for the heat kernel on a manifold with Rc ≥ 0 (see Corollary 3.1 in [121] and compare with Corollary 26.26 below), there exists a universal constant C < ∞ such that d(x,p0 )2 C C √ e− 5t ≤ t−n/2 H (x, p0 , t) ≤ c Vol B p0 , t for all p0 , x ∈ M and t ∈ (0, ∞). Thus  n H ln H dµ − ln (4πt) − n Wlin (f (t) , t) ≥ − 2 M n C ≥ − ln − ln (4π) − n c 2 for all t ∈ (0, ∞). This is the desired bound for Wlin . Conversely, suppose property (2) holds. We shall use the bound for Wlin to estimate volume ratios. Recall that the Li and Yau differential Harnack estimate says that n |∇H|2 + H ≥ 0. ∆H − H 2t Then    n |∇H|2 n  dµ ≤ ∆H + H dµ = . (19.15) H 2t 2t M M On the other hand, by Li and Yau’s lower bound for the heat kernel on a manifold with Rc ≥ 0 (see Theorem 4.1 in [121] and compare with Theorem 26.31 below), there exists a constant C (n) < ∞ depending only on n such that d(x,p0 )2 C (n) √ e− 3t H (x, p0 , t) ≥ Vol B p0 , t for all p0 , x ∈ M. Thus   H ln H dµ ≤ − − M

(19.16)



d(x,p0 )2 C (n) √ e− 3t H ln Vol B p0 , t M  √ ≤ C˜ (n) + ln Vol B p0 , t

 dµ (x)

2
0) dµ (x) is bounded from above by for some C˜ (n) < ∞, since M H d(x,p 3t a constant depending only on n (for this last fact, see [138] or it may be deduced from the methods in Chapter 26 in this volume).

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3. PERELMAN’S κ-SOLUTION ON THE n-SPHERE

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From (19.15) and (19.16) we obtain  √ n n Wlin (f (t) , t) ≤ + C˜ (n) + ln Vol B p0 , t − n − ln (4πt) 2 2 √

Vol B p0 , t n n = ln √ n − − ln (4π) + C˜ (n) 2 2 t for all p0 ∈ M and t ∈ (0, ∞). Since Wlin (f, t) ≥ −C, we conclude that √

Vol B p0 , t √ n ≥ −C − C˜ (n) ln t for all p0 ∈ M and t ∈ (0, ∞).




Remark 19.35. Assuming maximum volume growth, a precise relation between the asymptotic behaviors of volume growth and the linear entropy of a heat kernel is given by the following (see Corollary 4.3 of [139]) inf t∈(0,∞)

Wlin (f (t) , t) = lim Wlin (f (t) , t) = ln (AVR(ˆ g )) ≤ 0. t→∞

In the case of the (backward) Ricci flow g (τ ) on a closed manifold and the adjoint heat kernel u centered at (p0 , 0), recall from inequalities (16.106) and (16.105) in Part II that f (y, τ ) ≤  (y, τ ) , where u = (4πτ )−n/2 e−f and  is the reduced distance based at (p0 , 0). Integrating (4πτ )−n/2 e−f (y,τ ) ≥ (4πτ )−n/2 e−
(y,τ ) , we see that V˜ (τ ) ≤ 1, ˜ where V˜ is the reduced volume (see the next chapter for a discussion of V). Recall also that by Theorem 16.44 in Part II we have W (g (τ ) , u (τ ) , τ ) ≤ 0.

  It is possible that limτ →∞ W (g (τ ) , u (τ ) , τ ) is equal to ln limτ →∞ V˜ (τ ) or that in some other way these two invariants are related? Note that for solutions of the Ricci flow as compared to the static metric case, from this perspective, the reduced volume seems like a natural replacement for the asymptotic volume ratio (recall from Corollary 20.2 that any ancient solution with bounded Rm ≥ 0 must have AVR = 0). 3. Perelman’s κ-solution on the n-sphere A higher-dimensional analogue of the 2-dimensional King–Rosenau solution is Perelman’s κ-solution. This is the rotationally symmetric κ-solution on S 3 constructed in §1.4 of [153]. It is possible, as with other ancient solutions, that Perelman’s solution has physical significance.

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Theorem 19.36 (Perelman’s κ-solution). For any n ≥ 3, there exists a rotationally symmetric and reflectionally invariant Type II κ-solution on S n . Forward in time, this solution forms a Type I singularity since it shrinks to a round point. We remark that the fact that Perelman’s κ-solution is Type II is a consequence of the following result of one of the authors [141]. Theorem 19.37. If (Mn , g (t)) is a compact Type I κ-noncollapsed ancient solution to Ricci flow with positive curvature operator, then (Mn , g (t)) is isometric to a shrinking spherical space form. We expect that the following is true. Problem 19.38. Show that Perelman’s κ-solution has backward in time limits which are the Bryant soliton and the round cylinder S 2 × R, depending on how the sequence of points and times about which one rescales is chosen. Moreover, prove that these are the only backward in time limits of Perelman’s κ-solution.14 One may take a Z2 -quotient (generated by the antipodal map) of Perelman’s κ-solution to obtain on RP 3 , whose asymptotic

2

gradient a2 κ-solution shrinking Ricci soliton is S × R /Z2 , where Z2 ⊂ Isom S × R is generated by the map (x, y) → (−x, −y). Remark 19.39. By Theorem 17.13, Perelman’s κ-solution cannot be a finite time singularity model. In the rest of this section we give the details of the construction of Perelman’s ancient solution for all dimensions n ≥ 3. Let S n−1 (r) denote the round (n − 1)-sphere of radius r. To paraphrase Perelman,15 for any L ∈ (1, ∞) we shall construct a rotationally symmetric metric gL (0) on operator which metrically looks like a S n with weakly positive curvature   n−1 n and 2 (n − 2) × [−L, L] with two caps B+ long round cylinder S   n smoothly attached to the boundary components S n−1 2 (n − 2) × B− {−L, L}. Perelman’s ancient solution will be obtained by taking a rescaled and time translated limit, as L → ∞, of the solutions gL (t) of the Ricci flow with initial metrics gL (0). The work is to show that this limit exists. Nice properties (some are crucial) of this family {gL (t)}L∈(1,∞) (in fact, a sequence will do) of solutions include the following: (1) the solutions have positive curvature operator after the initial time, (2) the ratio of the spatial maximum over the spatial minimum of the eigenvalues of the curvature operator tend to infinity as one approaches the initial time, 14 15

In contrast, any forward in time limit is a round shrinking 3-sphere. See the bottom of p. 3 in § 1.4 of [153].

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3. PERELMAN’S κ-SOLUTION ON THE n-SPHERE

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(3) the initial scalar curvatures are uniformly bounded from above and below by positive constants, (4) the singularity times are uniformly bounded from above and below by positive constants, (5) the diameters tend to infinity. Step 1. Construction of the family of initial metrics. One way of constructing the rotationally symmetric initial metric gL (0) on S n is to mollify the warping function of the C 1 metric obtained by gluing two spherical caps to a round cylinder. In particular, we shall consider a rotationally symmetric metric of the form (19.17)

gL (0) = dr2 + ψ (r)2 gcan

on S n , where gcan denotes the standard metric on S n−1 (1) and the warping function ψ is to be defined below. By (20.53) and (20.54) in the notes and commentary at the end of Chapter 20, we have the standard formulas for the spherical and radial sectional curvatures:16 Ksph =

1 − (ψ  )2 , ψ2

To define ψ, we first consider R → [−1, 1] defined by17 ⎧ ⎨ sin r 1 f (r)  ⎩ −1

Krad = −

ψ  . ψ

the nondecreasing odd C 1 function f : if r ∈ (−π/2, π/2), if r ∈ [π/2, ∞), if r ∈ (−∞, −π/2].

Let η : R → [0, a] be the standard mollifier defined by  2 ae1/(r −1) if |r| < 1, η (r)  0 if |r| ≥ 1,
where a > 0 is chosen so that R η (r) dr = 1. Note that η is an even function. Given a small ε > 0, define the mollified function fε : R → [−1, 1] by  1 ρ dρ. f (r + ρ) η fε (r)  ε ε R Then fε is C ∞ , nondecreasing, and odd. In particular, fε (0) = 0. 16

The spherical sectional curvatures are the sectional curvatures of 2-planes contained in the tangent spaces of the spherical slices {r = const} whereas the radial sectional curvatures are the sectional curvatures of 2-planes containing the radial vector ∂/∂r. 17 An alternate construction of the warping function is as follows. It should not be difficult to prove that there exists a C ∞ function f˜ : [0, ∞) → [0, 1] such that  sin r if r ∈ [0, π/4], ˜ f (r)
1 if r ∈ [π/2, ∞), √ 0 ≤ f˜ ≤ C1 , and −C2 ≤ f˜ ≤ 0; we may take C1 = 1/ 2 since f˜ ≤ 0. We may use this function instead of fε in the definition of the warping function.

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Moreover, for all r ∈ [(π/2) + ε, ∞) we have fε (r) = 1 and for all r ∈ (−∞, − (π/2) − ε] we have fε (r) = −1. We have  1 ρ  dρ ∈ [0, 1] f  (r + ρ) η (19.18) fε (r) = ε ε R since f  ∈ [0, 1]; in particular, for ε ∈ (0, π/2),  1 ρ  dρ > 0. cos ρ η fε (0) = ε ε R We also have  1 ρ  dρ ∈ [−1, 0] f  (r + ρ) η (19.19) fε (r) = ε ε R for r ≥ 0 (this uses the fact that f  (r) ∈ [−1, 0] for r ≥ 0 and f  (r) is odd).18 Hence sup fε (r) = fε (0) .

(19.20)

r∈[0,∞)

Fix ε > 0 sufficiently small. We consider the function    ϕ : [0, ∞) → 0, 2 (n − 2) defined by  ϕ (r)  2 (n − 2)fε



r





. 2 (n − 2)fε (0)    2 (n − 2) has Rc ≡ 12 . If r ≥ 2 (n − 2)fε (0) (π/2 + ε), (Note that S n−1  then ϕ (r) = 2 (n − 2).) Clearly, (19.21) (19.22)

lim ϕ (r) = 0,

r→0

lim ϕ (r) = 1.

r→0

Since ϕ is the restriction to the nonnegative real numbers of a C ∞ odd function, we have both

near r = 0 (19.23) ϕ (r) = r + O r3 and (19.24) for all k ∈ N. Hence (19.25)

d2k ϕ (r) = 0 r→0 dr 2k lim



ϕ (r) = 1 + O r2

near r = 0.

Furthermore, by (19.20) we have (19.26)

sup ϕ (r) = 1. r∈[0,∞)

18

Note that f (r) is C ∞ except at r = ±π/2 and f  is a Lipschitz function on R.

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3. PERELMAN’S κ-SOLUTION ON THE n-SPHERE

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Now recall by Lemma 2.10 (the ‘cylinder-to-sphere rule’) in Volume One that a metric g = dr2 + ϕ (r)2 gcan on (0, ∞) × S n−1 , where ϕ : (0, ∞) → R+ , extends to a smooth metric on Rn if and only if (19.21)–(19.24)  hold.   We define ψ : [0, 2L ] → 0, 2 (n − 2) , where  L  L + 2 (n − 2)fε (0) (π/2 + ε) , by reflecting ϕ (r) about the line r = L , i.e.,  ϕ (r) if r ∈ [0, L ] , ψ (r)  ϕ (2L − r) if r ∈ [L , 2L ] . symmetric metric Then gL (0) given by (19.17) is a smooth   rotationally n n n−1 2 (n − 2) × [−L, L] isometrically on S and (S , gL (0)) contains S (we choose ε > 0 sufficiently small independent of L). Moreover, gL (0) is invariant under reflection about the spherical slice {r = L } ⊂ S n . Step 2. Curvature is uniformly bounded for the initial metrics. By (19.26), (19.25), and (19.23) we have that Ksph =

1 − (ψ  )2 ψ2

is nonnegative and bounded (for example, near r = 0 we have ψ  (r) = 2

1 + O r , so that Ksph (r) = O (1)).19 We also have that Krad = −

ψ  ψ

is also bounded and nonnegative (for example, by (19.19) we have fε (r) ≤ 0 for r ≥ 0, so that ϕ (r) ≤ 0 for r ≥ 0). In fact, (19.27)

|Rm| (gL (0)) ≤ C (n) ,

where C (n) < ∞ depends only on n (in particular, C (n) is independent of eigenvalues L). Note that the Riemann curvature operator has (n−1)(n−2) 2 equal to Ksph and n−1 eigenvalues equal to Krad . Therefore Rm (gL (0)) ≥ 0. Let (S n , gL (t)), t ∈ [0, TL ), be the solution of the Ricci flow with initial metric gL (0), where TL < ∞ is the maximal time of existence. By the strong maximum principle, since gL (0) has nonnegative curvature operator everywhere and positive curvature operator on the caps, we have that gL (t) has positive curvature operator for t > 0. Step 3. Normalizing the family of solutions. 19 The boundedness of the curvature also follows from Lemma 2.10 in Volume One on the smoothness of the metric on S n .

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

By Theorem 6.3 in Volume One and Theorem 11.2 in Part II (i.e., Hamilton’s convergence theorem for solutions of the Ricci flow with positive Ricci curvature on closed 3-manifolds [88], his 4-dimensional result [89], and B¨ ohm and Wilking’s corresponding result in all dimensions for solutions with 2-positive curvature operator [16], respectively), we know that under the Ricci flow, the metric gL (t) shrinks to a round point as t → TL . For any given δ > 0 sufficiently small, let tL ∈ (0, TL ) be a time such that the global sectional curvature ratio satisfies (19.28)

Λ (gL (t)) 

maxS n sect (gL (t)) ≤1+δ minS n sect (gL (t))

for t ∈ [tL , TL )

with equality (i.e., = 1 + δ) holding at the time tL . (Note that Λ (g) − 1 is a scale-invariant measure of how close the metric g is to constant sectional curvature.) The existence of such a time tL ∈ (0, TL ) follows from the fact that the solution g˜L (t) of the volume normalized Ricci flow with g˜L (0) = gL (0) converges to a round S n . Note that for any ε > 0 we have maxS n sect (gL (t)) =∞ t→0 minS n sect (gL (t)) lim

(the principal sectional curvatures, equal to 12 of the eigenvalues of the Rie1 or 0 on the cylinder mann curvature operator, at t = 0 are equal to 2(n−2) part). We consider the family of rescaled and time translated solutions 1 gL (TL + (TL − tL ) t) (19.29) gL (t)  TL − tL for TL + (TL − tL ) t ∈ [0, TL ), where L ∈ (1, ∞). The initial time for the solution gL (t) is (19.30)

t0 (L)  −

TL < −1 TL − tL

whereas the singularity time is 0. Step 4. The normalized family is uniformly κ-noncollapsed. Lemma 19.40. There exists κ >" 0 independent of L such that gL (t) is κ-noncollapsed at any scale 0 < r ≤ 4(TLTL−tL ) for all L ∈ (1, ∞). Proof. Since by (19.27), for the original solution gL (t), the curvature tensor satisfies |Rm| (gL (0)) ≤ C (n) for some C (n) < ∞ independent of L, we know that TL ≥ c (n) for some constant c (n) > 0 by Hamilton’s doubling time estimate for the norm of the curvature tensor (see Corollary 7.5 in Volume One or the slightly more precise Lemma 6.1 in [45]).20 20

In particular, |Rm| (gL (t)) ≤ 2C (n) for t ∈ [0, 1/ (16C (n))].

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On the other hand, since the initial scalar curvature satisfies R (gL (0)) ≥ c > 0, independent of L,21 by applying the weak maximum principle to the evolud R ≥ ∆R + n2 R2 for the scalar curvature, we obtain tion inequality dt R (gL (t)) ≥

n c

n . − 2t

This implies n . 2c   n and To summarize, gL (t) forms a singularity at a time TL ∈ c (n) , 2c there exists κ0 > 0 such that gL (0) is κ0 -noncollapsed for all L > 1. By Perelman’s weakened no local collapsing theorem (see §7.3 in Perelman [152] or Theorem 8.26 in Part I, which is restated as Theorem 19.52 below), we conclude that there exists κ > "0 independent of L such that gL (t) TL ≤

is κ-noncollapsed at any scale 0 < r ≤ T4L for all L > 1. Note that since we have Rm ≥ 0, weakly κ-noncollapsing (see p. 401 of Part I) is equivalent to κ-noncollapsing; moreover, from the proof of Theorem 8.26 in Part I, it is clear that the dependence of κ on supM×[0, T ] Rcg˜(t) can be replaced by supM×0,

1 16C(n)

 Rc

2

22 which is bounded independent of L here. Since gL (t) ,

the property of κ-noncollapsing is invariant " under scaling, the solution gL (t)

is κ-noncollapsed at any scale 0 < r ≤

TL 4(TL −tL )

for all L > 1.




Remark 19.41. The reason for using the weakened no local collapsing theorem (proved using the reduced volume monotonicity) instead of the no local collapsing theorem (proved using entropy monotonicity) is that by using the former result one can show that κ is independent of L. Step 5. The rescaled solutions tend to ancient as L → ∞. Claim 1. In definition (19.30) we have (19.31)

lim t0 (L) = −∞,

L→∞

that is, (19.32)

TL = ∞. L→∞ TL − tL lim

21 Note that for n = 2 this positive lower bound for R does not hold for the above construction since a 2-dimensional cylinder is flat. 22 To wit, the factor 12 in T2 can be replaced by any number in the interval (0, 1), in 1 . particular 16C(n)

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Proof. We divide the proof of this claim into a few steps.  of gL (t) has the upper (i) We first show that the scalar curvature R bound  (t) ≤ (n − 1) −1 − t0 (L) for t ∈ (t0 (L) , −1]. (19.33) R t − t0 (L) To see this inequality, at a fixed point we integrate in time Hamilton’s trace Harnack inequality (which holds since Rm ≥ 0; see [90] or Corollary 15.3 in Part II)   R ∂R ≥− ∂t t − t0 (L) from t to −1, to get (19.34)

 (x, −1) t − t0 (L) R ≥  −1 − t0 (L) R (x, t)

for any x ∈ S n and t ≤ −1. sphere metric of radius  We claim that gL (−1) is ε-close to the round n  2 (n − 1) and in particular R (−1) is close to 2 when ε is small (the idea is that gL (−1) has sectional curvatures pinched, with pinching ratio near 1, and it takes one unit of time for it to become singular under the Ricci flow). As a consequence,  (−1) ≤ n − 1 R (as ε → 0, we may take the upper bound approach n2 ). Combining this with (19.34) yields (19.33).  (x, t) satisfies min (t)  minx∈S n R We now prove the claim. Recall that R  2 2   2 d   Rmin ≥ 2 min Rc Rmin ,  ≥ x∈Ω dt n    (x, t) = R min (t) , and limt→0 R min (t) = ∞. where Ω = Ω (t)  x ∈ S n : R Hence, by integrating the inequality  −2 d d   −1 min ≤ − 2 , min Rmin R =− R dt dt n we have  τ −1  2 d   −1  Rmin (−1) = lim (t) dt ≤ − . − Rmin τ →0 −1 dt n That is, min (−1) ≤ n . R 2 On the other hand, for any η > 0, if (19.28) holds for δ > 0 sufficiently small depending only on η, then for t ∈ [−1, 0) 2     gL  ≤ η R 2 , Rc  − 1 R   n n

(19.35)

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 denotes the Ricci tensor of gL . We then compute where Rc  2 d   Rmax ≤ 2 max Rc  x∈Ω dt   2   1 1 2  gL  + R   − R = 2 max Rc  x∈Ω n n  2 2 max , (1 + η) R n    (x, t) = R max (t) . Integrating the inequalwhere Ω = Ω (t)  x ∈ S n : R ity  −1 −2 d d  max ≥ − 2 (1 + η) max Rmax R =− R dt dt n max (t) = ∞, we obtain from −1 to 0, while using limt→0 R n min (−1) ≥ . (19.36) R 2 (1 + η) ≤

By applying (19.28) at t = −1, i.e., gL (−1)) maxS n sect ( ≤1+δ minS n sect ( gL (−1)) to (19.35) and (19.36), i.e., n min (−1) ≤ n , ≤R 2 (1 + η) 2 1 ; we conclude that the sectional curvatures of gL (−1) are close to 2(n−1) intuitively, gL (−1) is metrically close to the round sphere metric of radius  2 (n − 1). (ii) Second we note the changing distances estimate for gL (t): ' −1 − t0 (L) ∂  dt (x, y) ≥ −4 (n − 1) for t ≤ −1 and x, y ∈ S n , (19.37) ∂t t − t0 (L)

where dt denotes the distance function for gL (t). To see this, recall that by Theorem 18.7(2), for a solution (Mn , g (t)) of the Ricci flow, if Rc(x, t0 ) ≤ (n − 1)K



−1/2 ∪ Bg(t0 ) x1 , K −1/2 , then the distance function for all x ∈ Bg(t0 ) x0 , K dg(t) (x0 , x1 ) satisfies the following differential inequality (at time t = t0 ):  √ ∂  d (x , x ) ≥ −4 (n − 1) K. 0 1 g(t) ∂t t=t0 To obtain (19.37), by Rm ≥ 0 and (19.33), it suffices to choose K (t) =

−1 − t0 (L) . t − t0 (L)

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

(iii) The diameter of gL (t0 (L)) does not exceed −8 (n − 1) t0 (L). By integrating (19.37) from t0 (L) to −1, we obtain d−1 (x, y) − dt0 (L) (x, y) =

 

≥

−1 t0 (L) −1

∂  dt (x, y)dt ∂t ' −4 (n − 1)

t0 (L)

−1 − t0 (L) dt t − t0 (L)

= −8 (n − 1) (−1 − t0 (L)) , so that dt0 (L) (x, y) ≤ d−1 (x, y) + 8 (n − 1) (−1 − t0 (L)) .

(19.38)

On the other hand, since the sectional curvatures of gL (−1) are close to that  of the round sphere of radius 2 (n − 1) (see the end of (i) in the proof), by Myers’s theorem, we have √ d−1 (x, y) ≤ 2 n − 1π for any x, y ∈ S n so long as ε is sufficiently small. Now (iii) follows from (19.38): √ dt0 (L) (x, y) ≤ 2 n − 1π − 8 (n − 1) − 8 (n − 1) t0 (L) ≤ −8 (n − 1) t0 (L) . We now finish the proof of Claim 1. Since diam (gL (0)) ≥ L, by our construction of gL (t) we know that the initial diameter has the lower bound diam( gL (t0 (L))) ≥ √

L . TL − tL

Hence √

TL L ≤ −8 (n − 1) t0 (L) = 8 (n − 1) , TL − tL TL − tL

and we have

 1 L TL − tL . 8 (n − 1)   n , we obtain the key estimate Recalling that TL ∈ c (n) , 2c TL ≥

lim (TL − tL ) = 0,

L→∞

that is, TL → −∞ TL − tL This finishes the proof of Claim 1. t0 (L) = −

as L → ∞.

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3. PERELMAN’S κ-SOLUTION ON THE n-SPHERE

103

Step 6. Taking a sequential limit to get Perelman’s κ-solution. The family of solutions { gL (t)}, t ∈ (t0 (L) , 0), satisfies the hypotheses of Hamilton’s compactness theorem for the Ricci flow (see [93] or Theorem 3.10 in Part I) with base time t = −1; specifically: (i) by the trace Harnack inequality and Rm ≥ 0, the curvatures are uniformly bounded on the interval −∞ < t ≤ −1/2 independent of L sufficiently large (provided δ > 0 is chosen sufficiently small), and (ii) there is a uniform injectivity radius lower bound at t = −1 as a consequence of Perelman’s no local collapsing theorem (or we may use Klingenberg’s injectivity radius theorem since we know that the sectional curvatures are sufficiently pinched and S n is simply connected). Therefore there exists a sequence gLi (t), with Li → ∞, which converges to a complete limit solution (Mn∞ , g∞ (t)), t ≤ − 12 . Moreover, the uniform bound on the diameters (independent of Li ) at time t = −1 implies that Mn∞ is compact and hence diffeomorphic to S n . Step 7. Properties of the limit solution. This limit solution is an ancient solution (since t0 (L) → −∞), is κnoncollapsed "at all scales (since gL (t) is κ-noncollapsed on scales less than or equal to

TL 4(TL −tL )

and by (19.32)), has nonnegative curvature operator,

and is nonflat. That is, by definition, (S n , g∞ (t)) is a κ-solution. Moreover, the limit solution is rotationally symmetric and invariant under a reflection. One way to see the rotational symmetry is as follows. Choose basepoints Oi on the center spheres (invariant under both rotation and the single reflectionally isometry) of (S n , gLi (t)). Then there exists a subsequence such that the pointed Cheeger–Gromov limit of (S n , gLi (t), Oi ), call it (S n , g∞ (t), O∞ ), exists. The group O (n) acts faithfully by isometries on each (S n , gLi (t)). By definition, there exist diffeomorphisms Φi : S n → S n such that Φ∗i gLi (t) converges to g∞ (t) in each C k norm. Since the group O (n) acts by isometries on each (S n , Φ∗i gLi (t)), we obtain that O (n) acts by isometries on each (S n , g∞ (t)). This action is faithful since the center spheres of Φ∗i gLi (t) have uniformly bounded (above and below) radii. Since at t = −1 the sectional curvatures of gL (t) oscillate exactly by the amount 1 + δ, the limit g∞ (−1) has sectional curvatures oscillating exactly by the amount 1 + δ. In particular, g∞ (t) is not a perfectly round sphere. Comments about the proof. The rescaling of the solutions via (19.28) is useful. By applying a curvature bound at time −1, the trace Harnack estimate, and the changing distances inequality (19.37) to the rescaled solutions, we obtain a good estimate for the difference of the distances at the initial time t0 (L) and the time −1, namely, inequality (19.38). Using the fact that the solutions at their initial times t0 (L) have relatively large diameter, one can then show that t0 (L) → −∞ and consequently the limit solution is an ancient solution.

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104

19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Note that the King–Rosenau solution is qualitatively different from Perelman’s κ-solution. This is related to the fact that the cigar soliton and the Bryant soliton are qualitatively different (for example, the curvature of the cigar soliton decays exponentially whereas the curvature of the Bryant soliton decays inverse linearly). For further discussions related to Perelman’s ancient solution on S n , see Example 20.12 and Optimistic Conjecture 20.26. Finally we remark that there is also a construction of Perelman’s rotationally symmetric noncompact (not ancient) standard solution on Rn (see §2 of [153]). 4. Equivalence of 2- and 3-dimensional κ-solutions with and without Harnack In this section we shall show that, in dimensions 2 and 3, the notion of κ-solution with Harnack is equivalent to the original notion of κ-solution; this result is due to one of the authors [142]. 4.1. Classification of 2-dimensional κ-solutions. The following classification result, due to Hamilton (see §26 of [92]), is also stated as Corollary 11.3 in [152] (see also Corollary 9.19 in [45] for a proof). Theorem 19.42 (Any 2-dimensional κ-solution is S 2 ). There is only one orientable 2-dimensional κ-solution: the shrinking round 2-sphere. We may prove the following generalization. Corollary 19.43 (Any 2-dimensional κ-solution with Harnack is S 2 ). There is only one orientable 2-dimensional κ-solution with Harnack: the round shrinking S 2 . Proof. By Theorem 19.42, it suffices to prove that any 2-dimensional κ-solution with Harnack has bounded curvature in space-time; we prove Suppose that there exists a κ-solution with Harnack

this 2by contradiction. M , g (t) , t ∈ (−∞, 0], which does not have bounded curvature in M × (−∞, 0]. Then sup R (x, 0) = ∞ x∈M

(using the fact that the trace Harnack estimate implies ∂R ∂t ≥ 0). Fix a point p ∈ M. Since supx∈M R (x, 0) = ∞, we may apply Proposition 18.12, so that there exists a sequence of points {xi }∞ i=1 and a sequence such that of positive numbers {ri }∞ i=1 (19.39)

dg(0) (xi , p) → ∞,

R (xi , 0) ri2 → ∞,

dg(0) (xi , p) /ri → ∞,

and (19.40)

sup

R (x, 0) ≤ 4R (xi , 0) .

Bg(0) (xi ,ri )

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4. 2- AND 3-DIMENSIONAL κ-SOLUTIONS

Let

105

  gi (t)  R (xi , 0) g R (xi , 0)−1 t

be the rescaled solutions. Since ∂R ∂t ≥ 0 and since g (t) is κ-noncollapsed at all scales, with the above choices of {xi } and {ri } and by Hamilton’s Cheeger–Gromov compactness theorem, a subsequence of (incomplete) so lutions Bg(0) (xi , ri ) , gi (t) , xi will converge to a complete limit ancient solution 2

M∞ , g∞ (t) , x∞ , t ∈ (−∞, 0], with (19.41)

Rg∞ (x∞ , 0) = 1 = 0

and supM∞ Rg∞ (·, 0) ≤ 4.23 On the other hand, by dimension reduction, this limit solution must contain a line (see Theorem 18.19) and it splits as



2 M∞ , g∞ (t) = R×W 1 , du2 + gW (t) . Since (W, gW (t)) is 1-dimensional, we have that (M∞ , g∞ (t)) is flat, which contradicts (19.41).
In contrast, recall that the cigar soliton solution has ASCR (g (0)) = 0, dimension reduces to a flat cylinder, and the cigar soliton is not κnoncollapsed at all scales for any κ > 0. 4.2. The equivalence of the notions of κ-solution and κ-solution with Harnack in dimension 3. In Corollary 19.43 we proved that a 2-dimensional κ-solution with Harnack is a κ-solution (i.e., has bounded curvature). This result extends to dimension 3 using the same idea of proof, namely dimension reduction, together with the fact that 3-dimensional κ-solutions with Harnack dimension reduce to round cylinders. We now prove the following. Proposition 19.44 (3-dimensional κ-solutions with or without Harnack

are equivalent). Any 3-dimensional κ-solution with Harnack M3 , g (t) , t ≤ 0, has bounded curvature on M × (−∞, 0]. Proof. Case (1). We first assume that (M, g (0)) does not have positive curvature operator. Then since Rm ≥ 0, it follows from Hamilton’s strong maximum principle that the universal cover of (M, g (0)) contains a line and splits as the product of a line with a 2-dimensional κ-solution with Harnack. It then follows from Corollary 19.43 that g (0) has bounded curvature. Case (2). Now suppose that (M, g (0)) has positive curvature operator. If (M, g (t)) does not have bounded curvature, then supx∈M R (x, 0) = ∞. 23 Note that since R (xi , 0) ri2 → ∞, the radii of the balls, that are defined with respect to gi (0), limit to infinity.

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Hence, by Proposition 18.12, fixing p ∈ M, there exist sequences {xi }∞ i=1 and {ri }∞ i=1 with ri ∈ (0, 1] satisfying (19.39), (19.40), and in particular R (xi , 0) → ∞.   Again let gi (t)  R (xi , 0) g R (xi , 0)−1 t . Since ∂R ∂t ≥ 0 and g (t) is κnoncollapsed at all scales, there exists a subsequence

Bg(0) (xi , ri ) , gi (t) , xi

which converges to a complete limit M3∞ , g∞ (t) , x∞ , t ≤ 0, with (19.42)

Rg∞ (x∞ , 0) = 1

and

sup Rg∞ (·, 0) ≤ 4 .

M∞

Since the trace Harnack estimate is preserved under Cheeger–Gromov limits,24 g∞ (t) satisfies the trace Harnack estimate. The limit (M∞ , g∞ (t)) must contain a line passing through x∞ (again see Theorem 18.19) and it splits as

(M∞ , g∞ (t)) = R×W 2 , du2 + gW (t) , where (W, gW (t)) is a 2-dimensional oriented κ-solution (it has positive bounded curvature by (19.42)). By Corollary 19.43, (W, gW (t)) is a round shrinking 2-sphere.25 Hence, for any ε > 0, there exists i0 ∈ N such that for i ≥ i0 , we have that Bg(0) xi , ε−1 cn R−1/2 (xi , 0) , g (0) is an embedded ε-neck in (M, g (0)) and xi lies on the center of the embedded ε-neck. Now since dg(0) (xi , p) → ∞, by choosing ε = ε (n) in Proposition 18.33, we have R (xi , 0) ≤ 144R (xi0 , 0) for all i sufficiently large. This contradicts R (xi , 0) → ∞ and the proposition is proved.
This proposition shows that it is not possible to have unbounded curvatures at centers of ε (n)-necks in noncompact manifolds with positive sectional curvatures. Remark 19.45. There is another approach, shown to us by Yu Ding, toward finishing the proof of Proposition 19.44 (see [54]). 5. Existence of an asymptotic shrinker In this section we derive a key estimate for the difference of the reduced distance at two points at the same time which was not proved earlier in this book series. After reviewing the reduced volume, we use this estimate to address an issue which we neglected to discuss in the proof of Lemma 8.38 in Part I. This lemma was used to prove the existence of an asymptotic shrinker. Since the convergence is in C ∞ on compact sets, the trace Harnack quadratics of the sequence converge to the trace Harnack quadratic of the limit solution. 25 Note that since M is noncompact and has positive sectional curvature, it is orientable, which implies that W is orientable. 24

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5. EXISTENCE OF AN ASYMPTOTIC SHRINKER

107

5.1. Two estimates for the reduced distance function. In this subsection we discuss two estimates for the reduced distance function which we shall use in the proof of the existence of an asymptotic shrinker. 5.1.1. A review of the reduced distance. Let (Mn , g (τ )), τ ∈ [0, T ), where 0 < T ≤ ∞, be a complete solution to the backward Ricci flow on a connected manifold. The genesis of Perelman’s L-geometry is the notion of the L-length of a piecewise C 1 -path γ : [τ1 , τ2 ] → M, where [τ1 , τ2 ] ⊂ [0, T ), which is defined by  2    τ2   dγ √ τ R (γ (τ ) , τ ) +  (τ ) dτ. (19.43) L (γ)  dτ τ1 g(τ ) Fixing a point p ∈ M, we define the L-distance from a point (x, τ ) ∈ M × (0, T ) to the ‘base point’ (p, 0) by (19.44)

L (x, τ )  L(p,0) (x, τ )  inf L (γ) , γ

where the infimum on the rhs is taken over all piecewise C 1 -paths γ : [0, τ ] → M joining p to x.26 This space-time distance-type function is fundamental to the study of the Ricci flow. The reduced distance from (x, τ ) to (p, 0) is defined by (19.45)

1  (x, τ )  √ L (x, τ ) . 2 τ

5.1.2. A lower estimate for the sum of  at two different points at the same time. We have the following estimate of Perelman relating the reduced distance at two different points at the same time (see Lemma 2.2 in Ye [191], (39.6) in Kleiner and Lott [110], or Lemma 9.25 on p. 192 ff. in Morgan and Tian [133]). Lemma 19.46. For any κ-solution (Mn , g (t)), t ≤ 0, and basepoint (p, 0), we have that the reduced distance of g (−τ ) satisfies (19.46)

− (q1 , τ¯) ≤  (q2 , τ¯) + 1 −

c (n) d2g(¯τ ) (q1 , q2 ) τ¯

for all τ¯ ∈ (0, ∞) and q1 , q2 ∈ M, where c (n) =  (q1 , τ¯) +  (q2 , τ¯) ≥

1 . 216(n+2)2

c (n) d2g(¯τ ) (q1 , q2 ) τ¯

In other words,

− 1.

26 More generally, for points p, q ∈ M and times 0 ≤ τ1 < τ2 < T , we define the L-distance by L(p,τ1 ) (q, τ2 ) = inf L (γ) , γ

where γ : [τ1 , τ2 ] → M joins p to q. If τ1 ≥ τ2 , then we define L(p,τ1 ) (q, τ2 ) = ∞.

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108

19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Proof. We follow the aforementioned references. Let γa be a minimal L-geodesic whose graph (γa (τ ) , τ ) joins (p0 , 0) to (qa , τ¯) for a = 1, 2. The distance between q1 and q2 at time τ¯ may be expressed as  τ¯

d dg(τ ) (γ1 (τ ) , γ2 (τ )) dτ dg(¯τ ) (q1 , q2 ) = 0 dτ   τ¯  ∂ (γ1 (τ ) , γ2 (τ )) dτ = d (19.47) ∂τ g(τ ) 0 ) 2  τ¯ (  dγa ∇a dg(τ ) (γ1 (τ ) , γ2 (τ )) , (τ ) dτ, + dτ 0 a=1

where ∇a dg(τ ) ( · , · ) denotes the gradient of dg(τ ) ( · , · ) with respect to the a-th variable. Recall from Riemannian geometry   ∇a dg(τ )  =1 (19.48) g(τ ) and by (7.54) in Part I (space-time Gauss lemma) dγa (τ ) = ∇ (γa (τ ) , τ ) . dτ Furthermore, the second inequality in Lemma 7.64 of Part I (see also the original (7.16) in Perelman [152]) says that for an ancient solution with bounded nonnegative curvature operator27 (19.49) Therefore

|∇|2 (q, τ ) + R (q, τ ) ≤

3  (q, τ ) . τ

   1/2  dγa   (τ ) = |∇| (γa (τ ) , τ ) ≤ 3  (γa (τ ) , τ )  .  dτ  τ

On the other hand, again using R ≥ 0, we have   1  (γa (τ ) , τ ) = √ L γa |[0,τ ] 2 τ 1 ≤ √ L (γa ) 2 τ  τ¯ 1/2 (19.50)  (qa , τ¯) = τ τ ) = qa ). Thus, for a = 1, 2 and τ ∈ (0, τ¯], (recall that γa (¯   √ 1/4  dγa   (τ ) ≤ 3 τ¯ 1/2 (qa , τ¯) .   dτ  τ 3/4 27

Note that for Euclidean space, |∇|2 (q, τ ) =

|q−p0 |2 4τ 2

=

1  (q, τ ). τ

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5. EXISTENCE OF AN ASYMPTOTIC SHRINKER

109

From this and (19.48) we conclude that the second pair of terms on the rhs of (19.47) is bounded by )  τ¯ ( dγa (τ ) dτ ∇a dg(τ ) (γ1 (τ ) , γ2 (τ )) , dτ 0   τ¯   dγa    ≤  dτ  (τ ) dτ 0 √ ≤ 4 3 τ¯1/2 1/2 (qa , τ¯) . Thus we have (19.51)

 ∂ dg(¯τ ) (q1 , q2 ) ≤ d (γ1 (τ ) , γ2 (τ )) dτ ∂τ g(τ ) 0   √ + 4 3 τ¯1/2 1/2 (q1 , τ¯) + 1/2 (q2 , τ¯) . 

τ¯ 

Now we bound the first term on the rhs of (19.51). The idea is to use estimate (18.15)–(18.16) for the time derivative of the distance function; namely, if for some r (τ ) and K (τ ) (19.52) then

Rc ≤ (n − 1) K (τ ) 

(19.53)

in B (γ1 (τ ) , r (τ )) ∪ B (γ2 (τ ) , r (τ )) ,

   1 ∂ d + K (τ ) r (τ ) ; (γ1 (τ ) , γ2 (τ )) ≤ 2 (n − 1) ∂τ g(τ ) r (τ )

we shall choose r (τ ) and K (τ ) below, so that (19.52) holds. Since √  √  |∇| 3   ∇  (q, τ ) = √ (q, τ ) ≤ √ , 2 τ 2  by applying the fundamental theorem of calculus along geodesics with respect to g (τ ), we have for any point q ∈ Bg(τ ) (γa (τ ) , r (τ )) √ √ √ 3  (q, τ ) ≤  (γa (τ ) , τ ) + √ r (τ ) 2 τ √  τ¯ 1/4 √ 3 (19.54)  (qa , τ¯) + √ r (τ ) ≤ τ 2 τ by (19.50). Since (19.46) is symmetric in q1 and q2 , without loss of generality we may assume that  (q1 , τ¯) ≤  (q2 , τ¯) . From Rc ≥ 0, (19.49), and this, we then have Rc (q, τ ) ≤ R (q, τ ) 3 ≤  (q, τ ) τ  2 √ 3 3  τ¯ 1/4 √  (q2 , τ¯) + √ r (τ )  (n − 1) K (τ ) ≤ τ τ 2 τ

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

for q ∈ Bg(τ ) (γa (τ ) , r (τ )). Then the rhs of (19.53) is  2 (n − 1)

 1 + K (τ ) r (τ ) r (τ ) ⎞ ⎛  2 √   1/4 √ 3 τ¯ n−1 3 +  (q2 , τ¯) + √ r (τ ) r (τ )⎠ . = 2⎝ r (τ ) τ τ 2 τ

Now let r (τ ) be of the form A A ≤ , r (τ ) = √ B  (q2 , τ¯) + B where below we choose A and B depending on τ and τ¯. Then   1 + K (τ ) r (τ ) 2 (n − 1) r (τ )  2 (n − 1) √  (q2 , τ¯) + B ≤ A  2 √ 3 A A 6  τ¯ 1/4 √ √  (q2 , τ¯) + √ + τ τ 2 τB  (q2 , τ¯) + B   2 (n − 1) √  (q2 , τ¯) + B = A 2  √ 3 A τ¯1/2 √ −1/4 A √ . (¯ ττ)  (q2 , τ¯) + + 6 3/2 2 B τ  (q2 , τ¯) + B First assume that √ 3 A (¯ τ τ )−1/4 = B. 2 B

(19.55) Then

 1 + K (τ ) r (τ ) r (τ )     τ¯1/2 √ n − 1 √  (q2 , τ¯) + B + 3A 3/2  (q2 , τ¯) + B . ≤2 A τ 

2 (n − 1)

Next assume equivalent to

2 A

1/2

= 3A ττ¯3/2 , that is, A =

"

2 −1/4 3/4 ¯ τ . 3τ

Then (19.55) is

1 −1/4 1/4 τ¯ τ . B= √ 4 2

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5. EXISTENCE OF AN ASYMPTOTIC SHRINKER

Thus

111

 1 + K (τ ) r (τ ) 2 (n − 1) r (τ )   √ √ 1 −1/4 1/4 1/4 −3/4 τ¯  (q2 , τ¯) + √ τ . ≤ 6 (n + 1) τ¯ τ 4 2 By applying this to (19.53), we conclude from (19.51) that    τ¯ √ √ 31/4 −1/4 1/4 1/4 −3/4 τ¯ τ  (q2 , τ¯) + √ τ¯ τ dτ dg(¯τ ) (q1 , q2 ) ≤ 6 (n + 1) 2 0  √ 1/2  1/2 τ  (q1 , τ¯) + 1/2 (q2 , τ¯) + 4 3¯ √ √ = 4 6 (n + 1) τ¯1/2  (q2 , τ¯) + 33/4 2 (n + 1) τ¯1/2  √ 1/2  1/2 + 4 3¯ τ  (q1 , τ¯) + 1/2 (q2 , τ¯)   √ ≤ 4 6 (n + 2) τ¯1/2 1/2 (q1 , τ¯) + 1/2 (q2 , τ¯) 

+ 33/4 2 (n + 1) τ¯1/2 . Therefore dg(¯τ ) (q1 , q2 ) 1 1 − 1/2 (q1 , τ¯) + 1/2 (q2 , τ¯) ≥ √ 2 τ¯1/2 4 6 (n + 2) since

31/4 2(n+1) √ 4 6(n+2)

≤ 12 . We finally obtain (19.46) from28   4 1/2 1 2 1/2  (q1 , τ¯) +  (q2 , τ¯) +  (q1 , τ¯) +  (q2 , τ¯) + 1 ≥ . 9 2


5.1.3. Estimates for  and R in large neighborhoods of (qτ , τ ). The following is Lemma 8.35 in Part I. Lemma 19.47. Let (Mn , g (t)), t ≤ 0, be a κ-solution with basepoint (p, 0). Let  (q, τ ) be the reduced distance of g (−τ ) and for each τ > 0 let qτ 28

For any a, b > 0 we have a2 + b2 + 1 ≥

since

4 9

 a+b+

1 2

2

2  1 a2 + b2 + 1 − c a + b + 2

c = (1 − c) a2 + (1 − c) b2 − 2cab − ca − cb + 1 − 4 c c 1 1 ≥ (1 − 2c) a2 − ca + 1− + (1 − 2c) b2 − cb + 1− . 2 4 2 4

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

be a point such that  (qτ , τ ) ≤ n/2. Then for any ε > 0 and A > 1, there exists δ > 0 such that for any τ > 0,  (q, τ˜) ≤ δ −1 and τ˜R (q, τ˜) ≤ δ −1    √  for all (q, τ˜) ∈ Bg(τ ) qτ , ε−1 τ × A−1 τ, Aτ .

(19.56)

5.2. Review of the reduced volume. In this subsection we recall some results which were proved in Chapter 8 of Part I. The following definition is partially motivated by the integral of the heat kernel and partially motivated as a space-time volume (see Definition 8.14 in Part I). Definition 19.48 (Reduced volume for Ricci flow). Let (Mn , g (τ )), τ ∈ [0, T ], be a complete solution to the backward Ricci flow with bounded curvature. The reduced volume functional is defined by  ˜ (4πτ )−n/2 exp (− (q, τ )) dµg(τ ) (q) (19.57) V (τ )  M

for τ ∈ (0, T ). The reduced volume is monotone nondecreasing under the Ricci flow (see Corollary 8.17 in Part I). Theorem 19.49 (Reduced volume monotonicity). Suppose (Mn , g(τ )), τ ∈ [0, T ], is a complete solution to the backward Ricci flow with the curvature bound |Rm (x, τ )| ≤ C0 < ∞ for (x, τ ) ∈ M × [0, T ]. Then (i) limτ →0+ V˜ (τ ) = 1. (ii) The reduced volume is nonincreasing: (19.58)

V˜ (τ1 ) ≥ V˜ (τ2 )

for any 0 < τ1 < τ2 < T, and V˜ (τ ) ≤ 1 for any τ ∈ (0, T ). (iii) Equality in (19.58) holds if and only if (M, g (τ )) is isometric to Euclidean space (Rn , gE ). The reduced volume monotonicity formula may be used to prove a weakened version of no local collapsing. Recall Definition 8.23 in Part I. Definition 19.50 (Strongly κ-collapsed). Let κ > 0 be a constant. We say that a solution (Mn , g˜ (t)), t ∈ [0, T ), to the Ricci flow is strongly κ-collapsed at (q0 , t0 ) ∈ M × (0, T ) at scale r > 0 if (1) (curvature bound in a parabolic cylinder ) |Rmg˜ (x, t)| ≤ r12 for all

x ∈ Bg˜(t0 ) (q0 , r) and t ∈ [max t0 − r2 , 0 , t0 ] and (2) (volume of ball is κ-collapsed ) Volg˜(t0 ) Bg˜(t0 ) (q0 , r) < κ. rn

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5. EXISTENCE OF AN ASYMPTOTIC SHRINKER

113

Given an r > 0, if for any t0 ∈ [r2 , T ) and any q0 ∈ M the solution g˜ (t) is not strongly κ-collapsed at (q0 , t0 ) at scale r, then we say that (M, g˜ (t)) is weakly κ-noncollapsed at scale r (note the slight difference between this and Definition 18.34). Recall the weakened no local collapsing estimate and theorem (Theorem 8.24 and Theorem 8.26 in Part I). Theorem 19.51 (Main estimate for weakened no local collapsing). Let (Mn , g˜ (t)) , t ∈ [0, T ), be a complete solution to the Ricci flow with T < ∞ and suppose supM×[0,t1 ] |Rm| < ∞ for any t1 < T. Then there exists c1 = c1 (n) ∈ (0, 12 ] depending only on n such that if for some κ1/n ≤ c1 (n) , the solution g˜ (t) is strongly κ-collapsed at (p∗ , t∗ ) at scale r, where t∗ > T2 √ and r < t∗ , then the reduced volume V˜∗ of g∗ (τ )  g˜ (t∗ − τ ) with basepoint p∗ has the upper bound

(19.59) V˜∗ εr2 ≤ φ (ε, n) , where ε  κ1/n and φ (ε, n) 

exp

1

6 n (n − n/2

(4π)

1)

ε

n/2

+ ωn−1 (n − 2)

n−2 2

e

− n−2 2

  1 exp − √ . 2 ε

Theorem 19.52 (Weakened no local collapsing). Let (Mn , g˜ (t)), t ∈ [0, T ), be a complete solution to the Ricci flow with T < ∞. Suppose (1) supM×[0,t1 ] |Rm| < ∞ for any t1 < T and (2) there exist r1 > 0 and v1 > 0 such that Volg˜(0) Bg˜(0) (x, r1 ) ≥ v1 for all x ∈ M. Then there exists κ > 0 depending only on r1 , v1 , n, T, and supM×[0,T/2] Rc g˜(t) such that g˜ (t) is  weakly κ-noncollapsed at any point (p∗ , t∗ ) ∈ M × (T /2, T ) at any scale r < T /2. 5.3. Proof of the existence of an asymptotic shrinker. An important aspect of the geometry of κ-solutions is the existence of an asymptotic gradient shrinking soliton, which is given by Perelman’s Proposition 11.2 in [152] (see also Theorem 8.32 in Part I). Theorem 19.53 (Existence of an asymptotic shrinking soliton for κ-solutions). Let (Mn , g˜ (t)), t ∈ (−∞, 0], be a κ-solution. Given a basepoint (p, t0 ) , let τ (t)  t0 − t and g (τ )  g˜ (t0 − τ ). (1) For any sequence τi → ∞, there exists qi ∈ M such that g (qi , τi ) ≤ C for all i for some C < ∞ (we may take C = n/2).

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

(2) There exists

still denoted by (qi , τi ), such that the an subsequence, −1 sequence M , τi g (τi τ ) , qi of dilated solutions of the backward Ricci flow converges in the Cheeger–Gromov sense to a complete nonflat gradient shrinking Ricci soliton (Mn∞ , g∞ (τ ) , q∞ ). (3) The trace Harnack estimate holds for g∞ (τ ); hence the limit solution (M∞ , g∞ (τ

with Harnack. )) is a κ-solution (4) If n = 3, then M3∞ , g∞ (τ ) , q∞ has bounded Rm∞ ≥ 0. For example, the (unique up to homothety) asymptotic shrinker of either (i) the Bryant soliton on Rn or (ii) Perelman’s ancient solution on S n is the shrinking cylinder S n−1 × R. We now complete our presentation of the proof of Theorem 19.53. The reader is assumed to have read §4 of Chapter 8 in Part I. Given a κ-solution (Mn , g (θ)), θ ∈ [0, ∞), to the backward Ricci flow and given τ > 0, define the rescaled solutions gτ (θ)  τ −1 g(τ θ), for θ ∈ [0, ∞). Step 1. Existence of a backward limit. Choosing any sequence τi → ∞ and qτi ∈ M so that n  (qτi , τi ) ≤ 2 (such qτi exist by (7.15) of Perelman [152]; see also (7.95) in Part I), recall that for a subsequence we have convergence in the C ∞ pointed Cheeger– Gromov sense (19.60)

(Mn , gτi (θ), qτi ) −→ (Mn∞ , g∞ (θ), q∞ ),

θ ∈ (0, ∞) ,

where g∞ (θ) is a complete solution to the backward Ricci flow which is κnoncollapsed on all scales, has Rmg∞ ≥ 0, and satisfies the trace Harnack estimate.29 In particular, there exist smooth embeddings Φi : (Ui , q∞ ) ⊂ (M∞ , q∞ ) → (M, qτi ) , where {Ui } is an exhaustion of M∞ , such that gi (θ)  Φ∗i gτi (θ) → g∞ (θ) in each C k norm, k ∈ N, on compact subsets of M∞ . Step 2. Convergence of the reduced distances. Fix a basepoint p0 ∈ M. We also have that the reduced distances of gτi , pulled back by Φi , converge to (19.61)

gτ

i  (p0i ,0) ◦ Φi → ∞ ,

where ∞ is a locally Lipschitz function on M∞ × (0, ∞) with ∇g∞ ∞ and ∂ g∞ ). ∂θ ∞ existing a.e. (a priori, it is possible that ∞ =    In Theorem 8.32 of Part I, we stated this for θ ∈ A−1 , A for arbitrary A > 1. The convergence for θ ∈ (0, ∞) follows from a standard diagonalization argument. 29

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5. EXISTENCE OF AN ASYMPTOTIC SHRINKER

115

Step 3. Limit of the reduced volumes. The reduced volume (with basepoint p0 ) of the solution gτi (θ) is  gτ ˜ (4πθ)−n/2 e−
i (q,θ) dµgτi (θ) (q) for θ ∈ (0, ∞) . Vgτi (θ)  M

Corresponding to the limit solution (Mn∞ , g∞ (θ), q∞ ) in (19.60), define the mock reduced volume by  (4πθ)−n/2 e−
∞ (q,θ) dµg∞ (θ) (q) for θ ∈ (0, ∞) , (19.62) Vˆ∞ (θ)  M∞

where ∞ is given by (19.61). The aforementioned issue we neglected to discuss (on p. 413 of Part I) is in regards to showing that Vˆ∞ (θ) is finite and (19.63)

Vˆ∞ (θ) = lim V˜g (τi θ). i→∞

The rest of this discussion is devoted to proving this equality. Fix

any θ ∈ (0, ∞), ε > 0, and consider the balls Bgτi (θ) qτi , 1ε ⊂ M and

Bg∞ (θ) q∞ , 1ε ⊂ M∞ . Clearly  gτ (4πθ)−n/2 e−
i (q,θ) dµgτi (θ) (q) Bgτ (θ) (qτi , 1ε ) i  (4πθ)−n/2 e−
i (q,θ) dµgi (θ) (q) = −1 1 Bg (θ) (Φi (qτi ), ε )  i (4πθ)−n/2 e−
∞ (q,θ) dµg∞ (θ) (q) (19.64) → 1 Bg∞ (θ) (q∞ , ε ) (note that Φi (q∞ ) = qτi ). Now we recall Perelman’s estimate for the reduced distance. By (19.46) we have that there exists a constant c (n) > 0 depending only on n such that for any q ∈ M and θ ∈ (0, ∞) (19.65) (19.66)

−gτi (q, θ) ≤ gτi (qτi , θ) + 1 − ≤

n + 2 c (n) − 2

c (n) d2gτ

i (θ)

(q, qτi )

θ

d2gτ (θ) (q, qτi ) i θ

.

This is the main estimate we shall use to prove (19.63). Pulling back by Φi and taking the limit, we have −∞ (q, θ) ≤

2 n + 2 c (n) dg∞ (θ) (q, q∞ ) − . 2 θ

From this and the volume comparison theorem (Rmg∞ (θ) ≥ 0), we obtain  (4πθ)−n/2 e−
∞ (q,θ) dµg∞ (θ) (q) = Vˆ∞ (θ). (19.67) lim 1 ε→0 B q , g∞ (θ) ( ∞ ε )

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

On the other hand, by (19.66) and the volume comparison theorem (again Rmgτi (θ) ≥ 0), we have for all i  gτ (4πθ)−n/2 e−
i (q,θ) dµgτi (θ) (q) M−Bgτ (θ) (qτi , 1ε ) i  c(n) 2 n+2 − d (q,qτi ) (4πθ)−n/2 e 2 e θ gτi (θ) dµgτi (θ) (q) ≤ M−Bgτ (θ) (qτi , 1ε ) i  c(n) 2 n+2 (4πθ)−n/2 e 2 e− θ |x| dx (19.68) ≤ Rn −B ( 1ε )  √  = δ n, ε θ ,   √ √  √  where δ n, ε θ depends only on n and ε θ and where δ n, ε θ → 0 as ε → 0 (for fixed θ > 0). Combining (19.64), (19.67), and (19.68), we conclude that lim V˜g (τi θ) = Vˆ∞ (θ),

i→∞

which is (19.63). This completes our discussion related to the proof of Theorem 19.53. Exercise 19.54. (1) Let g (τ ) = 2 (n − 1) (τ + 1) gS n , τ ∈ [0, ∞), where gS n is the standard metric on the unit n-sphere. Show that for θ ∈ (0, ∞) g∞ (θ)  lim gτi (θ) = 2 (n − 1) θ gS n , i→∞

where the limit is pointwise in C ∞ . (2) Let (Mn , gi (θ)), θ ∈ [0, ∞), be Einstein solutions of the backward Ricci flow with Rgi (0) ≡ n2 τi > 0. Assume gi (1) pointwise converges to a metric g∞ (1) on M. Show that n ∞ (q, θ)  lim gi (q, θ) ≡ i→∞ 2 for all q ∈ M and θ ∈ (0, ∞). Hint: See subsection 7.1 of Chapter 7 in Part I for a formula for  = gi . 6. The κ-gap theorem for 3-dimensional κ-solutions First we recall the following result of Perelman on the nonexistence of 3-dimensional κ-noncollapsed gradient shrinking solitons (see the original §1.2 of Perelman’s [153] or §9.6 of [45]). Theorem 19.55 (Nonexistence of 3-dimensional compact shrinkers with Rm > 0). There do not exist complete noncompact 3-dimensional gradient shrinking solitons which have bounded positive sectional curvatures and are κ-noncollapsed at all scales.

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6. THE κ-GAP THEOREM FOR 3-DIMENSIONAL κ-SOLUTIONS

117

Hence it follows from the strong maximum principle that the round cylinder S 2 × R1 and its Z2 -quotient under the action (p, u) → (−p, −u) are the only orientable noncompact gradient shrinking solitons which are also κ-solutions. (See also Theorem 9.79 in [45] for example.) The following κ-gap theorem for 3-dimensional noncompact κ-solutions was proved by Perelman in §11.9 of [152]. The version we state here is a slight extension. Theorem 19.56 (κ-gap theorem for 3-dimensional nonspherical space form ancient κ-solutions). There exists a universal constant κ0 > 0 such that any 3-dimensional nonspherical space form κ-solution M3 , g (t) , t ≤ 0, is actually κ0 -noncollapsed at all scales, i.e., (M, g (t)) is a κ0 -solution. By a nonspherical space form κ-solution we mean a κ-solution which is not isometric to a shrinking spherical space form. Remark 19.57. The nonspherical space form assumption in the above theorem is necessary. This may be seen from considering the spherical space forms S 3 /Γ, where Γ ⊂ SO (4), since the order of Γ is unbounded. Consequently, (noncompact) counterexamples to this theorem in dimension 4 are given by S 3 /Γ × R (compare with Example 19.10). Problem 19.58 (κ-gap theorem for 4-dimensional noncompact ancient κ-solutions with positive curvature operator). Does the 4-dimensional version of the κ-gap theorem hold, where the nonspherical space form assumption is replaced by the assumption that the ancient κ-solution is not isometric to a shrinking S 3 /Γ × R? It is unclear to us whether or not to expect a counterexample. Note that complete noncompact Riemannian manifolds with positive sectional curvature are diffeomorphic to Euclidean space. We now give the Proof of Theorem 19.56. Since we can shift time, it suffices to prove that there exists κ0 > 0 such that any 3-dimensional nonspherical space form κ-solution M3 , g (t) , t ∈ (−∞, 0], is κ0 -noncollapsed at all scales at t = 0. Step 1. Any asymptotic shrinking soliton is a round cylinder or its Z2 -quotient. Let gˆ (τ )  g (−τ ), τ ∈ [0, ∞), be the corresponding backward solution to the Ricci flow and define gˆτ (θ)  τ −1 · gˆ(τ θ),

where θ ∈ [0, ∞),

for any τ ∈ (0, ∞). Let  : M×(0, ∞) → R be the reduced distance function defined by the backward solution gˆ (τ ) with basepoint (p, 0), where p ∈ M is to be chosen later. By Theorem 19.53, there exist sequences τi → ∞ and qi = q (τi ) ∈ M with reduced distance 3  (qi , τi ) ≤ 2

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS



such that the sequence (M3 , gˆτi (θ), qi ) converges to a nonflat 3-dimensional gradient shrinking Ricci soliton (M3∞ , gˆ∞ (θ), q∞ ) for θ ∈ (0, ∞). The limit solution (M∞ , gˆ∞ (−θ)), θ ∈ (−∞, 0) (with time reversed), is a 3-dimensional κ-solution with Harnack. Therefore, by Proposition 19.44, (M∞ , gˆ∞ (−θ)) is a κ-solution, i.e., the curvature of gˆ∞ (−θ) is bounded. We claim that the limit manifold M∞ is noncompact. Suppose that M∞ is compact. Then (M∞ , gˆ∞ (θ)) is a compact nonflat shrinking gradient Ricci soliton with bounded curvature. By Perelman’s classification theorem,30 (M∞ , gˆ∞ (θ)) is a compact quotient of S 2 × R or S 3 . On the other hand, compact quotient solutions of S 2 × R may be ruled out

are not since they κ-noncollapsed on all scales. Since a backwards limit of M 3 , g (t) is a compact quotient of the shrinking round S 3 , this implies that M3 , g (t) itself is actually a compact quotient solution of the shrinking round S 3 (this follows from Hamilton’s convergence theorem for solutions with positive Ricci curvature on closed 3-manifolds; for example, one may use his ‘curvature pinching improves’ estimate31 ). However, such a compact quotient solution of the shrinking round S 3 is ruled out by the hypotheses of the theorem. Now we may apply Theorem 19.55 to conclude that (M∞ , gˆ∞ (−θ)) is a shrinking round cylinder S 2 × R1 or its Z2 -quotient. Step 2. Universal lower bound for the reduced volume and κ1 -noncollapsing. 1 (and A = 2 say) in Lemma 19.47, there exists a Choosing ε = 100 universal constant δ ∈ (0, 1) such that

 (q, τi ) ≤ δ −1 , τi Rgˆ (q, τi ) ≤ δ −1 √

for all q ∈ Bgˆ(τi ) qi , 10 τi . Hence Rgˆτi (q, 1) ≤ δ −1 for q ∈ Bgˆτi (1) (qi , 10) and thus (19.69)

Rgˆ∞ (q, 1) ≤ δ −1

for q ∈ Bgˆ∞ (1) (q∞ , 10) .

Since gˆ∞ (1) is a round cylinder metric on S 2 (ρ) × R or its Z2 -quotient, where ρ denotes the radius of S 2 (ρ), we obtain √ ρ ≥ 2δ. x∞ , y˜∞ ) in the universal cover of M∞ is a lift of q∞ , Thus, if q˜∞  (˜ then we have   √  √  q∞ , 10) ⊂ S 2 (ρ) × R ˜∞ , 2δ × BR y˜∞ , 7 2 ⊂ Bgˆ∞ (1) (˜ (19.70) BS 2 (ρ) x 30 31

See Lemma 1.2 in [152] (or Theorem 9.79 in [45]). See Theorem 6.30 in Volume One.

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6. THE κ-GAP THEOREM FOR 3-DIMENSIONAL κ-SOLUTIONS

119

since δ ∈ (0, 1).32 From (19.70) and    √  √  ˜∞ , 2δ × BR y˜∞ , 7 2 Vol BS 2 (ρ) x  √  √ ≥ 14 2δ Vol BS 2 (√2) ∗, 2 √ = 28 2δ (1 − cos 1) ≥ 18δ, we may estimate crudely that in M∞ (taking into account a possible Z2 quotient) Volgˆ∞ (1) Bgˆ∞ (1) (q∞ , 10) ≥ 9δ. Hence, for i sufficiently large, we have Volgˆτi (1) Bgˆτi (1) (qi , 10) ≥ 8δ. We now may estimate the reduced volume V˜ at τi from below:  ˜ (4πτi )−3/2 e−
(q,τi ) dµgˆ(τi ) (q) V (τi ) ≥ √ Bgˆ(τi ) (qi ,10 τi )

≥ (4πτi )−3/2 e−δ

(19.71)

−1

√ · Vol gˆ(τi ) Bgˆ(τi ) (qi , 10 τi )

= (4π)−3/2 e−δ

−1

· Volgˆτi (1) Bgˆτi (1) (qi , 10)

≥ (4π)−3/2 e−δ

−1

· 8δ.

Step 3. Finishing the proof of Theorem 19.56.

Let c1 (n) be the constant in Theorem 19.51. Suppose that M3 , g (t) is κ1 -collapsed at the scale r0 > 0 at some point (x0 , 0) ∈ M × (−∞, 0] for some κ1 ∈ (κ, c1 (3)). It then follows from the trace Harnack estimate (which bounds the curvature backward in time) that g (t) is strongly κ1 -collapsed at (x0 , 0) at scale r0 (see Definition 19.50). Choosing the basepoint p used in the above discussion to be x0 and applying Theorem 19.51 below, we obtain     1 e 1/3 1/2 2 −1/2 κ1 + ω2 e exp − 1/6 , (19.72) V˜ κ1 r0 ≤ (4π)3/2 2κ1 where ω2 = π is the volume of the unit Euclidean 2-ball. For i sufficiently 1/n large we have τi ≥ κ1 r02 , so that by the monotonicity of the reduced volume we have   1/n (19.73) V˜ (τi ) ≤ V˜ κ r2 1

0

for such i. Combining (19.72), (19.73), and (19.71), we conclude that   e 1 1/2 −3/2 −δ −1 −1/2 e 8δ ≤ κ1 + πe exp − 1/6 . 0 < (4π) (4π)3/2 2κ1 32

We still denote the lifted metric by gˆ∞ (1). Note also that

√ 2  √ 2 2 + 7 2 = 102 .

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

Since the rhs tends to zero as κ1 tends to zero, this inequality gives a
uniform positive lower bound for κ1 and thus the theorem is proved. In summary, a 3-dimensional noncompact κ-solution has an asymptotic shrinking soliton which is a round cylinder S 2 × R or its (noncompact) quotient S 2 × R /Z2 . This corresponds to ε-necks in the κ-solution at times ti = −τi → −∞, where the reduced distance and scalar curvature are controlled. This leads to a positive lower bound for the reduced volume at τi independent of i. 7. Notes and commentary §1. κ-solutions arise as singularity models in the finite time singularity analysis of the Ricci flow on closed 3-manifolds. We remark that in dimensions greater than or equal to 4, one may wish to study ancient solutions which are κ-noncollapsed on all scales and have curvature conditions which are no weaker than nonnegative scalar curvature and no stronger than nonnegative curvature operator. For, on one hand, ancient solutions must have nonnegative scalar curvature; note also that any Einstein solution with positive scalar curvature is κ-noncollapsed at all scales. On the other hand, B¨ohm and Wilking have shown that the singularity models of solutions of the Ricci flow on closed manifolds whose initial metrics have 2-nonnegative curvature operator are shrinking spherical space forms. §2. Regarding AVR and ASCR under Ricci flow, we refer the reader to §§18, 19, and 22 of Hamilton’s [92] and Chapter 8 of [45]. §3. In dynamical systems, a heteroclinic orbit is a path which joins two different stationary (equilibrium) points. In the space of metrics modulo diffeomorphisms and scalings on a closed manifold, a metric is a stationary point of the Ricci flow equation if and only if it is a Ricci soliton. Question 19.59. Are there examples of ancient solutions which are not heteroclinic orbits, where convergence of metrics is defined in the Cheeger– Gromov sense and where collapsing is allowed (in particular, the topology and dimension of the underlying manifold is allowed to change in the limit)? For example, one may think of the King–Rosenau ancient solution (a.k.a. sausage model) as joining a pair of cigar solitons, with a flat cylinder in between, at time −∞ to a round sphere at time 0 (the solution shrinks to a round point). Similarly, Perelman’s ancient solution on the n-sphere joins two Bryant solitons, with a round cylinder in between, at time −∞ to a round point at time 0. Note that physicists have defined other examples of ancient solutions. Remark 19.60. One may also ask the above question assuming the ancient solutions are κ-noncollapsed at all scales for some κ > 0. §5. Here we give a first variation calculation related to the L-length. ∂ g = 2 Rc. We consider Let g (τ ) be a solution to the backward Ricci flow ∂τ

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7. NOTES AND COMMENTARY

121

the effect on the L-length of a path γ : [0, τ¯] → M by a variation v (τ ) of g (τ ). When we compute the variations of associated quantities, we write δv . Suppose that v (τ ) satisfies the backward Lichnerowicz Laplacian heat equation ∂ vij = −∆L vij . ∂τ This implies V = g ij vij satisfies ∂ V = −∆V − 2 Rc ·v. ∂τ Fixing the path, considering the variation δg = v, and using δv R = div (div v) − ∆V − v · Rc, we compute



 τ¯ √  τ R (γ (τ ) , τ ) + |γ˙ (τ )|2g(τ ) dτ δv L(γ) = δv 0  τ¯ √ τ (div (div v) − ∆V − v · Rc +v (γ, ˙ γ)) ˙ dτ = 0  τ¯ √ τ (div (div v) + v · Rc −2 div v · γ˙ + v (γ, ˙ γ)) ˙ dτ = 0    τ¯ √ ∂ V + 2 div v · γ˙ dτ. τ + ∂τ 0 Now ∂ d (V (γ (τ ) , τ )) = V + ∇V · γ. ˙ dτ ∂τ So, from integrating by parts, we have  τ¯ √ τ (div (div v) + v · Rc −2 div v · γ˙ + v (γ, ˙ γ)) ˙ dτ δv L(γ) = 0    τ¯ √ d (V (γ (τ ) , τ )) + (−∇V + 2 div v) · γ˙ dτ τ + dτ 0    τ¯ √ 1 (19.74) = τ div (div v) + v · Rc −2 div v · γ˙ + v (γ, ˙ γ) ˙ − V dτ 2τ 0    τ¯ √ √ 1 ˙ τ ) , τ¯) + 2 τ div v − ∇V · γdτ, + τ¯V (γ (¯ 2 0 where the linear trace Harnack quadratic L (v, γ) ˙ defined by (17.91) appears in the third line. Note that, in the discussion of the Ricci–DeTurck flow in

1 V Volume One, div v − 2 ∇V = div v − 2 g is the 1-form on p. 79 in Volume One; moreover, essentially, the 1-form   ˜k Wj  gjk g pq Γk − Γ pq

pq

defined by (3.34) in Volume One has variation equal to div v − 12 ∇V .

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19. GEOMETRIC PROPERTIES OF κ-SOLUTIONS

As a special case we may take vij = 2Rij , so that V = 2R, which implies    τ¯ √ R 2 τ ∆R + 2 |Rc| − 2∇R · γ˙ + 2 Rc (γ, ˙ γ) ˙ − dτ δ2 Rc L(γ) = τ 0 √ τ ) , τ¯) . + 2 τ¯R (γ (¯ Suppose that the solution to the backward Ricci flow exists on the time interval [0, T ] and τ = T − t. Since Hamilton’s trace Harnack estimate says R ≥ 0, ˙ γ) ˙ + ∆R + 2 |Rc|2 − 2∇R · γ˙ + 2 Rc (γ, T −τ we have    τ¯ √ √ 1 1 δ2 Rc L(γ) ≥ − + R (γ (τ ) , τ ) dτ + 2 τ¯R (γ (¯ τ τ ) , τ¯) . T −τ τ 0

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http://dx.doi.org/10.1090/surv/163/04

CHAPTER 20

Compactness of the Space of κ-Solutions It never would come to me working on a mystery. – From “Runnin’ down a Dream” by Tom Petty

Compactness theorems are fundamental in geometric analysis. In the case of sequences of solutions of the Ricci flow with curvatures uniformly bounded and injectivity radii uniformly bounded from below, Hamilton’s Cheeger–Gromov compactness theorem yields a subsequence which converges in C ∞ on compact sets (see Chapters 3 and 4 of Part I). By Perelman’s no local collapsing theorem, such sequences occur in the study of finite time singular solutions on closed manifolds when we rescale about a sequence of points and times whose curvatures are comparable to their space-time maximums up to those times. On the other hand, we are interested in the geometry of finite time singular solutions in their high curvature regions, not just where the curvatures are comparable to their space-time maximums. For example, we are interested in rescaling a singular solution about arbitrary sequences of basepoints and times whose curvatures tend to infinity. For such a sequence, the rate of blow up of the curvatures at the basepoints may be slower than the rate of blow up of the corresponding spatial maximums of the curvatures. That is, for such a sequence, the curvatures are not comparable to their spacetime maximums. Moreover, the study of high curvature regions starts with singularity models and κ-solutions. In §1 we study the geometry at spatial infinity of n-dimensional (n ≥ 3) noncompact κ-solutions and in particular we show that they have asymptotic scalar curvature ratio ASCR = ∞ and asymptotic volume ratio AVR = 0. Moreover, we show that the result of having AVR = 0 does not require the κ-noncollapsed at all scales assumption. These results on ASCR and AVR are related to the fact that 3-dimensional noncompact κ-solutions are asymptotically cylindrical at infinity. In §2 we discuss two results: (1) solutions which are almost ancient and have bounded nonnegative curvature operator are collapsed at large scales (Proposition 20.4) and (2) a curvature estimate in noncollapsed balls (Proposition 20.6). As we shall see, either of these two results may be used to prove Perelman’s compactness modulo scaling theorem. 123 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

124

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

In §3, we prove that for any n ≥ 3 the collection of n-dimensional κsolutions with Harnack is compact modulo scaling. As a special case, using a result of §4 in Chapter 19, we obtain Perelman’s fundamental result that the collection of 3-dimensional κ-solutions is compact modulo scaling. Compactness is one of the cornerstones of 3-dimensional singularity analysis. It enables one to probe the geometry of κ-solutions by taking limits of pointed sequences. The proof of Perelman’s compactness theorem exploits the dichotomy between the properties of no local collapsing and AVR = 0. In §4 we discuss a rather easy but important consequence: scaled derivative of curvature estimates for 3-dimensional κ-solutions. We also state some optimistic conjectures concerning the classification of ancient solutions in low dimensions. 1. ASCR and AVR of κ-solutions In this section we consider the asymptotic scalar curvature ratio ASCR and the asymptotic volume ratio AVR of κ-solutions (see subsection 2.1 of Chapter 19 for the definitions of ASCR and AVR). To oversimplify, the reason we consider these invariants is that curvature, volume, and distance are important in the study of the Ricci flow. The AVR is an ‘infinite radius’ version of the volume ratio seen in the definition of κ-noncollapsing. The ASCR is related to the notion of scaled curvature bounds. 1.1. κ-solutions with Harnack have infinite ASCR and zero AVR. Let (Mn , g(t)) be a complete noncompact κ-solution with Harnack and fix a point p ∈ M (see Definition 19.27). Recall that the AVR and ASCR are defined by (19.7) and (19.8), respectively. Denote AVR(t)  AVR(g (t)) and

ASCR(t)  ASCR(g (t)).

The following result was proved by Perelman in §11.4 of [152]. Theorem 20.1 (κ-solution with Harnack has ASCR = ∞, AVR = 0). For any complete noncompact κ-solution with Harnack with n ≥ 2, we have ASCR(t) = ∞

and

AVR(t) = 0

for all t ∈ (−∞, 0]. As a consequence of the proof of Theorem 20.1 we have the following. Corollary 20.2 (Any ancient solution with bounded Rm ≥ 0 must have AVR = 0). Let (Mn , g(t)), t ≤ 0, be a complete noncompact nonflat ancient solution to the Ricci flow. Suppose g(t) has nonnegative curvature operator and | Rm | (x, t) < ∞. sup (x,t)∈M×(−∞,0]

Then the asymptotic volume ratio is zero, i.e., AVR(t) = 0 for all t.

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1. ASCR AND AVR OF κ-SOLUTIONS

125

Proof. We prove the corollary by contradiction. Suppose that there exists a complete noncompact nonflat ancient solution (Mn , g(t)) to the Ricci flow with bounded nonnegative curvature operator such that there exists t0 ≤ 0 with ωn AVR(t0 )  κ0 > 0. Then by the Bishop–Gromov volume comparison theorem we have that g(t0 ) is κ0 -noncollapsed at all scales (in fact, we get a lower bound for volume ratios without assuming curvature bounds on balls). By the proof of Theorem 20.1 we have AVR(t0 ) = 0; 1
this is a contradiction. Remark 20.3. The corresponding result in the K¨ ahler case, conjectured by H.-D. Cao, was proved by one of the authors (see Theorem 2 in [140]). The statement is as follows. Let (Mm , g(t)), t ∈ (−∞, 0], be a nonflat ∂ gαβ¯ = −Rαβ¯ with nonnegative ancient solution to the K¨ahler–Ricci flow ∂t bisectional curvature and sup(x,t)∈M×(−∞,0] R(x, t) < ∞. Then the asymptotic volume ratio satisfies AVR(t) = 0 for all t. We shall prove Theorem 20.1 by contradiction and by induction on the dimension n.2 First observe that Corollary 19.43 implies that when n = 2 there are no noncompact κ-solutions with Harnack, and hence we are done in this case. Then suppose that the theorem holds in dimension n − 1, where n ≥ 3. We shall prove by contradiction that in dimension n we have ASCR (t) = ∞ and AVR (t) = 0 for all t. In particular, if this last statement is false, then there exists t0 such that one of the following three mutually exclusive cases holds. Case A. ASCR(t0 ) = ∞ and AVR(t0 ) > 0. Case B. ASCR(t0 ) ∈ (0, ∞). Case C. ASCR(t0 ) = 0. Case A. ASCR(t0 ) = ∞ and AVR(t0 ) > 0 for some t0 ≤ 0. In this case we can perform dimension reduction and use mathematical induction. Since ASCR(t0 ) = ∞, by Theorem 18.10, there exist a sequence of points 2 {xi }∞ i=1 with dg(t0 ) (xi , p) → ∞ and radii ri > 0 such that R (xi , t0 ) ri → ∞, dg(t0 ) (xi , p) /ri → ∞, and sup

R (x, t0 ) ≤ 2R (xi , t0 ) .

Bg(t0 ) (xi ,ri )

Let (20.1)

 gi (t)  R(xi , t0 )g t0 +

t R(xi , t0 )

 ,

t ∈ (−∞, 0].

1

We leave it to the reader to check that in the proof below of Theorem 20.1 one only needs that g(t0 ) is κ0 -noncollapsed on all scales to conclude AVR(t0 ) = 0. 2 For another proof that AVR(t) = 0, by Hamilton, see Theorems 9.30 and 9.32 in [45]. See also Theorem 1 in [140] by one of the authors.

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126

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The assumption that g(t) is κ-noncollapsed at all scales3 implies that injectivity radii of gi (0) at xi have a uniform lower bound, i.e., injgi (0) (xi ) ≥ δ for some constant δ > 0. Applying Hamilton’s Cheeger–Gromov-type compactness theorem, we conclude  "    2 Bgi (0) xi , R(xi , t0 )ri , gi (t), xi → (Mn∞ , g∞ (t), x∞ ), where t ≤ 0 and Rg∞ (x∞ , 0) = 1. The limit g∞ (t) is a complete noncompact κ-solution with Rg∞ (x, t) ≤ 2. The limit (M∞ , g∞ (t) , x∞ ) must also contain a line by Theorem 18.17 and it splits as a product:

(Mn∞ , g∞ (t)) = R×W n−1 , du2 + gW (t) ,

where W n−1 , gW (t) is a κ-solution. Next we shall derive a contradiction by showing that (W, gW (t)) has positive AVR.4 Since AVR is independent of the choice of basepoint, by the Bishop–Gromov volume comparison theorem we have for any r ∈ (0, ∞) and i ∈ N, Vol gi (0) Bgi (0) (xi , r) rn

Vol g(t0 ) Bg(t0 ) xi , R(xi , t0 )−1/2 r

n = R(xi , t0 )−1/2 r ≥ ωn AVR (t0 ) . Taking the limit as i → ∞, we obtain all r ∈ (0, ∞); hence

Volg∞ (0) Bg∞ (0) (x∞ ,r) rn

≥ ωn AVR(t0 ) for

AVR(g∞ (0)) ≥ AVR(t0 ) > 0. Denote x∞ = (0, xW ) ∈ R×W. For the product metric g∞ (0) = du2 + gW (0), we have for any r ∈ (0, ∞) (−r, r) × BgW (0) (xW , r) ⊃ Bdu2 +gW (0) (x∞ , r) ,

Voldu2 +gW (0) (−r, r) × BgW (0) (xW , r) = 2r VolgW (0) BgW (0) (xW , r) . Hence Voldu2 +gW (0) Bdu2 +gW (0) (x∞ , r) VolgW (0) BgW (0) (xW , r) ≥ . rn−1 2rn This implies that ωn AVR(g∞ (0)) > 0. AVR(gW (0)) ≥ 2ωn−1 On the other hand, by the induction assumption, any (n − 1)-dimensional κ-solution (W, gW (t)) must have zero AVR. This is a contradiction, so Case A is ruled out. 3

Actually in Case A, the property that g(t0 ) is κ-noncollapsed at all scales is automatic since we assume AVR(t0 ) > 0. This is all we need to obtain the injectivity radius estimate. 4 Aside: AVR > 0 implies the manifold is noncompact.

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1. ASCR AND AVR OF κ-SOLUTIONS

127

Case B. 0 < ASCR (t0 ) < ∞ for some t0 ≤ 0. In this case a semi-global blow-down limit is a piece of the nonflat asymptotic cone which contradicts the strong maximum principle implying that it is flat. By the definition of ASCR, there exists a sequence of points xi ∈ M such that as i → ∞, dg(t0 ) (xi , p) → ∞,

R(xi , t0 ) d2g(t0 ) (xi , p) → ASCR(t0 ).

Define the rescaled solutions {(Mn , gi (t))}i∈N as in (20.1). Note that this is a blow-down sequence since R(xi , t0 ) → 0 as i → ∞. We have as i → ∞,  (20.2) dgi (0) (xi , p) = R(xi , t0 )1/2 dg(t0 ) (xi , p) → ASCR(t0 ) ∈ (0, ∞) .  Let b and B be any two constants with 0 < b < ASCR(t0 ) < B < ∞ and let Ni (b, B)  Bgi (0) (p, B) \ Bgi (0) (p, b) ⊂ M be the annulus, with respect to gi (0), with inner radius b and outer radius B. Then by (20.2) we have xi ∈ Ni (b, B) for i sufficiently large. ∂ Rgi ≥ 0, and since By the trace Harnack estimate, which implies that ∂t ASCR(t0 ) ∈ (0, ∞), we have the curvature bound Rgi (x, t) ≤ Rgi (x, 0) ≤

200 ASCR(t0 ) 2 ASCR(t0 ) ≤ 2 81b2 dgi (0) (x, p)

9b 11B for all x ∈ Ni ( 10 , 10 ) and t ∈ (−∞, 0], provided i is sufficiently large. Again (as in Case A), the assumption that g(t) is κ-noncollapsed at all scales implies that injgi (0) (xi ) ≥ δ for some δ > 0. Applying the (local) compactness theorem, 3.16 in

Part I, to the sequence i.e.,9bTheorem 11B of (incomplete) solutions (Ni ( 10 , 10 ), gi (t), xi ) i∈N , we obtain that for a subsequence,

(Ni (b, B) , gi (t) , xi ) → (N∞ (b, B) , g∞ (t) , x∞ ) as i → ∞. Here (N∞ (b, B) , g∞ (t)) is a nonflat (incomplete) solution to the Ricci flow with bounded nonnegative curvature operator. Since gi (0) = R(xi , t0 )g(t0 ), since R(xi , t0 ) → 0 as i → ∞, and since the sectional curvatures of gi (0) are nonnegative, by Theorem I.26 in Appendix I we have that (Mn , gi (0) , p) → (CW, d∞ , p∞ ) converges in the pointed Gromov–Hausdorff topology as i → ∞ to a unique limit, where CW  (R≥0 × W)/(0 × W) is a Euclidean metric cone over a metric space (W, dW ). The limit metric space (CW, d∞ , p∞ ) is called the Gromov–Hausdorff asymptotic cone5 of (M, g (t0 )). 5

See §1 of Appendix H.

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128

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

Recall that (Ni (b, B) , gi (0)) ⊂ (M, gi (0)) and note that "  dgi (0) (p, xi ) = R(xi , t0 ) · d2g(t0 ) (xi , p) ≤ 2 ASCR(t0 ) for i sufficiently large. It follows easily from the uniqueness of the above pointed Gromov–Hausdorff limit that if we change the basepoint p to a sequence of basepoints yi , a uniformly bounded distance away, then a corresponding Gromov–Hausdorff limit is isometric to the original limit. Because of this, as a metric space, the limit (N∞ (b, B) , g∞ (0)) of the sequence of pointed Riemannian manifolds {(Ni (b, B) , gi (0) , xi )} is isometric to an open (top-dimensional) submanifold of the limit (CW, d∞ , p∞ ) of the sequence {(M, gi (0) , p)}. Hence W has a smooth (n − 1)-manifold structure and there exists a Riemannian metric gW (0) on W such that the Euclidean metric cone structure (CW, d∞ ) is given by the Riemannian metric g∞ (0) = dr2 + r2 gW (0) .

(20.3)

The Riemann curvature tensor Rmg∞ (0) of the metric g∞ (0) = dr2 + 6 W (0) on N∞ (b, B) satisfies ) (   ∂ ∂ ∂ ∂ , j = 0, (20.4) Rmg∞ (0) , ∂r ∂y ∂y j ∂r n−1 where y j j=1 are local coordinates on W. Hence for each 1 ≤ j ≤ n − 1,

r2 g

∂ ∧ ∂y∂ j lies in the null space of the curvature operator of g∞ (0). the 2-form ∂r Recall that Hamilton’s strong maximum principle for Rm (i.e., Theorem 12.50 and Theorem 12.53 in Part II) says that the null space of Rm∞ (0) is invariant under parallel translation. Let ∇ denote the covariant derivative of g∞ (0). From7

∇∂

∂r

∂ =0 ∂r

and

∇

∂ ∂y i

∂ ∂ ∂ =∇∂ = r−1 i , i ∂r ∂r ∂y ∂y

∂ ∧ ∂y∂ j , for 1 ≤ we have that the (infinitesimal) parallel translates of ∂r j ≤ n − 1, span all of Λ2 N∞ (b, B). Indeed, in local coordinates where 6 See, for example, equation (20.56) in the notes and commentary at the end of this chapter or Proposition 9.106 of [15]. 7 Let r = y 0 so that g∞ (0)00 = 1, g∞ (0)ij = r2 gW (0)ij , and g∞ (0)i0 = 0 for i, j ≥ 1. We have

∇

∂ ∂y i

n−1  k ∂ ∂ ∂ Γi0 k + Γ0i0 = ∂r ∂y ∂r k=1

1 k ∂ ∂ ∂ g gi

= r−1 i . 2 ∂r ∂y k ∂y   ∂ ∂ = 0. since ∂y i , ∂r =

Note also that ∇

∂ ∂y i

∂ ∂r

=∇

∂ ∂r

∂ ∂y i

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2. ALMOST κ-SOLUTIONS

∇

∂ ∂y i

∂ ∂y j

129

= 0 at a point, we have at that point,       ∂ ∂ ∂ ∂ ∂ ∂ ∧ ∧ j + ∧ ∇ ∂ = ∇ ∂ ∇ ∂ j ∂r ∂y j ∂y ∂r ∂y i ∂y i ∂r ∂y i ∂y ∂ ∂ = r−1 i ∧ j ∂y ∂y

and

 ∇∂

∂r

∂ ∂ ∧ j ∂r ∂y



   ∂ ∂ ∂ ∂ ∧ j + ∧ ∇∂ = ∇∂ ∂r ∂r ∂r ∂y j ∂y ∂r ∂ ∂ ∧ i. = r−1 ∂r ∂y 

Since null (Rm (g∞ (0))) is invariant under parallel translation, we have null (Rm (g∞ (0))) = Λ2 N∞ (b, B) and so the curvature operator of g∞ (0) is identically zero, which contradicts Rg∞ (x∞ , 0) = 1. Hence Case B cannot happen. Case C. ASCR(t0 ) = 0 for some t0 ≤ 0 with n ≥ 3. In this case a gaptype splitting theorem yields a 2-dimensional κ-solution with ASCR = 0, a contradiction. By a result of Petrunin and Tuschmann (see Theorem B on p. 777 of n n−2 × [157]), we have

that the universal cover of (M , g (t08)) is isometric to E 2 Σ , gΣ (t0 ) and that (Σ, gΣ (t0 )) has zero ASCR. It then follows from n Hamilton’s strong maximum that the universal 2 principle

cover of (M , g (t)) n−2 2 × Σ , gΣ (t) , where Σ , gΣ (t) is a κ-solution (as in is isometric to E

2 the proof of Theorem 18.17). 19.42, Σ , gΣ (t0 ) is a round From Theorem

sphere and hence En−2 × Σ2 , gΣ (t0 ) cannot have zero ASCR. We have a contradiction. Theorem 20.1 is now proved. Synopsis. In some sense the main result of this section hinges on

dimen

n−1 , k × R, where P n−1 , k sion reduction and the fact that a product P is a closed Riemannian manifold, has AVR = 0. (A) The case ASCR(t0 ) = ∞ enables one to do dimension reduction. (B) The case 0 < ASCR (t0 ) < ∞ leads to the contradiction of an asymptotic cone being both nonflat and flat. (C) The case ASCR(t0 ) = 0 yields the product of a surface and Euclidean space as the universal cover; however orientable κ-solutions on surfaces are round 2-spheres, which implies ASCR(t0 ) > 0 and we have a contradiction. 2. Almost κ-solutions In this section we prove two propositions. These propositions are the consequence of Theorem 20.1 and they shall be used in proving Perelman’s compactness theorem (Theorem 20.9). 8

Here En−2 denotes Rn−2 with the flat Euclidean metric.

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130

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

2.1. Volume collapsing on large space-time scales. The following is Corollary 11.5 in Perelman’s [152]. We may think of this result as a quantitative or approximate version of the fact that nonflat ancient solutions (not necessarily κ-noncollapsed) with bounded Rm ≥ 0 have AVR = 0 (Corollary 20.2). As is usual for such a result, this is proved by contradiction. Proposition 20.4 (Almost ancient solutions with bounded Rm ≥ 0 are collapsed at large scales). For every ε > 0 there exists A < ∞ with the following property: Suppose we have a sequence of (not necessarily complete) solutions (Mnk , gk (t)), t ∈ [tk , 0], n ≥ 2, with nonnegative curvature operator and with xk ∈ Mk and rk ∈ (0, ∞) such that (1) (relatively compact balls) for each k the ball Bgk (0) (xk , rk ) is compactly contained in Mk , (2) (curvature bounds on parabolic cylinders) 1 Rg (x, t) ≤ Rgk (xk , 0)  Qk 2 k for all (x, t) ∈ Bgk (0) (xk , rk ) × (tk , 0], and (3) (parabolic cylinders are large) as k → ∞,

(20.5)

(20.6)

tk Qk → −∞,

rk2 Qk → ∞.

Then the volume ratios satisfy −1/2

(20.7)

Volgk (0) Bgk (0) (xk , AQk   −1/2 n AQk

)

≤ε

provided k is sufficiently large. That is, we have small volume ratios at large scales. Remark 20.5. Properties (20.5) and (20.6) say that the dilated solutions g˜k (t)  Qk · gk (Q−1 k t),

t ∈ [tk Qk , 0],

are ‘almost ancient’ (since tk Qk → −∞) and have uniformly bounded cur1/2 vatures on larger and larger balls (since rk Qk → ∞) Rg˜k (x, t) ≤ 2 1/2

for all (x, t) ∈ Bg˜k (0) (xk , rk Qk ) × (tk Qk , 0]. Proof. We prove the proposition by contradiction. If the proposition is false, then by diagonalization there exist n ≥ 2 and sequences (Mnk , gk (t)), t ∈ [tk , 0], with Rmgk (t) ≥ 0, xk ∈ Mk and rk ∈ (0, ∞) satisfying hypotheses (1)–(3) and there exist ε0 > 0 and a sequence Ak → ∞ such that −1/2

Vol gk (0) Bgk (0) (xk , Ak Qk   −1/2 n Ak Qk

)

≥ ε0

for all k.

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2. ALMOST κ-SOLUTIONS

131

In particular the dilated solutions g˜k (t) = Qk · gk (Q−1 k t) satisfy Vol g˜k (0) Bg˜k (0) (xk , Ak ) ≥ ε0 (Ak )n

(20.8)

for all k.

Since Rm g˜k (0) ≥ 0, by the Bishop–Gromov volume comparison theorem we have Vol g˜k (0) Bg˜k (0) (xk , 1) ≥ ε0 for all k such that Ak ≥ 1. 1/2 From the curvature bound |Rg˜k (x, 0)| ≤ 2 on Bg˜k (0) (xk , rk Qk ) and 1/2

since rk Qk ≥ 1 for k large enough, we know by Proposition 19.3 (i.e., the local injectivity radius estimate) that there exists δ0 (ε0 ) > 0 such that inj g˜k (0) (xk ) ≥ δ0 for k large enough. We can apply Hamilton’s local compactness theorem (see Theorem 3.16 in Part I) to conclude that 1/2

(Bg˜k (0) (xk , Qk rk ), g˜k (t), xk ),

t ∈ (tk Qk , 0],

converges to a solution (Mn∞ , g˜∞ (t), x∞ ), t ∈ (−∞, 0]. The limit is complete since Bgk (0) (xk , rk ) is compactly contained in Mk 1/2

and Qk rk → ∞. Clearly the limit is a nonflat solution of the Ricci flow and has nonnegative bounded curvature operator (the boundedness of the curvature follows from (20.5)). Now (20.8) and the Bishop volume comparison theorem imply the asymptotic volume ratio has a positive lower bound: ε0 > 0. (20.9) AVR (˜ g∞ (0)) ≥ ωn However this contradicts Corollary 20.2, which says that AVR (˜ g∞ (0)) = 0, and the proposition is proved.
2.2. Curvature bound under the noncollapsing assumption. In this subsection we give a proof of Corollary 11.6 in Perelman [152] regarding a curvature estimate. 2.2.1. Statement of the main result. Proposition 20.6 (Curvature estimate in noncollapsed balls). For every ω > 0 there exist B = B(ω) < ∞, C = C(ω) < ∞, and τ0 = τ0 (ω) > 0 with the following properties: Let (Mn , g (t)), t ∈ [t0 , 0], where t0 ∈ (−∞, 0), be a (not necessarily complete) solution to the Ricci flow with nonnegative curvature operator. Let r0 > 0 be a constant, let x0 ∈ M, and suppose that the metric ball Bg(0) (x0 , r0 ) is compactly contained in M. (a) (Noncollapsed on a time interval) If for each time t ∈ [t0 , 0], Volg(t) Bg(t) (x0 , r0 ) ≥ ωr0n ,

(20.10) then we have (20.11)

R(x, t) ≤ Cr0−2 + B(t − t0 )−1

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132

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

for all (x, t) ∈ M × (t0 , 0] satisfying dg(t) (x, x0 ) ≤ 14 r0 . (b) (Noncollapsed at one time) If we only assume Volg(0) Bg(0) (x0 , r0 ) ≥ ωr0n ,

(20.12)

then when −t0 ≥ τ0 r02 , we have R(x, t) ≤ Cr0−2 + B(t + τ0 r02 )−1

(20.13)

for all (x, t) ∈ M × (−τ0 r02 , 0] satisfying dg(t) (x, x0 ) ≤ 14 r0 . Remark 20.7. Note that statement (a) in the above proposition is independent of τ0 . We devote the rest of this section to proving this proposition. Since we can rescale the solution g (t) to r0−2 g r02 t , it suffices to prove the proposition with r0 = 1. 2.2.2. Proof of Proposition 20.6(a) with r0 = 1. Suppose that the proposition is false. Then there exist ω0 > 0, sequences {Bk } → ∞ and {Ck } → ∞, and a sequence of (not necessarily complete) pointed solutions {(Mnk , gk (t), x0k ) : t ∈ [t0k , 0]} , where t0k ∈ (−∞, 0), to the Ricci flow with nonnegative curvature operator such that (1) the metric ball Bgk (0) (x0k , 1) is compactly contained in Mk , (2) at each time t ∈ [t0k , 0] (20.14)

Vol gk (t) Bgk (t) (x0k , 1) ≥ ω0

for all k,

(3) there exist points (xk , t1k ) satisfying dgk (t1k ) (xk , x0k ) ≤ (t0k , 0], and (large curvature points) (20.15)

1 4,

t1k ∈

Rgk (xk , t1k ) > Ck + Bk (t1k − t0k )−1 .

With Proposition 18.25 regarding curvature control in a parabolic cylinder in mind, we take σ = 23 , (N n , h (t)) = (Mnk , gk (t)), (t0 , t1 ] = (t0k , t1k ], and p = x0k and we define (i)    

1 1 2 −1  Ck + Bk (t1k − t0k ) (t1k − t0k ) , · min Ak  Ak 3 3 2304 

and (ii) Dk  min

1/2

Ak 100(n−1) ,

1 (t −t0k )Ck +Bk 3 1k

2

 . Clearly Ak > 1 for k large

and Ak → ∞ and Dk → ∞. By Lemma 18.24 and Proposition 18.25, there exist t∗k ∈ (t0k , t1k ], with t∗k − t0k ≥ 13 (t1k − t0k ), and y∗k ∈ Bgk (t∗k ) x0k , 13 such that (I) Rgk (y∗k , t∗k ) > Ck + Bk (t∗k − t0k )−1 and

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2. ALMOST κ-SOLUTIONS

133

(II) for any time t ∈ [t∗k − Dk Rg−1 (y , t ), t ] and for any point y ∈  k ∗k ∗k ∗k  1/2 −1/2 1 Ak Rgk (y∗k , t∗k ) , we have Bgk (t∗k ) y∗k , 10 Rgk (y, t) ≤ 2Rgk (y∗k , t∗k ). Note that9 1 1/2 −1/2 1 Ak Rgk (y∗k , t∗k ) ≤ . 10 480

(20.16)

Hence, from dgk (t∗k ) (y∗k , x0k ) < 13 we have     1 1/2 −1/2 161 . Bgk (t∗k ) y∗k , Ak Rgk (y∗k , t∗k ) ⊂ Bgk (t∗k ) x0k , 10 480 Since Rc ≥ 0, (20.16), and dgk (t∗k ) (y∗k , x0k ) < 13 , we have by the Bishop– Gromov volume comparison theorem that10   1 1/2 −1/2 Volgk (t∗k ) Bgk (t∗k ) y∗k , 10 Ak Rgk (y∗k , t∗k ) n  1 1/2 −1/2 A R (y , t ) gk ∗k ∗k 10 k

Volgk (t∗k ) Bgk (t∗k ) y∗k , 12 1 n ≥ 2

 n Volgk (t∗k ) Bgk (t∗k ) x0k , 16 1 1 n ≥ · 3 6

≥ 3−n Volgk (t∗k ) Bgk (t∗k ) (x0k , 1) ≥ 3−n ω0 ,

where we used (20.14) with t = t∗k to obtain the last inequality. On the other hand, wecan apply Proposition 20.4to g˜k (t)  gk (t + t∗k ), 1 1/2 −1/2 Ak Rgk (y∗k , t∗k ) , and the time interval the geodesic ball Bgk (t∗k ) y∗k , 10 −1/2

[−Dk Rgk 9

(y∗k , t∗k ), 0], to obtain that for every ε > 0 there exists A˜ (ε) < ∞

Indeed, by the definition of Ak we have 1 1/2 −1/2 A Rgk (y∗k , t∗k ) 10 k  1/2 1 1  ≤ Ck + Bk (t1k − t0k )−1 Rg−1/2 (y∗k , t∗k ) 10 2304 k  1  1 −1 1/2 Rg−1/2 (y∗k , t∗k ) ≤ Ck + Bk (t∗k − t0k ) 10 2304 k 1 . ≤ 480

  Since Bgk (t∗k ) x0k , 161 ⊂ Bgk (t∗k ) (x0k , 1) ⊂ Bgk (0) (x0k , 1), which is compactly 480 contained in Mk , this enables us to apply the volume comparison theorem. 10

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134

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

such that for sufficiently large k Volgk (t∗k ) Bgk (t∗k ) (y∗k , A˜ (ε) Rg−1/2 (y∗k , t∗k )) k n  ≤ ε. −1/2 ˜ (y∗k , t∗k ) A (ε) Rg k

If k is large enough, then comparison theorem, we

1 1/2 10 Ak have11

> A˜ (ε) and, by the Bishop–Gromov volume

  1 1/2 −1/2 Volgk (t∗k ) Bgk (t∗k ) y∗k , 10 Ak Rgk (y∗k , t∗k ) n  3−n ω0 ≤ 1 1/2 −1/2 A R (y , t ) gk ∗k ∗k 10 k   Volgk (t∗k ) Bgk (t∗k ) y∗k , A˜ (ε) Rg−1/2 (y∗k , t∗k ) k n  ≤ ≤ ε. −1/2 ˜ (y∗k , t∗k ) A (ε) Rg k

−n

We obtain a contradiction if we choose say ε = 3 2 ω0 . Proposition 20.6(a) is proved. 2.2.3. Proof of Proposition 20.6(b) with r0 = 1. Let (Mn , g (t)) be a solution of the Ricci flow satisfying the assumptions of Proposition 20.6(b) with r0 = 1. Let τ0 (ω) > 0 be a constant depending only on ω which will be specified later (see (20.20) below). Suppose that12 −t0 ≥ τ0 (ω) and define t1 ∈ [t0 , 0] to be the smallest constant such that13 dg(t) (y, x0 ) ≤



for all y ∈ Bg(0) x0 , 15 and t ∈ [t1 , 0]. Step 1. We shall prove the following. Claim 1. We have (20.17)

19 80

Volg(t) Bg(t) (x0 , 1) ≥ 5−n ω

for t ∈ [t1 , 0]. It follows from the nonnegativity of the Ricci curvature and the Bishop– Gromov volume comparison theorem that for t ∈ [t1 , 0], 1 Volg(t) Bg(t) (x0 , 1) ≥ Volg(t) Bg(0) (x0 , ) 5 1 ≥ Volg(0) Bg(0) (x0 , ) 5  n 1 Volg(0) Bg(0) (x0 , 1). ≥ 5 From the assumption (20.12), with r0 = 1, we obtain (20.17). 11

Here again, we use Bgk (t∗k ) y∗k ,

1/2 −1/2 1 A Rgk (y∗k , t∗k ) 10 k

⊂ Bgk (0) (x0k , 1) is com-

pactly contained in Mk , as we remarked above. 12 Otherwise there is nothing to prove. 13 Clearly there exists such a t1 .

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2. ALMOST κ-SOLUTIONS

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Step 2. We shall prove the following. Claim 2. For τ0 (ω) defined by (20.20) below, we have t1 ≤ −τ0 (ω) . When t1 ≤ −1, Claim 2 is true since τ0 (ω) < 1. When t1 > −1, we will prove Claim 2 by contradiction. If Claim 2 is not

true, i.e., if |t1 | ≤ τ0 (ω), then there exists y0 ∈ Bg(0) x0 , 15 and t2 ∈ [t1 , 0] such that 9 19 (20.18) dg(t2 ) (y0 , x0 ) = , dg(t1 ) (y0 , x0 ) = , 40 80 and 19 9 ≤ dg(t) (y0 , x0 ) ≤ for t ∈ [t1 , t2 ] . 40 80 Applying Proposition 20.6(a) to g (t), t ∈ [t1 , 0], we get



(20.19) R (y, t) ≤ C 5−n ω + B 5−n ω (t − t1 )−1

for y ∈ Bg(t) x0 , 14 and t ∈ [t1 , 0]. Now we apply Theorem 18.7(2) to g (t), t ∈ [t1 , t2 ], with the

Ricci curva1 14 ture upper bound on Bg(t) (x0 , r˜0 )∪Bg(t) (y0 , r˜0 ) ⊂ Bg(t) x0 , 4 to estimate 1 is to be chosen below. We get dg(t) (y0 , x0 ), where r˜0 ≤ 80     −n B (5−n ω) 2 1 ∂ d (y0 , x0 ) ≥ −2 (n − 1) C 5 ω + r˜0 + . ∂t g(t) 3 t − t1 r˜0 Since −t1 ≤ 1, we may choose r˜0 = ∂ d (y0 , x0 ) ∂t g(t)  ≥ −2 (n − 1)

(t−t1 )1/2 80

C (5−n ω) (t − t1 )1/2 + 120



≤

1 80

to obtain

  B (5−n ω) + 80 (t − t1 )−1/2 . 120

Integrating this inequality in time from t1 to t2 , we have dg(t2 ) (y0 , x0 ) − dg(t1 ) (y0 , x0 )     B (5−n ω) C (5−n ω) 3/2 1/2 |t1 | + + 160 |t1 | , ≥ −2 (n − 1) 180 60 where we used t2 − t1 ≤ |t1 |. Define ⎛ (20.20)

14

τ0 (ω)  ⎝

320 (n − 1)

⎞2 

1 C(5−n ω) 180

+

B(5−n ω) 60

⎠ . + 160

The Ricci curvature bound follows from (20.19).

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20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

It follows from the contradiction assumption |t1 | ≤ τ0 (ω) that we have     B (5−n ω) C (5−n ω) |t1 |3/2 + + 160 |t1 |1/2 2 (n − 1) 180 60     −n B (5−n ω) C (5 ω) 1/2 1/2 |t1 | + + 160 |t1 | ≤ 2 (n − 1) 180 60   C (5−n ω) B (5−n ω) ≤ 2 (n − 1) + + 160 τ0 (ω)1/2 180 60 1 . = 160 Hence we have 37 19 1 = < . dg(t1 ) (x0 , y0 ) ≤ dg(t2 ) (x0 , y0 ) + 160 160 80 This contradicts the choice of y0 , i.e., (20.18). We have proved t1 ≤ −τ0 (ω) when t1 > −1. Step 3. Finishing the proof of the proposition. Since τ0 (ω) < 1, we have proved that 19 80

1 for all y ∈ Bg(0) x0 , 5 and t ∈ [−τ0 (ω) , 0]. From (20.17) we have dg(t) (y, x0 ) ≤

Volg(t) Bg(t) (x0 , 1) ≥ 5−n ω for t ∈ [−τ0 (ω) , 0]. Now that we have volume noncollapsing on a time interval instead of at just one time, Proposition 20.6(b) follows from Proposition 20.6(a). 3. The compactness of κ-solutions An important step in understanding the high curvature regions of finite time singular solutions is to understand the compactness, modulo scaling, of κ-solutions. That is, given a sequence of κ-solutions and a sequence of spacetime points in this sequence, one would like to obtain a limit of rescalings which is a κ-solution. Remarkably, in dimension 3, this is possible. That is, the collection of 3-dimensional κ-solutions is precompact modulo scaling (see Corollary 20.10 below). 3.1. Statement of the main theorem. We say that a collection of ancient solutions of Ricci flow with positive scalar curvature is precompact modulo scaling if for any sequence of solutions (Mnk , gk (t)), t ≤ 0, in the collection, for any points xk ∈ Mk and for any times tk ≤ 0 the following is true. A subsequence of (Mnk , g¯k (t) , xk ), t ≤ 0, where   t , (20.21) g¯k (t)  Rgk (xk , tk ) gk tk + Rgk (xk , tk )

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3. THE COMPACTNESS OF κ-SOLUTIONS

137

converges in the C ∞ Cheeger–Gromov topology to a limit solution (Mn∞ , g¯∞ (t) , x∞ ) , t ≤ 0. If the limit (Mn∞ , g¯∞ (t) , x∞ ) is always contained in the collection, we say that the collection is compact modulo scaling. Note that Rg¯k (xk , 0) = 1.

(20.22)

We may think of (xk , 0) as being the space-time origin, where the solution g¯k (t) is normalized so that its scalar curvature at the origin is equal to 1. For this reason we call g¯k (t) a curvature normalized sequence of solutions. Remark 20.8. Note that an ancient solution with bounded curvature either has positive scalar curvature or is Ricci flat. (See Lemma 2.18 in [45] for example.) Given κ > 0 and n ∈ N − {1}, let MHarn n,κ denote the collection of ndimensional κ-solutions with Harnack and let Mn,κ denote the collection of n-dimensional κ-solutions. As we mentioned before, Mn,κ ⊂ MHarn n,κ

and

M3,κ = MHarn 3,κ .

Perelman proved the following remarkable compactness result (see §11.7 of [152] and §1.1 of [153]). Theorem 20.9 (Compactness of κ-solutions with Harnack). For any κ > 0 and n ≥ 2, the collection MHarn n,κ is compact modulo scaling. In dimension 3, by Proposition 19.44 we have the following. Corollary 20.10 (Compactness of 3-dimensional κ-solutions). For any fixed κ > 0 the collection of 3-dimensional κ-solutions is compact modulo scaling. By Theorem 19.56 and Corollary 20.10 we have the following. Fix κ > 0 and let Mncpt 3,κ denote the collection of noncompact 3-dimensional κ-solutions and let Mnsph 3,κ denote the collection of nonspherical space form 3-dimensional κ-solutions. Corollary 20.11 (Compactness of 3-dimensional κ-solutions improved). (i) The collection

.

Mncpt 3,κ ,

κ>0

which is the collection of all 3-dimensional noncompact solutions which are κ-solutions for some κ > 0, is compact modulo scaling. That is, for any sequence

. ncpt 3 M3,κ Mk , gk (t) ∈ κ>0

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20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

and space-time points (xk , 0) with Rgk (xk , 0) = 1, there exists a convergent subsequence



3 Mk , gk (t) , xk → M3∞ , g∞ (t) , x∞ and the limit is also a noncompact κ-solution for some κ > 0. / (ii) The collection κ>0 Mnsph 3,κ , i.e., the collection of all 3-dimensional nonspherical space form κ-solutions for some κ > 0, is precompact modulo scaling. Proof of Corollary 20.11. (i) By Theorem 19.56, there exists κ0 such that / ncpt ncpt κ>0 M3,κ = M3,κ0 ,

3 i.e., the Mk , gk (t) are all κ0 -noncollapsed. Now Corollary 20.10 implies that Mncpt 3,κ0 ⊂ M3,κ0 is compact modulo scaling. Thus there exists a converging subsequence

(M3k , gk (t), xk ) → M3∞ , g∞ (t) , x∞ / such that the limit is a κ0 -solution. Since (Mk , gk (t)) ∈ κ>0 Mncpt 3,κ , clearly the limit manifold M∞ is noncompact. (ii) By Theorem 19.56, there exists κ0 such that / nsph nsph κ>0 M3,κ = M3,κ0 . Now Corollary 20.10 implies that Mnsph 3,κ0 ⊂ M3,κ0 is compact modulo scaling. Thus there exists a subsequence which converges

(M3k , gk (t), xk ) −→ M3∞ , g∞ (t) , x∞ and for which the limit is a κ0 -solution. Note, however, that the limit
solution (M∞ , g∞ (t)) could be spherical (see the following remark). Example 20.12. (1) For any sequence of solutions whose elements are pointed n-dimensional Bryant solitons, the only possible pointed limits are the ndimensional Bryant soliton itself or a round cylinder S n−1 × R; see Example 19.12 and Exercises 19.13 and 19.14. (2) It seems clear that one should be able to prove that the only types of limits of Perelman’s κ-solution on S n are (a) the shrinking constant sectional curvature S n , (b) Perelman’s κ-solution itself, (c) the Bryant soliton, and (d) the round cylinder. Case (a) corresponds to when the sequence of times approaches the singularity time, case (b) corresponds to when the sequence of times stays bounded away from both −∞ and the singularity time, and cases (c) and (d) both correspond to when the sequence of times tends to −∞, depending on whether or not (respectively) the sequence of spatial points stays a bounded distance from the tips after rescaling the curvature at the tips to be constant.

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3. THE COMPACTNESS OF κ-SOLUTIONS

139

Remark 20.13 (Heuristic). A priori, it is conceivable that it is possible for a sequence of pointed κ-solutions (Mnk , gk (t) , xk ), t ∈ (−∞, 0], to the Ricci flow to have at t = 0 uniformly bounded curvature at the basepoints xk but also to have curvature blowing up at points yk with dgk (0) (yk , xk ) ≤ C. For example, one can imagine a ‘conical-type singularity’ forming near the points yk ; that is, a Gromov–Hausdorff limit (Mn∞ , d∞ (t) , x∞ ) of (Mnk , gk (t) , xk ) as k → ∞ may exist which is a Euclidean metric cone based at y∞ and such that M∞ − y∞ is a smooth n-manifold. Note that for such a limit, were it to exist, we would have R∞ (z) d2∞ (z, y∞ ) ≤ const < ∞. Needless to say, by the above compactness theorem, this cannot occur for n-dimensional κ-solutions with Harnack. We shall give two proofs of Theorem 20.9. 3.2. First proof of the compactness of κ-solutions — via Proposition 20.4. In this subsection we give a proof of the compactness Theorem 20.9 using Proposition 20.4. This is the proof which Perelman gave in §11.7 of [152]. Harn Let {(Mnk , gk (t))}∞ k=1 be a sequence in Mn,κ , xk ∈ Mk and tk ≤ 0. Let {¯ gk (t)} be the curvature normalized sequence of solutions defined by (20.21):   t . g¯k (t)  Rgk (xk , tk ) gk tk + Rgk (xk , tk ) Let15 A k  sup Rg¯k (z, 0)d2g¯k (0) (xk , z) z∈Mk

= sup Rgk (z, tk )d2gk (tk ) (xk , z). z∈Mk

Note that if Mk is noncompact, then gk (0)) = ASCR (gk (tk )) , A k ≥ ASCR (¯ where the asymptotic scalar curvature ratio ASCR is defined by (19.8). We first choose a subsequence {(Mk , gk (t))} such that either (a) A k ≤ 1 for all k or (b) A k > 1 for all k. We shall extract a convergent subsequence in MHarn n,κ in either (a) or (b). One of the main ideas in the proof of this theorem is to obtain uniform curvature bounds so that one can simply apply Hamilton’s (local) Cheeger– Gromov-type compactness theorem. Case (a). A k ≤ 1 for all k. In this case we have for each k for which Mk is noncompact that ASCR (¯ gk (0)) ≤ 1. 15 More generally, given a pointed Riemannian manifold (Mn , g, O), we may define the ‘maximum scalar curvature ratio’ as A (g)
supx∈M R (x) d2 (x, O).

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140

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

Hence, by Theorem 20.1, which says ASCR (¯ gk (0)) = ∞ if Mk is noncompact, we have that the manifold Mk is compact for all k. We claim there exists a constant C0 < ∞ independent of k such that (20.23)

max Rg¯k (z, 0) ≤ C0

z∈Mk

for all k ;

that is, we have a uniform curvature bound for all of the rescaled metrics g¯k (0). If not, then there would exist a subsequence {(Mk , gk (t))} and a corresponding sequence of points {yk ∈ Mk } such that (20.24)

0 < Rg¯k (yk , 0) = max Rg¯k (z, 0) → ∞ z∈Mk

as k → ∞. Note that since A k ≤ 1, this implies d2g¯k (0) (xk , yk ) → 0 as k → ∞. (Already the reader may sense that this is counterintuitive; we shall confirm this by rescaling and taking a limit to obtain a contradiction.) We rescale about the maximum curvature points so as to define   t . (20.25) gˇk (t)  Rg¯k (yk , 0) g¯k Rg¯k (yk , 0) Then Rgˇk (z, 0) ≤ Rgˇk (yk , 0) = 1 for all z ∈ Mk and k ∈ N. So by the trace ∂ Rgˇk ≥ 0, we have Harnack estimate, which implies that ∂t Rgˇk (z, t) ≤ 1 for all z ∈ Mk , t ≤ 0, and k ∈ N. On the other hand, it follows from (20.22), i.e., Rg¯k (xk , 0) = 1, and (20.24) that Rgˇk (xk , 0) → 0. Since we are assuming Case (a), we have Rg¯k (yk , 0)d2g¯k (0) (xk , yk ) ≤ 1 and hence by (20.25) we have the uniform distance bound (20.26)

d2gˇk (0) (xk , yk ) ≤ 1.

For each k, since (Mnk , gk (t)) has nonnegative curvature operator and is κ-noncollapsed at all scales, we have that (Mnk , gˇk (t)) has uniformly bounded (independent of k, by our rescaling) nonnegative curvature operator and is κ-noncollapsed at all scales. Hence we may apply Hamilton’s Cheeger– Gromov-type compactness theorem to get a subsequence {(Mnk , gˇk (t) , yk )} which converges to a complete limit solution

n ˇ , gˇ∞ (t) , y∞ M ∞ with bounded nonnegative curvature operator and such that xk → x∞ for ˇ ∞ . Here we used (20.26) and passed to a subsequence. some x∞ ∈ M

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3. THE COMPACTNESS OF κ-SOLUTIONS

141

The limit also satisfies Rgˇ∞ (y∞ , 0) = 1 and Rgˇ∞ (x∞ , 0) = 0. However this contradicts the strong maximum principle (see Corollary 12.43 in Part II) and hence we have proved sup Rg¯k (z, 0) ≤ C0

z∈Mk

for some C0 < ∞ as claimed. Since now we have a uniform curvature bound, i.e., (20.23), we may apply Hamilton’s Cheeger–Gromov-type compactness theorem to obtain a subsequence {(Mnk , g¯k (t) , xk )} converging to a complete nonflat limit solution (Mn∞ , g¯∞ (t), x∞ ) with bounded nonnegative curvature operator. We have that (M∞ , g¯∞ (t)) is κ-noncollapsed at all scales and satisfies the trace Harnack estimate since these properties are preserved under C ∞ Cheeger– Gromov convergence. This finishes the proof of Case (a). Remark 20.14. To crystallize one of the ideas in Case (a), we note the following. Suppose that (Mn , g) is a closed Riemannian manifold, x ∈ M with R (x) = 1, and R (y) d (y, x)2 ≤ C for all y ∈ M. Given w ∈ M, define the rescaled metric g˜ = R (w) g. Then ˜ (x) = R (w)−1 , R ˜ (w) = 1, R d˜(w, x) = R (w)1/2 d (w, x) ≤ C 1/2 . Case (b). A k > 1 for all k. Step 1. Point picking. There exists zk ∈ Mk such that zk is a point which is closest to xk in the nonempty set of all points z satisfying (20.27)

Rg¯k (z, 0)d2g¯k (0) (xk , z) = 1.

Let (20.28)

rk  dg¯k (0) (xk , zk ) = 

1 . Rg¯k (zk , 0)

Note that (20.27) implies Rg¯k (z, 0)d2g¯k (0) (xk , z) < 1   1 √ . Note also that we have for all z ∈ Bg¯k (0) (xk , rk ) = Bg¯k (0) xk ,

(20.29)

Rg¯k (zk ,0)

bounded curvature in an annulus, i.e., −2 Rg¯k (z, 0) < d−2 g¯k (0) (xk , z) ≤ 4rk = 4Rg¯k (zk , 0),

for z ∈ Bg¯k (0) (xk , rk ) − Bg¯k (0) xk , 12 rk . Step 2. Relative curvature bound in balls centered at zk and containing Bg¯k (0) (xk , rk ).

(20.30)

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20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

Claim. There exists a constant C1 < ∞ independent of k such that (20.31)

Rg¯k (y, 0) ≤ C1 Rg¯k (zk , 0)

for all y ∈ Bg¯k (0) (zk , 2rk ) .

To see the claim, we choose yk ∈ Mk such that 3 dg¯k (0) (xk , yk ) = rk . 4



1 Since Bg¯k (0) yk , 4 rk ⊂ Bg¯k (0) (xk , rk ) − Bg¯k (0) xk , 12 rk , by (20.30) we have Rg¯k (y, 0) < 4rk−2 for any



y ∈ Bg¯k (0) yk , 14 rk .

This local curvature bound and the κ-solution assumption (in particular, being κ-noncollapsed at all scales16 ) imply that



n Volg¯k (0) Bg¯k (0) yk , 14 rk ≥ κ 14 rk . For any y ∈ Bg¯k (0) (zk , 2rk ), we have



Bg¯k (0) (y, 4rk ) ⊃ Bg¯k (0) (zk , 2rk ) ⊃ Bg¯k (0) (xk , rk ) ⊃ Bg¯k (0) yk , 14 rk

and hence17 (20.32)

Volg¯k (0) Bg¯k (0) (y, 4rk ) ≥ κ

1 n 16

(4rk )n .

Now consider the rescaled metric hk  Rg¯k (zk , 0) g¯k (0) = rk−2 g¯k (0) , which satisfies Rhk (zk ) = 1, together with the ball Bhk (zk , 2) = Bg¯k (0) (zk , 2rk ) . Now if the claimed local curvature bound (20.31) is not true, then there exists a subsequence such that sup w∈Bhk (zk ,2)

Rhk (w) → ∞

as k → ∞. Applying the point picking Lemma 18.15 to this sequence of metrics and balls, we obtain sequences of points and radii wk ∈ Bg¯k (0) (zk , 2rk )

and

sk ∈ (0, 3rk )

such that Rg¯k (wk , 0) · s2k → ∞ Note that since Rm ≥ 0, we have |Rm| ≤ R. To wit, by ‘radii picking’ (the zk determine the radii dg¯k (0) (xk , zk )) we have curvature control in balls (centered at yk ), which by κ-noncollapsing imply volume (lower) bounds for those balls, which in turn by containment imply volume bounds for larger balls with centers not too far away (for which we do not yet have curvature control). 16 17

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143

as k → ∞ and for w ∈ Bg¯k (0) (wk , sk ),18 Rg¯k (w, 0) ≤ 2Rg¯k (wk , 0) . Then applying the crucial Proposition 20.4 to the sequence of solutions g¯k (t), t ∈ (−∞, 0], on Bg¯k (0) (wk , sk ), we have that for every ε > 0 there exists A (ε) < ∞ independent of k such that Volg¯k (0) Bg¯k (0) (wk , A (ε) Rg¯k (wk , 0)−1/2 )  n ≤ε (20.33) A (ε) Rg¯k (wk , 0)−1/2 1 n when k is large enough. Let ε0  κ2 16 . From Rg¯k (wk , 0) · (3rk )2 ≥ Rg¯k (wk , 0) · s2k → ∞, we know that the radius in (20.33) with ε = ε0 satisfies r˜k  A (ε0 ) Rg¯k (wk , 0)−1/2 ≤ 4rk for k sufficiently large.19 Thus, by the Bishop–Gromov volume comparison theorem (Rm ≥ 0 implies Rc ≥ 0) and (20.33) we have Volg¯k (0) Bg¯k (0) (wk , r˜k ) Volg¯k (0) Bg¯k (0) (wk , 4rk ) κ 1 n ≤ ≤ ε = . 0 n (4rk ) r˜kn 2 16 Since (20.32) holds for y = wk , i.e., 1 n Volg¯k (0) Bg¯k (0) (wk , 4rk ) ≥ κ 16 , n (4rk ) we obtain a contradiction. The claim is proved. Step 3. Uniform bound for Rg¯k (zk , 0). Now that we have proved (20.34)

Rg¯k (y, 0) ≤ C1 Rg¯k (zk , 0)

for y ∈ Bg¯k (0) (zk , 2rk ) ,

where C1 < ∞ independent of k, we next show by a standard argument using the trace Harnack estimate and Shi’s local derivative estimate that Rg¯k (zk , 0) has a uniform upper bound. The integrated form of the trace Harnack estimate for the ancient solution g¯k (t) with t1 = −ark2 and t2 = 0 (see Exercise 15.6 in Part II) implies ⎛ 2 ⎞ dg¯ −ar2 (xk , zk ) Rg¯k (xk , 0) k( k) ⎠,

≥ exp ⎝− (20.35) 2 2ark2 Rg¯k zk , −ark 18 In particular, for k large, the point wk has large curvature and is the center of a large (compared to the curvature at the center) ball where the curvatures are bounded relative to the value at the center. 19 That is, the large scale A (ε0 ), relative to the curvature at wk , for the radius r˜k is ‘embedded’ in the fixed scale 4, relative to the curvature at zk , for the radius 4rk defined in (20.28).

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20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

where a < 1 is a small positive constant to be chosen later. On the other ∂Rg¯ hand, by (20.34) and ∂t k (x, t) ≥ 0 we have the curvature bound Rg¯k (y, t) ≤ C1 Rg¯k (zk , 0) = C1 rk−2   in Bg¯k (0) (zk , 2rk ) × −2rk2 , 0 . This and Rcg¯k ≤ Rg¯k imply

(20.36)

d2g¯ (−ar2 ) (xk , zk ) ≤ e2aC1 d2g¯k (0) (xk , zk ) = e2aC1 rk2 . k k Hence by (20.22) and (20.35),



e2aC1 1 = Rg¯k (xk , 0) ≥ exp − 2a that is, (20.37)



 Rg¯k (zk , −ark2 )

≤ exp

Rg¯k (zk , −ark2 ),

e2aC1 2a

 .

Recall that Shi’s local derivative estimate says the following (see Theorem 14.14 in Part II). For any α, K, r, n, and m ∈ N there exists a constant C = C (α, m, n) < ∞ depending only on α, m, and n such that if Mn is a manifold, p ∈ M, and g (t), t ∈ [0, τ ], 0 < τ ≤ α/K, is a solution to ¯g(0) (p, r) as a the Ricci flow on an open neighborhood U of p containing B compact subset, and if |Rm (x, t)| ≤ K

for all x ∈ U and t ∈ [0, τ ],

then



C max K, r−2 (20.38) |∇ Rm (y, t)| ≤ tm/2

for all y ∈ Bg(0) p, min r, K −1/2 /2 and t ∈ (0, τ ]. Note that the dependence of the constant on the rhs of (20.38) is more explicit in terms of K and r than in Theorem 14.14 in Part II; however, this dependence follows easily from the latter result for K = 1 and r = 1 and from rescaling the solution (in space and time) by max K, r−2 . Since by (20.36) we have the bound m

|Rmg¯k | ≤ C1 Rg¯k (zk , 0) = C1 rk−2   for the curvature of g¯k (t) in Bg¯k (0) (zk , 2rk ) × −2rk2 , 0 , we may apply Shi’s local derivative estimate to obtain20    m  −m−2 Rm ∇  g¯k g ¯k  (y, t) ≤ C (C1 , m, n) rk   in Bg¯k (−2r2 ) (zk , rk ) × −rk2 , 0 . In particular, k ∆Rg¯k (y, t) ≤ C (C1 , n) rk−4 = C (C1 , n) Rg¯k (zk , 0)2 20

Note that Bg¯k (−2r2 ) (zk , 2rk ) ⊂ Bg¯k (0) (zk , 2rk ). k

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3. THE COMPACTNESS OF κ-SOLUTIONS

145

  for (y, t) ∈ Bg¯k (−2r2 ) (zk , rk ) × −rk2 , 0 . Using |Rcg¯k | ≤ Rg¯k , we obtain that k   for t ∈ −rk2 , 0 , ∂Rg¯k (zk , t) ≤ ∆Rg¯k (zk , t) + 2Rg2¯k (zk , t) ∂t ≤ C (C1 , n) Rg2¯k (zk , 0) for some constant C (C1 , n) < ∞ depending only on C1 and n. Choosing   1 1 , < 1, a = min 2C (C1 , n) 2 we have ∂Rg¯k 1 2 1 −4 (zk , t) ≤ Rg¯k (zk , 0) = r ∂t 2a 2a k   for t ∈ −rk2 , 0 . Integrating the above equation from −ark2 to 0, we obtain 1 Rg¯ (zk , 0) ≤ Rg¯k (zk , −ark2 ). 2 k Combining (20.37) and (20.39), we get the following uniform bound at (zk , 0):  2aC1  e −2 . (20.40) rk = Rg¯k (zk , 0) ≤ 2 exp 2a

(20.39)

Now by combining Steps 2 and 3 we obtain the following. Step 4. Curvature bounds in balls centered at xk . Specifically, as a consequence of combining (20.31) and (20.40), we conclude that there exists C3 > 0 independent of k such that Rg¯k (y, 0) ≤ C3−2

for y ∈ Bg¯k (0) (xk , C3 ) .

Hence the volume lower bound (20.41)

Volg¯k (0) Bg¯k (0) (xk , C3 ) ≥ κC3n

follows from the κ-noncollapsing assumption. Step 5. Bounds of the curvature, independent of k, in balls centered at xk with arbitrarily large radii. Given any r > 0, for any y ∈ Bg¯k (0) (xk , r) we have Bg¯k (0) (xk , C3 ) ⊂ Bg¯k (0) (y, r + C3 ) and hence Volg¯k (0) Bg¯k (0) (y, r + C3 ) κC3n ≥ . (r + C3 )n (r + C3 )n We shall show that there exists a constant C4 < ∞, depending on r and on the sequence but independent of k, such that (20.42)

Rg¯k (y, 0) · (r + C3 )2 ≤ C4

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146

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

for any y ∈ Bg¯k (0) (xk , r). This estimate and the κ-noncollapsing condition enable us to apply Hamilton’s compactness theorem to get a convergent subsequence (Mnk , g¯k (t) , xk ) → (Mn∞ , g¯∞ (t), x∞ ). Clearly the limit is a (complete) κ-solution with Harnack. Therefore Theorem 20.9 is proved modulo the proof of (20.42). We now give the details of the proof of (20.42). We prove the following lemma regarding uniform bounded curvature at points with uniformly bounded distance from the origin for a sequence of solutions of the Ricci flow. This result is also known by the catchphrase ‘bounded curvature at bounded distance’. The idea of the proof is similar to the argument in the second half of the proof of the claim stated at the beginning of Step 2 above. Lemma 20.15. Let {(Mnk , gk (t) , xk ) , t ∈ (−αk , 0]}k∈N , be a sequence of complete solutions of the Ricci flow with nonnegative curvature operator and ∂ Rgk (x, t) ≥ 0 for any (x, t) ∈ Mk × (−αk , 0). Suppose that with ∂t Rgk (xk , 0) = 1

Volgk (0) Bgk (0) (xk , r0 ) ≥ v0

and

for some constants r0 > 0 and v0 > 0 independent of k. Then, for any r > 0, either (i) there exists a constant C0 < ∞ independent of k such that Rgk (x, 0) ≤ C0

sup

αk

x∈Bgk (0) (xk ,r)

or (ii) there exists a constant C1 < ∞ independent of k such that Rgk (x, 0) ≤ C1 .

sup x∈Bgk (0) (xk ,r)

Proof. We will prove the lemma by contradiction. If the lemma is false, then there exist r∗ > 0 and a sequence of complete solutions {(Mnk , gk (t) , xk ) , t ∈ (−αk , 0]} which satisfies the following conditions: (1) The curvature operator is nonnegative, i.e., Rmgk ≥ 0, and the ∂ Rgk (x, t) ≥ 0 for all scalar curvature is pointwise nondecreasing, i.e., ∂t (x, t) ∈ Mk × (−αk , 0). (2) There exist constants r0 > 0 and v0 > 0 such that Rgk (xk , 0) = 1

and

Volgk (0) Bgk (0) (xk , r0 ) ≥ v0 .

We have (3) αk

Rgk (x, 0) → ∞

sup x∈Bgk (0) (xk ,r∗ )

and (4) sup

Rgk (x, 0) → ∞.

x∈Bgk (0) (xk ,r∗ )

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3. THE COMPACTNESS OF κ-SOLUTIONS

147

By Lemma 18.15 there exists wk ∈ Bgk (0) (xk , r∗ ) and sk ∈ (0, r∗ + 1) such that (a) Rgk (wk , 0) s2k → ∞ as k → ∞ and (b) Rgk (x, 0) ≤ 2Rgk (wk , 0)  2Qk for all x ∈ Bgk (0) (wk , sk ). From the proof of Lemma 18.15 it is clear that Rgk (wk , 0) ≥ (r∗ + 1)−2

sup

Rgk (x, 0) .

x∈Bgk (0) (xk ,r∗ )

Hence αk Rgk (wk , 0) → ∞. Now we can apply Proposition 20.4 on the parabolic cylinder Bgk (0) (wk , sk ) × (−αk , 0] to obtain that for any ε > 0 there exists A (ε) ∈ (0, ∞) such that −1/2

Volgk (0) Bgk (0) (wk , A (ε) Qk   −1/2 n A (ε) Qk

)

≤ε

for k large enough. On the other hand, for any wk ∈ Bgk (0) (xk , r∗ ) we have Bgk (0) (xk , r0 ) ⊂ Bgk (0) (wk , r∗ + r0 ) and hence

Volgk (0) Bgk (0) (wk , r∗ + r0 ) v0 ≥ . (r∗ + r0 )n (r∗ + r0 )n −1/2

v0 ≤ r∗ + r0 when Choose ε  12 (r∗ +r n . Since Qk → ∞, we have A (ε) Q k 0) k is large enough. By the Bishop–Gromov volume comparison theorem we have −1/2

ε≥

Volgk (0) Bgk (0) (wk , A (ε) Qk   −1/2 n A (ε) Qk

)

Volgk (0) Bgk (0) (wk , r∗ + r0 ) ≥ 2ε. (r∗ + r0 )n This is a contradiction and the lemma is proved. ≥




3.3. Second proof of the compactness of κ-solutions — via Proposition 20.6. In this subsection we give another proof of Theorem 20.9, this time using Proposition 20.6 (note that this result also depends on Proposition 20.4). This variation on Perelman’s proof is due to one of the authors [142]. Let (Mnk , gk (t)) be a sequence in MHarn n,κ and, given xk ∈ Mk and tk ≤ 0, let   t g¯k (t) = Rgk (xk , tk ) gk tk + Rgk (xk , tk ) be defined by (20.21). Again, the solution g¯k (t) is ‘normalized’ by the property Rg¯k (xk , 0) = 1.

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148

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

First we prove a claim by contradiction. (This claim corresponds to (20.41) in our first proof of Theorem 20.9.) Claim 20.16 (Volume noncollapsing only assuming curvature bounded at center). There exists a constant c0 = c0 (κ, n) > 0 such that Volg¯k (0) Bg¯k (0) (xk , 1) ≥ c0 for all k ≥ 1. Proof of Claim 20.16. Suppose that the claim is false. Then, for some subsequence, Volg¯k (0) Bg¯k (0) (xk , 1) → 0

(20.43)

Volg¯

Bg¯

(0) (xk ,s)

k k = ωn , where ωn is the volume as k → ∞. Since lims→0+ sn of the unit Euclidean n-ball, for k large enough we can find δk ∈ (0, 1) such that Volg¯k (0) Bg¯k (0) (xk , δk ) 1 = ωn . (20.44) δkn 2 (0)

Since Volg¯k (0) Bg¯k (0) (xk , 1) ≥ Volg¯k (0) Bg¯k (0) (xk , δk ), it follows from (20.43) and (20.44) that δk → 0 as k → ∞. Consider the blown-up sequence of solutions

(20.45) g˜k (t)  δk−2 g¯k δk2 t . Then (Mnk , g˜k (t)) is a κ-solution with Harnack with 1 Volg˜k (0) Bg˜k (0) (xk , 1) = ωn . 2 For any r > 1 we have Volg˜k (0) Bg˜k (0) (xk , 4r) ≥ Volg˜k (0) Bg˜k (0) (xk , 1) ωn 1 (4r)n . = ωn = 2 2 (4r)n ωn Applying Proposition 20.6(b), with ω = 2(4r) n and r0 = 4r, to the ball Bg˜k (0) (xk , 4r), we get for z ∈ Bg˜k (0) (xk , r)     ωn ωn C B n n 2(4r) 2(4r)   + (20.46) Rg˜k (z, 0) ≤ ωn 16r2 16r2 τ0 n 2(4r)

for some B = B(ω) < ∞, C = C(ω) < ∞, and τ0 (ω) > 0. Note that the bound on the rhs of (20.46) depends only on r and n. We can now apply Hamilton’s Cheeger–Gromov-type compactness theorem (Corollary 3.18 in Part I) to get a convergent subsequence: (Bg˜k (0) (xk , r) , g˜k (t) , xk ) → (Mn∞ , g˜∞ (t) , x∞ )

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4. DERIVATIVE ESTIMATES AND SOME CONJECTURES

149

with 1 Volg˜∞ (0) Bg˜∞ (0) (x∞ , 1) = ωn . 2 n The limit solution (M∞ , g˜∞ (t)) is complete with nonnegative curvature operator and is κ-noncollapsed at all scales. On the other hand, from Rg˜k (xk , 0) = Rg¯k (xk , 0)δk2 = δk2 we have that (20.47)

Rg˜∞ (x∞ , 0) = 0 and hence by Hamilton’s strong maximum principle (M∞ , g˜∞ (t)) is a flat solution. Since (M∞ , g˜∞ (t)) is flat, it follows from κ-noncollapsing that (M∞ , g˜∞ (t)) has maximum volume growth, i.e., Volg˜∞ (0) Bg˜∞ (0) (x∞ , r) ≥ κrn for all r > 0. Since a complete flat metric with maximum volume growth must be Euclidean space, this contradicts (20.47). The claim is proved. For any r > 1, we have Volg¯k (0) Bg¯k (0) (xk , 4r) ≥ Volg¯k (0) Bg¯k (0) (xk , 1) c0 (4r)n . ≥ c0 = (4r)n Applying Proposition 20.6(b) to Bg¯k (0) (xk , 4r) , we get for z ∈ Bg¯k (0) (xk , r) and r ≥ 1,     c0 C B c0 n n (4r) (4r)  . + Rg¯k (z, 0) ≤ 2 16r 16r2 τ0 c0 n (4r)

This and the κ-noncollapsing assumption enable us to apply Hamilton’s compactness theorem and we get a subsequence (Mk , g¯k (t) , xk ) converging to (Mn∞ , g¯∞ (t), x∞ ). Clearly the limit is a κ-solution with Harnack and Theorem 20.9 is proved.
4. Derivative estimates and some conjectures It is remarkable that compactness can be used to obtain derivative estimates rather than the other way around. 4.1. Derivative estimates for κ-solutions. A simple consequence of Perelman’s compactness theorem, i.e., Theorem 20.9, is the following derivative estimate for κ-solutions with Harnack. Theorem 20.17 (Derivative estimates for κ-solutions satisfying trace Harnack). Given κ > 0 and n ≥ 2, there exists a constant η (κ, n) < ∞ such that for any κ-solution (Mn , g (t)), t ≤ 0, with Harnack we have the estimates     ∂R 3/2  (x, t) ≤ η (κ, n) R (x, t)2 (20.48) |∇R (x, t)| ≤ η (κ, n) R (x, t) ,  ∂t for any x ∈ M and t ≤ 0.

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150

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

Proof. If the theorem is not true, then for some κ > 0 and n ≥ 2 there exist a sequence of κ-solutions (Mnk , gk (t)), t ≤ 0, with Harnack and space-time points (xk , tk ) ∈ Mk × (−∞, 0] such that we have either |∇Rgk (xk , tk )| ≥ kRgk (xk , tk )3/2 for all k ≥ 1 or

  ∂   Rg (xk , tk ) ≥ k · Rg (xk , tk )2 k  ∂t k  for all k ≥ 1. Define the rescaled solutions (Mnk , gk (t), xk ), t ∈ (−∞, 0], by   t . gk (t) = Rgk (xk , tk ) · gk tk + Rgk (xk , tk ) Then Rgk (0) (xk ) = 1 and we have either   ∇Rg (xk , 0) ≥ k for all k or k

  ∂   Rg (xk , 0) ≥ k for all k. k  ∂t  On the other hand, by Theorem 20.9 there exists a subsequence (Mnk , gk (t), xk ) → (Mn∞ , g∞ (t), x∞ )

converging in the Cheeger–Gromov C ∞ -topology. In particular,     ∇Rg (xk , 0) → ∇Rg (x∞ , 0) k

∞

and

∂ ∂ Rgk (xk , 0) → Rg∞ (x∞ , 0). ∂t ∂t convergences since either the sequence  to have both of these  It is impossible ∇Rg (xk , 0) or the sequence  ∂ Rg (xk , 0) is divergent. The theorem is k k ∂t proved.
Remark 20.18 (Estimates for derivatives of any order). It is clear that the above proof also implies the following. For any κ > 0, , m, and n ∈ N − {1}, there exists a constant η (κ, , m, n) < ∞ such that for any κsolution (Mn , g (t)), t ≤ 0, with Harnack we have the estimates  
 ∂ m m  ≤ η (κ, , m, n) R (x, t)1+
+ 2  ∇ R (x, t) (20.49)   ∂t


∂ for any x ∈ M and t ≤ 0. (The same result holds for compositions of ∂t and ∇, in any order, of either Rm, Rc, or R with the form of the rhs unchanged.) Note that estimate (20.49) is scale invariant, that is, if g satisfies (20.49), then C · g satisfies (20.49) for any C ∈ (0, ∞).

Combining Theorem 19.56 and Theorem 20.17, we obtain the following. Corollary 20.19 (Universal scaled derivative bounds in dimension 3). There exists a universal constant

(independent of κ) such that for η0 < ∞ any 3-dimensional κ-solution M3 , g (t) , t ≤ 0, we have the estimates     ∂R 3  ≤ η0 R (x, t)2  2 (x, t) (20.50) |∇R (x, t)| ≤ η0 R (x, t) ,   ∂t

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4. DERIVATIVE ESTIMATES AND SOME CONJECTURES

151

for x ∈ M and t ≤ 0. More generally, for any  and m there exists η (, m) < ∞ (independent of κ) such that  
 ∂ 1+
+ m m   2  ∂t
∇ Rm (x, t) ≤ η (, m) R (x, t) for any x ∈ M and t ≤ 0. We may call these  scaled derivative bounds since the quantities  universal  ∂ m −(1+
+ m ) 2 R (x, t)  ∂t ∇ Rm (x, t) are scale-invariant. Proof. Using the improved compactness result of Corollary 20.11(ii),

/ we can prove Corollary 20.19 above for any M3 , g (t) ∈ κ>0 Mnsph 3,κ just as we proved the derivative estimate of Theorem 20.17. This is possible / / because κ>0 M3,κ equals the union of κ>0 Mnsph 3,κ with the collection of spherical space form solutions and because the estimates clearly hold for the spherical space form solutions with η0 = 1.21 Hence the derivative estimates /
hold for the collection κ>0 M3,κ . Remark 20.20 (Derivatives estimate fails on the cigar soliton). Recall that the cigar soliton is Σ ∼ = R2 with the metric gΣ = ds2 + tanh2 s dθ2 and scalar curvature RΣ = 4 sech2 s. From this it is easy to see that the first space derivatives estimate does not hold on the cigar soliton, i.e., supΣ R−3/2 |∇R| = ∞. Moreover, dilating the cigar by curvature about a sequence of points tending to infinity, one obtains as the limit a half-line with basepoint equal to its vertex. 4.2. Some applications of the derivative estimates. Let (Mn , g (t)), t ∈ (−∞, 0], be a noncompact κ-solution with Harnack; note that R > 0. Fix the time and a point p ∈ M. Given any x ∈ M, let α : [0, s¯] → M be a minimal unit speed geodesic joining p to x. By Theorem 20.17, we have 0  1 1  η d −1/2 R (α (s) , t) = ∇ R−1/2 , α (s) ≤ R−3/2 |∇R| ≤ . ds 2 2 Integrating this along α implies −2  η . (20.51) R (x, t) ≥ R−1/2 (p, t) + dg(t) (x, p) 2 In particular, we obtain Corollary 20.21. For a noncompact κ-solution with Harnack lim inf dg(t) (x,p)→∞

R (x, t) dg(t) (x, p)2 ≥

4 . η2

Note that the shrinking round 3-sphere g (t) = (1 − 4t)gS 3 (1) has R (x, t) ≡ and |∇m Rm| ≡ 0 for m ≥ 1. 21

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6 1−4t

152

20. COMPACTNESS OF THE SPACE OF κ-SOLUTIONS

Problem 20.22. Can one show that for any complete noncompact κsolution with Harnack we have lim inf dg(t) (x,p)→∞

R (x, t) dg(t) (x, p)2 = ∞

for all t ∈ (−∞, 0] ? Note that a steady gradient Ricci soliton (Mn , g (t) , f (t)) with Rc ≥ 0 automatically satisfies the trace Harnack estimate since for any tangent vector X we have ∂R + 2 ∇R, X + 2 Rc (X, X) = 2 Rc (X + ∇f, X + ∇f ) ≥ 0. ∂t The following calculations occur at t = 0. Therefore, by Theorem 20.17, on a nonflat steady gradient Ricci soliton with Rm ≥ 0 which is κ-noncollapsed at all scales, we have 1 ∂R η 1 ∇R, ∇f  = ≤ R2 . 2 2 ∂t 2 Let σ : [0, ∞) → M be an integral curve to −∇f , i.e., σ˙ (u) = − (∇f ) (σ (u)). Then 

 d −1 R (σ (u)) = −∇f, ∇ R−1 = R−2 ∇R, ∇f  ≤ η. du This implies (20.52)

Rc (∇f, ∇f ) =

Corollary 20.23. For a nonflat steady gradient Ricci soliton with Rm ≥ 0 which is κ-noncollapsed at all scales we have R (σ (u)) ≥

1 R−1 (σ (0))

+ ηu

.

If n = 3, then one can show from this that c R (x) ≥ 1 + d (x, p) for some positive constant c. Furthermore, if one invokes Perelman’s ‘canonical neighborhood’ theorem for 3-dimensional κ-solutions (see §11 of [152]), then one may obtain the corresponding upper bound for R. For every ε > 0 there exists a compact set Kε in M3 such that every point x ∈ M − Kε is the center of an ε-neck. Then for any δ > 0, there exists ε > 0 such that |∆R| ≤ δR2

and

|Rc|2 ≥

1−δ 2 R 2

in M − Kε (in the model (round) cylinder case we have ∆R ≡ 0 and |Rc|2 ≡ 1 2 2 R ). Then ∇R, ∇f  = ∆R + 2 |Rc|2 ≥ (1 − 2δ) R2 . This implies that if σ : [0, ∞) → M − Kε is an integral curve to −∇f , then

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4. DERIVATIVE ESTIMATES AND SOME CONJECTURES

R (σ (u)) ≤

R−1 (σ (0))

153

1 . + (1 − 2δ) u

From this we may obtain Corollary 20.24. For a 3-dimensional nonflat steady gradient Ricci soliton with Rm ≥ 0 which is κ-noncollapsed at all scales we have C R (x) ≤ 1 + d (x, p) for some C < ∞. ∞.

Finally, by (20.52) this implies Rc (∇f, ∇f ) ≤

C (1+d(x,p))2

for some C
0 and |Rm (x)| d (x, O)2 ≤ C for all x ∈ M. Such a manifold is a static eternal solution to the Ricci flow which is κ-noncollapsed at all scales. (Nice examples 4are given by Kronheimer’s (Ricci flat) hyper-K¨ ahler ale 4-manifolds M , g .) Consider any sequence of points xi → ∞ (with |Rm (xi )| = 0) and consider the corresponding sequence of pointed blown-down manifolds {(Mn , gi , xi )}, where gi  |Rm (xi )| g. Note that |Rmgi (xi )| = 1, |Rmgi (O)| = |Rm (xi )|−1 |Rmg (O)| → ∞, √ and dgi (xi , O) = |Rm (xi )|1/2 d (xi , O) ≤ C. Hence such a sequence does not have a subsequence which converges in the pointed Cheeger–Gromov sense; that is, compactness fails.24

24

We thank Bing Wang for suggesting these examples to us.

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http://dx.doi.org/10.1090/surv/163/05

CHAPTER 21

Perelman’s Pseudolocality Theorem You speak to me in riddles and you speak to me in rhymes. – From “Possession” by Sarah MacLachlin

One mantra is that the Ricci flow smooths out metrics. Among the results which support this contention are the following: (1) The fact that many geometric quantities derived from the metric, such as the various curvature tensors, satisfy heat-type equations. See Chapter 6 of Volume One. (2) Bando’s Bernstein-type global derivative estimates, assuming a bound on the curvature, which provide bounds for tm/2 |∇m Rm| for m ∈ N. Here the bounds for the derivatives of curvature improve as t increases at least for short time. See Chapter 7 of Volume One and Chapter 14 of Part II, respectively. (3) The long time existence and convergence results in all dimensions, usually under a positive curvature hypothesis, which exhibit smoothing to a best possible type of metric, namely a constant positive sectional curvature metric. See Chapters 5 and 6 of Volume One for the case of dimensions 2 and 3 and see Chapter 11 in Part II for higher dimensions. (4) The entropy and reduced volume monotonicity formulas of Perelman (we think of monotonicity as evidence of smoothing), which imply the no local collapsing theorem and affect the classification of singularity models. See Chapters 6 and 8 in Part I, respectively. (5) The differential inequalities satisfied by the reduced distance function and the structure theorems for κ-solutions. (Heuristically such results would not be possible without smoothing.) See Chapters 7 and 8 of Part I. Recall from Problem 17.33 that one goal in the study of Ricci flow and geometric analysis is to localize various formulas. The localization of the aforementioned Bando global derivative estimates is Shi’s local derivative estimates. For the heat equation on a Riemannian manifold with a lower bound on the Ricci curvature, the Li–Yau differential Harnack estimate was originally proved in localized form, which yielded a sharp global estimate in the case of Rc ≥ 0 (see Theorem 25.8 below). Another example of a local estimate for the heat equation is the parabolic mean value inequality (see Theorem 25.2). 157 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

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21. PERELMAN’S PSEUDOLOCALITY THEOREM

In contrast, Hamilton’s matrix Harnack estimate for the Ricci flow is apparently difficult to localize in an effective way. More generally, fine (as opposed to coarse) estimates for Ricci flow appear more difficult to localize; see the discussion of fine versus coarse estimates in §1 of Chapter 14 in Part II. In this chapter we present an additional local version of this smoothing property of Ricci flow; we discuss Perelman’s pseudolocality theorem for smooth complete solutions of the Ricci flow on both compact and noncompact manifolds (see the original Theorem 10.1 of Perelman [152] as well as the more analytically technical version of Chau, Tam, and Yu [26]). Perelman’s pseudolocality theorem is, in spirit, related to the idea of localizing the ‘doubling time’ estimate for the norm of the Riemann curvature tensor. In particular, see §10.3 of Perelman [152]. Pseudolocality is proved by localizing the (integral) entropy monotonicity formula via a global (pointwise) differential Harnack formula; see the proof of part (1) of Lemma 22.13. Perhaps the idea is that fine integral monotonicity formulas are easier to localize than fine pointwise estimates (since in the former case we have the benefit of integration by parts). In §1 we present the statement of Perelman’s pseudolocality theorem together with an intuitive discussion and example. In the remaining sections of this chapter we present Perelman’s proof of his pseudolocality theorem modulo some technical tools, such as point picking results and the logarithmic Sobolev inequality via the isoperimetric inequality, which are presented in the next chapter.

1. Statement and interpretation of pseudolocality We recall that heat-type equations have infinite speed of propagation. For example, for the heat equation, at any positive time the fundamental solution, which starts off as a δ-function, is positive everywhere in space. The solution at one point at the initial time instantaneously affects the solution at all other points, although for small times and for points at large distances, this effect is small. Perelman’s pseudolocality theorem limits the amount (but not the speed) of propagation of the Ricci flow. Perhaps this is one reason Perelman calls it pseudolocality; that is, for short times the effect of the Ricci flow is principally local. 1.1. Statement of the pseudolocality theorem. First we recall some concepts which are related to the statement of the pseudolocality theorem. By a regular domain in a Riemannian manifold we mean a bounded domain whose boundary is sufficiently regular; unless otherwise specified, this shall mean that the boundary is C 1 . Given a regular domain Ω in a complete Riemannian manifold (N n , h), let Area(∂Ω) denote the (n − 1)-dimensional volume of ∂Ω and let Vol (Ω) denote the

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1. STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY

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n-dimensional volume of Ω. We have the following sharp Euclidean isoperimetric inequality (see § 8.1.2 (in particular, inequality (7) on p. 69) of Burago and Zalgaller [20] or Theorem II.2.2 of Chavel [28]).1 Proposition 21.1 (Euclidean isoperimetric inequality). For any regular domain in Euclidean space Ω ⊂ En we have (21.1)

(Area(∂Ω))n ≥ nn ωn (Vol(Ω))n−1 ,

where ωn denotes the volume of the Euclidean unit n-ball. Perelman’s pseudolocality theorem says the following (see Theorem 10.1 in [152]). Theorem 21.2 (Perelman’s pseudolocality). For every α > 0 and n ≥ 2 there exist δ > 0 and ε0 > 0 depending  only on α and n with the following 2 n property. Let (M , g (t)), t ∈ 0, (εr0 ) , where ε ∈ (0, ε0 ] and r0 ∈ (0, ∞), be a complete solution of the Ricci flow with bounded curvature and let x0 ∈ M be a point such that (21.2)

R (x, 0) ≥ −r0−2

for x ∈ Bg(0) (x0 , r0 )

and ‘Bg(0) (x0 , r0 ) is δ-almost isoperimetrically Euclidean’:

n

n−1 (21.3) Area g(0) (∂Ω) ≥ (1 − δ)cn Volg(0) (Ω) for any regular domain Ω ⊂ Bg(0) (x0 , r0 ), where cn  nn ωn is the Euclidean isoperimetric constant. Then we have the interior curvature estimate 1 α (21.4) | Rm |(x, t) ≤ + t (ε0 r0 )2 for x ∈ M such that dg(t) (x, x0 ) < ε0 r0 and t ∈ (0, (εr0 )2 ].2 Remark 21.3. The size of the time interval uses ε whereas both the size of the ball and the curvature bound use ε0 . Note that if x1 ∈ M is such that dg(0) (x1 , x0 ) < (1 − η) r0 , where η ∈ (0, 1), then Bg(0) (x1 , ηr0 ) ⊂ Bg(0) (x0 , r0 ), so that under the hypotheses of the pseudolocality theorem we have R (x, 0) ≥ −r0−2 ≥ − (ηr0 )−2

in Bg(0) (x1 , ηr0 )

and Bg(0) (x1 , ηr0 ) is δ-almost isoperimetrically Euclidean. Hence the pseudolocality theorem, using the point x1 and radius ηr0 , implies that 1 α | Rm |(x, t) ≤ + t (ε0 ηr0 )2 1

In [20], the Euclidean isoperimetric inequality is considered as a consequence of the Brunn–Minkowski inequality for compact subsets of Euclidean space. See also Gardner [66]. 2 That is, the curvature estimate (21.4) holds in the small parabolic ‘cylinder’  t∈(0,(εr0 )2 ] Bg(t) (x0 , ε0 r0 ) × {t}.

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21. PERELMAN’S PSEUDOLOCALITY THEOREM

for any x ∈ Bg(t) (x1 , ε0 ηr0 ) and t ∈ (0, (εηr0 )2 ]. In particular, applying this curvature estimate at x = x1 , we obtain the following. Corollary 21.4 (Interior curvature estimate). Given α, n as above, there exist δ, ε0 > 0 such that if g (t) is a complete solution of Ricci flow with bounded curvature for t ∈ 0, (εr0 )2 , where ε ∈ (0, ε0 ] and r0 ∈ (0, ∞), and if R (x, 0) ≥ −r0−2 for x ∈ Bg(0) (x0 , r0 ) where Bg(0) (x0 , r0 ) is δ-almost isoperimetrically Euclidean, then we have the estimate 1 α |Rm| ≤ + t (ε0 ηr0 )2 in the parabolic cylinder Bg(0) (x0 , (1 − η) r0 ) × (0, (εηr0 )2 ] for any η ∈ (0, 1). Remark 21.5 (Scale-invariance of the statement). The statement of the theorem is scale invariant, i.e., if C ∈ (0, ∞),

and r˜0  C 1/2 r0 , g˜ (t)  Cg C −1 t then we have the following: r0−2 in (1) If Rg (x, 0) ≥ −r0−2 in Bg(0) (x0 , r0 ), then Rg˜ (x, 0) ≥ −˜ Bg˜(0) (x0 , r˜0 ). (2) If for some δ > 0 the metric g (0) satisfies (21.3) for Ω ⊂ Bg(0) (x0 , r0 ), then for the same δ > 0 the metric g˜ (0) satisfies (21.3) for Ω ⊂ Bg˜(0) (x0 , r˜0 ). (3) If |Rmg | (x, t) ≤ αt + (ε0 1r0 )2 provided dg(t) (x, x0 ) < ε0 r0 and t ∈ (0, (εr0 )2 ], then |Rmg˜ | (x, t) ≤ r0 ε0 r˜0 and t ∈ (0, (ε˜

α t

+

1 (ε0 r˜0 )2

provided dg˜(t) (x, x0 )
0 may be chosen arbitrarily small; this is balanced by the fact that the ε0 factor in the other term (ε0 1r0 )2 on the rhs of (21.4) depends on α. This theorem may appear to be somewhat reminiscent of, but not quite like, Shi’s Bernstein-type local derivative estimates. Note that a difference between the pseudolocality theorem and Shi’s estimate is that, in the latter 3









Note that
Rmg˜(t)
= C −1
Rmg (C −1 t)
, so that if
Rmg(t)
≤

α , t



then
Rmg˜(t)
≤

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α . t

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161

result, one requires a curvature bound on a whole parabolic cylinder in order to get estimates for the derivatives of curvature. On the other hand, in the pseudolocality theorem, one has local geometric hypotheses only at the initial time. The isoperimetric ratio condition in Theorem 21.2 is, in a sense, deceptively strong. One reason for this is that there is no restriction on the ‘scale’ of Ω (i.e., how small Ω is) in the ball B (x0 , r0 ). For example, curvature bumps in the sense of Hamilton are ruled out. In particular, it is not possible for the initial metric to have the properties that there is a point  g (0) √  p0 ∈ B (x0 , r0 ) with B p0 , 1/ K ⊂ B (x0 , r0 ) and the smallest eigenvalue  √  satisfies λ1 (Rm) ≥ K in B p0 , 1/ K . On the other hand, curvature spikes (i.e., regions of very large curvature where the size of the region is very small relative to the size of the curvature) are possible, so that the isoperimetric ratio condition does not imply a curvature bound at the initial time. Note that in the case of a complete, simply-connected manifold with constant nonpositive curvature, we have Area(∂Ω)n ≥ cn Vol(Ω)n−1 (this is also conjectured for variable nonpositive curvature and has been proven in dimensions 2, 3, and 4; see Weil [186], Kleiner [109], and Croke [48]). On the other hand, if the curvature is constant and positive, then Area(∂Ω)n  cn Vol(Ω)n−1 . So, roughly speaking, (21.2) rules out too much negative curvature, whereas (21.3) rules out too much positive curvature. In §0.2 of [152] Perelman writes: “In this picture, t corresponds to the scale parameter; the larger is t, the larger is the distance scale and the smaller is the energy scale; to compute something on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale.” The statement that the larger t is, the larger is the distance scale is consistent with the philosophy that Ricci flow makes metrics more homogeneous. Given a solution to the Ricci flow, the metric g (t) and its observables (invariants) at a time t are some sort of average of the metrics and its corresponding observables at earlier times; i.e., the Ricci flow is the ‘heat equation for metrics’. Given a closed Riemannian manifold (Mn , g), we may formally define its distance scale ρ (g) ∈ [0, ∞] to be the square root of the supremum time of existence of the backward Ricci flow with initial metric g. Intuitively, one expects that ‘most’ Riemannian manifolds have distance scale 0. On the other hand, any Riemannian manifold which is the time slice of an ancient solution has distance scale equal to ∞. Note that ρ (cg) = c1/2 ρ (g). It may

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21. PERELMAN’S PSEUDOLOCALITY THEOREM

be interesting to consider the following class of metrics: Sδ  {(Mn , g) : ρ (g) ≥ δ} for δ > 0. In view of Perelman’s work, one may ask if this space has some form of compactness modulo scaling in dimension 3. Then in §10 of [152] Perelman writes: “Thus, under the Ricci flow, the almost singular regions (where curvature is large) cannot instantly significantly influence the almost Euclidean regions. Or, using the interpretation via renormalization group flow, if a region looks trivial (almost Euclidean) on higher energy scale, then it cannot suddenly become highly nontrivial on a slightly lower energy scale.” In terms of the statement of pseudolocality, this says that if a ball of a given radius has a scalar curvature lower bound and is almost Euclidean at an earlier time, then there is a curvature estimate in the corresponding parabolic cylinder up to a slightly later time and with a slightly smaller radius. Moreover, in the above quote, one should emphasize the word ‘significantly’ since the influence can be instant but for short times this influence is small. One can imagine that at the initial time there is an almost Euclidean ball and next to it is a thin neck (almost singular region) which is about to pinch. The pseudolocality theorem says that for a short time, one has a curvature estimate in a smaller ball. In particular, a singularity forming in a relatively short time (such as a neck pinch) in a region close to the ball does not affect the curvature in a smaller ball much. 1.3. Intuition related to neckpinch singularities. As an intuitive consistency check, consider the pseudolocality theorem in the following singularity formation scenario. Given a rotationally and reflectionally symmetric 3-dimensional neckpinch (see §5 of Chapter 2 in Volume One or §2 of Appendix D in Part II), we can take x0 on the center sphere. Suppose that the radius of the center sphere is equal to ρ0 at time t = 0. Choosing r0  ρ0 , the ball B (x0 , r0 ) is almost Euclidean at t = 0. −2 At t = 0 we have |Rm| ≈ ρ−2 0  r0 . We also have | Rm |(x, t) 

ρ20

1 , − 2t ρ2

with the singularity occurring at time t ≈ 20 . Note that this is consistent 1 ≤ (ε0 1r0 )2 for t ≤ (εr0 )2 with Perelman’s pseudolocality theorem since ρ2 −2t 0 √

provided ε ≤ ε0 ≤ ρ0 / 3r0 . That is, if we have an almost Euclidean ball at the center of a neck forming a singularity, the ball must be very small compared to the radius of the neck and Perelman’s curvature estimate

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applies to a time interval much smaller than the time elapsed before the singularity forms. We also compare this with the situation of the heat equation on Euclidean space. There, the equation is linear and singularities do not form. 2 The fundamental solution H (x, y, t) = (4πt)−n/2 e−|x−y| /4t tends to δy (x) as t → 0 and satisfies the estimate 0 < H (x, y, t) ≤ (4πt)−n/2 for all t > 0. 2 Moreover, for |x − y| ≥ c > 0, we have H (x, y, t) ≤ (4πt)−n/2 e−c /4t . Theorem 21.2 is also similar in some respects to ε-regularity results such as that of Ecker and Huisken [56] for the mean curvature flow (see §10.6* of [152]). 1.4. Topping’s example. We now give an example (due to Peter Topping [180]) which shows that Theorem 21.2 is false if we do not assume that (Mn , g (t)) is complete. Consider the cylinder S 1 (r) × [−1, 1] with the flat product metric, where S 1 (r) denotes the circle of radius r (i.e., circumference 2πr) and r > 0 is small. Topologically we cap each of the two ends of the cylinder with a disc D 2 . Metrically we use a cutoff function 2 (r) in to smoothly blend the cylinder metric with the round hemisphere S+ thin collars about their boundaries to construct a rotationally symmetric Riemannian metric g0r with nonnegative curvature on a surface Σ2 which is diffeomorphic to the 2-sphere. In our construction we may assume 22 4πr ≤ Areag0r (Σ) ≤ πr 5 provided r is sufficiently small.4 Let 2 r

Σ , g (t) , t ∈ [0, T r ), be the maximal solution of the Ricci flow with g r (0) = g0r . We may estimate the maximal time T r as follows. By the Gauss–Bonnet theorem we have  d Areagr (t) (Σ) = − Rgr (t) dµgr (t) = −8π, dt Σ so that Areagr (t) (Σ) = Areag0r (Σ) − 8πt. In particular, 1 11 Areag0r (Σ) ≤ r. 8π 20 2 In fact, by Hamilton’s result that metrics on S with nonnegative curvature 1 Volg0r (Σ). shrink to round points under the Ricci flow, we have T r = 8π r Note that limr→0+ T = 0. (21.5)

4

Tr ≤

  Note that 4πr = Area S 1 (r) × [−1, 1] .

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21. PERELMAN’S PSEUDOLOCALITY THEOREM

Now we can define Topping’s incomplete solution of the Ricci flow. Let π : R → S 1 (r) denote the standard covering map given by π (x) = [x], the equivalence class of x mod 2πr. Example 21.6 (Topping’s incomplete solution). Let     3 3 3 3 2 × − , M  − , 5 5 5 5 and define the (into) local covering map φ : M → S 1 (r) × [−1, 1] ⊂ Σ2 by φ (x, y) = (π (x) , y) .

r (t) to the Ricci flow, where Consider the incomplete solution M2 , gM r (t)  φ∗ g r (t) . gM

Since Σ2 , g r (t) converges to a round point as t T r , we have   r (x, t) = ∞. (21.6) limr inf RgM



tT

x∈M

is the standard flat metric on M2 . In particular, for Note also that ¯ ((0, 0) , ρ) is compact. ρ ∈ 0, 35 , the closed ball B r (0) gM

Theorem 21.7 (Topping). There exist counterexamples to Theorem 21.2 if we do not assume that the solution (Mn , g (t)) in its hypothesis is complete.

r (t) be the incomplete solution in Example 21.6, Proof. Let M2 , gM ¯ (x0 , r0 ) is compact. Suppose let x0 = (0, 0), and let r0 = 12 . Note that B that Theorem 21.2 holds for this example and let ε0 > 0 be the constant given in this theorem. By (21.5) we may choose r > 0 small enough so that 1 T r < ε20 = (ε0 r0 )2 4 5 2 (i.e., choose r < 11 ε0 ). Then for any T ∈ [ 12 T r , T r ), by choosing √ √ ε = 2 T < 2 T r < ε0 , we would have from (21.4) (21.7)

|R|(x0 , t) ≤

4 α + 2 t ε0

for any t ≤ (εr0 )2 = T . Since T may be chosen arbitrarily close to T r , by (21.7) we have for any t ∈ [ 12 T r , T r ) |R|(x0 , t) ≤

2α 1 2α + 1 + r = . r T T Tr

However this contradicts (21.6).

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1. STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY

165



r (t) , Remark 21.8. By taking products of Topping’s example M2 , gM t ∈ [0, T r ), with flat tori T k , we obtain higher-dimensional counterexamples to pseudolocality without assuming completeness. 1.5. Entropy monotonicity and the idea of the proof of pseudolocality. 1.5.1. Entropy monotonicity. In §5 and §6 of Chapter 6 in Part I, we saw that an entropy (i.e., W functional) lower bound at scale τ implies no local collapsing at spatial scale √ τ . We formulated this result in terms of the infimum of W (g, ·, τ ), i.e., the µ-invariant µ (g, τ ); see Propositions 6.64, 6.70, and 6.72 in Part I. On the other hand, we may also investigate the geometric applications of an upper bound for µ (g, τ ). Recall that Perelman’s differential Harnack estimate says that, for a fundamental solution H (x, t) = (4πτ (t))−n/2 e−f (x,t) to the adjoint heat equation coupled to a solution (Mn , g (t)) to the Ricci flow, we have     v (x, t) = τ R + 2∆f − |∇f |2 + f − n H ≤ 0 (see (16.81) and (16.82) in Part II). Since W (g (t) , f (t) , τ (t)) =

 M

v (t) dµg(t) ,

this implies W (g (t) , f (t) , τ (t)) ≤ 0. We also have W (g (t) , f (t) , τ (t)) = 0 if and only if the solution is stationary Euclidean space. Thus, there is an important distinction between the cases W = 0 and W < 0 for the fundamental solution; this distinction is a basis for the proof of pseudolocality. Furthermore, note that, in some sense, being Euclidean space is the opposite extreme from collapsing. 1.5.2. The idea of the proof of pseudolocality. In §10.1 of [152], before proving the result, Perelman first wrote the following synopsis: ‘It is an argument by contradiction. The idea is to pick a point (¯ x, t¯) not far from (x0 , 0) and consider the solution u to the conjugate heat equation, starting as δ-function at (¯ x, t¯), and the corresponding nonpositive function v as in 9.3. If the curvatures at (¯ x, t¯) are not small compared to t¯−1 and
are larger than at nearby points, then one can show that v at time t is bounded away from zero for (small) time intervals t¯− −1 x, t¯). By monotonicity we conclude t of the
order of | Rm | (¯ that v is bounded away from zero at t = 0. In fact, using (9.1) and an appropriate cut-off function, we can show that at t = 0 already the integral of v over B(x0 , r) is bounded away from zero, whereas the integral of u over this ball is close to 1,

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21. PERELMAN’S PSEUDOLOCALITY THEOREM

where r can be made as small as we like compared to r0 . Now using the control over the scalar curvature and isoperimetric constant in B(x0 r0 ), we can obtain a contradiction to the logarithmic Sobolev inequality.’ Below we give a technical summary of the proof. The reader may wish to return to this after reading the detailed proof which begins in §2. 1.5.3. A summary of the proof of pseudolocality. The method of proof of the pseudolocality theorem is by contradiction. Assuming that the theorem is false, there exists a sequence of counterexamples (Mni , gi (t) , x0i ) corresponding to δi → 0+ and ε0i → 0+ (see Counterstatement A in subsection 2.1). One picks large curvature points (¯ xi , t¯i ) ‘not far from (x0i , 0)’ and one considers the corresponding adjoint heat kernels xi , t¯i ) and their associated nonpositive entropy integrands Hi centered at (¯   vi = τi (2∆fi − |∇fi |2 + Rgi ) + fi − n Hi ≤ 0, where fi is related to Hi by Hi = (4πτi )−n/2 e−fi with τi = t¯i − t. The points (¯ xi , t¯i ) are ‘well-chosen’ in the sense that one has good enough semi-global curvature control to extract a complete limit of dilations of the solutions xi , t¯i ). gi (t) based at (¯ If the pseudolocality theorem is not true (i.e., if we do not have a suitable local curvature bound), then the sequence of Riemannian manifolds (Mni , gi (0)), with isoperimetric constants In (gi (0)) = (1 − δi )cn , has the ¯ )−1 gi (0). Then there exist functions ψi following property. Let  gˇi (0)  (2ti
with supp ψi ⊂ Bgˇi (0) x0i , (2t¯i )−1/2 such that Mi ψi2 dµgˇi (0) ≤ 1 and (21.8)     2

2 2 2 ψi2 dµgˇi (0) 2|∇ψi |gˇi (0) − ψi log ψi − sn ψi dµgˇi (0) ≤ −β ≤ −β Mi

Mi

for some β independent of i. This contradicts the logarithmic Sobolev in  −1/2 since the In (gi (0)) equality (22.102) applied to gˇi (0) on Bgˇi (0) x0i , (2t¯i ) tend to their Euclidean value cn . In regards to the hypotheses of the pseudolocality theorem we make the following remarks. The lower bound on the scalar curvature on the initial ball is used to compare Perelman’s entropy with the almost Euclidean logarithmic Sobolev inequality; see (22.83). The α/t term in the curvature estimate is used to obtain curvature control backwards in time from (¯ xi , t¯i ) on a (scaled) time interval of length bounded from below; see Claim 2 in §1 of Chapter 22 as well as (21.26) and (21.40). 2. Setting up the proof by contradiction and point picking We shall prove the following theorem, i.e., the r0 = 1 case of Theorem 21.2. Observe that Theorem 21.2 follows directly from this theorem by scaling the solution of the Ricci flow (see Remark 21.5).

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2. SETTING UP THE PROOF BY CONTRADICTION AND POINT PICKING

167

Theorem 21.9 (r0 = 1 case of pseudolocality). For every α > 0 and n ≥ 2 there exist δ >0 and ε0 > 0 depending only on α and n such that if (Mn , g (t)), t ∈ 0, ε2 , where ε ∈ (0, ε0 ], is a complete solution of the Ricci flow with bounded curvature and if x0 ∈ M is a point such that R (x, 0) ≥ −1

(21.9) and



(21.10)

Areag(0) (∂Ω)

n

in Bg(0) (x0 , 1)

n−1 ≥ (1 − δ)cn Volg(0) (Ω)

for any regular domain Ω ⊂ Bg(0) (x0 , 1), where cn = nn ωn is the Euclidean isoperimetric constant, then5 | Rm |(x, t) ≤

(21.11)

1 α + 2 t ε0

for x ∈ M such that dg(t) (x, x0 ) < ε0 and t ∈ (0, ε2 ].6 Remark 21.10 (Monotonicity of δ, ε0 , and α). Note that, in the above theorem, the dependence of δ and ε0 on α are monotone in the following sense. If δ > 0 and ε0 > 0 work for some α > 0, then for any 0 < δ  ≤ δ, 0 < ε0 ≤ ε0 , and α ≤ α < ∞, we have that δ  and ε0 work for α . Since the proof of Theorem 21.9 is somewhat long, we divide it into steps which we present in the remaining sections. The proofs of some of the technical results are deferred to the next chapter. 2.1. If the pseudolocality Theorem 21.9 is false — a sequence of counterexamples. We begin the proof of the pseudolocality Theorem 21.9. From Remark 21.10, clearly we only need to prove the theorem for α sufficiently small, say 1 √ (this is used in the proof of Claim 2 in §1 of Chapter 22). 0 < α < 13(n−1) n Step 1. If the theorem is false, then there exists a bad sequence of solutions, points, times, and scales (by ‘bad ’ we mean having ‘large curvature’). Suppose that the theorem is false. We then have the following.   1 √ and sequences of Counterstatement A. There exist α ∈ 0, 13(n−1) n positive numbers δi → 0+ , ε0i → 0+ , and smooth complete pointed solutions 5

max

 it as an exercise to show that one can replace the rhs of (21.11) by We leave α 1 . , t ε2

6

0

Observe the following elementary fact: For any f (t) which is o (1) near t = 0 and for any α > 0 there exists ε0 > 0 such that α f (t) ≤ t t for t ∈ (0, ε20 ].

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  (Mni , gi (t) , x0i ), t ∈ 0, ε2i , where εi ∈ (0, ε0i ], for i ∈ N, of the Ricci flow with bounded curvatures such that (21.12) (21.13)

Rgi (x, 0) ≥ −1 in Bgi (0) (x0i , 1) ,

n

n−1 Areagi (0) (∂Ω) ≥ (1 − δi )cn Volgi (0) (Ω)

for all regular domains Ω ⊂ Bgi (0) (x0i , 1), and there exist ‘bad ’ points and times xi ∈ Mi and ti ∈ (0, ε2i ] satisfying (21.14)

dgi (ti ) (xi , x0i ) < ε0i

and (21.15)

| Rm gi |(xi , ti ) >

α 1 + ti ε20i

for all i ∈ N. In view of the above counterstatement, we make the following: Definition 21.11. Given a continuous   family of complete smooth Riemannian manifolds (Mn , g (t)), t ∈ 0, ε2 , where ε > 0, and given α ∈ (0, ∞), let  α (21.16) Mα  (x, t) ∈ M × (0, ε2 ] : | Rm |(x, t) > t be the set of α-large curvature points. Hence for (Mi , gi (t)) and α as in Counterstatement A, the set  α (21.17) Miα  (Mi )α = (x, t) ∈ Mi × (0, ε2i ] : | Rm gi |(x, t) > t is nonempty for all i. We shall show that Counterstatement A leads to a contradiction, from which Theorem 21.9 follows. 2.2. Adjusting the bad sequence of points and times — getting local curvature control. For later use in carrying out the proof of Theorem 21.9 (see Lemma 22.13, in particular (22.57), below), we observe the following. Lemma 21.12 (An adjustment for the sequences εi and (xi , ti )). For each i ∈ N, we may assume, by adjusting εi > 0 smaller if necessary, that (21.18) 2 α | Rm gi |(x, t) ≤ + 2 whenever t ∈ (0, ε2i ] and dgi (t) (x, x0i ) ≤ ε0i , t ε0i and we may also adjust (xi , ti ) accordingly, so that Counterstatement A still holds for the adjusted εi and (xi , ti ), for all i ∈ N.7 7

Note that we adjust neither x0i nor δi .

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2. SETTING UP THE PROOF BY CONTRADICTION AND POINT PICKING

169

Remark 21.13. Call εi the ‘size’ of the parabolic cylinder, which tends to 0. The lemma says that in Counterstatement A we may adjust the sizes and choices of {(xi , ti )} so that we have curvature control, relative to the size, inside the parabolic cylinder based at (x0i , 0) while the curvature at the bad sequence of points (xi , ti ) is still large relative to the size. A consequence of this curvature control is the local bound: | Rm gi |(x, t) < 2| Rm gi | (xi , ti ) whenever t ∈ [ t2i , ε2i ] and dgi (t) (x, x0i ) ≤ ε0i . We think of εi as being the ‘right’ sizes for the contradiction argument. Proof. Suppose that i ∈ N is such that (21.18) does not hold. Define   2 α | Rm gi | (x, t) − − 2 < ∞ max (21.19) µi (ε)  sup ¯g (t) (x0i ,ε0i ) t ε0i t∈(0,ε2 ] x∈B i for ε ∈ (0, εi ]. (Clearly the supremum in (21.19) is finite and attained.) For such an i, the function µi (ε) is both continuous and monotonically nondecreasing in ε and, since (21.18) does not hold, we also have µi (εi ) > 0. Furthermore, for this i, since  limε→0 µi (ε) = −∞ (because gi (t) has bounded curvature on Mi × 0, ε2i ), there exists εi ∈ (0, εi ] such that

µi εi = 0. That is, the desired inequality (21.18) holds with εi replaced by εi and there ¯  (x0i , ε0i ) such that exist ti ∈ (0, (εi )2 ] and xi ∈ B gi (t ) i



α 2 | Rm gi | xi , ti =  + 2 . ti ε0i Thus clearly there exists xi ∈ Bgi (t ) (x0i , ε0i ) i



α 1 | Rm gi | xi , ti >  + 2 . ti ε0i
Assume from now on that for Counterstatement A the εi and (xi , ti ) have been adjusted as in Lemma 21.12. Step 2. Further improving the bad sequence of points and times by point picking. We shall further use point picking to obtain a new bad sequence of points and times with curvature control in larger and larger parabolic cylinders, so that we may obtain a complete limit when dilating about these new points and times.

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170

21. PERELMAN’S PSEUDOLOCALITY THEOREM

Let8 (x1i , t1i ) ∈ Miα denote the bad points given by Counterstatement A and Lemma 21.12. In particular, we have dgi (t1i ) (x1i , x0i ) < ε0i and | Rm gi |(x1i , t1i ) >

α 1 + 2 . t1i ε0i

We apply the ‘point picking’  Claim 1 in §1 of Chapter 22 to the solutions (Mni , gi (t) , x0i ), t ∈ 0, ε2i , Ai 

(21.20)

1 , 100nε0i

and (x1i , t1i ) ∈ Miα as in Counterstatement A. Note that Ai → ∞ as i → ∞. Then we obtain ‘well-chosen’ α-large curvature space-time points: (¯ xi , t¯i ) ∈ Miα , ¯ i  | Rm g |(¯ ¯ Q i xi , t i )

(21.21) (21.22) satisfying (21.23)

x ¯i ∈ Bgi (t¯i ) (x0i , (2Ai + 1)ε0i )

and

t¯i ∈ (0, ε2i ]

and ¯i | Rm gi |(x, t) ≤ 4Q

(21.24) for all (x, t) ∈ Miα with (21.25)

0 < t ≤ t¯i

¯− 2 . dgi (t) (x, x0i ) ≤ dgi (t¯i ) (¯ xi , x0i ) + Ai Q i 1

and

Furthermore, by Claim 2 in §1 of Chapter 22 (which uses α < the (¯ xi , t¯i ) in (21.21) also satisfy (21.26)

1 √ ), 13(n−1) n

¯i | Rm gi |(x, t) ≤ 4Q

for all (x, t) such that α ¯ −1 Ai ¯ −1/2 ¯ Q ¯i ) ≤ . t¯i − Q i ≤ t ≤ ti and dgi (t¯i ) (x, x 2 10 i That is, we have curvature control in backward parabolic cylinders centered at (¯ xi , t¯i ) with relatively large spatial radii (since Ai → ∞). Note that since (¯ xi , t¯i ) ∈ Miα , by (21.17) we have ¯ −1 , ¯ i > α , i.e., t¯i > αQ (21.27) Q i t¯i ¯ −1 > t¯i /2. in particular, t¯i − α2 Q i 8 The subscript ‘1’ is introduced here to mesh better with the notation of Claim 1 in §1 of Chapter 22.

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3. LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS

171

3. Local entropies are nontrivial near bad points With the aim of obtaining a contradiction, we study the local entropies centered at the bad points. In this section we shall prove the following. Step 3. A uniform negative integral upper bound for vi over a ball. 3.1. Local geometry at the bad points via adjoint heat kernels and their local entropies. To understand the (local) geometry of the solutions (Mni , gi (t)) based at the well-chosen points (¯ xi , t¯i ) given by (21.21), we consider the (globally defined) adjoint heat kernels centered at (¯ xi , t¯i ) and their corresponding entropy functionals. 3.1.1. The adjoint heat kernels centered at (¯ xi , t¯i ). Recall that the adjoint heat kernel centered at (¯ xi , t¯i ) Hi : Mi × [0, t¯i ) → R+ is defined to be the minimal positive fundamental solution of the adjoint heat equation   ∂ ∗ (21.28a)
i Hi  − − ∆gi + Rgi Hi = 0, ∂t (21.28b) lim Hi (·, t) = δx¯i tt¯i

(see Definition 16.43 in Part II). Since the solution (Mi , gi (t)) is complete with bounded curvature, such a solution to (21.28) exists; see Chapter 23. By (26.20) in Chapter 26, we have  Hi (x, t) dµgi (t) (x) ≡ 1 (21.29) Mi

for all t ∈ [0, t¯i ). Define fi : Mi × [0, t¯i ) → R by (21.30)

Hi (x, t)  (4π (t¯i − t))−n/2 e−fi (x,t) .

Define Perelman’s differential Harnack quantity vi on Mi × [0, t¯i ) by     (21.31) vi  vHi = (t¯i − t) Rgi + 2∆gi fi − |∇gi fi |2 + fi − n Hi . By Perelman’s differential Harnack estimate for complete solutions of the Ricci flow with bounded curvatures (for closed manifolds, see §9 of [152] or Theorem 16.44 in Part II; for noncompact manifolds, see Theorem 7.1 and Corollary 7.1 in Chau, Tam, and Yu [26]), we have (21.32) vi (x, t) ≤ 0 in Mi × [0, t¯i ). Thus the entropies of the adjoint heat kernels are nonpositive (compare with (17.44)):  vi (x, t) dµgi (t) (x) ≤ 0 W (gi (t) , fi (t) , τi (t)) = Mi

for t ∈ [0, t¯i ), where τi (t)  t¯i − t.

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172

21. PERELMAN’S PSEUDOLOCALITY THEOREM

3.1.2. A uniform negative upper bound for the local entropies. We shall prove the following, which says that the local entropies of the adjoint heat kernels, based at the points (¯ xi , t¯i ) given by (21.21), have a 1 √ be as certain amount of concentration (negativity). Let α ∈ 0, 13(n−1) n in Counterstatement A. Claim 3. For the sequence of Harnack quantities vi defined in (21.31), by passing to a subsequence (still indexed by i), we have that for all i ∈ N there exists times  α ¯ −1 ¯  , ti t˜i ∈ t¯i − Q 2 i at which the ‘¯ xi -centered local entropies’ of the adjoint heat kernels based at (¯ xi , t¯i ) have a certain amount of negativity: 

v t˜i dµgi (t˜i ) ≤ −β1 , (21.33)   i 1/2 ¯i ,(t¯i −t˜i ) Bg (t˜ ) x i i ¯ i is defined by (21.22), where β1 > 0 is a constant independent of i, where Q and where (¯ xi , t¯i ) is given by (21.21). First we note that the above statement is invariant under the following space-time rescalings:   ˆ i gi x, t¯i + Q ˆ −1 t , (21.34a) gi (x, t) −→ gˆi (x, t)  Q i   ˆ i (x, t)  Q ˆ −n/2 Hi x, t¯i + Q ˆ −1 t , (21.34b) Hi (x, t) −→ H i i   ˆ −1 t , (21.34c) fi (x, t) −→ fˆi (x, t)  fi x, t¯i + Q i   ˆ −n/2 vi x, t¯i + Q ˆ −1 t , (21.34d) vi (x, t) −→ vˆi (x, t)  Q i i ˆ i > 0 are any constants.9 where the ‘dilation factors’ Q For the two cases in the proof of Claim 3 below, we shall make different ˆ i. choices for Q 

 ˆ i t¯i , Q ˆ i ε2 − t¯i Remark 21.14. Note that gˆi (t) is defined for t ∈ −Q i   −1 2 ¯ , ε ], so that in particular gˆi (t) is defined for t ∈ −αQ ˆ iQ ¯ −1 , 0 . and t¯i ∈ (αQ i i i ˆ ˆ ˆ ¯ On the other hand, Hi (t), fi (t), and vˆi (t) are defined for t ∈ [−Qi ti , 0). By the above rescalings, Claim 3 is equivalent to the following statement. Claim 3 . There exist β1 > 0 and a subsequence such that for all i ∈ N we have 

ˆi t˘i dµgˆi (t˘i ) ≤ −β1 (21.35)  v 1/2 ¯i ,(−t˘i ) Bgˆ (t˘ ) x i i   ˆ i. ˆ iQ ¯ −1 , 0 and some sequence of dilation factors Q for some t˘i ∈ − α2 Q i 9

ˆ n/2 dµ ¯ ˆ −1 . Note that dµgˆi (t) = Q i gi (ti +Q t) i

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3. LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS

173

3.2. Proof that the local entropies are nontrivial near (¯ xi , t¯i ).  Proof of Claim 3 . The idea of the proof of the claim is that if it is not true, then we should have a subsequence which converges to a limit with Perelman’s differential Harnack quantity identically equal to zero, leading to a contradiction. In particular, one obtains a shrinking gradient Ricci soliton g¯∞ (t),which is either nonflat or has finite injectivity radius, on a time interval − α2 , 0 . Moreover, the ‘scale’ goes to zero as t → 0, and g¯∞ (0) is C ∞ . This implies that g¯∞ (t) is the flat Euclidean space, which is a contradiction. We divide the discussion into two cases, depending on whether or not we have an injectivity radius estimate at (¯ xi , t¯i ). The choice of the rescaling ˆ factor Qi in (21.34a) shall depend on the case. Case 1 (Noncollapse — uniform lower bound for the rescaled injectivity radii). Suppose that the sequence of points and times {(¯ xi , t¯i )}i∈N satisfies   ¯ 1/2 injg (t¯ ) (¯ xi ) > ι0 (21.36) lim sup Q i i i i→∞

for some ι0 > 0 independent of i. In this case we may choose a subsequence with ¯ −1/2 xi ) ≥ ι0 Q injgi (t¯i ) (¯ i

(21.37)

for all i ∈ N. Applying the above rescalings (21.34), with rescaling factor ¯ i, ˆi  Q Q we obtain the sequence (21.38) 

 n ¯ i (t) , f¯i (t) , v¯i (t)  Mn , gˆi (t) , H ˆ i (t) , fˆi (t) , vˆi (t) , Mi , g¯i (t) , H i ¯ i (t), f¯i (t) and v¯i (t) are where g¯i (t) is defined for t ∈ [−α, 0], whereas H 10 defined for t ∈ [−α, 0), with xi ) ≥ ι0 > 0. injg¯i (0) (¯

(21.39)

By (21.26), we have the curvature bound (21.40)

|Rm g¯i (x, t)| ≤ 4

for (x, t) ∈ Bg¯i (0)



Ai x ¯i , 10



 α  × − ,0 . 2

By (21.32) we have (21.41)

v¯i (x, t) ≤ 0

in Mi × [−α, 0).

From (21.40) and (21.39), we can apply Hamilton’s (local) compactness theorem (see Theorem 3.16 and Corollary 3.18 in Part I) to the pointed sequence of (incomplete) solutions      α  Ai , g¯i (t) , (¯ xi , 0) , t ∈ − ,0 , ¯i , Bg¯i (0) x 10 2 10

¯ i t¯i > α by (21.27). Here we have used Q

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174

21. PERELMAN’S PSEUDOLOCALITY THEOREM

to obtain a subsequence converging, in the C ∞ pointed Cheeger–Gromov sense, to a smooth limit solution  α 

n ¯ , g¯∞ (t) , (¯ x∞ , 0) , t ∈ − , 0 . (21.42) M ∞ 2 1 Since Ai = 100nε0i → ∞, this limit is complete and satisfies x∞ , 0) = 1 | Rm g¯∞ |(¯

(21.43)

  ¯ ∞ × −α, 0 . on M 2 Note that by the definition of C ∞ pointed Cheeger–Gromov convergence (slightly modifying Definition 3.6 in Part I) we have established the following.

and

| Rm g¯∞ |(x, t) ≤ 4

Lemma 21.15. There exist ¯ ∞ by open sets with x ¯∞ ∈ Ui and (1) an exhaustion {Ui }i∈N of M (2) a sequence of diffeomorphisms (21.44)

Φi : Ui → Vi  Φi (Ui ) ⊂ Mi ,   i x∞ ) = x ¯i and Φi (Ui ) ⊂ Bg¯i (0) x ¯i , A with Φi (¯ 10 ,

such that the sequence in C ∞  converges − α2 , 0 .

 

Ui , Φ∗i g¯i (t)|Vi

¯ ∞ , g¯∞ (t) uniformly on compact sets in M ¯∞ × to M

Now from Lemma 22.9 in the next chapter we have the following. Lemma 21.16. For the above subsequence of (21.38) converging to (21.42) in the sense of Lemma 21.15, there exists a further subsequence such that we have the convergences to certain C ∞ limit functions ¯ ∞ , f¯i ◦ Φi −→ f¯∞ , and v¯i ◦ Φi −→ v¯∞ ¯ i ◦ Φi −→ H (21.45) H

¯ n∞ × − α , 0 uniformly on compact sets in the C ∞ -topology and we on M 2 ¯ ∞ = (−4πt)−n/2 e−f¯∞ is a positive solution to the adjoint heat have that H equation   ¯ ∞  − ∂ − ∆g¯∞ + Rg¯∞ H ¯ ∞ = 0. (21.46)
∗∞ H ∂t By Lemma 22.9, we also have     2  ¯∞ ≤ 0 (21.47) v¯∞ = −t Rg¯∞ + 2∆g¯∞ f¯∞ − ∇g¯∞ f¯∞  + f¯∞ − n H



¯ ∞ × −α, 0 ¯ ∞ × − α , 0 , and on M on M 2 2  2   1 ∗ g¯∞ g¯∞ ¯  ¯∞ (21.48)
∞ v¯∞ = 2t Rcg¯∞ +∇ ∇ f∞ + g¯∞  H 2t g¯∞ ≤ 0.

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3. LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS

175

By applying the strong maximum principle (since v¯∞ ≤ 0, we may apply Theorem 12.40 in Part II without assuming c (x, t) ≤ 0) to the above backward heat inequality   ∂ − − ∆g¯∞ + Rg¯∞ v¯∞ ≤ 0, ∂t we have that there exists a time t¯ ∈ (− α , 0] such that 2

¯ ∞ × (− α , t¯) v¯∞ < 0 on M 2 11  (by definition, [0, 0)  ∅). We shall show that if Claim 3 is false, then t¯ < 0, leading to a contradiction. (21.49)

¯ ∞ × [t¯, 0) and v¯∞ ≡ 0 on M

¯ ∞ × (− α , 0)) Exercise 21.17. Show that either t¯ = 0 (i.e., v¯∞ > 0 on M 2 ¯ ∞ × (− α , 0)). or t¯ = − α2 (i.e., v¯∞ ≡ 0 on M 2 Note also that  (21.50) Bg¯ (t˜ ) (¯ xi ,r) i



v¯i t˜ dµg¯i (t˜ ) →

 Bg¯∞ (t˜ ) (¯ x∞ ,r)



v¯∞ t˜ dµg¯∞ (t˜ )

for any t˜ ∈ (− α2 , 0) and r ∈ (0, ∞). Now suppose that Claim 3  is false in Case 1. Then the above discussion 1/2 in (21.50), for any t˜ ∈ (− α2 , 0) implies that, by taking r  −t˜  

v ¯ t˜ dµg¯∞ (t˜ ) = 0. (21.51)   ∞ 1/2 ¯∞ ,(−t˜ ) Bg¯∞ (t˜ ) x

Since as a consequence of (21.47), v¯∞ t˜ ≤ 0 for all t˜ ∈ (− α2 , 0), it follows from (21.51) that we have

(21.52) v¯∞ x, t˜ = 0  1/2  ¯∞ , −t˜ and t˜ ∈ (− α2 , 0). We have shown that for all x ∈ Bg¯∞ (t˜ ) x

v¯∞ t˜ vanishes in a ball. By the strong maximum principle (21.49), we have that   ¯ ∞ × −α, 0 . (21.53) v¯∞ ≡ 0 on all of M 2

n ¯ , g¯∞ (t) satisfies Applying equation (21.48), we conclude that M ∞

1 (21.54) Rcg¯∞ +∇g¯∞ ∇g¯∞ f¯∞ + g¯∞ ≡ 0 2t α ¯ on M∞ ×(− 2 , 0), and hence g¯∞ (t) is a shrinking gradient soliton. However, the solution g¯∞ (t) is smooth for t ∈ (− α2 , 0], and in particular its curvature is uniformly bounded on this interval. This leads to a contradiction unless 11 Compare with the proof of Lemma 6.57 on p. 246 of [45]. Note that in our present situation, we have a nonpositive subsolution to a backward heat-type equation.

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176

21. PERELMAN’S PSEUDOLOCALITY THEOREM

g¯∞ (t) is flat.12 However, g¯∞ (t) being flat contradicts (21.43), which says x∞ , 0) = 1. Hence we have proved that Claim 3 is true under that | Rmg¯∞ |(¯ the assumption of Case 1. Now we consider Case 2 (Collapse — the rescaled injectivity radii tend to zero). Suppose that the sequence of points and times {(¯ xi , t¯i )} given in (21.21) satisfies   ¯ 1/2 injg (t¯ ) (¯ x ) = 0. (21.55) lim sup Q i i i i i→∞

ˆ i for (21.34) defined so that the In this case we use the dilation factor Q n rescaled solution (Mi , gˆi (t)) in (21.34a) satisfies (21.56)

xi ) = 1. injgˆi (0) (¯

This implies13 (21.57)

¯ −1 Q ˆ i = ∞. lim Q i

i→∞

Using the above two facts along with    Ai ¯ −1/2 ˆ 1/2 α ¯ −1 ˆ  −1 ¯ ˆ on Bgˆi (0) x Qi Qi , 0 ¯ i , Qi × − Q | Rm gˆi |(x, t) ≤ 4Qi Qi 10 2 i 1 → ∞, by applying Hamilton’s (local) comfrom (21.26) and Ai = 100nε 0i pactness theorem to the pointed sequence    Ai ¯ −1/2 ˆ 1/2 α ¯ −1 ˆ  xi , 0)) on Bgˆi (0) x Qi Qi , 0 , ¯ i , Qi × − Q (ˆ gi (t) , (¯ 10 2 i

we obtain a subsequence converging to a smooth pointed solution (21.58)

x∞ , 0)) (ˆ g∞ (t) , (¯

ˆ n × (−∞, 0]. on M ∞

ˆ ∞ is well defined. Moreover, this limit In particular, the metric gˆ∞ (0) on M solution is complete and satisfies   Rm gˆ (t)  ≡ 0 (21.59) ∞ and x∞ ) = 1. injgˆ∞ (0) (¯ 12

Since g¯∞ (t) is a shrinking gradient soliton with extinction time t = 0, we have



t0 sup
Rmg¯∞ (t0 )
, sup
Rmg¯∞ (t)
= t ¯ ¯ M∞ M∞






so that limt→0− supM ¯ ∞ Rmg ¯ ∞
Rmg ¯∞ (t) = ∞ if supM ¯∞ (t0 ) = 0. −2 13 ˆ Indeed, Qi = injgˆi (0) (¯ xi ) , so that

−2 ¯ −1 ¯ 1/2 inj ˆ Q xi ) → ∞. i Qi = Qi g ˆi (0) (¯

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3. LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS

177

ˆ ∞ may be either compact or noncompact. We Note that, a priori, M shall rule out both of these possibilities simultaneously.14 Similarly to (21.45), there exists a subsequence such that the functions in (21.34b)–(21.34d) converge in C ∞ on compact sets to some C ∞ functions ˆ ∞, ˆi → H H where (21.60)

 vˆ∞ =

fˆi → fˆ∞ ,

and vˆi → vˆ∞ ,

   2    ˆ ∞. −t Rgˆ∞ + 2∆gˆ∞ fˆ∞ − ∇gˆ∞ fˆ∞  + fˆ∞ − n H

Just as in (21.48), we have (21.61)


∗∞ vˆ∞

 2   1 gˆ∞ gˆ∞ ˆ  ˆ∞ = 2t Rcgˆ∞ +∇ ∇ f∞ + gˆ∞  H 2t ∞ ≤ 0.

Furthermore, again by the strong maximum principle, since vˆ∞ ≤ 0, there exists t¯ ∈ (−∞, 0] such that (21.62)

ˆ ∞ × [t¯, 0) and vˆ∞ ≡ 0 on M

vˆ∞ < 0

ˆ ∞ × (−∞, t¯) on M

(this is analogous to (21.49)). ˆ n we have that for all Claim 4. For the flat solution gˆ∞ (t) on M ∞ ∈ (−∞, 0)  

v ˆ tˆ dµgˆ∞ (tˆ ) < 0. (21.63)   ∞ 1/2 ¯∞ ,(−tˆ ) Bgˆ∞ (tˆ ) x

tˆ

Proof of Claim 4. If the claim is false, then there exists tˆ ∈ (−∞, 0) such that  

v ˆ tˆ dµgˆ∞ (tˆ ) = 0.   ∞ 1/2 ¯∞ ,(−tˆ ) Bgˆ∞ (tˆ ) x Since vˆ∞ ≤ 0, by the strong maximum principle (21.62), ˆ ∞ × [tˆ , 0). on M   ˆ n , gˆ∞ (t) , t ∈ [tˆ , 0], is a By (21.61) and (21.59), this implies that M ∞ complete shrinking gradient Ricci soliton with zero curvature: vˆ∞ ≡ 0

(21.64)

1 ∇gˆ∞ ∇gˆ∞ fˆ∞ + gˆ∞ = 0 2t

for t ∈ [tˆ , 0)

and (21.65)

Rmgˆ∞ = 0

for t ∈ (−∞, 0].

14 Although the compact case is even easier to rule out, in the sense that it has a worse isoperimetric constant than the Euclidean isoperimetric constant.

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178

21. PERELMAN’S PSEUDOLOCALITY THEOREM

Fix a time in the interval (tˆ , 0). Since, by (21.64), fˆ∞ is a proper strictly conˆ ∞ is diffeomorphic to Rn , and hence by ˆ ∞ ,15 we have M cave function on M   ˆ ∞ , gˆ∞ (t) , t ∈ (−∞, 0], is isometric to the stationary Euclidean (21.65), M x∞ ) = 1. This proves Claim space En ,16 contradicting the fact that injgˆ∞ (0) (¯  4, which in turn implies that Claim 3 is true for Case 2. In conclusion, both
Claim 3 and the equivalent Claim 3 are proven. 4. Contradicting the almost Euclidean logarithmic Sobolev inequality Now we complete the proof of Theorem 21.9 by obtaining a contradiction with the almost Euclidean logarithmic Sobolev inequality.   1 √ and Recall what we have so far accomplished: Let α ∈ 0, 13(n−1) n δi , εi → 0 and   (Mni , gi (t) , x0i ) , t ∈ 0, ε2i , for i ∈ N, be as in Counterstatement A and satisfying (21.18). In particular, by (21.13), the balls Bgi (0) (x0i , 1) are δi -almost Euclidean isoperimetrically. Let (¯ xi , t¯i ) be the well-chosen α-large curvature space-time points given by 1 . By Claim 3, (21.21) and satisfying (21.23)–(21.25), where Ai  100nε 0i passing to a subsequence, there exists β1 > 0 such that for all i ∈ N 

v t˜i dµgi (t˜i ) ≤ −β1 (21.66)   i 1/2 ¯i ,(t¯i −t˜i ) Bg (t˜ ) x i i 

  α ¯ −1 , t¯i ⊂ 0, ε2 , where Q ¯ i is defined by (21.22). for some t˜i ∈ t¯i − 2 Q i i Step 4. Contradicting the almost Euclidean logarithmic Sobolev inequality at time zero. By Lemma 22.13 below we shall relate the local entropy bound (21.66), which is at a time t˜i , to the corresponding bound at time 0. This will then contradict the isoperimetric assumption at time 0 in Theorem 21.9. Using (21.18) and (21.23)–(21.25) and taking i large enough so that Ai ≥ 67n, for each i we may apply Lemma 22.13(2) below to (Mni , gi (t) , x0i ), xi , t¯i ), and t˜i , to obtain the existence of a nonnegative function t ∈ 0, ε2i , (¯ ˜ i on Mi satisfying H (21.67)      ˜ i dµg (0) ≥ β1 1 − 1 − 1 − ε2 ≥ β1 −t¯i |∇f˜i |2 − f˜i + n H i i 2 A2i A2i Mi 15

s∈R

ˆ ∞ is a unit speed geodesic, then If γ : R → M

d2 ˆ f ds2 ∞

1 (γ (s)) = − 2t , so that for all

2

s fˆ∞ (γ (s)) = − + C1 s + C2 4t for some C1 , C2 ∈ R. This shows that fˆ∞ is strictly concave and proper. 16 Alternatively, since the reduced volume of gˆ∞ (t) is independent of t, by Corollary ˆn 8.17 on p. 392 in Part I, M ˆ∞ (t) is the stationary Euclidean space. ∞, g

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5. NOTES AND COMMENTARY

179


˜ i ⊂ Bg (0) (x0i , 20Ai ε0i ) ⊂ ˜ i dµg (0) ∈ (0, 1] and with supp H with M H i i Bgi (0) (x0i , 1). Here the quantities in the integral on the lhs of (21.67) are at time t = 0. Let gˇi (0)  (2t¯i )−1 gi (0) and let

˜ i (x, 0) = (2π)−n/2 e−fi (x,0) . ψi2 (x)  (2t¯i )n/2 H  
We have supp ψi ⊂ Bgˇi (0) x0i , 20 (2t¯i )−1/2 Ai ε0i , Mi ψi2 dµgˇi (0) ∈ (0, 1], and f˜i = − n2 log (2π) − log ψi2 . Hence, using sn  n2 log(2π) + n, inequality (21.67) implies    2 2 n β1 1 2 − |∇ log ψi |gˇi (0) + log (2π) + log ψi + n ψi2 dµgˇi (0) ≥ , 2 2 2 Mi ˜

so that (21.68)     2

β1 β1 2 2 2 ψ 2 dµgˇi (0) . 2|∇ψi |gˇi (0) − ψi log ψi − sn ψi dµgˇi (0) ≤ − ≤ − 2 2 Mi i Mi On the other hand, since



 Mi

ψi2 dµgˇi (0) ≤ 1

 and since supp ψi ⊂ Bgˇi (0) x0i , (2t¯i )−1/2 from the choice of Ai , by using the isoperimetric inequality (21.13) in the logarithmic Sobolev Theorem 22.19   −1/2 ) we have (applied to gˇi (0) on Bgˇi (0) x0i , (2t¯i )     ψi2 dµgˇi (0) . 2|∇ψi |2gˇi (0) − ψi2 log ψi2 dµgˇi (0) ≥ (sn + log(1 − δi )) Mi

Mi

This contradicts (21.68) since β1 > 0 is fixed, whereas δi → 0. Remark 21.18. The assumption that R (x, 0) ≥ −1 in Bg(0) (x0 , 1) in Theorem 21.9 is used in the proof of (22.82). 5. Notes and commentary The original reference for pseudolocality and its variants is §10 of Perelman [152]; with regret, in this chapter we only discuss §10.1 and we do not discuss §§10.2–10.5. Regarding further work on Perelman’s pseudolocality, one may also see the papers by Chau, Tam, Yu [26], Kleiner and Lott [110], and Topping [179]. See also Chen and Yin [35], Hsu [101], Y. Wang [184], and one of the authors [124]. §1. For large time, the bound |Rm| (x, t) ≤ αt can be a strong assumption. By the proof of Theorem 16.2 in Hamilton [92] (see also Lemma 8.9 in [45]), we have the following.

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180

21. PERELMAN’S PSEUDOLOCALITY THEOREM

Lemma 21.19 (Curvature gap for Type III solutions). If an eternal solution (Mn , g (t)), t ∈ (0, ∞), on a compact manifold satisfies α (21.69) max |Rm| (x, t) ≤ x∈M t for t sufficiently large, where α < in the sense of Gromov as t → ∞.

1 2(n−1) ,

then g (t) approaches almost flat

Proof. Throughout the proof we shall assume that t is sufficiently large so that (21.69) holds. This curvature bound implies t max |Rc| (x, t) ≤ (n − 1) α. x∈M

Since the length of a unit speed path γ : [a, b] → M evolves by 

d L g(t) (γ) = − Rc γ  (s) , γ  (s) ds, dt γ where ds is the arc length element, we have (n − 1) α d Lg(t) (γ) ≤ Lg(t) (γ) . dt t Since M is compact, this implies that the diameter satisfies (n − 1) α d diam (g (t)) ≤ diam (g (t)) , dt t so that diam (g (t)) ≤ C t(n−1)α for some constant C < ∞. The inequality α
0 and ε0 > 0 such that Pn,α (δ, ε0 ), where Pn,α is a statement. Its negation is (i) for every δ > 0 and ε0 > 0, not Pn,α (δ, ε0 ). This implies that (ii) for every pair of sequences δi → 0+ and εi → 0+ , not Pn,α (δi , εi ). This in turn implies that (iii) there exist sequences δi → 0+ and εi → 0+ such that not Pn,α (δi , εi ). On the other hand, by the monotonicity property in Remark 21.10, we have that (i) and (iii) are equivalent. §3. Regarding the complete flat stationary solution x∞ , 0)) , (Mn∞ , gˆ∞ (t) , (¯

t ∈ (−∞, 0],

in (21.58), recall the following:

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5. NOTES AND COMMENTARY

181

(1) (Bieberbach theorem) A closed flat n-manifold is isometric to the quotient of a flat n-torus; see Theorem 3.3.1 and Corollary 3.4.6, both in Wolf [189]. (2) (Splitting and soul theorems) Let (F n , h) be a complete noncompact flat n-manifold. (a) Then a finite cover of (F n , h) is isometric to Rk × T n−k with 0 < k ≤ n and where T n−k is a flat torus. (b) Moreover, (F n , h) is also isometric to a rank k flat real vector bundle over a closed flat (n − k)-manifold, where 0 < k ≤ n. (c) There is a real analytic deformation retraction of (F n , h) onto a compact totally geodesic submanifold; see Theorem 3.3.3 of [189].

For example, we have the M¨ obius band: F 2 = S 1 × R /Z2 , where the Z2 -action is generated by the antipodal map. Then F 2 is nonorientable and is isometric to a flat line bundle over RP 1 ∼ = S 1 .  n ˆ n , gˆ∞ (t) in (21.58). In The results in (2) apply to (F , h) = M ∞   n ˆ particular, a finite cover of M∞ , gˆ∞ (t) is isometric to Rk × T n−k with 0 < k < n and where T n−k is a flat torus. Remark 21.20. Let Γ denote the group of deck transformations of the covering space π : Rk × T n−k → Mn∞ . ˜ ∞ denote the ¯∞ , i.e., π (˜ x∞ ) = x ¯ ∞ . Let H Let x ˜∞ ∈ Rk × T n−k be a lift of x

k n−k ×(−∞, 0) fundamental solution to the adjoint heat equation on R × T based at (˜ x∞ , 0) and let v˜∞ denote the corresponding Perelman’s differential Harnack quantity (defined analogously to (21.60)). We then have  ˜ ∞ (γ (x) , t) ˆ ∞ (π (x) , t) = H H γ∈Γ

×T and t ∈ (−∞, 0) (the sum is finite since |Γ| < ∞). for all x ∈ However we do not have the analogous formula relating vˆ∞ and v˜∞ since ˜ ∞ , respectively. ˆ ∞ and H they depend nonlinearly on H Rk

n−k

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http://dx.doi.org/10.1090/surv/163/06

CHAPTER 22

Tools Used in Proof of Pseudolocality We know you’ve got to blame someone for your own confusion. – From “Lunatic Fringe” by Red Rider

This chapter is a companion chapter to the previous chapter on pseudolocality. Here we present the proofs of tools used in the proof of the pseudolocality theorem. We have formulated some of these tools in more generality with a view towards their independent interest. In §1 we give the proofs of the point picking Claims 1 and 2. In §2 we discuss the convergence of heat kernels for a sequence of solutions to the Ricci flow which converges in the C ∞ Cheeger–Gromov sense. In §3 we prove a uniform negative upper bound for the local entropies centered at the well-chosen bad points at time zero. In §4 we prove a sharp form of the logarithmic Sobolev inequality, relating to the isoperimetric inequality. 1. A point picking method We now discuss some space-time point picking methods, which were used to adjust the sequence of points {(xi , ti )} given by Counterstatement A in the proof of the pseudolocality Theorem 21.9. Given a solution to the Ricci flow, the main aim of point picking methods is to find ‘nice’ (under the circumstances) sequences of points in space-time so that one can obtain limits of subsequences of solutions rescaled about these sequences of points. Roughly speaking, a thematic way in which point picking is employed is to assume that a curvature estimate (one that we want to prove) does not hold. This implies there exists a sequence of points with ‘large curvature’. From this sequence we wish to obtain a better sequence of points with large curvature; often ‘better’ means sufficient to obtain a contradiction.1 The purpose of point picking is to accomplish this. First we describe a general point picking method. This in general helps us to obtain uniformly bounded curvature at bounded distances away from the point we have picked. Second we apply point picking to a situation which will be used in the proof of Theorem 21.9. 1

One way of obtaining a contradiction is to show that under the circumstances, small scales look like large scales, whereas they do not look alike. Note, in this regard, the dichotomy of no local collapsing and AVR = 0. 183 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

184

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

1.1. Point picking method for points with α-large curvature. In this subsection (Mn , g (t)), t ∈ [0, T ], shall be a smooth family of complete Riemannian manifolds with (22.1)

sup |Rm| < ∞.

M×[0,T ]

1.1.1. Defining small and large curvature points. Definition 22.1 (α-small and α-large curvature points). Given α ∈ (0, ∞), we say that a point (x, t) is an α-small curvature point if α (22.2) | Rm |(x, t) ≤ . t On the other hand, recall from Definition 21.11 that (x, t) is an α-large curvature point if α (22.3) | Rm |(x, t) > . t Note that inequality (22.2) is scale invariant in the following sense. If ˜ ˜ |(x, t) ≤ α and C ∈ (0, ∞), then t | Rm |(x, t ) ≤ α, where g t| Rm gC C t˜   g ˜ Cg t and t˜  Ct. C

Note also that for an expanding soliton with bounded curvature and singular initial time 0, we have supM Rmg(t)  = const t . For (x , t ) ∈ M × [0, T ], we define the parabolic cylinder



P (x , t , r , η)  (x, t) : dg(t ) x, x < r , t ∈ [t , t + η]

= Bg(t ) x , r × [t , t + η]. Here r > 0, whereas η ∈ R; our notation is such that [a, b] denotes the closed interval of real numbers between a and b, where we allow b < a. 1.1.2. Finding large curvature points with local curvature control. Now fix ε ∈ (0, 1) and let (x0 , t0 ) ∈ M × [T /2, T ] be a point with ‘large enough’ norm of curvature (relative to the length of the time interval), i.e., 8 . Tε Assume that (x0 , t0 ) is an α-large curvature point, i.e., | Rm |(x0 , t0 ) > tα0 . Let H  T8 Q. We claim that we can find another ‘better chosen’ α-large curvature point (ˆ x0 , tˆ0 ), with tˆ0 ≥ T4 , such that (22.4)

Q  | Rm |(x0 , t0 ) >

8 ˆ  | Rm |(ˆ Q x0 , tˆ0 ) ≥ Tε ˆ ˆ −1 ) we have the and such that for every (x, t) ∈ P (ˆ x0 , tˆ0 , (εQ)−1/2 , −(εQ) local curvature bound (22.5)

ˆ | Rm |(x, t) ≤ 2Q.

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1. A POINT PICKING METHOD

185

We call the above properties the ‘requirements of (ˆ x0 , tˆ0 )’. Essentially, (22.5) says that we have a good relative curvature bound in a parabolic cylinder centered at (ˆ x0 , tˆ0 ). Now we give a proof of the above claim. If the point (x0 , t0 ) itself satisfies the requirements of (ˆ x0 , tˆ0 ), then we are done. Otherwise, there −1/2 , −(εQ)−1 ) such that exists (x1 , t1 ) ∈ P (x0 , t0 , (εQ) Q1  | Rm |(x1 , t1 ) > 2Q, so that in particular, (x1 , t1 ) is an α-large curvature point.2 Note that by (22.4), we have Q1 > T16ε and 3 3 t1 ≥ t0 − HQ−1 ≥ t0 ≥ T. 4 8 x0 , tˆ0 ), then we are done. Now if (x1 , t1 ) satisfies the requirements of (ˆ −1/2 , −(εQ1 )−1 ) such that Otherwise, we can find (x2 , t2 ) ∈ P (x1 , t1 , (εQ1 ) Q2  | Rm |(x2 , t2 ) > 2Q1 > 22 Q. Note that 5T . 16 Moreover, we have the fact that (x2 , t2 ) is an α-large curvature point.3 If (x2 , t2 ) satisfies the requirements of (ˆ x0 , tˆ0 ), then we are done. Otherwise we repeat the process. Notice that this process must stop at some point since supM×[0,T ] | Rm |(x, t) < ∞. The last point (xj , tj ) satisfies the requirements of (ˆ x , tˆ ) with tˆ0 ≥ t0 − 2HQ−1 ≥ T2 − T4 = T4 (more precisely,

0 0 T −j tj ≥ 4 1 + 2 ). −1 − HQ−1 2−1 ≥ t2 ≥ t1 − (εQ1 )−1 ≥ t1 − HQ−1 1 ≥ t0 − HQ

Exercise 22.2. Fill in the details of the above argument to confirm the existence of the desired point (ˆ x0 , tˆ0 ). In conclusion, we have proved the following general point picking result. Lemma 22.3 (Large curvature point with local curvature control). Given α ∈ (0, ∞) and ε ∈ (0, 1), if there exists a point (x0 , t0 ) ∈ M × [T /2, T ] with   α 8 , , | Rm |(x0 , t0 ) > max t0 T ε then there exists a point (ˆ x0 , tˆ0 ) ∈ M × [T /4, T ] with   α 8 ˆ  | Rm |(ˆ , Q x0 , tˆ0 ) > max tˆ0 T ε 2 3

Indeed, | Rm |(x1 , t1 ) > 2| Rm |(x0 , t0 ) > We have

2α t0

≥

α . t1

| Rm |(x2 , t2 ) > 2| Rm |(x1 , t1 ) > (since t2 ≥ t1 − HQ−1 2−1 = t1 −

T 16

2α α ≥ t1 t2

≥ 12 t1 ).

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186

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

ˆ −1/2 , −(εQ) ˆ −1 ) we have and for every (x, t) ∈ P (ˆ x0 , tˆ0 , (εQ) ˆ | Rm |(x, t) ≤ 2Q. Note that, assuming supM×[0,T ] | Rm |(x, t) < ∞, the above discussion ˆ is replaced by finding a point holds when the property | Rm |(x, t) ≤ 2Q ¯ ¯ (¯ x, t ) with | Rm |(x, t) ≤ θ| Rm |(¯ x, t ) for some θ > 1. Later we will see that a refinement of this method will yield (¯ x, t¯) with other properties and we can also drop the bounded curvature assumption (22.1). 1.1.3. Proof of Claim 1. The following is the original Claim 1 on p. 24 of Perelman [152], which is used in the proof of Theorem 21.9. Recall that Mα is defined in (21.16). Claim 1 (Picking a well-chosen large curvature  2  point). Let ε0 > n 0, α > 0, and A > 0. Suppose that (M , g (t)), t ∈ 0, ε , where ε ∈ (0, ε0 ], is a continuous family of complete smooth Riemannian manifolds, x0 ∈ M, and there exists a time t1 ∈ (0, ε2 ] and a ‘nearby’ point x1 ∈ Bg(t1 ) (x0 , ε0 ) with large curvature: α 1 + 2. (22.6) | Rm |(x1 , t1 ) > t 1 ε0 Then there exists a ‘not far ’ α-large curvature point (¯ x, t¯) ∈ Mα , with (22.7)

x ¯ ∈ Bg(t¯) (x0 , (2A + 1)ε0 )

and

t¯ ∈ (0, ε2 ],

such that we have curvature control : (22.8)

| Rm |(x, t) ≤ 4| Rm |(¯ x, t¯)

for all (22.9) 1 x, x0 ) + A| Rm |− 2 (¯ x, t¯), (x, t) ∈ Mα with 0 < t ≤ t¯ and dg(t) (x, x0 ) ≤ dg(t¯) (¯ ¯ is i.e., for all α-large curvature points not too much farther from x0 than x 4 from x0 . Proof of Claim 1. We shall define a sequence of points {(xk , tk )}k≥1 with (xk , tk ) ∈ Mα and xk ∈ Bg(tk ) (x0 , (2A + 1)ε0 ) inductively and show at some finite step  < ∞ that (¯ x, t¯)  (x
, t
) is a point for which the claim is true. Let (x1 , t1 ) be a point as in the hypothesis of the claim. Suppose that, inductively, for some m ∈ N the points (xk , tk ) ∈ Mα with xk ∈ Bg(tk ) (x0 , (2A + 1)ε0 ) and tk > 0 have been defined in the fashion described below for 1 ≤ k ≤ m. Moreover, suppose that the point (¯ x, t¯) in the claim x, t¯) can be taken to be (xm , tm ), then cannot be taken to be (xm , tm ). (If (¯ 4

Note that in (22.9) the distance from x to x0 is measured with respect to g (t) whereas the distance from x ¯ to x0 is measured with respect to g (t¯). The point x ¯ is ‘not far’ from x0 and x1 whereas all we know (from the proof below) about t¯ is that it is in (0, t1 ].

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1. A POINT PICKING METHOD

187

we are done.) Then we may define (xm+1 , tm+1 ) ∈ Mα to be a point such that (22.10)

| Rm |(xm+1 , tm+1 ) > 4| Rm |(xm , tm ),

(22.11) 1 0 < tm+1 ≤ tm and dg(tm+1 ) (xm+1 , x0 ) ≤ dg(tm ) (xm , x0 )+A| Rm |− 2 (xm , tm ). By induction we also have that (22.10) and (22.11) hold with m replaced by k ∈ [1, m − 1]. Thus, for all 1 ≤ k ≤ m + 1 we have 0 < tk ≤ t1 ≤ ε2 and the curvatures at (xk , tk ) are increasing at least geometrically: (22.12)

| Rm |(xk , tk ) > 4k−1 | Rm |(x1 , t1 ),

so that (22.13)

| Rm |− 2 (xk , tk ) < 1

1

| Rm |− 2 (x1 , t1 ). 1

2k−1

Hence, by (22.11) and (22.13), for all 1 ≤ k ≤ m + 1 the distance of xk to x0 at time tk has the following upper bound: dg(tk ) (xk , x0 ) ≤ dg(tk−1 ) (xk−1 , x0 ) + A| Rm |− 2 (xk−1 , tk−1 ) 1

≤ dg(t1 ) (x1 , x0 ) + A| Rm |− 2 (x1 , t1 ) 1

+ · · · + A| Rm |− 2 (xk−1 , tk−1 )   1 1 1 ≤ ε0 + A 1 + + · · · + k−2 | Rm |− 2 (x1 , t1 ). 2 2 1

(22.14)

Since (22.6) implies | Rm |− 2 (x1 , t1 ) < ε0 , we have 1

dg(tk ) (xk , x0 ) < (2A + 1)ε0 for 1 ≤ k ≤ m + 1. In particular, xm+1 ∈ Bg(tm+1 ) (x0 , (2A + 1)ε0 ). By (22.12), if the point (¯ x, t¯) in the claim cannot be taken to be (xk , tk ) for any k ∈ N, then we have lim | Rm |(xk , tk ) = ∞,

k→∞

which contradicts the fact / that the metrics g (t) are continuous in time so ¯g(t) (x0 , (2A + 1)ε0 ) × {t} is compact and that the space-time set t∈[0,ε2 ] B |Rm| has a uniform upper bound on this set. Hence there exists  ∈ N such that (¯ x, t¯) can be taken to be (x
, t
). This completes the proof of Claim 1.
Remark 22.4. From the proof of Claim 1 we see that it suffices   to assume ¯g(t) (x0 , (2A + 1)ε0 ) is compact in M for each t ∈ 0, ε2 instead of that B assuming that M is complete.

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188

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

We note the following elementary variant of Perelman’s Claim 1.   Exercise 22.5. Let (Mn , g (t)), t ∈ 0, ε2 , where ε ∈ (0, ε0 ], be a continuous family of complete smooth Riemannian manifolds, let x0 ∈ M, and let S ⊂ M × (0, ε2 ]. Show that if there exists (x1 , t1 ) ∈ S with x1 ∈ x, t¯) ∈ S with Bg(t1 ) (x0 , ε0 ) and | Rm |(x1 , t1 ) > ε12 , then there exists (¯ 0

(22.15)

x ¯ ∈ Bg(t¯) (x0 , (2A + 1)ε0 )

and

t¯ ∈ (0, t1 ],

such that | Rm |(¯ x, t¯) >

1 ε20

(22.16)

| Rm |(x, t) ≤ 4| Rm |(¯ x, t¯)

and

for all (22.17) 1 x, x0 ) + A| Rm |− 2 (¯ x, t¯). (x, t) ∈ S with 0 < t ≤ t¯ and dg(t) (x, x0 ) ≤ dg(t¯) (¯ In preparation for Claim 2 on p. 24 of [152] we note the following. Remark 22.6. We wish to show that all points in a parabolic cylinder centered at (¯ x, t¯) (not centered at (x0 , t¯)) satisfy (22.8). In view of the changing distances estimate, we consider the more ideal case where  2 Rc (g (t)) ≤ 0 for t ∈ 0, ε , so that distances are nondecreasing in time. Let t ∈ [0, t¯]. 1 ¯) ≤ A| Rm |− 2 (¯ x, t¯), then by the triangle inequality, (1) If dg(t¯) (x, x dg(t) (x, x0 ) ≤ dg(t¯) (x, x0 ) ¯) + dg(t¯) (¯ x, x0 ) ≤ dg(t¯) (x, x x, x0 ) + A| Rm |− 2 (¯ x, t¯). ≤ dg(t¯) (¯ 1

¯) ≤ A| Rm |− 2 (¯ x, t¯), then (2) Alternatively, if dg(t) (x, x 1

¯) + dg(t) (¯ x, x0 ) dg(t) (x, x0 ) ≤ dg(t) (x, x x, x0 ) + A| Rm |− 2 (¯ x, t¯). ≤ dg(t¯) (¯ 1

Hence Claim 1 implies that (22.8) holds for all points (x, t) ∈ Mα satisfying the hypothesis of either (1) or (2). 1.2. The existence of a well-chosen large curvature point. Now we show that the point picking method in the previous subsection gives a well-chosen large curvature point, i.e., the curvature estimate (22.8) holds inside a parabolic cylinder of (¯ x, t¯). The following is Claim 2 on p. 24 of [152]. Claim 2 (The well-chosen large curvature point (¯ x, t¯) has controlled curvature in a backward parabolic cylinder centered at (¯ x, t¯)). Under the assumptions of Claim 1, further assume A ≥ 1 and 1 √ . For the point (¯ x, t¯) shown to exist in Claim 1, we have α < 13(n−1) n (22.18)

| Rm |(x, t) ≤ 4| Rm |(¯ x, t¯)

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1. A POINT PICKING METHOD

for all (x, t) such that α ¯ −1 ≤ t ≤ t¯ and (22.19) t¯ − Q 2 ¯  | Rm |(¯ where Q x, t¯).

dg(t¯) (x, x ¯) ≤

189

A ¯− 1 Q 2, 10

Remark 22.7. The following proof of the claim also easily implies (exercise) that | Rm |(x, t) ≤ 4| Rm |(¯ x, t¯) α −1 ¯ ¯ −1/2 (where we ¯) ≤ AQ for all (x, t) such that t¯− 2 Q ≤ t ≤ t¯ and dg(t) (x, x use dg(t) instead of dg(t¯) ). We shall use the following in the proof of the claim. Lemma 22.8. If (x, t) satisfies (22.19) and (x, t) ∈ Mα , then (22.20)   2   3 . A A α ¯g(t) x0 , dg(t¯) (¯ ¯g(t¯) x x, x0 ) + 1 × {t} . B ¯, × t¯ − ¯ , t¯ ⊂ B ¯ 12 ¯2 2Q 10Q Q ¯ t∈(0,t] ¯ > αt¯−1 and Proof of Claim 2. By definition, (¯ x, t¯) ∈ Mα implies Q α ¯ −1 1¯ ¯ hence t − 2 Q > 2 t. Suppose that (x, t) satisfies (22.19). (1) If (x, t) ∈ / Mα , then  α ¯ −1 −1 −1 ¯ < 2αt¯−1 < 2| Rm |(¯ x, t¯), (22.21) | Rm |(x, t) ≤ αt ≤ α t − Q 2 ¯ −1 ≤ t and the last inequalwith the second inequality following from t¯− α2 Q ity following from (¯ x, t¯) ∈ Mα . Thus we have proved that Claim 2 holds for those points (x, t) ∈ / Mα . (2) Now suppose that (x, t) ∈ Mα . By Lemma 22.8, we have that if      A ¯ −1/2 α ¯ −1 ¯ ¯g(t¯) x ,t , ¯, Q × t¯ − Q (x, t) ∈ Mα ∩ B 10 2 then (x, t) satisfies condition (22.9) and we may apply Claim 1 to conclude | Rm |(x, t) ≤ 4| Rm |(¯ x, t¯). This completes the proof of Claim 2 modulo establishing the lemma.
  ¯ −1 , t¯ and x ∈ M Proof of Lemma 22.8. Suppose that t ∈ t¯ − α2 Q 1 ¯ −1/2 . We need to show that ¯) ≤ 10 AQ with dg(t¯) (x, x (22.22)

¯− 2 . x, x0 ) + AQ dg(t) (x, x0 ) ≤ dg(t¯) (¯ 1

By the triangle inequality, we have 1 ¯− 1 AQ 2 . 10 Therefore at t = t¯ we have the compact containment     9 ¯− 1 1 ¯− 1 ¯ 2 2 x, x0 ) + AQ ¯, AQ ⊂ Bg(t¯) x0 , dg(t¯) (¯ . Bg(t¯) x 10 10 x, x0 ) + dg(t¯) (x, x0 ) ≤ dg(t¯) (¯

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22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

¯ −1 which is the first time, going Suppose that there exists t1 > t¯ − 12 αQ 1 ¯ − 12 ) inter¯g(t¯) (¯ x, 10 AQ backwards in time from t¯, such that the closed ball B 9 ¯ − 12 ); we shall show that ¯g(t ) (x0 , dg(t¯) (¯ x, x0 ) + 10 AQ sects the boundary of B 1 this leads to a contradiction. Let x∗ be such a point of intersection, so that 9 ¯− 1 2. x, x0 ) + AQ (22.23) dg(t1 ) (x∗ , x0 ) = dg(t¯) (¯ 10 ¯g(t¯) (¯ ¯ − 21 ), we have x, 1 AQ Note that since x∗ ∈ B 10

1 ¯− 1 AQ 2 . 10 x, x0 ) + By the choice of t1 and x∗ we have dg(t) (x∗ , x0 ) ≤ dg(t¯) (¯ 1 −1/2 ¯ ¯ for any t ∈ [t1 , t ]. Hence if dg(t) (x, x∗ ) ≤ 10 AQ , then x, x0 ) + dg(t¯) (x∗ , x0 ) ≤ dg(t¯) (¯

(22.24)

9 ¯ −1/2 10 AQ

dg(t) (x, x0 ) ≤ dg(t) (x, x∗ ) + dg(t) (x∗ , x0 ) 9 ¯− 1 1 ¯− 1 2 + AQ 2 , x, x0 ) + AQ ≤ dg(t¯) (¯ 10 10 so that (22.25) (22.26)

  1 ¯− 1 ¯ − 12 ), ¯g(t) (x0 , dg(t¯) (¯ ¯ x, x0 ) + AQ Bg(t) x∗ , AQ 2 ⊂ B 10   ¯ − 21 ⊂ B ¯ − 12 ) ¯g(t) (x0 , dg(t¯) (¯ ¯g(t) x0 , 1 AQ x, x0 ) + AQ B 10

for t ∈ [t1 , t¯] (with the second inclusion being obvious). Note that by combining (22.9) and (22.21) to cover the cases where / Mα , respectively, we have (x, t) ∈ Mα and (x, t) ∈  α ¯ −1 ¯ − 12 ¯ ¯ ¯ ¯ x, x0 ) + AQ ) × t − Q , t . | Rm |(x, t) ≤ 4Q in Bg(t) (x0 , dg(t¯) (¯ 2 In particular, by (22.25) and (22.26),     1 ¯− 1 1 ¯− 1 ¯ ¯ ¯ Rc(x, t) ≤ 4nQ for x ∈ Bg(t) x∗ , AQ 2 ∪ Bg(t) x0 , AQ 2 10 10 and t ∈ [t1 , t¯]. Hence we may apply Theorem 18.7(2) on how distances ¯ −1/2 )5 to ¯ and r0 = 1√ Q change under the Ricci flow (with K = 4nQ 10 n conclude that √ 104 d ¯ 12 for t ∈ [t1 , t¯] . dg(t) (x∗ , x0 ) ≥ − (n − 1) nQ dt 5 5

This implies

  − 2 (n − 1) Kr0 + r0−1   √ 1/2 1 ¯ −1/2 ¯ ¯ √ = −2 (n − 1) 4nQ + 10 nQ Q 10 n √ 1/2 104 ¯ . (n − 1) nQ =− 5

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2. HEAT KERNELS UNDER CHEEGER–GROMOV LIMITS

191

Integrating the above inequality on the interval [t1 , t¯], we obtain (note that ¯ −1 ) t¯ − t1 < α2 Q √ 1 α −1 104 ¯ ¯2 · Q (n − 1) nQ dg(t1 ) (x∗ , x0 ) ≤ dg(t¯) (x∗ , x0 ) + 5 2 √ 1 ¯ − 1 52 ¯ − 12 2 + (n − 1) nαQ ≤ dg(t¯) (¯ x, x0 ) + AQ 10 5 9 ¯− 1 (22.27) < dg(t¯) (¯ x, x0 ) + AQ 2 , 10 where we used (22.24) in the second inequality and we used A ≥ 1 and 1 √ in the last inequality. However, inequality (22.27) contradicts α < 13(n−1) n (22.23). This completes the proof of Lemma 22.8 and hence Claim 2.
2. Heat kernels under Cheeger–Gromov limits In this section we discuss the convergence of heat kernels for a sequence of solutions to the Ricci flow which converges in the C ∞ Cheeger–Gromov sense. 2.1. Pointed C ∞ convergence of solutions. A motivating setup for our discussion in the next subsections is the following (in view of Lemma 21.15 on Cheeger–Gromov convergence). Let (Mn∞ , g∞ (t) , x∞ ), t ∈ [−T, 0], be a pointed complete solution to the Ricci flow. Suppose that {Ui }i∈N is an exhaustion of M∞ by open sets with x∞ ∈ Ui for all i and suppose that gi (t), t ∈ [−T, 0], are (in general, incomplete) solutions to the Ricci flow on Ui such that the sequence {(Ui , gi (t))}i∈N converges in C ∞ to (Mn∞ , g∞ (t)) uniformly on compact sets in M∞ × [−T, 0]. Note that here the convergence is pointwise, i.e., we do not pull back by diffeomorphisms.  i ⊃ Ui Let gi (t), t ∈ [−T, 0], be complete solutions to the Ricci flow on M such that gi (t) = gi (t) on Ui , i.e., gi (t) are complete extensions of gi (t). For each i, let  i × [−T, 0) → R+ i : M H be the minimal positive fundamental solution to i ∂H  i + Rg (t) H i = −∆gi (t) H (22.28) i ∂t  i (·, t) = δx∞ . with limt0 H The above discussion relates to Cheeger–Gromov convergence, as in Lemma 21.15, as follows. We have the following two setups. n , g (t) , x ), the exhaustion {(U , g (t))} (1) The limit (M ∞ i i ∞ ∞ i∈N , and  n   are as above (where Mi ⊃ Ui ⊂ the solutions M , gi (t) i

M∞ ).

i∈N

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22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

(2) The limit (Mn∞ , g¯∞ (t) , x ¯∞ ), the exhaustion {Ui }i∈N , the solutions n ¯i )}i∈N , the embeddings Φi : (Ui , x ¯∞ ) → (Mi , x ¯i ), {(Mi , g¯i (t) , x and their images Vi = Φi (Ui ) are as in Lemma 21.15. The way that we obtain setup (1) from setup (2) is to let Ui ⊂ M∞ be the same and to let ¯∞ , (i) g∞ (t) = g¯∞ (t), x∞= x (ii) gi (t) = Φ∗i g¯i (t)|Vi on Ui ,  i = (Mi  Ui ) / ∼, where ∼ identifies x ∈ Ui with Φi (x) ∈ Mi (iii) M  i → Mi (this way we have both a natural diffeomorphism Ii : M  i ), and and a natural inclusion of Ui into M ∗ gi (t)). (iv) gi (t) = Ii (¯ 2.2. Estimates for the adjoint heat kernel. In this subsection, based on estimates for the adjoint heat kernel, we give a proof of Lemma 21.16. For convenience, we formulate a general result concerning Cheeger–Gromov limits and heat kernels. Let {(Mni , gi (τ ) , (xi , 0))} , τ ∈ [0, ω] , be a pointed sequence of complete solutions of the backward Ricci flow with bounded curvature; here the bounds for the curvatures of the solutions gi (τ ) on Mi may depend on i. Suppose that the above sequence converges, in the C ∞ pointed Cheeger–Gromov sense, to a smooth complete limit solution (Mn∞ , g∞ (τ ) , (x∞ , 0)) ,

τ ∈ [0, ω] .

By Definition 3.6 in Part I, this means that there exist an exhaustion {Ui }i∈N of M∞ by open sets with x∞ ∈ Ui and a sequence of diffeomorphisms Φi : Ui → Vi  Φi (Ui ) ⊂ Mi with Φi (x∞ ) = xi such that (Ui , g˜i (τ )), where   g˜i (τ )  Φ∗i gi (τ )|Vi , converges pointwise in C ∞ to (M∞ , g∞ (τ )) uniformly on compact subsets of M∞ × [0, ω]. Let Hi : Mi × (0, ω] → (0, ∞) be the adjoint heat kernel centered at (xi , 0), i.e., Hi is the minimal positive function such that   ∂ ∗ − ∆gi + Rgi Hi = 0, (22.29)
i Hi  ∂τ lim Hi ( · , τ ) = δxi . τ 0

The existence of Hi will be proved in Chapter 23. As in (21.30) and (21.31), let Hi  (4πτ )−n/2 e−fi and let     vi  τ Rgi + 2∆gi fi − |∇gi fi |2 + fi − n Hi

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2. HEAT KERNELS UNDER CHEEGER–GROMOV LIMITS

193

be Perelman’s differential Harnack quantity. Recall that by (21.32), since gi (τ ) has bounded curvature, we have vi ≤ 0

on Mi × (0, ω].

Lemma 21.16 is a consequence of the following property of heat kernels under pointed Cheeger–Gromov convergence of the underlying manifolds. Lemma 22.9 (Convergence of heat kernels under Cheeger–Gromov limits). Under the above setup, for a subsequence we have the following C ∞ convergences to C ∞ functions on M∞ : ˜ i  Hi ◦ Φi −→ H∞ , H f˜i  fi ◦ Φi −→ f∞ , v˜i  vi ◦ Φi −→ v∞ . Moreover, H∞ = (4πτ )−n/2 e−f∞ is a positive solution to the adjoint heat equation   ∂ ∗ − ∆g∞ + Rg∞ H∞ = 0 (22.30)
∞ H∞  ∂τ and (22.31)

    v∞ = τ Rg∞ + 2∆g∞ f∞ − |∇g∞ f∞ |2 + f∞ − n H∞ ≤ 0

satisfies the equation (22.32)


∗∞ v∞

 2   1 g∞ g∞  = −2τ Rcg∞ +∇ ∇ f∞ − g∞  H∞ . 2τ g∞

We now give a proof of this lemma. ˜ i to a nonnegative C ∞ function (1) To prove the C ∞ convergence of H H∞ for a subsequence, we shall show that for any δ ∈ (0, ω/2] we have for all i

(22.33) Hi ≤ C in Bgi (0) xi , δ −1 × [δ, ω − δ] , ˜ i to H∞ on where C < ∞ is independent of i. The C ∞ convergence of H M∞ × (0, ω] then follows from the Bernstein local derivative estimates for heat-type equations (see Ladyˇzenskaja, Solonnikov, and Ural´ceva [113]). (2) To show that H∞ > 0, we shall show that for any δ ∈ (0, ω] we have for all i (22.34)

Hi (xi , τ ) ≥ c

for τ ∈ [δ, ω] ,

where c > 0 is independent of i. Then the positivity of H∞ follows from the strong maximum principle. Proof of (1). First note that, by Corollary 26.15 below, we have  Hi (x, τ ) dµgi (τ ) (x) ≡ 1 (22.35) Mi

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194

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

for τ ∈ (0, ω]. By the pointed Cheeger–Gromov convergence of gi (τ ) to g∞ (τ ), given any R > 0, there exists C˜0 (R) < ∞ such that C˜0 (R)−1 gi (x, 0) ≤ gi (x, τ ) ≤ C˜0 (R) gi (x, 0) for all x ∈ Bgi (0) (xi , R), τ ∈ (0, ω], and i ∈ N. Since dµgi (τ ) ≥ C˜0 (R)−n/2 dµgi (0) in Bgi (0) (xi , R), we have by (22.35)   n/2 ˜ Hi (x, τ ) dµgi (0) (x) ≤ C0 (R) Bgi (0) (xi ,R)

Bgi (0) (xi ,R)

Hi (x, τ ) dµgi (τ ) (x)

≤ C˜0 (R)n/2

(22.36)

for τ ∈ (0, ω]. We may now obtain (22.33) from either (i) the mean value inequality (Theorem 25.2) or (ii) the Li–Yau estimate. We illustrate the proof using method (ii). By (22.36) we have that, for each τ ∈ (0, ω], there exists xτ ∈ Bgi (0) (xi , 1) such that C˜0 (1)n/2 . Hi (xτ , τ ) ≤ Volgi (0) Bgi (0) (xi , 1) By the Li–Yau estimate (i.e., Corollary 25.13), for any

(x, τ ) ∈ Bgi (0) xi , δ −1 × [δ, ω − δ] we have



2 C˜0 δ −1 dgi (0) (x, xτ +δ ) exp Hi (x, τ ) ≤ Hi (xτ +δ , τ + δ)e 4(1 − ε) δ  



2 n C˜0 δ −1 δ −1 + 1 C˜0 (1)n/2 C12 δ 2−3ε e , 2 exp ≤ Volgi (0) Bgi (0) (xi , 1) 4(1 − ε) δ     where C12 depends only n, δ, ε, and the local bounds on Rcgi (τ ) , ∇Rgi (τ ) , and ∆Rgi (τ ) (and we may choose ε = 1/2 for example); hence we obtain (22.33). Regarding the use of Corollary 25.13 (and hence the use of Theorem 25.9), note the following: (1) Let γ : [τ, τ + δ] → Mi be a minimal geodesic joining x to xτ +δ , with respect to gi (0). Then γ (τ  ) ∈ Bgi (0) xi , δ −1 + 2 for each τ  ∈ [τ, τ + δ]. Hence γ (τ  ) ∈ Bgi (τ  ) (xi , Ri ) for each τ  ∈ [τ, τ + δ], 

1/2 −1

δ +2 . where Ri  C˜0 δ −1 + 2 (2) Since gi (τ ) is a solution to the backward  Ricci flow on [0, ω], by  Shi’s local derivative estimate, ∇Rgi (τ )  and ∆Rgi (τ ) are bounded in Bgi (0) (xi, 2 Ri ) ×[0, ω − δ] only in terms of δ and bounds on the curvatures Rmgi (τ )  in Bgi (0) (xi , 2 Ri +1)×[0, ω] (in particular, the bounds are independent of i). (3) Volgi (0) Bgi (0) (xi , 1) → Volg∞ (0) Bg∞ (0) (x∞ , 1) > 0 as i → ∞. 

C12 δ

τ +δ τ



n 2−3ε



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2. HEAT KERNELS UNDER CHEEGER–GROMOV LIMITS

195

Exercise 22.10. Use method (i), i.e., mean value inequality, to prove (22.33). Proof of (2). We prove a general statement, which we apply to Hi at the very end. Let (Mn , g (τ )), τ ∈ [0, ω], be a complete solution to the backward Ricci flow. Suppose that x0 ∈ M is such that Rcg(τ ) ≥ (n − 1) K

and

Rg(τ ) ≤ L

in Bg(0) (x0 , R)

for τ ∈ [0, ω], where K ≤ 0, L ≥ 0, and where R > 0 is chosen sufficiently small so that Bg(0) (x0 , R) is regular. Let HR : Bg(0) (x0 , R) × [0, ω] → R+ be the adjoint Dirichlet heat kernel (for its existence, see §5 of Chapter 24), i.e.,   ∂ ∗ − ∆g(τ ) + Rg(τ ) HR = 0,
HR  ∂τ HR |∂Bg(0) (x0 ,R) = 0 for τ ∈ (0, ω], lim HR ( · , τ ) = δx0 .

τ 0

Then (22.37)





∂ − ∆g(τ ) eLτ HR ≥ eLτ
∗ HR = 0, ∂τ

lim eLτ HR ( · , τ ) = δx0 . τ 0

For comparison, let



¯0 , R × [0, ∞) → R+ HnK  ,R : B x

be the Dirichlet heat kernel centered at x ¯0 for the (static) simply-connected n space form MK  of constant sectional curvature K  . Here K√ is to be chosen below and when K  > 0, we shall choose at least R < π/ K  . Note that ¯0 and we shall also write it as HnK  ,R ( · , τ ) is rotationally symmetric about x a function of the distance to x ¯0 . Let r = dg(τ ) (y, x0 ), let HK (r) denote the mean curvature of the disx0 , r) in MnK , and let H (y) denote the mean curvature at tance sphere SK (¯ y of the sphere S (x0 , r) in M with respect to g (τ ). Transplanting HnK  ,R to M, we have  

∂ − ∆g(τ ) HnK  ,R dg(τ ) (y, x0 ) , τ ∂τ    ∂ ∂ n d H   (r, τ ) (y, x0 ) (22.38) = HK  (r) − H (y) + ∂τ g(τ ) ∂r K ,R   ∂ n H   (r, τ ) ≤ (HK  (r) − HK (r) + (n − 1) Kr) ∂r K ,R

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22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

∂ by ∂r HnK  ,R ≤ 0, by the mean curvature of distance spheres comparison theorem, which says H (y) ≤ HK (r) (see Lemma 1.126 in [45] for example), ∂ dg(τ ) (y, x0 ) ≥ (n − 1) Kdg(τ ) (y, x0 ). Hence, choosing K  suffiand by ∂τ ciently large and R > 0 sufficiently small, both depending only on K, we obtain  

∂ − ∆g(τ ) HnK  ,R dg(τ ) (y, x0 ) , τ ≤ 0 (22.39) ∂τ

and (22.40)



Bg(τ ) x0 , R ⊂ Bg(0) (x0 , R)

for all τ ∈ [0, ω]. Indeed, without loss of generality we may assume K < 0. Then we have    |K|r HK (r) = (n − 1) |K| coth   3

1 |K| + r+O r , = (n − 1) r 3 so it suffices to take K  = 5K to obtain (22.39) from (22.38) for R sufficiently small. We remark that in the calculation of (22.38) we have used the fact that for any function f , in spherical coordinates centered at x0 , we have ∆g(τ ) f (y) =

∂ 2f ∂f + ∆S(p,r) f. + H (y) ∂r2 ∂r

We also have

lim HnK  ,R dg(τ ) ( · , x0 ) , τ = δx0 τ 0  Lτ and by (22.40), e HR ∂B (x0 ,R ) ≥ 0 for τ ∈ (0, ω]. g(τ )

One can now apply the same argument (a form of the maximum principle) as in the proof of Lemma 16.49 in Part II to (22.37) and (22.39) to obtain

eLτ HR (y, τ ) ≥ HnK  ,R dg(τ ) (y, x0 ) , τ for y ∈ Bg(τ ) (x0 , R ) and τ ∈ (0, ω]. Therefore the adjoint heat kernel H of (Mn , g (τ )) centered at (x0 , 0) satisfies (22.41)

H (x0 , τ ) ≥ HR (x0 , τ ) ≥ e−Lτ HnK  ,R (0, τ ) .

Applying this generally formulated estimate to the adjoint heat kernels Hi defined by (22.29), we obtain (22.34). This completes the proof of (2). ˜ i converges to H∞ > 0 in C ∞ , we have f˜i converges (3) Since by (1), H

to f∞ in C ∞ and the equality H∞ = (4πτ )−n/2 e−f∞ . Clearly
∗∞ H∞ = 0 follows from
∗i Hi = 0. This also implies v˜i converges to v∞ in C ∞ and we have the equality in (22.31). Finally, v∞ ≤ 0 follows from vi ≤ 0 and vi → v∞ ; equation (22.32) for
∗∞ v∞ follows from taking the limit of the analogous equations for
∗i vi . This finishes the proof of Lemma 22.9.

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3. UPPER BOUND FOR THE LOCAL ENTROPY


B

v dµ

197

The following problem is not necessary for our discussion but is of independent interest. Mini-Problem 22.11 (H∞ is the adjoint heat kernel). Prove that H∞ in Lemma 22.9 satisfies lim H∞ ( · , τ ) = δx∞ ,

τ →0+

i.e., H∞ is the minimal positive fundamental solution to the adjoint heat equation. Note that when the curvatures of (Mni , gi (t)) are uniformly bounded (independent of i), this was proved by S. Zhang [196]. 3. Upper bound for the local entropy


B

v dµ

In this section we show that if there is a negative upper bound for the local entropy (i.e., the integration of Perelman’s Harnack quantity v in a ball) at some time t˜, then there is a bound for a corresponding local entropy at time 0. The idea of the proof is to localize the entropy monotonicity formula. 3.1. A nice time-dependent cutoff function. We shall calculate a local form of the entropy monotonicity formula (17.12) via multiplying by a suitably nice time-dependent cutoff function h, which we now proceed to define. Let φ : R → [0, 1] be a smooth function which is strictly decreasing on the interval [1, 2] and which satisfies  1 if s ∈ (−∞, 1], (22.42) φ (s) = 0 if s ∈ [2, ∞), and



(22.43a)

φ (s)

2

≤ 10φ(s),



φ (s) ≥ −10φ(s)

(22.43b)

for s ∈ R.   Let (Mn , g (t) , x0 ), t ∈ 0, ε2 , where ε ∈ (0, ε0 ], be a complete pointed solution to the Ricci  Given positive constants a and b, define the cutoff  flow. function h : M × 0, ε2 → [0, 1] by  √  dg(t) (x, x0 ) + a t . (22.44) h(x, t) = φ b Lemma 22.12. Suppose (22.45) | Rm |(x, t) ≤

2 α + 2 t ε0

whenever t ∈ (0, ε2 ] and dg(t) (x, x0 ) ≤ ε0 ,

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198

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

where α ≤ 1. If (22.46a)

a = 200n,

(22.46b)

b ≥ 5ε0 A,

where A ≥ 41n, then h is a subsolution to the following linear heat equation:     10 ∂ − ∆g(t) h (x, t) ≤ 2 h(x, t) on M × 0, ε2 (22.47)
h (x, t)  ∂t b in the weak sense.

√ dg(t) (x,x0 )+a t , b

so that h(x, t) = φ (s (x, t)). We Proof. Let s (x, t)  compute the heat operator acting on h as    ∂ a φ (s (x, t)) (22.48) − ∆g(t) dg(t) (x, x0 ) + √
h (x, t) = b ∂t 2 t  2 φ (s (x, t))  ∇dg(t) (x, x0 )g(t) . − 2 b

∂ − ∆g(t) dg(t) . First We shall now apply Perelman’s lower bound for ∂t   note that at any point (x, t) ∈ M × 0, ε2 where φ (s (x, t)) = 0, we have s (x, t) ∈ [1, 2], which implies √ √ (22.49) b − a t ≤ dg(t) (x, x0 ) ≤ 2b − a t ≤ 2b. In particular, (22.50)

supp (h (·, t)) ⊂ Bg(t) (x0 , 2b) .

Since (22.46) and A ≥ 41n imply that a and b satisfy b ≥ (a + 1) ε0 ,

(22.51) we have

√ √   b − a t ≥ t for all t ∈ 0, ε2 .  √

 Then h(x, t) = 1 for x ∈ Bg(t) x0 , t and t ∈ 0, ε2 . The hypothesized curvature bound (22.45) implies (22.52)

Rc ≤ (n − 1) K where r0 =

√

in Bg(t) (x0 , r0 ) ,

t ≤ ε and

K=

2 α + 2. t ε0

By Theorem 18.7(1) (see 8.3(a) in Perelman [152]),  the original Lemma  also 2  if a point (x, t) ∈ M × 0, ε is such that φ (s (x, t)) = 0, then       2 √ 2 α 1 ∂ √ ; − ∆g(t) dg(t) (x, x0 ) ≥ −(n − 1) + 2 t+ ∂t 3 t ε0 t note that (22.52) implies x ∈ M − Bg(t) (x0 , r0 ).

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3. UPPER BOUND FOR THE LOCAL ENTROPY


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v dµ

199

  Since t ≤ ε2 , this implies that for (x, t) ∈ M× 0, ε2 where φ (s (x, t)) = 0,

 (n − 1) (2α + 7) ∂ √ − ∆g(t) dg(t) (x, x0 ) ≥ − ∂t 3 t 100n (22.53) ≥− √ , t which holds in the weak sense; for the last inequality we used α ≤ 1. Using a = 200n and  by substituting (22.53) into (22.48), we have that for all (x, t) ∈ M × 0, ε2 (independent of whether or not φ (s (x, t)) is equal to zero) 2 φ (s (x, t))   (x, x ) ∇d
h (x, t) ≤ − 0 g(t) g(t) b2 10 ≤ 2 h(x, t) b  in the weak sense, since φ (s) ≤ 0, −φ (s) ≤ 10φ(s), and   ∇dg(t) (x, x0 )2 = 1 g(t) 

a.e. This completes the proof of the lemma.




3.2. A negative upper bound for the local entropy at time zero.   Let (Mn , g (t)), t ∈ 0, ε2 , be a complete solution of the Ricci flow with bounded curvature and let (¯ x, t¯) ∈ M × (0, ε2 ]. Define the function (22.54)

H (x, t)  (4π (t¯ − t))−n/2 e−f (x,t)

on M × [0, t¯) to be the minimal positive fundamental solution of the adjoint heat equation centered at (¯ x, t¯), i.e.,   ∂ ∗ (22.55a)
H  − − ∆g + Rg H = 0, ∂t (22.55b) lim H (·, t) = δx¯ . tt¯

We have

 M

H (x, t) dµg(t) = 1.

Define Perelman’s Harnack quantity v on M × [0, t¯) by     (22.56) v (x, t)  (t¯ − t) Rg + 2∆g f − |∇g f |2 + f − n H. Here is the main result of this section.

  Lemma 22.13. Let (Mn , g (t) , x0 ), t ∈ 0, ε2 , where ε ∈ (0, ε0 ], be a complete pointed solution to the Ricci flow with bounded curvature such that 2 α (22.57) | Rm |(x, t) ≤ + 2 whenever t ∈ (0, ε2 ] and dg(t) (x, x0 ) ≤ ε0 , t ε0

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200

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

where α ≤ 1. Given A ≥ 67n, suppose that there exists (¯ x, t¯) ∈ M × (0, ε2 ] with x, x0 ) ≤ (2A + 1)ε0 , dg(t¯) (¯

(22.58)

α ¯  | Rm |(¯ Q x, t¯) > t¯

(22.59) and such that

¯ | Rm |(x, t) ≤ 4Q

(22.60)

for all (x, t) ∈ M × (0, t¯] satisfying α ¯ − 12 . and dg(t) (x, x0 ) ≤ dg(t¯) (¯ x, x0 ) + AQ | Rm |(x, t) > t Further assume that there exists  α ¯ −1 ¯ ,t (22.61) t˜ ∈ t¯ − Q 2 such that  (22.62)  √  v dµg (t˜) ≤ −β1 ¯, Bg (t˜) x

t¯−t˜

for some β1 > 0. Let h be the cutoff function defined by (22.44) with b  10ε0 A. (1) Then      1  v h dµg(t)  ≤ −β1 1 − 2 . (22.63) A M t=0 (2) Let ˜  H h and f˜  f − log h, H where h is the same cutoff function as above, so that ˜ ˜ H(x, t)  (4π (t¯ − t))−n/2 e−f (x,t) .

If we further suppose that A ≤ R (·, 0) ≥ −1 then   (22.64) M

1 20ε0

and that 6

in Bg(0) (x0 , 1),

   1 1 2 ˜ ˜ ˜ ¯ −t|∇f| − f + n H dµg(0) ≥ β1 1 − 2 − 2 − ε2 , A A

where all the quantities on the lhs are evaluated at t = 0. Moreover  ˜ (x, 0) dµg(0) (x) ≤ 1. H M

  ¯g(0) (x0 , 1). ˜ ( · , 0) ⊂ B ¯g(0) (x0 , 20ε0 A) ⊂ B Note that supp H 1 Note that A ≥ 67n and A ≤ 20ε imply that for the lemma to be nonvacuous, we 0 1 1 need ε0 ≤ 1340n . In practice, we take A = 100nε and ε0 → 0 (see (21.20)). 0 6

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3. UPPER BOUND FOR THE LOCAL ENTROPY


B

v dµ

201

3.3. Proof of part (1) of Lemma 22.13. We localize the entropy monotonicity formula. Recall that v defined in (22.56) satisfies  2   1 ∗  ¯ gij  H ≤ 0 (22.65)
v = −2 (t − t) Rij + ∇i ∇j f − 2 (t¯ − t) by Lemma 6.8 in Part I. By (22.65) and the fact that h ≥ 0 has compact support, we compute using Green’s second identity (compare with Lemma 26.1)   d (−v) h dµg(t) = (−v
h + h
∗ v) dµg(t) dt M M ≤ (−v)
h dµg(t) M  10 (22.66) ≤ 2 (−v) h dµg(t) , b M where b = 10ε0 A and in the last inequality we have used (22.47) and Perelman’s Harnack estimate, i.e., v ≤ 0. Integrating (22.66) in time, we have        10 (−v) h dµg(t)  ≥ exp − 2 tˆ (−v) h dµg(t)  b M M t=0 t=tˆ     10ε2 (22.67) ≥ exp − 2 (−v) h dµg(t)  b M t=tˆ  2 for any tˆ ∈ 0, ε . Now let"(¯ x, t¯) and t˜ be as in the hypothesis of Lemma 22.13. Note that  √ α ¯ −1 < √1 t¯ ≤ √ε . We shall later show that b = 10ε0 A t¯ − t˜ ≤ Q 2

2

2

satisfies

     ¯, t¯ − t˜ ⊂ Bg(t˜) x0 , b − 200n t˜ . Bg(t˜) x  √

Since h ·, t˜ = 1 on Bg(t˜) x0 , b − 200n t˜ and since v ≤ 0, using assumption (22.62), we then have  −β1 ≥  √  v dµg (t˜)

(22.68)

 ≥  ≥

¯, Bg (t˜) x

t¯−t˜

 √  v dµg (t˜) Bg (t˜) x0 ,b−200n t˜

v h dµg(t˜)   10ε2 v h dµg(0) ≥ exp b2 M M

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202

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

by (22.67). Since e−x ≥ 1 − x, this implies      10ε2 1 v h dµg(0) ≤ −β1 1 − 2 = −β1 1 − , (22.69) b 10A2 M which is slightly stronger than (22.63). Now we establish the inclusion (22.68), i.e.,   x, x0 ) + t¯ − t˜ ≤ b − 200n t˜, (22.70) dg(t˜) (¯   ¯ −1 , t¯ satisfies (22.62). To prove (22.70), we shall apply where t˜ ∈ t¯ − α2 Q   the changing distances estimate on the time interval t˜, t¯ to the inequality (22.71)

x, x0 ) ≤ (2A + 1)ε0 . dg(t¯) (¯

It follows from (22.60) that if (x, t) ∈ M×(0, t¯] is such that dg(t) (x, x0 ) ≤ ¯ − 12 , then x, x0 ) + AQ dg(t¯) (¯  α ¯ . , 4Q | Rm |(x, t) ≤ max t   ¯ −1 > α Q ¯ −1 ; that Moreover, if t ∈ t˜, t¯ , then by (22.61) we have t ≥ t¯ − α2 Q 2 is, α ¯ < 2Q. t   Hence, if (x, t) ∈ M × t˜, t¯ is such that   ¯ − 12 , ¯g(t) x0 , dg(t¯) (¯ x, x0 ) + AQ x∈B then 1 ¯ Rc(x, t) ≤ | Rm |(x, t) ≤ 4Q. n−1 ¯ Now by the changing distancesTheorem    18.7(2), if Rc(x, t) ≤ 4(n − 1)Q 1 1 ¯ − 2 ∪ Bg(t) x ¯ − 2 , then ¯, Q for all x ∈ Bg(t) x0 , Q (22.72)

∂ ¯ 21 . dg(t) (¯ x, x0 ) ≥ −10(n − 1)Q ∂t

Hence, as long as ¯− 2 , x, x0 ) ≤ dg(t¯) (¯ x, x0 ) + (A − 1) Q dg(t) (¯       ¯ − 21 ∪ Bg(t) x ¯ − 12 , ¯ − 21 ⊂ B ¯g(t) x0 , dg(t¯) (¯ x, x0 ) + AQ ¯, Q so that Bg(t) x0 , Q we have (22.72). On the other hand, suppose t ∈ (t˜, t¯] is such that (22.73) holds on the interval [t , t¯]. Then by integrating (22.72) in time, we have

¯ 21 t¯ − t . x, x0 ) ≤ dg(t¯) (¯ x, x0 ) + 10(n − 1)Q (22.74) dg(t ) (¯ (22.73)

1

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3. UPPER BOUND FOR THE LOCAL ENTROPY


B

v dµ

203



¯ t¯ − t˜ , then (22.74) Now provided A is chosen so that A ≥ 1 + 10(n − 1)Q ¯ −1 , implies that (22.73) holds on [t − σ, t¯] for some σ > 0. Since t¯ − t˜ ≤ α2 Q by choosing A such that A ≥ 1 + 5(n − 1)α,

(22.75)

  we conclude that (22.74) holds on the whole interval t˜, t¯ . In particular, taking t = t˜ in (22.74), we obtain

¯ 12 t¯ − t˜ x, x0 ) ≤ dg(t¯) (¯ x, x0 ) + 10(n − 1)Q dg(t˜) (¯ !  α ¯ ˜ t − t. ≤ (2A + 1)ε0 + 10(n − 1) 2

Thus (22.70) will follow from showing that !   α ¯ ˜ ¯ ˜ t − t + t − t ≤ b − 200n t˜. (22.76) (2A + 1)ε0 + 10(n − 1) 2 Now since b = 10ε0 A, A ≥ 67n, and α ≤ 1, we have !   α + 1 + 200n ε0 . (22.77) b ≥ (2A + 536n) ε0 ≥ 2A + 1 + 10(n − 1) 2  √ This implies (22.76) since t¯ − t˜ ≤ ε and t˜ ≤ ε. The proof of (22.63) is complete. 3.4. Proof of part (2) of Lemma 22.13. We now prove (22.64). ˜ and f˜ to satisfy (22.64). Step 1. A sufficient inequality for H By (22.69), we have    1 (−v)h dµg(0) ≤ β1 1 − 10A2 M      ˜ dµg(0) . (22.78) =− t¯ R + 2∆f − |∇f |2 + f − n H M

On the other hand, 

˜ dµg(0) 2∆f − |∇f |2 H M        ˜ dµg(0) = 2∆ f˜ + log h − |∇ f˜ + log h |2 H M    ∆h 2 ˜ ˜ dµg(0) − 2|∇ log h| H (22.79) = 2∆f + 2 h M      ˜2 2 ˜ ˜ dµg(0) . − ∇f  + 2∇f , ∇ log h + |∇ log h| H M

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204

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Regarding the ∆h term above, we have (using ∇H = −H ∇f )   ∆h ˜ H dµg(0) = (∆h)H dµg(0) M h M = ∇h, ∇f H dµg(0) M  ˜ dµg(0) = ∇ log h, ∇f H M   ˜ ˜ ˜ dµg(0) . (22.80) = ∇ log h, ∇f H dµg(0) + |∇ log h|2 H M

M

Applying (22.80) to (22.79), we obtain      2

 ˜ 2 ˜ 2 ˜ ˜ dµg(0) . 2∆f − |∇f | H dµg(0) = 2∆f − ∇f  − |∇ log h| H M

M

Hence (22.78) yields (22.81)   1 β1 1 − 10A2       2   ˜ dµg(0) t¯ R + 2∆f˜ − ∇f˜ − |∇ log h|2 + f˜ + log h − n H ≤− M       ˜2 ˜ ˜ dµg(0) ¯ = −t ∇f  − f + n H M      |∇h|2 ¯ − h log h H dµg(0) , + t −R h + h M where we used integration by parts. In view of (22.81), the desired inequality (22.64) follows directly from showing that      1 |∇h|2 ¯ − h log h H dµg(0) ≤ 2 + ε2 . t −R h + (22.82) h A M Step 2. A sufficient inequality to prove (22.82). Since (22.50) and b = 10ε0 A, we have that h (·, 0) ∈ [0, 1] vanishes outside of Bg(0) (x0 , 20ε0 A). By the assumption that R ≥ −1 in Bg(0) (x0 , 1) and 20ε0 A ≤ 1, we have   ¯ ¯ R h H dµg(0) ≤ t H dµg(0) ≤ ε2 , (22.83) −t M

Bg(0) (x0 ,1)

where we used t¯ ≤ ε2 .

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3. UPPER BOUND FOR THE LOCAL ENTROPY


B

v dµ

205

Using the definition (22.44) of h and the inequality (φ (s))2 ≤ 10φ (s), we compute   2 |∇h| (φ (s (x, 0)))2 1 H dµg(0) = t¯ H dµg(0) t¯ 2 h M M φ (s (x, 0)) b 10ε2 1 (22.84) ≤ 2 = . b 10A2
It remains to estimate the integral M −h log h H dµg(0) . We have by definition h log h = 0 when h = 0 or h = 1 and we have −h log h ≤ e−1 ≤ 1 for all h ≥ 0. Recall that if h (x, ·) ∈ (0, 1), then by (22.49), 10ε0 A ≤ dg(0) (x, x0 ) ≤ 20ε0 A. Thus



 −

M

h log h H dµg(0) ≤

H dµg(0) Bg(0) (x0 ,20ε0 A)−Bg(0) (x0 ,10ε0 A)



≤1−

(22.85)

H dµg(0) . Bg(0) (x0 ,10ε0 A)

The result now follows from the claim that   ˜ (x, 0) H dµg(0) (x) ≥ 1 − H dµg(0) ≥ h (22.86) Bg(0) (x0 ,10ε0 A)



˜ : M × 0, ε where h

 2

M

9 , 10A2

→ [0, 1] is defined, similarly to (22.44), as  √  dg(t) (x, x0 ) + 200n t ˜ t) = φ , h(x, ˜b

˜ · , 0) ⊂ Bg(0) (x0 , 10ε0 A). Indeed, (22.82) where ˜b  5ε0 A. Note that supp h( follows from combining (22.83), (22.84), (22.85), and (22.86). Step 3. Completion of the proof of part (2) of Lemma 22.13. Now it remains to prove (22.86). Assuming that A ≥ 67n, by (22.47) we have ˜ t) ˜ (x, t) ≤ 10 h(x,
h ˜b2   on M × 0, ε2 . We compute     d ˜ dµg(t) = ˜ −h ˜
∗ H dµg(t) Hh H
h dt M M ˜ dµg(t) = H
h M  10 ˜ dµg(t) , ≤ Hh ˜b2 M

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206

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

where the first equality is by integration by parts. Hence     2   10ε2 − 10ε 2 ˜ ˜ ˜  H h dµg(t)  ≥e b H h dµg(t)  ≥ e− ˜b2 M

M

t=0

since t¯ ≤ ε2 and since

t=t¯

 M

˜ dµg(t¯) = h ˜ (¯ Hh x, t¯) = 1,

where for the first equality we used that H satisfies (22.55b) and for the second equality we used A ≥ 67n. We have proved    10ε2 2 ˜ dµg(t)  Hh ≥ 1 − ≥1− ,  ˜b2 5A2 M

t=0

which implies (22.86) as desired. This completes the proof of (22.64). 4. Logarithmic Sobolev inequality via the isoperimetric inequality Motivated by Theorem 22.15 below, we make the following definition. Definition 22.14 (Logarithmic Sobolev inequality). We say that a Riemannian manifold (Mn , g) satisfies the logarithmic Sobolev inequality if there exists a constant CP > −∞ such that    1 2 |∇f | + f − n u dµ ≥ CP (22.87) M 2
for all u  (2π)−n/2 e−f such that M u dµ = 1. Given any ψ ∈ W 1,2 (M), define f by   −n/2 −f 2 2 e ψ dµ = u ψ  (2π) M

ψ 2 dµ. M

From this it is easy to see that (22.87) is equivalent to   

2 2 2 2 (22.88) ψ dµ ψ 2 dµ 2|∇ψ| − ψ log ψ dµ + log M M M  n log(2π) + n + CP ψ 2 dµ. ≥ 2 M The following sharp logarithmic Sobolev inequality was proved for Euclidean space by Gross [79] (for Euclidean space we have CP = 0 in (22.87); see (6.70) in Part I). Theorem 22.15 (Euclidean logarithmic Sobolev inequality). For any function ϕ on Rn , we have   

ϕ2 dµRn ϕ2 dµRn 2|∇ϕ|2 − ϕ2 log ϕ2 dµRn + log n n n R R R  n log(2π) + n (22.89) ≥ ϕ2 dµRn . 2 Rn

W 1,2

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4. LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY

207

In this section we prove a sharp form of the logarithmic Sobolev inequality, relating to the isoperimetric inequality. The method of proof is to apply spherical symmetrization (also called Steiner symmetrization) and to reduce the inequality to (22.89). The following result is due to Perelman; see also Proposition 4.1 in [139] by one of the authors for the proof in the In = cn case. Theorem 22.16 (Logarithmic Sobolev via isoperimetric inequality). Let be a complete Riemannian manifold which satisfies an isoperimetric inequality; that is, there exists a constant In ∈ (0, ∞) such that (Mn , g)

(Area(∂Ω))n ≥ In (Vol(Ω))n−1

(22.90)

for any compact domain Ω ⊂ M whose boundary is C 1 . Then on (M, g) we have the logarithmic Sobolev inequality   

2 2 2 2 ψ dµ ψ 2 dµ 2|∇ψ| − ψ log ψ dµ + log M M M     In (22.91) ψ 2 dµ, ≥ sn + log cn M where ψ is any W 1,2 function on M, sn  n2 log(2π) + n, and cn = nn ωn is the isoperimetric constant of Euclidean space Rn . Remark 22.17. A general idea relating isoperimetric inequalities and Sobolev inequalities is to consider the level sets of functions as the boundaries of the superlevel sets of these functions. Technically this is facilitated by the co-area formula, which holds for Lipschitz functions. Proof. Observe that the isoperimetric inequality (22.90) is invariant  2/n g, we under scalings of the metric g. For the rescaled metric g˜  cInn see that (22.91) is equivalent to7 (22.92)  2/n     2

cn 2 2 2 ˜ 2 |∇ψ| d˜ µ− ψ log ψ d˜ ψ d˜ µ ψ 2 d˜ µ µ + log In M M M M  ψ 2 d˜ µ, ≥ sn M

where d˜ µ denotes the volume form of g˜. By replacing ψ by |ψ| and using an approximation argument, it suffices to prove (22.92) under the assumption that ψ is a nonnegative function in C 1 with compact support. More precisely, we may first approximate a 7

This follows easily from the facts that if g˜
d˜ µ=

cn dµ In

and



cn In

2/n

g, then  −2/n ˜ 2
|∇ψ| ˜ 2g˜ = cn |∇ψ| |∇ψ|2 . In

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208

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

W 1,2 function by a C 1 function ψ, then replace ψ by |ψ| since their weak derivatives are a.e. the same up to a sign, and then approximate |ψ| by a nonnegative C 1 function with compact support. For s > 0 we define Ms  {x ∈ M : ψ (x) ≥ s} , Γs  ∂Ms , and

F (s)  Volg˜ (Ms ) . Since ψ has compact support, there exists r0 < ∞ such that Volg˜ ({x ∈ M : ψ (x) > 0}) = ωn r0n = Vol (B (r0 )) , where B (r0 ) ⊂ Rn is the ball of radius r0 centered at the origin. Clearly the function F : (0, ∞) → [0, ωn r0n ] is nonincreasing. Let h : Rn → R be a nonnegative rotationally symmetric function such that (22.93)

Vol ({y ∈ Rn : h (y) ≥ s}) = F (s)  Volg˜ ({x ∈ M : ψ (x) ≥ s})

for all s > 0 and h (y) = 0 when |y| ≥ r0 . It is clear that there exists a unique such function and that h (|y|)  h (y) is nonincreasing in |y|. We define Ms  {y ∈ Rn : h (y) ≥ s} and Γs  ∂Ms . By definition, we have

Volg˜ (Ms ) = F (s) = Vol Ms for all s > 0. Since Ms is a round ball in Euclidean space, we have

n−1

n = cn Vol(Ms ) Area Γs = cn (Volg˜ (Ms ))n−1 cn ≤ (Areag˜(Γs ))n , In where we have used the isoperimetric inequality (22.101) below to obtain the last inequality. We have shown for s > 0 that  1/n 

cn Areag˜ (Γs ). (22.94) Area Γs ≤ In Since integration by parts holds for Lipschitz functions (see Lemma 7.113 in Part I), for any Lipschitz function λ : [0, ∞) → R with λ (0) = 0 we have  ∞  ∞ dF dλ (s) F (s) ds = − (s) ds, λ (s) (22.95) ds ds 0 0 where we have used F (s) = 0 for s > supM ψ. Applying the co-area formula (see Lemma 5.4 in [45] for example)   ∞     ˜  H ∇f  d˜ µ= H d˜ σ ds M

−∞

{f =s}

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4. LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY

   ˜ −1 to Mt with H = ∇ψ  and f = ψ, we have   ∞

F (t) = Volg˜ (Mt ) =

{ψ=s}

t

209

    ˜ −1 σ ds, ∇ψ  d˜

where d˜ σ is the volume (n − 1)-form of {x ∈ M : ψ (x) = s}, and    dF  ˜ −1 (s) = σ for a.e. s > 0. − ∇ψ  d˜ ds {ψ=s} Hence we have (22.96)



∞

− 0

dF (s) ds = λ (s) ds



λ (s) {ψ=s}

0

Again by the co-area formula   λ (ψ) d˜ µ= (22.97) {x∈M : ψ(x)>0}



∞

   ˜ −1 σ ds. ∇ψ  d˜



∞

λ (s) {ψ=s}

0

   ˜ −1 σ ds. ∇ψ  d˜

Combining (22.95), (22.96), and (22.97), we have   ∞ dλ (s) F (s) ds = λ (ψ) d˜ µ. ds 0 {x∈M : ψ(x)>0} For the same reason, we have (see Exercise 22.18 below)   ∞ dλ (s) F (s) ds = λ (h) dµRn , (22.98) ds 0 {y∈Rn : h(y)>0} and hence



 λ (ψ) d˜ µ=

(22.99) {x∈M : ψ(x)>0}

{y∈Rn : h(y)>0}

λ (h) dµRn .

Choosing λ (s) = log s2 s2 + sn s2 , where sn = n2 log(2π) + n, the above inequality implies 



log ψ 2 ψ 2 + sn ψ 2 d˜ µ M 



log ψ 2 ψ 2 + sn ψ 2 d˜ µ = {x∈M : ψ(x)>0} 



log h2 h2 + sn h2 dµRn = {y∈Rn : h(y)>0} 



= log h2 h2 + sn h2 dµRn .


Note that we have Claim. 

Rn

Mψ

cn In

2 d˜ µ

=

2/n  M


Rn

h2 dµRn by choosing λ (s) = s2 in (22.99).

    ˜ 2 µ≥ ∇ψ  d˜

Rn

|∇h|2 dµRn .

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210

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Assuming this claim and using  2




Mψ

2 d˜ µ

=


Rn

h2 dµRn , we have

2/n 



˜ 2 d˜ |∇ψ| µ− ψ 2 log ψ 2 d˜ µ M M    2 2 ψ d˜ µ ψ d˜ µ − sn ψ 2 d˜ µ + log M M M  

|∇h|2 dµRn − h2 log h2 dµRn ≥2 n Rn  R  2 2 + log h dµRn h dµRn − sn h2 dµRn cn In

Rn

≥ 0,

Rn

Rn

where we have used (22.89) to obtain the last inequality. Hence (22.92) is proved modulo the claim. Finally we give a proof of the claim. By the co-area formula, we have  ∞  ∞ 1 1   d˜ dσRn ds. σ ds = F (t) =   |∇h|  ˜  t t Γs ∇ψ Γs Hence  (22.100) Γs

1   σ=  ˜  d˜ ∇ψ 

 Γs

1 dσRn . |∇h|

Since h is rotationally symmetric and Γs is a round sphere centered at the origin, we have by (22.94),   1 dσRn |∇h| dσRn · |∇h|   Γs Γs

2 = Area Γs  2/n cn ≤ (Areag˜ (Γs ))2 In  2/n     cn 1 ˜    d˜ ≤ σ, ∇ψ d˜ σ ·    In ˜  Γs Γs ∇ψ where we used the H¨older inequality to obtain the last inequality. By this and (22.100), we have 

 Γs

|∇h| dσRn ≤

cn In

2/n  Γs

  ˜  σ. ∇ψ  d˜

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5. NOTES AND COMMENTARY

211

The claim now ensues from the following consequences of the co-area formula:  ∞    ˜  2 ˜ |∇ψ| d˜ µ= σ ds, ∇ψ  d˜ 0 M Γs  ∞  2 n |∇h| dµR = |∇h| dσRn ds. Rn

0

Γs

This also completes the proof of the theorem.




Exercise 22.18. Prove (22.98). The following result has been used in deriving the contradiction in the proof of the pseudolocality Theorem 21.9. Theorem 22.19. Let (Mn , g) be a Riemannian manifold and suppose ¯ B (x0 , ρ) is compact. If there exists a constant In ∈ (0, ∞) such that (22.101)

(Area(∂Ω))n ≥ In (Vol(Ω))n−1

for any compact domain Ω ⊂ B (x0 , ρ) whose boundary is C 1 , then for any C 1 function ψ compactly supported in B (x0 , ρ) we have   

2 2 2 2 ψ dµ ψ 2 dµ 2|∇ψ| − ψ log ψ dµ + log M M M     In (22.102) ψ 2 dµ. ≥ sn + log cn M Proof. By approximation, we may assume that ψ is a nonnegative C 1 function compactly supported in B (x0 , ρ). With this, the proof of Theorem 22.16 applies without change. Note that since supp (ψ) ⊂ B (x0 , ρ), we have
Ms ⊂ B (x0 , ρ) for s > 0. 5. Notes and commentary §4. For a classical application of spherical symmetrization to the proof of the Faber–Krahn inequality (originally conjectured by Rayleigh), see §III.3 of Chavel [28]. Backward uniqueness and unique continuation. Recently, related to work of Alexakis [2] (see also Alexakis–Ionescu–Klainerman [3] and Wong–Yu [190]) and using Carleman-type estimates, Kotschwar [112] proved the following. Theorem 22.20 (Backward uniqueness for solutions of Ricci flow). If g1 (t) and g2 (t) are two complete solutions of the Ricci flow with bounded curvature on a manifold Mn and time interval [0, T ] such that g1 (T ) = g2 (T ), then g1 (t) = g2 (t) for all t ∈ [0, T ]. As a consequence, we have the following answer to a question of Arthur Fischer (see also Problem 4.21 in [45]).

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212

22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Corollary 22.21 (Isometry group is preserved under Ricci flow). If t ∈ [0, T ], is a complete solution to the Ricci flow with bounded curvature, then Isom (g (t)) = Isom (g (0)) for all t > 0.

(Mn , g (t)),

Indeed, Theorem 22.20 implies that Isom (g (t)) ⊂ Isom (g (0)) for t > 0, whereas the uniqueness theorems of Hamilton (for closed manifolds) and Chen–Zhu [37] (for complete solutions with bounded curvature on noncompact manifolds) imply Isom (g (0)) ⊂ Isom (g (t)) for t > 0. Recall that, by result of Bando [10], a complete solution (Mn , g (t)), t ∈ [0, T ), to the Ricci flow with bounded curvature is real analytic in the space variables for any t ∈ (0, T ) (see §2 of Chapter 13 in Part II). This implies that if (Mni , gi (t)), i = 1, 2, are complete solutions with bounded curvature such that g1 (t ) and g2 (t ) restricted to some pair of open sets  are isometric for some  t ∈(0, T ), then the universal covers with the lifted  4n , g˜2 (t ) are isometric (see the original §4 of 4n , g˜1 (t ) and M metrics M 1 2 Myers [136] or Corollary 6.4 of Kobayashi and Nomizu [111]; the case of Einstein metrics is Corollary 5.28 in Besse [15]). By forward and backward uniqueness, this implies g˜1 (t) and g˜2 (t) are isometric for all t ∈ [0, T ). Regarding the property of unique continuation, one may also ask the following question. Can one define a ‘canonical’ notion of solution of the Ricci flow with surgery on closed 3-manifolds whose properties include the following? Criterion 22.22 (For Ricci flow with surgery on closed 3-man canonical ifolds). If M31 (t) , g1 (t) and M32 (t) , g2 (t) are solutions of the Ricci flow with surgery on the time interval [0, T ], where the manifolds M31 (t) and M32 (t) are closed at all nonsurgery times, where M31 (0) and M32 (0) are  g2 (t ) connected, and such that for some t ∈ (0, T ] we have that  g1 (t ) and  43 (t) , g1 (t) and restricted to some pair of open sets are isometric, then M 1   3 4 M2 (t) , g2 (t) must be isometric for all t ∈ [0, T ]. (Note that g1 and g2 are singular metrics at the surgery times.) This would have the following consequence. If Ricci flow with surgery evolves a dumbbell shaped 3-sphere into two (disjoint) 3-spheres with a single surgery occurring at some time t∗ ∈ (0, T ), then the metric at any time t > t∗ on an open subset of one of the two 3-spheres ‘uniquely determines’ the metric at all times after the surgery time on the other 3-sphere. Finally, we remark that on p. 3 of [152], when speculating on the ‘Wilsonian picture’ of renormalization group (RG) flow, Perelman wrote: “In this picture, t corresponds to the scale parameter; the larger is t, the larger is the distance scale and the smaller is the energy scale; to compute something on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale. In other words, decreasing of t should correspond to looking at our

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5. NOTES AND COMMENTARY

Space through a microscope with higher resolution, where Space is now described not by some (Riemannian or any other) metric, but by an hierarchy of Riemannian metrics, connected by the Ricci flow equation. Note that we have a paradox here: the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale; moreover, if we allow Ricci flow through singularities, the regions that are in different connected components at larger distance scale may become neighboring when viewed through microscope.”

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213

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http://dx.doi.org/10.1090/surv/163/07

CHAPTER 23

Heat Kernel for Static Metrics Got a good reason for taking the easy way out now. – From “Day Tripper” by The Beatles

In this chapter we discuss the heat kernel on a compact manifold with a fixed metric. In the next chapter we consider heat-type kernels on compact and noncompact manifolds with time-dependent metrics. The two main issues which we shall discuss are that of existence and asymptotic expansions for short time. Let

(23.1) R2> = (t, u) ∈ R2 : t > u . Definition 23.1 (Fundamental solution to heat equation). Let (Mn , g) be a complete Riemannian manifold. We say a function h : M × M × R2> → R is a fundamental solution (to the heat equation) if (1) h is continuous, C 2 in the first two space variables, and C 1 in the last two time variables, (2) changing notation by h (x, t; y, u)  h (x, y, t, u),     ∂ ∂ − ∆x h ( · , · ; y, u) = 0, + ∆y h (x, t; · , · ) = 0, (23.2) ∂t ∂u (3) (23.3)

(23.4) (23.5)

lim h ( · , t; y, u) = δy ,

lim h (x, t; · , u) = δx ,

tu

ut

i.e., for any continuous function f on M with compact support we have  h (x, t; y, u) f (x) dµ (x) = f (y) , lim tu M  h (x, t; y, u) f (y) dµ (y) = f (x) . lim ut M

Remark 23.2. In fact, for this definition, we do not need to assume that (M, g) is complete; however, for the applications we are interested in, (M, g) shall be complete. 215 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

216

23. HEAT KERNEL FOR STATIC METRICS

We say that a fundamental solution H to the heat equation is the minimal positive fundamental solution if H is positive and if for every positive fundamental solution h we have h ≥ H. The minimal positive fundamental solution, which we shall show always exists, is also called the heat kernel. By definition, the minimal positive fundamental solution is unique. In this chapter we shall present a proof of the following classical result. Theorem 23.3 (Existence of the heat kernel on a closed manifold). If (Mn , g) is a closed Riemannian manifold, then there exists a fundamental solution H (x, t; y, u) to the heat equation. Moreover, H (x, t; y, u) is unique, positive, C ∞ , symmetric in x and y, and  H (x, t; y, u) dµ (x) ≡ 1. (23.6) M

In particular, H (x, t; y, u) is the heat kernel. Remark 23.4. For this theorem we only discuss existence and smoothness. Uniqueness, positivity, and symmetry were discussed earlier in Theorem E.9 and Lemma E.11, both in Part II. In the first three sections we discuss the existence of the heat kernel on a closed Riemannian manifold. In §1 we start with a good approximation to the heat kernel. In §2 we construct the heat kernel by establishing the convergence of the ‘convolution series’. In §3 we prove some results on differentiating convolutions used in the previous section. In §4 we discuss aspects of the asymptotics of the heat kernel. In §5 we recall some elementary facts used earlier. 1. Construction of the parametrix for the heat kernel on a Riemannian manifold In this section we discuss the construction of the ‘parametrix’ (good approximation) for the heat kernel on a closed oriented Riemannian manifold. In the next section we use this to discuss the existence of the heat kernel. The manner of our presentation is intended to facilitate the later adaptation to the evolving metric case. In this section (Mn , g) shall denote a closed Riemannian manifold and d (x, y) shall denote the Riemannian distance between points x and y in M. 1.1. First approximation to the heat kernel on a manifold — transplanting the Euclidean heat kernel. Consider the function E : M × M × R2> → (0, ∞)

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

(R2> ⊂ R2 is given by (23.1)) defined by



d2 (x, y) exp − (23.7) E (x, y, t, u)  (4π (t − u)) 4 (t − u) Often we shall switch notation for the variables and write −n/2

217

 .

E (x, t; y, u) = E (x, y, t, u) . = En is Euclidean space, the function E is equal In the case where to the Euclidean heat kernel. Let inj (g) = inf x∈M injg (x) ∈ (0, ∞) denote the injectivity radius of g and let (Mn , g)

(23.8)

Minj(g)  {(x, y) ∈ M × M : d (x, y) < inj (g)} .

Note that E restricted to Minj(g) × R2> is C ∞ since d2 (x, y) is C ∞ in Minj(g) (while, on all of M × M × R2> , we only have that E is Lipschitz). In general, we think of E as a ‘transplanted heat kernel’, which, for (x, y) near the diagonal of M×M and for t−u small, is a first approximation to the heat kernel of (M, g) that we are seeking to construct. In the next subsection we construct a good approximation. 1.2. Constructing a good approximation to the heat kernel — multiplying E by a finite series. To improve our approximation to the sought after heat kernel on (Mn , g), we multiply E by a finite series in the time variable with coefficients which are functions on Minj(g) . In particular, for a given N ∈ N with N > n/2, we shall define a function HN : Minj(g) × R2> → R of the form1 (23.9) HN (x, y, t, u)  HN (x, t; y, u)  E (x, t; y, u)

N 

φk (x, y) (t − u)k ,

k=0

where the functions φk : Minj(g) → R, for k = 1, . . . , N, are to be defined below. On Minj(g) × R2> , we shall show that the function HN is a good approximation to the heat kernel in a sense which we make precise in subsections 1.4 and 3.1 below. Note that since N > n/2,   2 d (x, y) N −n/2 N −(n/2) (t − u) exp − 0 < (t − u) E (x, t; y, u) = (4π) 4 (t − u) ≤ (4π)−n/2 (t − u)N −(n/2) tends to zero as t − u → 0, uniformly in x, y ∈ M. 1 Since the Laplacian is with respect to a fixed metric, in the following discussion one may take u = 0 for convenience.

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218

23. HEAT KERNEL FOR STATIC METRICS

Fix y ∈ M. We shall show the existence of the functions {φk }N k=0 in Lemma 23.7 with φ0 (y, y) = 1

(23.10)

and so that HN satisfies the defining equation:   ∂HN (23.11) ∆x HN − (x, t; y, u) = E (x, t; y, u) (∆x φN ) (x, y) (t − u)N ∂t in Minj(g) × R2> , where ∆x denotes the Laplacian with respect to the x variable. Remark 23.5. The reason for the choice of the form of the rhs of (23.11) will become apparent from the derivation of (23.23b) (note, in particular, the case k = N of (23.23b)). We now compute the ode for the φk which we derive from equation (23.11) with the initial values (23.10).2 In doing so, we shall see that φk is C ∞ for 0 ≤ k ≤ N . Since E is a radial function in the space variable x centered at y, we consider the equations for φk (derived from the equation for HN ) on geodesics emanating from y. Given our y ∈ M, let r (x)  d (x, y) and define the normal coordinates volume density (or Jacobian) α : Minj(g) → (0, ∞) by



(23.12)

α (x, y) 

det g S (x)

r (x)n−1

,

  S (x) and where det g S (x) = det gij

S  g ∂/∂θi , ∂/∂θj (23.13) gij

n are the components of the metric in geodesic spherical coordinates θi i=1 centered at y.3 Note that (23.14)

lim α (x, y) = 1  α (y, y) ,

x→y

so that α is well defined along the diagonal of Minj(g) . n Let xi i=1 be exponential normal coordinates centered at y. We have  (23.15) α (x, y) = det (gk
) (x), By the Ansatz (23.9), the expansion in powers of t − u for (23.11) termwise vanishes. For a discussion of geodesic spherical coordinates, see subsection 10.2 of Chapter 1 in [45]. 2 3

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

219



where gk
 g ∂/∂xk , ∂/∂x
. Therefore α is C ∞ on Minj(g) . Moreover, within the cut locus of y, the volume form may be expressed in positively oriented normal coordinates as dµ (x) = α (x, y) dx1 ∧ · · · ∧ dxn . Remark 23.6 (Gauss lemma). Let x ∈ M be a point within the cut ∂ locus of y. If we let ei = ∂x i (y), which is an orthonormal basis of Ty M,

∂ then we may write ∂xi (x) = d expy exp−1 (x) (ei ). The Gauss lemma says y that ∂γ (r (x)) , ∇r (x) = ∂r where γ : [0, r (x)] → M is the unique minimal unit speed geodesic from y to x. That is, 6 ( ) 5 n n  ∂r ∂ ∂γ ∂ xj ∂ xj xi = (r (x)) , gji , (x) = , = = r ∂xi ∂r ∂xi r ∂xj ∂xi r j=1

j=1

or equivalently, xi =

(23.16)

n 

xj gji .

j=1

Recall that the Laplacian, in the x variable, of a radial function f (x, y)  f (r (x)) is given inside the cut locus Cut (y) (in particular, for points x ∈ M such that d (x, y) < inj (g)) by  d2 f ∂ df log det g S + 2 dr ∂r dr   n − 1 ∂ log α df d2 f + , = 2 + dr r ∂r dr

∆x f = (23.17)

∂ is the unit radial vector field, in the x variable, defined in Minj(g) where ∂r  ∂ (note that ∂r log det g S is the same as the mean curvature of the sphere of radius r centered at y; see (1.132) and (1.135) in [45] for example). Since   2 ∂ n−1 ∂ ∂ − E ( · , · , y, u) = 0 + ∂r2 r ∂r ∂t

for fixed (y, u) ∈ M × (−∞, ∞), inside the cut locus of y the heat operator on M applied to E ( · , · , y, u) is given by   ∂ log α ∂E r ∂ log α ∂ E= =− E. (23.18) ∆x − ∂t ∂r ∂r 2 (t − u) ∂r

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23. HEAT KERNEL FOR STATIC METRICS

Note that for r < inj (g) we have that α is bounded and that  ∂ n−1 ∂ log α = log det g S − = o (1) (23.19) ∂r ∂r r for r near 0.4 Applying the heat operator to (23.9) and using the defining equation (23.11) and also (23.18) yields (23.20) E · ∆x φN · (t − u)N ∂HN = ∆x HN − ∂t     N ∂ E φk (t − u)k = ∆x − ∂t k=0

 ∂ log α  r E φk (t − u)k + 2 (t − u)k ∇x E, ∇φk  =− 2 (t − u) ∂r N

N

k=0

+E

N 

k=0

∆x φk · (t − u)k − E

k=0

N 

kφk (t − u)k−1 .

k=0

Note that ∇x E = −

(23.21)

∂ r E ∈ Tx M. 2 (t − u) ∂r

We may thus rewrite (23.20) as, after factoring out E and cancelling the term on the lhs,  ∂φk r ∂ log α  φk (t − u)k−1 − (t − u)k−1 r 2 ∂r ∂r N

0=−

N

k=0

+

N −1 

k=0

∆x φk · (t − u)k −

k=0

N 

kφk (t − u)k−1 ;

k=0

that is, by grouping like terms, we have   1 ∂ log α ∂φ0 −1 (23.22) 0 = − (t − u) r φ0 + 2 ∂r ∂r   N  r ∂ log α ∂φk − φk − kφk + ∆x φk−1 (t − u)k−1 . −r + ∂r 2 ∂r k=1

Thus we have established the first part of Lemma 23.7 (Recursive odes for φk ). Let (Mn , g) be a closed Riemannian manifold and let y ∈ M. 4

By definition, ϕ (x) = o (1) if limx→y

ϕ(x) r(x)

= 0.

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

221

(1) The defining equation (23.11) implies the following first-order linear ordinary differential equations along geodesics emanating from y (corresponding to the coefficients in (23.22) being zero): (23.23a) (23.23b)

1 ∂ log α ∂φ0 ( · , y) + φ0 ( · , y) = 0, ∂r  2 ∂r  r ∂ log α ∂φk ( · , y) + + k φk ( · , y) = ∆x φk−1 ( · , y) r ∂r 2 ∂r

for 1 ≤ k ≤ N . (2) With the initial data (23.10) and the assumption that the φk ’s are finite along the diagonal of M × M, the odes (23.23a)–(23.23b) on Minj(g) may be solved for smooth φk recursively in k. Then the function HN defined by (23.9) satisfies equation (23.11). We now prove part (2) of the lemma. Equations (23.23a) and (23.10) imply (23.24)

φ0 (x, y) = α−1/2 (x, y) ,

where we also used (23.14). Since α is C ∞ (and nonzero), we have that

φ0 ∈ C ∞ Minj(g) . Next we consider k ≥ 1. We rewrite the ode (23.23b) as ∂  k 1/2  r α φk = α1/2 rk−1 ∆x φk−1 , ∂r so that, for 1 ≤ k ≤ N , we have the recursive formula (using the requirement that φk is finite along the diagonal)  r(x) α1/2 rk−1 ∆x φk−1 dr, (23.25) φk (x, y) = r (x)−k α−1/2 (x, y) 0

where the integral is along the unique unit speed minimal geodesic joining x to y and where d (x, y) < inj (g). Remark 23.8. Note that φk is independent of N ≥ k. Given (x, y) ∈ Minj(g) , let V ∈ S n−1 ⊂ Ty M be the unit vector tangent to the unique minimal geodesic from y to x. Making the change of variables ρ  r/r (x), we may rewrite (23.25) as Lemma 23.9 (Recursive formula for the φk ). If for 1 ≤ k ≤ N the φk ’s are finite along the diagonal of M × M, then the solutions to (23.23b) are given by (23.26)  1  

φk (x, y) = α−1/2 (x, y) ρk−1 α1/2∆x φk−1 expy (ρ r (x) V ) , y dρ 0

for (x, y) ∈ Minj(g) .

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23. HEAT KERNEL FOR STATIC METRICS

By induction on k ≥ 0, we see that the functions φk defined recursively by (23.26) are C ∞ on Minj(g) for all 1 ≤ k ≤ N . This completes the proof of part (2) of Lemma 23.7. From (23.26) we see that along the diagonal of Minj(g) (23.27)

φk (y, y) =

1 (∆x φk−1 ) (x, y)|x=y k

for 1 ≤ k ≤ N . Note that formula (23.27) also follows directly from (23.23b) when assuming the finiteness of φk and |∇φk |.5 Now that we have constructed a good approximation, in the next subsection we establish some of its properties. 1.3. Properties of the good approximation to the heat kernel — bounds on its heat operator and derivatives. Let
x  ∆x −

∂ . ∂t

Since E restricted to Minj(g) × R2> is C ∞ and since φk : Minj(g) → R is C ∞ for 0 ≤ k ≤ N , we conclude from definition (23.9) that HN = E

N 



φk · (t − u)k ∈ C ∞ Minj(g) × R2> .

k=0



Furthermore, since φN ∈ C ∞ Minj(g) , by (23.11) and N > n/2, we may extend the function
x HN , which is defined on Minj(g) × R2> , continuously to a function
x HN defined on Minj(g) × R2> and taking the value 0 on

Minj(g) × ∂ R2> . In fact, by (23.11), since |∆x φN | is bounded, 

|
x HN | = O (t − u)

(23.28) where N −

N− n 2

n 2

  2 d (x, y) , exp − 4 (t − u)

> 0.

Next

nwe consider bounds for the space-time derivativesk of
x HN . Let i U , x i=1 be any local coordinate system in M. Let ∂x = ∂x ◦ · · · ◦ ∂x

∂ denote some k-th partial derivative in these coordinates and let ∂t
 ∂t

denote the -th time derivative for k,  ∈ N ∪ {0}. Differentiating (23.11),

we have in (U × M) ∩ Minj(g) × R2> (not explicitly writing the coefficients

Note that a model case for the ode (23.23b) is r dφ + kφ = 0, whose solutions are of dr the form φ (r) = Cr−k , where C ∈ R. In particular, the only finite solution is φ (r) ≡ 0. 5

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

223

in the second line) (23.29) ∂t
∂xk (
x HN ) (x, t; y, u) =

k
 

  coeff ·∂tq ∂xp E (x, t; y, u) ∂xk−p (∆x φN (x, y)) ∂t
−q (t − u)N

q=0 p=0

= E (x, t; y, u) (t − u)N −k−2
Fk,
(x, y, t − u)   2 d (x, y) −n/2 N −(n/2)−k−2
Fk,
(x, y, t − u) (t − u) exp − = (4π) 4 (t − u) for N >

n 2

+ k + 2, where Fk,
∈ C ∞





(U × M) ∩ Minj(g) × [0, ∞) .

The power N − (n/2) − k − 2 of t − u in (23.29) is due to the fact that each space derivative of E introduces a (t − u)−1 factor and each time derivative of E introduces a (t − u)−2 or (t − u)−1 factor. (Recall that d2 and φN are both C ∞ functions on Minj(g) .) Given T ∈ (0, ∞), let (23.30)



R2T = (t, u) ∈ R2 : 0 < t − u ≤ T ⊂ R2> .

In particular, formula (23.29) implies that for any k,  ∈ N∪{0}, any compact subset K ⊂ U , and any T ∈ (0, ∞), we have (23.31)   2   d (x, y)  
k N −(n/2)−k−2
exp − ∂t ∂x (
x HN ) (x, t; y, u) ≤ C (t − u) 4 (t − u) ≤ C (t − u)N −(n/2)−k−2


for all (x, t; y, u) ∈ (K × M) ∩ Minj(g) × R2T . Note that as a special case (when k =  = 0), we have (23.28). The same types of formulas as (23.29) and (23.31) are true on all of Minj(g) × R2T , for any T ∈ (0, ∞), with the partial derivatives ∂xk replaced by covariant derivatives ∇kx . Lemma 23.10 (Covariant derivatives of heat operator of HN ). For any k,  ∈ N ∪ {0} and T ∈ (0, ∞), (23.32)

  2   d (x, y)  
k N −(n/2)−k−2
exp − ∂t ∇x (
x HN ) ≤ C (t − u) 4 (t − u)

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224

23. HEAT KERNEL FOR STATIC METRICS

on Minj(g) × R2T . Moreover, if N >

n 2

+ k + 2, then

(23.33) ∂t
∇kx (
x HN ) (x, t; y, u) −n/2

= (4π)

(t − u)

N −(n/2)−k−2


  2 d (x, y) ˜ Fk,
(x, y, t − u) , exp − 4 (t − u)

where F˜k,
is a C ∞ covariant k-tensor on Minj(g) × [0, ∞). Proof. Since there exists a finite collection of local coor is closed, M n

m dinate charts Uα , xiα i=1 α=1 and compact subsets Kα ⊂ Uα such that m .

Kα = M. In each chart Uα , xiα we may rewrite the components α=1   of the covariant derivatives ∂t
∇kx (
x HN ) ∂i1 , . . . , ∂ik in terms of the ∂xα

∂xα

partial derivatives ∂t
∂xj α (
x HN ) for 1 ≤ j ≤ k and the Christoffel symbols and their derivatives. Now: (1) From (23.29) we may deduce (23.33). (2) From (23.31) in each Kα we deduce (23.32).




The next step is to multiply the good approximation by a cutoff function to obtain the so-called parametrix. 1.4. Existence of a parametrix for the heat operator — multiplying the good approximation by a cutoff function. We formally define what it means for a space-time function to be a good approximation to the heat kernel. Definition 23.11 (Parametrix for the heat operator). We say that a C ∞ function P : M × M × R2> → R is a parametrix for the heat operator ∂ if ∆ − ∂t



∂ ∂ P and ∆y + ∂u P both extend continu(1) the functions ∆x − ∂t ously to M × M × R2> and (2) limtu P ( · , t; y, u) = δy and limut P (x, t; · , u) = δx , that is, for any function f ∈ C 0 (M),  (23.34a) P (x, t; y, u) f (x) dµ (x) = f (y) , lim tu M  (23.34b) P (x, t; y, u) f (y) dµ (y) = f (x) . lim ut M

Let (Mn , g) be a closed Riemannian manifold; note that since M is closed, inj (g) ≤ diam (g) < ∞. Let HN : Minj(g) × R2> → R be as defined in the previous section. Now we multiply the locally defined function HN by

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

225

a cutoff function to obtain a parametrix for the heat operator. Define the function PN : M × M × R2> → R by (23.35)

PN (x, t; y, u)  η (d (x, y)) HN (x, t; y, u) ,

where η : [0, ∞) → [0, 1] is a C ∞ radial cutoff function with  1 if s ≤ inj(g) 4 , (23.36) η (s) = 0 if s ≥ inj(g) 2 . Note that η (x, y)  η (d (x, y)) is C ∞ on all of M × M. Moreover, we may assume that η is such that there exists a constant C < ∞ depending only on inj (g) such that √ |∇x η| ≤ C η ≤ C and |∆x η| ≤ C on M × M. We also have bounds for all of the higher derivatives of η, i.e., there exist Ck < ∞ such that on M × M    k  ∇x η  ≤ Ck for k ≥ 2. Since HN is C ∞ on Minj(g) × R2> and the support of η (d (x, y)) is con

tained in Minj(g) , we conclude that PN ∈ C ∞ M × M × R2> . Note that  |PN (x, t; y, u)| dµ (y) ≤ C (23.37) M

for some constant C < ∞. Proposition 23.12 (Existence of a parametrix for the heat operator). ∂ . If N > n/2, then PN is a parametrix for the heat operator ∆ − ∂t

Proof. (0) We have already shown that PN ∈ C ∞ M × M × R2> . (1) We have (23.38)


x PN = η
x HN + (∆x η) HN + 2 ∇x η, ∇HN  .

To show that
x PN extends continuously to M × M × R2> , we estimate the rhs of (23.38) on M×M×R21 , where R2T is defined in (23.30). Observe that we may estimate
x PN in three relatively open sets covering this region as follows.

(i) In the exterior region M × M − Minj(g)/2 × R2> we have PN ≡ 0

and


x PN ≡ 0.

(ii) In the interior region Minj(g)/4 × R2> we have
x PN =
x HN . Recall that
x HN satisfies (23.28) and extends continuously to a function

defined on Minj(g) × R2> , which takes the value 0 on Minj(g) × ∂ R2> .

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226

23. HEAT KERNEL FOR STATIC METRICS



(iii) In the intermediate region Minj(g) − Minj(g)/8 × R21 , by (23.38) and (23.11), we have |
x PN | (x, t; y, u) ≤ |
x HN | + C |HN | + 2C |∇HN | ≤ CE (x, t; y, u) (t − u)N + CE (x, t; y, u) C E (x, t; y, u) + t−u   2 n (inj (g) /8) (t − u)−1 ≤ C (4π (t − u))− 2 exp − 4 (t − u)   inj (g)2 ≤ C exp − 257 (t − u)

(23.39) (23.40)

since d (x, y) ≥ inj (g) /8 and 257 > 256. The rhs of (23.39), which is independent of x and y, tends to zero as t  u. Thus we see, from considering all three cases (i)–(iii), that we may extend to M × M × R2> so that it takes the value 0 on M ×
x PN continuously 2

M × ∂ R> .

∂ P extends continuously to M × Similarly, we may show that ∆y + ∂u 2 M × R> . We leave this as an exercise. (2)(a) Let X be a topological space and let f ∈ C 0 (X × M). We shall establish (23.34a) for P = PN by showing that  PN (x, t; y, u) f (p, x) dµ (x) = f (p, y) , (23.41) lim tu M

where the convergence is uniform in (p, y) on compact subsets of X × M. We have  PN (x, t; y, u) f (p, x) dµ (x) M

=

N   k=0

 η (x, y) (4π (t  inj(g) B y, 2



r2 (x) × exp − 4 (t − u) Recall that (23.42)



− u))− 2

n



φk (x, y) (t − u)k f (p, x) dµ (x) . 

|¯ x − y¯|2 (4π (t − u))−n/2 exp − 4 (t − u) Rn

 dµE (¯ x) = 1

for any y¯ ∈ Rn and (t, u) ∈ R2> . Given 0 ≤ k ≤ N and factoring out (t − u)k , using (23.42), we note that the integral  2 n − r (x) η (x, y) (4π (t − u))− 2 e 4(t−u) φk (x, y) f (p, x) dµ (x) ≤ C B(y,inj(g)/2)

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1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL

227

is bounded independent of (p, y) in any given compact subset K of X × M. Hence, using (23.42) again, we obtain  PN (x, t; y, u) f (p, x) dµ (x) lim tu M  2 n − r (x) η (x, y) (4π (t − u))− 2 e 4(t−u) φ0 (x, y) f (p, x) dµ(x) = lim tu B(y,inj(g)/2)

= f (p, y) since limx→y φ0 (x, y) = 1, where the convergence is uniform in (p, y) ∈ K. (2)(b) We may similarly prove (23.34b).
Exercise 23.13. Prove that

∂ P extends continuously to M × M × R2> , (1) ∆y + ∂u (2) PN satisfies (23.34b). Again, by considering the three cases (i)–(iii) as in part (1) of the proof of Proposition 23.12, from (23.29), we have Lemma 23.14 (Derivatives of heat operator of PN ). For any k,  ∈ N ∪ {0}, (23.43) ∂t
∇kx (
x PN )(x, t; y, u) = (t − u)N − 2 −k−2
e n

d2 (x,y)

− 5(t−u)

Gk,
(x, y, t − u) ,

where ∇kx  ∇x ◦ · · · ◦ ∇x (k times) and Gk,
is a C ∞ covariant k-tensor on M × M × [0, ∞) (note that Gk,
has support in Minj(g) × [0, ∞)).6 In particular (k =  = 0),    2 d (x, y) N− n . (23.44) |
x PN | (x, t; y, u) = O (t − u) 2 exp − 5 (t − u) Proof. By (23.33) we have the representation (23.43) for (x, y, t, u) ∈ M inj(g) × R2> ; recall that η (s) = 1 if s ≤ inj(g) 4 . Recall also that PN ≡ 0 in 4

2 M × M − Minj(g)/2 × R> . Thus we only need to consider the case where (x, y) ∈ Minj(g) − Minj(g)/8 . In this region, one uses (23.38) and the fact that   (inj (g) /8)2 · (t − u)−p → 0 exp − 20 (t − u) as t − u → 0 for any p ∈ R. Since the φk are bounded in Minj(g) − Minj(g)/8 for k = 0, 1, . . . , N , by definition (23.9) we have that for (x, y, t, u) ∈ 6

Since we only need a little bit of an exponential to dominate a polynomial, in the rhs of (23.43) we may replace the factor 5 in the denominator of the exponential by 4 + ε for any ε > 0.

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228

23. HEAT KERNEL FOR STATIC METRICS



Minj(g) − Minj(g)/8 × R2T ,

   (inj (g) /8)2 d2 (x, y) exp − exp − |HN | (x, t; y, u) ≤ C (t − u) 5 (t − u) 20 (t − u)   2 d (x, y) (23.45) ≤ const · (t − u)M −(n/2) exp − 5 (t − u) −(n/2)



for any M ∈ N, where the dependence of const < ∞ includes M and T . Similarly, we have estimates of the same form (with different constants) for all the higher

covariant derivatives and time derivatives of HN in Minj(g) − Minj(g)/8 × R2T . For example, for any M ∈ N and T ∈ (0, ∞) there exists const < ∞ such that |∇x HN | (x, y, t, u)

   N    (d∇x d) (x, y)  k φk (x, y) (t − u)  ∇x φk (x, y) − = E (x, t; y, u)    2 (t − u) k=0    2 1 d (x, y) 1+ ≤ C (t − u)−n/2 exp − 4 (t − u) t−u   2 d (x, y) ≤ const · (t − u)M −(n/2) exp − 5 (t − u)

2 for (x, y, t, u) ∈ Minj(g) − Minj(g)/8 ×RT ; similarly we may estimate |∂t HN |. An easy induction leads to   2   d (x, y)  
k M −(n/2) exp − ∂t ∇x HN  (x, y, t, u) ≤ const · (t − u) 5 (t − u)

for (x, y, t, u) ∈ Minj(g) − Minj(g)/8 ×R2T , where the dependence of const < ∞ includes k, , M , and T . With this, we may then take covariant derivatives of the equation (23.38) and use the estimates for the covariant deriva-  tives of HN and η (see Lemma 23.10) to obtain bounds for ∂t
∇kx (
x PN )
which yield the existence of Gk,
satisfying (23.43). 2. Existence of the heat kernel on a closed Riemannian manifold via parametrix Let (Mn , g) be a closed Riemannian manifold. Now that we have constructed a parametrix for its heat operator, we may use a space-time convolution to obtain a fundamental solution to the heat equation. We begin with some preliminaries. 2.1. Space-time convolution and its general properties. Given two functions F, G ∈ C 0 (M × M × (0, ∞)) ,

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2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD

229

their (space-time) convolution is given by  t F (x, z, s) G (z, y, t − s) dµ (z) ds (23.46) (F ∗ G) (x, y, t)  0

M

as long as the integral is well defined. Clearly (F, G) → F ∗ G is bilinear. Since we are assuming M is closed, all of the integrals regarding convolutions of C 0 functions on M × M × [0, ∞) are finite. When one of the functions in the convolution is a parametrix, the finiteness of the resulting integrals and their first and second derivatives is an issue which we shall address. The convolution operation is associative; that is, if we are given three functions F, G, H ∈ C 0 (M × M × [0, ∞)), then (F ∗ G) ∗ H = F ∗ (G ∗ H) .

(23.47) Indeed, we compute (23.48)

((F ∗ G) ∗ H) (x, y, t)  t (F ∗ G) (x, z, s) H (z, y, t − s) dµ (z) ds = 0 M  t s  F (x, w, r) G (w, z, s − r) H (z, y, t − s) dµ (w) dµ (z) drds, = 0

0

M

M

whereas (23.49) (F ∗ (G ∗ H)) (x, y, t)  t F (x, w, v) (G ∗ H) (w, y, t − v) dµ (w) dv = 0 M  t  t−v   F (x, w, v) G(w, z, u) H(z, y, t − v − u) dµ(z) dµ(w) dudv = 0 0 M M  t  s  F (x, w, s − u) G(w, z, u) H(z, y, t − s) dµ(z) dµ(w) duds = 0 0 M M  t  s  F (x, w, r) G(w, z, s − r) H(z, y, t − s) dµ(z) dµ(w) drds. = 0

0

M

M

The associativity formula (23.47) now follows from comparing (23.48) and (23.49) while using Fubini’s theorem to change the order of integration with respect to the variables w and z. Since convolution is associative, given F ∈ C 0 (M × M × [0, ∞)) and k ∈ N, we may uniquely define F ∗k  F ∗ · · · ∗ F

(k many F ’s),

so that F ∗1 = F and F ∗2 = F ∗ F .

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230

23. HEAT KERNEL FOR STATIC METRICS

For basic properties and results about general convolution transforms, see Hirschman and Widder [100]. 2.2. The parametrix convolution series. Since PN depends only on x, y, t − u, accordingly, we also write (23.50)

PN (x, y, t − u)  PN (x, y, t, u) .

Recall from Proposition 23.12 that PN ∈ C ∞ (M × M × (0, ∞)) for N > n2 ∂ defined in (23.35) is a parametrix for the heat operator
 ∆ − ∂t . Applying the heat operator to (23.46), while using the formulas (23.93) and (23.109) below, we have for G ∈ C 0 (M × M × [0, ∞))
x (PN ∗ G) (x, y, t) = −G (x, y, t) +

 t 0

M

(
x PN ) (x, z, t − s) G (z, y, s) dµ (z) ds.

Therefore we have the following formula relating convolution and the heat operator. Lemma 23.15 (Heat operator of a convolution with the parametrix). For any G ∈ C 0 (M × M × [0, ∞)) (23.51)


x (PN ∗ G) = (
x PN ) ∗ G − G,

where PN is the parametrix defined by (23.35). Now we construct the heat kernel using convolution. We look for a heat kernel of the form (23.52)

H  PN + PN ∗ G,

where G ∈ C 0 (M × M × [0, ∞)) is a function to be determined. By (23.51) we have (23.53)


x H =
x PN + (
x PN ) ∗ G − G.

Hence
x H = 0 is equivalent to (23.54)


x PN +
x PN ∗ G − G = 0.

Making the analogy of this equation with the equation x + xy − y = 0, whose solution is ∞  x = xk , y= 1−x k=1

we see that the parametrix convolution series (23.55)

GN 

∞ 

(
x PN )∗k

k=1

is a formal solution of (23.54).

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2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD

Theorem 23.16. For N >

and GN as in (23.55), the function

H  P N + P N ∗ GN

(23.56) is contained in (23.57)

n 2

231

C ∞ (M

× M × (0, ∞)), is independent of N , and satisfies


x H =
x PN + (
x PN ) ∗ GN − GN = 0.

Moreover, H is the unique fundamental solution to the heat equation. Proof. Step 1. H solves the heat equation. Let N > n2 . By Corollary ∞ ∗k in (23.55) converges abso23.21 below, the series GN = k=1 (
x PN ) lutely, where the convergence is uniform on compact sets. In particular, GN ∈ C 0 (M × M × [0, ∞)) . Since
x PN ∈ C 0 (M × M × [0, ∞)), this implies (see Exercise 23.22 below) ∞ ∞   (
x PN )∗k = (
x PN )∗k . (
x PN ) ∗ k=1

k=2

Hence
x H =
x PN + (
x PN ) ∗ GN − GN =
x PN + (
x PN ) ∗

∞  k=1

(
x PN )∗k −

∞ 

(
x PN )∗k

k=1

= 0, which is (23.57). Step 2. H is C ∞ . Furthermore, given , m ∈ N ∪ {0} with 2 + m < N − n2 , by Lemma 23.23 below we have that ∂t
∂xm GN exists, is continuous, and ∞    ∂t
∂xm (
x PN )∗k , (23.58) ∂t
∂xm GN = k=1

where the convergence of the series on the rhs of (23.58) is absolute and uniform on compact sets. In particular, GN is m +  times differentiable in the time and first space variables on M × M × [0, ∞) for 2 + m < N − n2 . In the next step we shall show that it follows, for any N > n2 , that the space-time function ∞  (
x PN )∗k (23.59) H = P N + P N ∗ GN = P N + P N ∗ k=1

is a fundamental solution to the heat equation. Since the fundamental solution to the heat equation is unique (see Theorem E.9 in Part II), we have that the expression on the rhs of (23.59) is independent of N for N > n2 . Since we may take N arbitrarily large, from this independence of N , from PN ∈ C ∞ (M × M × (0, ∞)), and from PN ∗ GN being m +  times differentiable in space-time when 2 + m < N − n2 (exercise), we conclude that H is C ∞ .

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232

23. HEAT KERNEL FOR STATIC METRICS

Step 3. H is the fundamental solution. Recall from Proposition 23.12 that we have for any f ∈ C 0 (M)  PN (x, t; y, u) f (x) dµ (x) = f (y) . lim tu M

By (23.59), that H is the fundamental solution follows from the claim that    ∞  (
x PN )∗k (x, t; y, u) f (x) dµ (x) = 0. PN ∗ (23.60) lim tu M

k=1

Now we prove (23.60). For simplicity we take u = 0. By (23.69) and (23.70) below, we have (23.61)   n ∞    d2 (x,y) C Vol (M) tN − 2 +1  ∗k  N− n e− 5t . (
x PN )  (x, y, t) ≤ Ct 2 exp n N − 2 +1 k=1

Therefore (23.62)    ∞      (
x PN )∗k (x, y, t)  PN ∗   k=1    t  ∞     ∗k (x, z, s) (
P ) (z, y, t − s) P ≤  dµ (z) ds  N z N   0 M k=1   N− n  t  2 +1 C Vol(M)(t−s) d2 (z,y)  n n   − N− N − 2 +1 e 5(t−s)  dµ (z) ds, ≤ PN (x, z, s) C (t − s) 2 e   0 M which tends to zero as t  0 uniformly in (x, y) since N > n2 . Thus we obtain (23.60) for u = 0. The theorem now follows from Lemma 26.4.
2.3. The heat kernel expansion. We have the following. Theorem 23.17 (Heat kernel expansion). As t → 0, we have  ∞  2 d (x, y)  j −n/2 exp − t uj (x, y). (23.63) H(x, y, t) ∼ (4πt) 4t j=0

More precisely, equation (23.63) means that there exists a sequence of functions uj ∈ C ∞ (M × M) such that  N  2 d (x, y)  k −n/2 exp − t uk (x, y)  wN (x, y, t), (23.64) H(x, y, t)−(4πt) 4t k=0

with u0 (y, y) = 1 and

  n wN (x, y, t) = O tN +1− 2

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2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD

233

as t → 0, uniformly for all x, y ∈ M. Exercise 23.18. Show that for k, ˙ ≥ 0,     n  
k (23.65) ∂t ∇x wN  (x, y, t) = O tN +1− 2 −k−2
as t → 0, uniformly for all x, y ∈ M. Proof. Let uk (x, y)  η (d (x, y)) φk (x, y) . By (23.64) and (23.59), wN = H − P N = P N ∗

∞ 

(
x PN )∗k .

k=1

Moreover, by (23.62), we have    ∞      ∗k (
x PN ) (x, y, t)  PN ∗   k=1



t

≤C 0

(t − s) 

t

≤ const

N− n 2

e

N +1− n 2 C Vol(M)(t−s) N +1− n 2

 M

(t − s)

N− n 2

e

N +1− n 2 C Vol(M)(t−s) N +1− n 2

|PN (x, z, s)| dµ (z) ds

ds

0

≤ const e

N− n 2 +1 C Vol(M)t N− n +1 2

n

tN +1− 2 N + 1 − n2

n

≤ const ·tN +1− 2
for 0 < t ≤ 1 since M |PN (x, z, s)| dµ (z) ≤ const. Hence we obtain the
desired estimate for wN . 2.4. Convergence of the parametrix convolution series.  ∗k Let PN be the parametrix in (23.50). Consider the series ∞ k=1 (
x PN ) . From (23.11), (23.38), and the proof of Proposition 23.12, we can show the following. Lemma 23.19. We have (23.66)

ˆ (x, y, t) qN (x, y, t) ,
x PN (x, y, t) = tN E

where −n 2

ˆ (x, y, t)  (4πt) E



d2 (x, y) exp − 5t



and qN ∈ C ∞ (M × M × [0, ∞)).

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234

23. HEAT KERNEL FOR STATIC METRICS

Proof. Since qN ∈ C 0 (M × M × (0, ∞)) and qN (x, y, t) is C ∞ in x and t, the issue is at t = 0. Note that, regarding the first term on the rhs of (23.38), η
x HN ∈ C ∞ (M × M × [0, ∞)). From (23.9) and ∇x HN = E

N 

∇x φk tk −

k=0

r ∂ · HN , 2t ∂r

we have that the last two terms on the rhs of (23.38) may be expressed as (23.67) (∆x η) HN + 2 ∇x η, ∇x HN      2 N  r ∂φk ˆ · exp − d (x, y) − φk (∆x η) φk + 2η  · tk , =E 20t ∂r 2t k=0   which has support in M inj(g) − M inj(g) × (0, ∞). Now the fact that 2

4

qN (x, y, t) in (23.66) is C 0 at t = 0 follows from the easy fact that for any M ∈ N there exists C < ∞ independent of t such that   N      2  r ∂φk d (x, y)     − φk (∆x η) φk + 2η · tk  exp −   20t ∂r 2t k=0  

inj2 (g) −1 t + tN ≤ C exp − 320t ≤ CtM

on

  M inj(g) − M inj(g) × (0, 1), where in the last line the dependence of 2

4

C < ∞ includes M . Finally, that qN (x, y, t) in (23.66) is C ∞ at t = 0 follows from similar estimates for the derivatives of equation (23.67).
We now proceed to show that (23.66) is sufficient to imply that the series (23.55) converges. Lemma 23.20 (Estimates for the convolutions of
x PN ). Given N > n/2 and T ∈ (0, ∞), let (23.68)

C

sup M×M×[0,T ]

|qN | < ∞,

where qN ∈ C ∞ (M × M × [0, ∞)) is as in (23.66). Then for any k ∈ N we have n   C k Vol (M)k−1 tk(N − 2 +1)−1 − d2 (x,y)  ∗k  5t (23.69) (
x PN )  (x, y, t) ≤

k−1 e (k − 1)! N − n2 + 1 on M × M × (0, T ]. As a consequence of the lemma, since the series (23.70)   n ∞ N− n +1  2 n C Vol (M) t C k Vol (M)k−1 tk(N − 2 +1)−1 N−

k−1 = Ct 2 exp n N − n2 + 1 (k − 1)! N − + 1 k=1 2

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2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD

235

converges for all t ∈ R+ , we have the following. Corollary 23.21 (Convergence of the parametrix convolution series). The series ∞  d2 (x,y) e 5t · (
x PN )∗k k=1

converges absolutely and uniformly on M × M × (0, T ] for any T < ∞. Hence the series ∞  (
x PN )∗k k=1

converges absolutely and uniformly on M × M × [0, T ] for any T < ∞. We consider the convergence of the derivatives in the next subsection. Proof of Lemma 23.20. We shall prove by induction that for any k ∈ N we have (23.69). (1) k = 1. By (23.66) we have |
x PN | (x, y, t) ≤ CtN − 2 e− n

(23.71)

d2 (x,y) 5t

on M × M × [0, T ], where C is given by (23.68). (2) Suppose that (23.69) holds for some k ≥ 1. We proceed to show this estimate for k replaced by k + 1. Note that7 (d (x, z) + d (z, y))2 d2 (x, y) d2 (x, z) d2 (z, y) + ≥ ≥ . s t−s t t

(23.72)

Using this, (23.46), (23.71), and the induction hypothesis, we compute that     ∗k+1 P ) (x, y, t) (
  x N   t    (
x PN )∗k (x, z, s)
z PN (z, y, t − s) dµ (z) ds =  0

≤

 t 0

≤

 t 0

M

d2 (x,z)

C k Vol (M)k−1 sk(N − 2 +1)−1 e− 5s |
z PN (z, y, t − s)| dµ (z) ds

k−1 (k − 1)! N − n2 + 1 n

M

C k Vol(M)k−1 sk(N − 2 +1)−1 e− M (k − 1)! N − n

C k+1 Vol (M)k t(k+1)(N − 2 +1)−1 e− ≤

k k! N − n2 + 1 n

2

d2 (x,z) 5s

n 2

d2 (x,y) 5t

2

C(t − s)N − 2 e

k−1 +1 n

d2 (z,y)

− 5(t−s)

dµ(z)ds

, 2

For x, y ∈ R and a, b > 0 we have xa + yb ≥ (x+y) ; indeed, multiplying this by a + b a+b and cancelling terms, we see that this inequality is equivalent to ab x2 + ab y 2 ≥ 2xy. 7

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236

23. HEAT KERNEL FOR STATIC METRICS

where we also used (t − s)N − 2 ≤ tN − 2 and8 n  t tk(N − 2 +1) k (N − n +1)−1 2

. s ds = k N − n2 + 1 0 n

n




Estimate (23.69) now follows from induction.

We leave the reader with the following elementary exercise used in the proof of Theorem 23.16. Exercise 23.22 (Convolution and sum for
x PN commute). Show that (
x PN ) ∗

∞ 

(
x PN )∗k =

k=1

∞ 

(
x PN )∗k .

k=2

2.5. Convergence of the derivatives of the heat kernel series. We shall show that the sum of the derivatives of the terms of the series (23.55) converges. First recall some elementary facts about differentiating convolutions. Assuming that F is C m in the first space variable on M × M × [0, ∞), by taking space derivatives of (23.46), we have  t m ∂xm F (x, z, t − s) G (z, y, s) dµ (z) ds ∂x (F ∗ G) (x, y, t) = 0

(23.73)

M

= ((∂xm F ) ∗ G) (x, y, t) .

On the other hand, assuming that F is C 1 in the time variable on M × M × [0, ∞), by taking a time derivative we have  F (x, z, 0) G (z, y, t) dµ (z) + (∂t F ∗ G) (x, y, t) . ∂t (F ∗ G) (x, y, t) = M

If F (x, z, 0) = 0, then we obtain (23.74)

∂t (F ∗ G) (x, y, t) = ((∂t F ) ∗ G) (x, y, t) .

Therefore, given m and , we have for F differentiable to sufficiently high order,    (23.75) ∂t
∂xm (F ∗ G) (x, y, t) = ∂t
∂xm F ∗ G (x, y, t) provided (∂tp ∂xm F ) (x, z, 0) = 0 for 0 ≤ p ≤  − 1. 8

Another way to estimate (actually, evaluate) the time integrals of the powers of s and t − s, here and below, is to use the following formula (see p. 15 of [61]). For a, b > 0 we have  1 Γ (a) Γ (b) , (1 − σ)a−1 σ b−1 dσ = Γ (a + b) 0 where Γ is the Gamma function. Making the change of variable s = tσ, this implies  t Γ (a) Γ (b) a+b−1 t (t − s)a−1 sb−1 ds = . Γ (a + b) 0

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2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD

237

From this we see that for F satisfying the above hypothesis and k ≥ 2, we have      ∂t
∂xm F ∗k (x, y, t) = ∂t
∂xm F ∗ F ∗(k−1) (x, y, t) . The derivatives of the series (23.55) absolutely converge locally uniformly. Lemma 23.23. Let K be any compact subset of any local coordinate sys

tem U , xi and let T < ∞. Given , m ∈ N∪{0} and any N > n2 +2+m, the series ∞    ∂t
∂xm (
x PN )∗k k=1

converges absolutely and uniformly on K × K × [0, T ]. Hence ∞    ∂t
∂xm GN = ∂t
∂xm (
x PN )∗k k=1

exists and is continuous, where GN is defined in (23.55). Proof. Recall from (23.43) that N − 2 −2
−m − e ∂t
∇m x (
x PN ) (x, y, t) = t n

(23.76)

d2 (x,y) 5t

Gm,
(x, y, t) ,

where Gm,
is a C ∞ covariant k-tensor on M × M × [0, ∞). In particular, lim ∂t
∇m x (
x PN ) (x, y, t) = 0

t0

provided 2 + m < N − n2 . Thus, by (23.75) we have     (23.77) ∂t
∂xm (
x PN )∗k = ∂t
∂xm
x PN ∗ (
x PN )∗(k−1) . Now (23.76) implies there exists Cm,
< ∞ such that    d2 (x,z) n  
m  ∂t ∂x
x PN (x, z, s) ≤ Cm,
sN − 2 −2
−m e− 5s on K × K × [0, T ]. Thus, from applying (23.69) to the rhs of (23.77), we see that    
m ∗k  ∂t ∂x (
x PN )  (x, y, t)   t      ∗(k−1)
m  (z, y, t − s) dµ (z) ds ∂t ∂x
x PN (x, z, s) (
x PN ) = 0

M

C k−1 Vol (M)k−2 t(k−1)(N − 2 +1)−1

k−2 (k − 2)! N − n2 + 1  t   2   d2 (x,z) − d (z,y) N− n −2
−m −  5(t−s) × Cm,
s 2 e 5s e dµ (z) ds n

≤

0

M

2 Cm,
C k−1 t(k−1)(N − 2 +1)−1 tN − 2 −2
−m+1 k−1 − d (x,y) 5t Vol (M) e . ≤



k−2 (k − 2)! N − n2 + 1 N − n2 − 2 − m + 1 n

n

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238

23. HEAT KERNEL FOR STATIC METRICS

Hence

∞ 

  ∂t
∂xm (
x PN )∗k (x, y, t)

k=1

converges absolutely and uniformly on K × K × [0, T ].




From the above lemma we also obtain Lemma 23.24. Given T < ∞, , m ∈ N ∪ {0}, and N > series of covariant m-tensors ∞    ∗k ∂t
∇m P ) (
x N x

n 2

+ 2 + m, the

k=1

converges absolutely and uniformly on M × M × [0, T ]. Hence ∞    ∗k
m ∂t
∇m ∂ t ∇ x GN = x (
x PN ) k=1

exists and is continuous. 3. Differentiating a convolution with the parametrix In the last section, when applying the convolution to the proof of the existence of the heat kernel (see the derivation of (23.51)), we used some basic properties about differentiating a convolution with the parametrix. In this section we present these properties; an issue in proving them is that the parametrix in (23.50) is singular when t − u = 0. 3.1. Continuity of integration against the parametrix. Let (Mn , g) be a closed Riemannian manifold and let PN be the parametrix defined by (23.50). Let R2T be as in (23.30). Lemma 23.25 (Continuity of integration against the parametrix). If f : M × [0, T ] → R is a continuous function, where T ∈ (0, ∞), then IN : M × R2T → R, defined by

 IN (x, t, u) 

M

PN (x, z, t − u) f (z, u) dµ (z) ,

is a continuous function on all of its domain M × R2T and (23.78)

lim IN (x, t, u) = f (x, u)

tu

uniformly with respect to x and u. Moreover, IN is C ∞ with respect to x and t; for any  and m, 
m ∂t
∂xm PN (x, z, t − u) f (z, u) dµ (z) . (23.79) ∂t ∂x IN (x, t, u) = M

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3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX

239

We leave the proof as an exercise or see Theorem 1 on pp. 4–6 of [61] where the analogous result is proved for bounded domains in Euclidean space. 3.2. First space derivatives of a convolution with the parametrix. In the next two lemmas we shall show that we can differentiate in space (at least up to order two) a space-time convolution with a parametrix under the integral sign. The first partial derivatives of a convolution with the parametrix PN are given by the following (we obtain the same answer as (23.73) with m = 1). Lemma 23.26

n space derivatives of a convolution with the para (First metrix). Let U , xi i=1 be a local coordinate system on M. If G ∈ C 0 (M × M × [0, ∞)) , then PN ∗ G is C 1 with respect to the space variables and for x ∈ U , the first space derivatives of PN ∗ G are given by (23.80)  t ∂PN ∂ (PN ∗ G) (x, y, t) = (x, z, t − s) G (z, y, s) dµ (z) ds. i i ∂x 0 M ∂x Proof. Recall from (23.35) and (23.9) that (23.81)

PN (x, y, t)  η (d (x, y)) E (x, y, t)

N 

φk (x, y) tk ,

k=0

 2  r (x) and ry (x)  d (x, y). In particuwhere E (x, y, t) = (4πt)−n/2 exp − y4t lar,  t PN (x, z, s) G (z, y, t − s) dµ (z) ds (23.82) (PN ∗ G) (x, y, t) = 0

M

is an improper integral, with the integrand having a singularity at (z, s) = (x, 0). We address this issue below while showing that the first space derivatives of PN ∗ G exist. For any n ≥ 2, r, s > 0, and α ∈ (0, 1) we have  2  2  2  n2 −α r r −n/2 −α 2α−n r =s r exp − exp − s 4s s 4s ≤ Cs−α r2α−n , so that (23.83)

PN (x, z, s) ≤ CE (x, z, s) ≤ Cs−α rx (z)2α−n ,

where C < ∞ is independent of x, z ∈ M and s > 0. Note also that |G| is bounded on compact sets and PN (x, · , s) has compact support in

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240

23. HEAT KERNEL FOR STATIC METRICS

¯ (x, inj (g) /2). Hence, regarding the space integral in (23.82), by (23.83) B and α > 0, we have that 

 M

|PN (x, z, s) G (z, y, t − s)| dµ (z) ≤

¯ B(x,inj(g)/2) −α



inj(g)/2

≤ Cs

≤ Cs−α ,

Cs−α rx (z)2α−n dµ (z) r2α−1 dr

0

where we also used Vol (∂B (x, r)) ≤ Crn−1 for r ∈ (0, inj (g) /2]. This implies that the improper integral in (23.82) converges absolutely since
t −α ds < ∞ by α < 1. 0 s Now let  PN (x, z, t − s) G (z, y, s) dµ (z) , (23.84) JN (x, y, s, t)  M

so that  (23.85)

t

(PN ∗ G) (x, y, t) =

JN (x, y, s, t) ds. 0

By Lemma 23.25, JN (x, y, s, t) is C ∞ in the variable x and continuous in the variable s. n

Let U , xi i=1 be a local coordinate system and let x ∈ U be as in the hypothesis of the lemma. By (23.79) we have (23.86)

∂JN (x, y, s, t) = ∂xi

 M

∂PN (x, z, t − s) G (z, y, s) dµ (z) ∂xi

for x, y ∈ M and s > 0. We shall estimate this integral. Observe that9 (23.87)

∂PN (x, z, t − s) ∂xi   N  ∂ rz2 η ∂η − φk (x, z) (t − s)k =E i i ∂x 4 (t − s) ∂x k=0

+ ηE

N  k=0

∂φk (x, z) (t − s)k . ∂xi

9

In our estimates, essentially we can choose to ignore the term with support is away from the singularity.

∂η ∂xi

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since its

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX

241

In particular, (23.88)     ∂PN    ∂xi (x, z, t − s)     rz (x) rz2 (x) −n/2 ≤C 1+ (t − s) exp − t−s 4 (t − s)   n2 −α  2  n +1−α    2α−n  2 rx (z) 2 1 rx (z) rx2 (z) Crx (z) + exp − = (t − s)α t−s rx (z) t − s 4 (t − s) ≤ C (t − s)−α rx (z)2α−n−1 , where C < ∞ is independent of x, z ∈ M and s ∈ (0, t] and where we used the fact that rx (z) is bounded.

Hence, by (23.88) we have, provided α ∈ 12 , 1 ,         ∂PN  ∂JN   dµ (z)   (x, y, s, t) ≤ (x, z, t − s) G (z, y, s)     ∂xi i M ∂x  −α rx (z)2α−n−1 |G (z, y, s)| dµ (z) ≤C   (t − s) B x,

inj(g) 2

≤ C (t − s)−α ≤ C (t − s)−α

(23.89)



inj(g)/2

r2α−2 dr

0

    inj(g) N (x, · , t − s) ⊂ B x, , where C < ∞ is independent since supp ∂P 2 ∂xi of x, y ∈ M and s ∈ (0, t]. In particular, the improper integral  t 0

M

∂PN (x, z, t − s) G (z, y, s) dµ (z) ds = ∂xi



t 0

∂JN (x, y, s, t) ds ∂xi

on the rhs of (23.80) converges absolutely. By (23.85) and (23.86), proving (23.80) is equivalent to showing that the partial derivative (23.90)  t  t ∂ ∂JN ∂ (PN ∗ G) (x, y, t)  JN (x, y, s, t) ds = (x, y, s, t) ds i ∂xi ∂xi 0 0 ∂x exists; we do this now. Let γi : (−ε, ε) → M be a smooth path with γi (0) = x,

xj (γi (u)) = const for j = i,

and

d i x (γi (u)) = 1, du

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242

so that

23. HEAT KERNEL FOR STATIC METRICS d du γi

=

∂ . ∂xi

By the mean value theorem, we have

(23.91)
t
t  t ∂JN 0 JN (γi (h) , y, s, t) ds − 0 JN (γi (0) , y, s, t) ds − (x, y, s, t) ds i h 0 ∂x   t ∂JN ∂JN (γ (h ) , y, s, t) − (x, y, s, t) ds = i ∗ ∂xi ∂xi 0 for some h∗ contained in the interval from 0 to h. Note that, in regards to the integral on the rhs of (23.91), by (23.89) we have that for any ε > 0, there exists δ (ε) > 0 such that    t  ∂JN 

    ∂xi x , y, s, t  ds < ε, t−δ(ε)

x

∈ M. On the other hand, given δ (ε) > 0, there exists independent of η > 0 such that if |h| < η, then    ε  ∂JN ∂JN <  (γ (h ) , y, s, t) − (x, y, s, t) i ∗  t  ∂xi ∂xi for s ∈ [0, t − δ (ε)] since we are away from the singularity of PN . We con
t
t−δ(ε)
t + t−δ(ε) clude from splitting the integral on the rhs of (23.91) as 0 = 0 that if |h| < η, then 
t  J (γ (h) , y, s, t) ds −
t J (γ (0) , y, s, t) ds i  0 N i 0 N   h   t  ∂JN  < 3ε. (x, y, s, t) ds −  i 0 ∂x Taking the limit as h → 0, we obtain (23.90). This completes the proof of Lemma 23.26.
3.3. Second space derivatives of a convolution with the parametrix. The second partial derivatives of a convolution with the parametrix PN are given by the following (we obtain the same answer as (23.73) with m = 2). Lemma 23.27 (Second space derivatives of a convolution with the parametrix). Under the same hypotheses as Lemma 23.26, for x ∈ U such that   1 B x, inj (g) ⊂ U , 2 we have (PN ∗ G) (x, y, t) is C 2 with respect to the space variable x and the second space derivatives of PN ∗ G are given by (23.92)  t ∂ 2 PN ∂2 (P ∗ G) (x, y, t) = (x, z, t − s) G (z, y, s) dµ (z) ds, N i j ∂xi ∂xj 0 M ∂x ∂x Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX

243

where we assumed that xi are geodesic coordinates centered at x. 2 Since ∆ = g ij ∂x∂i ∂xj at the center of geodesic coordinates xi , an immediate consequence of Lemma 23.27 is the following. Corollary 23.28 (Laplacian of a convolution with the parametrix). If G ∈ C 0 (M × M × [0, ∞)), then (23.93)  t ∆x PN (x, z, t − s) G (z, y, s) dµ (z) ds. ∆x (PN ∗ G) (x, y, t) = 0

M

Proof of Lemma 23.27. By (23.79) we have  ∂ 2 PN ∂ 2 JN (x, y, s, t) = (x, z, t − s) G (z, y, s) dµ (z) . (23.94) i j ∂xi ∂xj M ∂x ∂x Now differentiating (23.87), we have ∂ 2 PN (x, z, t − s) ∂xj ∂xi     N  ∂ rz2 ∂ rz2 η ∂η 1 − φk (x, z) (t − s)k =E − 4 (t − s) ∂xj ∂xi 4 (t − s) ∂xi k=0  2  N  ∂φk ∂ rz 1 +E − (x, z) (t − s)k η j i 4 (t − s) ∂x ∂x k=0   N  ∂η ∂ rz2 ∂ 2η 1 +E − φk (x, z) (t − s)k j i j i ∂x ∂x 4 (t − s) ∂x ∂x k=0  2  N 2  ∂ r 1 η j zi +E − φk (x, z) (t − s)k 4 (t − s) ∂x ∂x k=0  2  N  ∂φk ∂ rz ∂η η +E − (x, z) (t − s)k ∂xi 4 (t − s) ∂xi ∂xj k=0   N N  ∂η  ∂φk ∂ 2 φk k k +E (x, z) (t − s) + η (x, z) (t − s) . ∂xj ∂xi ∂xj ∂xi k=0

k=0

From this we may estimate   2   ∂ PN    ∂xj ∂xi (x, z, t − s)     rz2 (x) rz2 (x) 1 −n/2 + exp − (t − s) ≤C 1+ t − s (t − s)2 4 (t − s) n       rz2 (x) 2 rz2 (x) rz2 (x) −1 −n exp − ≤ C (t − s) rz (x) 1 + t−s t−s 4 (t − s) ≤ C (t − s)−1 rz−n (x)

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244

23. HEAT KERNEL FOR STATIC METRICS

since t − s is bounded from above. However, from this estimate (contrast with (23.88)) we cannot conclude that the integral
t
∂ 2 PN (x, z, t − s) G (z, y, s) dµ (z) ds (23.95) i j 0 M ∂x ∂x converges absolutely. The idea for solving this issue is to integrate by parts, i.e., to use the divergence theorem. To accomplish this, we first need to ‘move’ the partial derivatives from x variable onto the z variable. Recall that we  the n  are assuming U , xi i=1 are geodesic coordinates centered at x ∈ U and B (x, inj (g)) ⊂ U ; in particular, supp (PN (x, · , s)) ⊂ U . Let

∂ 2 PN

∂ 2 PN (x, z, · ) ∂xi ∂xj ∂xix ∂xjx denote the originalsecond partial derivatives with respect to the x variable  in the coordinates xi and let (x, z, · ) =

∂ 2 PN ∂xiz ∂xjz

(x, z, · )

denote the second partial derivative with respect to the z variable in the  i same coordinates x centered at x, which is well defined since z ∈ U . Since supp (PN (x, · , s)) ⊂ U , we may write the expression in (23.94) as 
 2 ∂ 2 PN ∂ PN − (23.96) (x, z, t − s) G (z, y, s) dµ (z) ∂xix ∂xjx ∂xiz ∂xjz M
∂ 2 PN (x, z, t − s) G (z, y, s) dµ (z) . + j M ∂xiz ∂xz (I) We shall estimate the first term in (23.96) to show that the integral 
t
 2 ∂ 2 PN ∂ PN − (x, z, t − s) G (z, y, s) dµ (z) ds (23.97) ∂xix ∂xjx ∂xiz ∂xjz 0 M converges absolutely. (I-1) First and second derivatives of the distance function. Observe that for (x, z) ∈ Minj(g) such that x = z ∂ ∂ d (x, z) + i d (x, z) = 0. i ∂xx ∂xz  n 2 Indeed, using d2 (x, z) = j=1 xj (z) , first we compute ⎛ ⎞ n ∂ ⎝ j ∂ 2 d (x, z) = x (z)2 ⎠ = 2xi (z) , ∂xiz ∂xiz (23.98)

j=1

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3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX

245

so that xi (z) ∂ . d (x, z) = ∂xiz d (x, z) On the other hand, consider the unit speed minimal geodesic β : [0, d (x, z)] → M from z to x. The unit tangent vector of β at x (and any other point along β) is n  xj (z) ∂ β˙ (d (x, z)) = − . d (x, z) ∂xj j=1

Thus, by the first variation of arc length formula ( ) ∂ ∂ ˙ d (x, z) = β (d (x, z)) , i ∂xix ∂x x 5 n 6  xj (z) ∂ ∂ = − , d (x, z) ∂xj ∂xi j=1

=−

xi (z) d (x, z)

x

,

so that ∂ ∂ 2 d (x, z) = 2d (x, z) i d (x, z) i ∂xx ∂xx = −2xi (z) . Clearly (23.98) follows. Next we claim that we have (23.99)

∂2

d2 (x, z) − j

∂xix ∂xx

∂2 ∂xiz ∂xjz



d2 (x, z) = O d2 (x, z) .

To see this, note that ∂2 ∂xiz ∂xjz

d2 (x, z) =

∂ j 2x (z) = 2δij . i ∂xz

n On the other hand, since xi i=1 are geodesic normal coordinates centered n ∂ at x, we have Γkij (x) = 0. Letting ei = ∂x i (x), so that {ei }i=1 is an orthonormal frame at x, we compute (23.100)

∂2 ∂xix ∂xjx



d2 (x, z) = ∇ei ∇ej d2 (x, z) = 2δij + O d2 (x, z) .

Indeed, ∇ei ∇ej d2 (x, z) = 2d∇∇d (ei , ej ) + 2 ei , ∇d ej , ∇d

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246

23. HEAT KERNEL FOR STATIC METRICS

and ∇∇d (ei , ej ) = ∇∇d (ei − ei , ∇d ∇d, ej − ej , ∇d ∇d) 1 = ei − ei , ∇d ∇d, ej − ej , ∇d ∇d + O (d) d 1 = (δij − ei , ∇d ej , ∇d) + O (d) , d where we used ∇∇d (X, Y ) = d1 X, Y  + O (d) for unit vectors X and Y (see (1.133) on p. 62 of [45] for example), which imply (23.100). This completes the derivation of (23.99). (I-2) Expanding

∂ 2 PN ∂xix ∂xjx

−

∂ 2 PN . ∂xiz ∂xjz

Now differentiating (23.81) twice with

respect to both the x and z variables and taking their difference while using (23.98), we have (let u  t − s)10 (23.101) ∂ 2 PN ∂xix ∂xjx

−

∂ 2 PN ∂xiz ∂xjz

 N  d2 (x,z)  ∂d2 ∂φk ∂d2 ∂φk η (d (x, z)) −n − 4u 2 (4πu) e + (x, z) uk =− 4u ∂xix ∂xjx ∂xjx ∂xix k=0  2  N d2 (x,z)  η (d (x, z)) ∂d2 ∂φk ∂d ∂φk −n − 4u 2 (4πu) e + + j (x, z) uk i 4u ∂xiz ∂xjz ∂x ∂xz z k=0   N d2 (x,z) η (d (x, z)) ∂ 2 d2 (x, z) ∂ 2 d2 (x, z)  −n − (4πu) 2 e 4u − − φk (x, z) uk j j i i 4u ∂xx ∂xx ∂xz ∂xz k=0   N 2 2 2  d (x,z) n ∂ φ (x, z) φ (x, z) ∂ k k + η (d (x, z)) (4πu)− 2 e− 4u − uk i ∂xj i ∂xj ∂x ∂x x z x z k=0   N d2 (x,z)  n ∂d ∂φk ∂d ∂φk + η  (d (x, z)) (4πu)− 2 e− 4u + j (x, z) uk j i ∂xx ∂xx ∂xx ∂xix k=0  N  d2 (x,z)  n ∂d ∂φk ∂d ∂φk − η  (d (x, z)) (4πu)− 2 e− 4u + (x, z) uk j ∂xi ∂xiz ∂xjz ∂xz z k=0   N d2 (x,z)  ∂ 2 d (x, z) ∂ 2 d (x, z) −n 2 e− 4u + η  (d (x, z)) − φk (x, z) uk . (4πu) j j i i ∂xx ∂xx ∂xz ∂xz k=0

10

Again the reader may choose to ignore the terms involving derivatives of η since its support is away from the singularity. In (23.101) we have ordered the rhs so that the terms involving η  are at the end; by (23.98) no terms involving η  appear.

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3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX ∂ 2 PN ∂xix ∂xjx

(I-3) Estimating

−

∂ 2 PN . ∂xiz ∂xjz

247

We shall show that

(23.102)      ∂ 2P d2 (x,z) d2 (x, z) ∂ 2 PN   − 4(t−s) N −n 2 e (x, z, t − s) ≤ C 1 + (t − s) − ,    ∂xix ∂xjx ∂xiz ∂xjz  t−s where the function on the lhs of (23.102) has support in Minj(g) × (0, ∞). This implies the absolute convergence of the integral   t  2 ∂ 2 PN ∂ PN − (x, z, t − s) G (z, y, s) dµ (z) ds. ∂xix ∂xjx ∂xiz ∂xjz 0 M We now show that the absolute value of each of the seven lines on the rhs of (23.101) is bounded by the rhs of (23.102): (I-3a) Since η  (d (x, z)) ≡ 0 for d (x, z) < last three lines is bounded by C (t − s)

−n 2

e

inj(g) 4 ,

d2 (x,z) − 4(t−s)

the absolute value of the

.

(I-3b) The same is true for the fourth line since are bounded.

∂ 2 φk (x,z) ∂xix ∂xjx

and

∂ 2 φk (x,z) ∂xiz ∂xjz

In the first three lines of the rhs of (23.101) we need to control the u1 factor.   2 2  ∂ 2 d2 (x,z)  − (I-3c) Since  ∂ di (x,z)  ≤ Cd2 (x, z) by (23.99), the absolute ∂x ∂xj ∂xi ∂xj x

x

z

z

(x,z) (t − s)− 2 e value of the third line is bounded by C d t−s 2

uk

n

−

d2 (x,z) 4(t−s)

.

(I-3d) For the first two lines, when k ≥ 1 in the summation, the factor controls the u1 factor, so that these terms are bounded in absolute value

by C (t − s)− 2 e n

−

d2 (x,z) 4(t−s)

.

(I-3e) For k = 0 in the first two lines, we now show that the combination 2 ∂φ 2 0 of the ∂d ∂x ∂x terms are bounded by Cd . Let β (x)  φ0 (x, x), which is identically equal to 1. Then   ∂φ0  ∂φ0  ∂β (x) = (x, z) + (x, z) , (23.103) 0=   ∂xj ∂xjx x=z ∂xjz z=x which implies ∂φ0

(23.104)

∂xjx

(x, z) +

∂φ0 ∂xjz

(x, z) = O (d (x, z))

since the functions on the lhs of (23.104) are smooth. From this and (23.98) we see that the k = 0 terms of2 the first two lines are bounded in absolute (x,z) (t − s)− 2 e value by C d t−s 2

n

−

d (x,z) 4(t−s)

. This completes the proof of (23.102).

(II) Next we estimate the second term in (23.96). Since PN (x, ·, t − s) has compact support in B (x, inj (g)) ⊂ U , we may integrate by parts to

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248

23. HEAT KERNEL FOR STATIC METRICS

obtain (23.105) 

∂ 2 PN

(x, z, t − s) G (z, y, s) dµ (z) ∂xiz ∂xjz  ∂G ∂PN (x, z, t − s) i (z, y, s) dµ (z) =− j ∂x M ∂xz z    ∂  ∂PN (x, z, t − s) G (z, y, s) det gk
dx1z · · · dxnz , − j i ∂x M ∂xz z   √   where  ∂x∂ i det gk
 ≤ C in B (x, inj (g)). As in the proof of Lemma 23.26, z by applying (23.88) to (23.105), we have    2    ∂ PN −α (x, z, t − s) G (z, y, s) (23.106)  dµ (z) ≤ C (t − s) ,  j i   M ∂xz ∂xz 1

where α ∈ 2 , 1 . Thus the integral on the rhs of (23.92), i.e.,  t 2  t ∂ 2 PN ∂ JN (x, z, t − s) G (z, y, s) dµ (z) ds = (x, y, s, t) ds j i ∂xj i ∂x 0 0 M ∂xx ∂xx M

is the sum of two integrals each of which converges absolutely. Now by Lemma 23.26, i.e., (23.90), proving Lemma 23.27 is equivalent to showing that  t 2  t ∂JN ∂ JN ∂ (x, y, s, t) ds = (x, y, s, t) ds (23.107) i j i j ∂x 0 ∂x 0 ∂x ∂x exists. Let the path γi be as above. By the mean value theorem, we have (23.108)
t ∂JN
t ∂JN  t 2 (γ (h) , y, s, t) ds − ∂ JN i j 0 ∂x 0 ∂xj (γi (0) , y, s, t) ds − (x, y, s, t) ds i j h 0 ∂x ∂x   t 2 ∂ 2 JN ∂ JN (γ (h ) , y, s, t) − (x, y, s, t) ds = i ∗ ∂xi ∂xj ∂xi ∂xj 0 for some h
∗ contained in the interval from 0 to h. t ∂ 2 JN Since 0 ∂x i ∂xj (x, y, s, t) ds is a sum of integrals which converge absolutely, for any ε > 0, there exists δ (ε) > 0 such that   2  t  ∂ JN 

    i j x , y, s, t  ds < ε t−δ(ε) ∂x ∂x independent of x ∈ M. On the other hand, given δ (ε) > 0, there exists η > 0 such that if |h| < η, then   2  ε  ∂ JN ∂ 2 JN <  (γ (h ) , y, s, t) − (x, y, s, t) i ∗  t  ∂xi ∂xj ∂xi ∂xj

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3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX

249

for s ∈ [0, t − δ (ε)]. We conclude from splitting the rhs of (23.108) as
t
t−δ(ε)
t + t−δ(ε) that if |h| < η, then 0 = 0 
t   0  

∂JN ∂xj

(γi (h) , y, s, t) ds − h


t

∂JN 0 ∂xj

(γi (0) , y, s, t) ds 

t

− 0

  ∂ 2 JN  < 3ε. (x, y, s, t) ds  ∂xi ∂xj

Taking the limit as h → 0, we obtain (23.107). This completes the proof of Lemma 23.27.
3.4. Time derivative of a convolution with the parametrix. Now that we have formulas for the first two spatial derivatives of a convolution with a parametrix, we consider the time derivative (compare with (23.74)). Lemma 23.29 (Time derivative of a convolution with the parametrix). If G ∈ C 0 (M × M × [0, ∞)), then PN ∗ G is C 1 with respect to the time variable and (23.109)

∂ (PN ∗ G) (x, y, t) ∂t

 t

= G (x, y, t) + 0

M

∂ PN (x, z, t − s) G (z, y, s) dµ (z) ds. ∂t

Proof. Recall from (23.85) that 

t

(PN ∗ G) (x, y, t) =

JN (x, y, s, t) ds, 0

where

 JN (x, y, s, t) =

M

PN (x, z, t − s) G (z, y, s) dµ (z) .

The time-difference quotient associated to PN ∗ G is (PN ∗ G) (x, y, t + h) − (PN ∗ G) (x, y, t) h  t+h   t 1 JN (x, y, s, t + h) ds − JN (x, y, s, t) ds = h 0 0   t JN (x, y, s, t + h) − JN (x, y, s, t) 1 t+h ds JN (x, y, s, t + h) ds + = h t h 0    t  ∂ 1 t+h JN  JN (x, y, s, t + h) ds + (x, y, s, t) ds, = h t 0 ∂t t=t∗

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250

23. HEAT KERNEL FOR STATIC METRICS

where t∗ ∈ (t, t + h) also depends on x, y, s. We claim that by taking the limit as h  0, we have the fact that the right derivative is  t ∂ ∂ JN (x, y, s, t) ds (PN ∗ G) (x, y, t) = lim JN (x, y, t∗ , t + h) + + h0 ∂t 0 ∂t  (23.110) PN (x, z, t + h − t∗ ) G (z, y, t∗ ) dµ (z) = lim h0 M  t ∂ PN (x, z, t − s) G (z, y, s) dµ (z) ds + 0 M ∂t = G (x, y, t)  t ∂ PN (x, z, t − s) G (z, y, s) dµ (z) ds, + 0 M ∂t where. t∗ ∈ (t, t + h). (Exercise: Prove that we have the same formula for ∂ (PN ∗ G) (x, y, t).) ∂t− By (23.79) we have (23.111)  t  t ∂ ∂ JN (x, y, s, t) ds = PN (x, z, t − s) G (z, y, s) dµ (z) ds, ∂t ∂t 0 0 M which yields the second equality in (23.110). Since PN satisfies (23.34a), we also have the third equality in (23.110). Thus we only need to justify the first equality in (23.110), i.e.,   t  t  ∂ ∂  JN  JN (x, y, s, t) ds (x, y, s, t) ds = (23.112) lim h0 0 ∂t 0 ∂t t=t∗ ∂ JN (x, y, s, t) is uniformly continuous for s converges absolutely. First, ∂t away from t. We also have that for every ε > 0 there exists tε < t such that   t  ∂  JN (x, y, s, t) ds < ε.   ∂t tε

Second, recall by (23.44) that we have      2   n ∂P N  (x, z, t − s) = O (t − s)N − 2 exp − d (x, z)  ∆ x PN − .  ∂t  5 (t − s) Thus        ∂ ∂   JN (x, y, s, t) =  P (x, z, t − s) G (z, y, s) dµ (z) N     ∂t M ∂t     ∆x PN (x, z, t − s) G (z, y, s) dµ (z) ≤   M   2    d (x, z) N− n  G (z, y, s) dµ (z) (t − s) 2 exp − +C 5 (t − s) M ≤ C (t − s)−α + C,

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4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC

251



where α ∈ 12 , 1 and where we used (23.102) and (23.106). From the above we conclude (23.112).
This concludes our discussion of the existence of the heat kernel on a closed Riemannian manifold.

4. Asymptotics of the heat kernel for a static metric In this section, we continue to consider the case of a fixed Riemannian metric g on a closed manifold M. We wish to compute asymptotic expansions along the diagonal for the functions φk (x, y) in the finite series (23.9) for HN . We also consider an aspect of the heat kernel asymptotics related to §9.6 in Perelman [152]. In the next chapter we shall consider the Ricci flow case.

4.1. The first few terms of the asymptotic expansion of the heat kernel. We start with the computation of (the asymptotics of) φ0 in Minj(g) and of φ1 along the diagonal of M × M. By (23.24), i.e., φ0 (x, y) = α−1/2 (x, y) , and the k = 1 case of (23.27), we have     φ1 (y, y) = (∆x φ0 ) (x, y)|x=y = ∆x α−1/2 (x, y)

x=y

,

n where, in geodesic normal coordinates xk k=1 centered at y ∈ M, we have  (23.15), i.e., α (x, y) = det gk
(x). Again let r (x)  d (x, y). Recall that in the geodesic normal coordinates above, the components of the metric g have the following expansion near y (see formula (3.4) on p. 211 of [168]) (23.113) 1 1 gk
(x) = δk
− Rkpq
xp xq − ∇r Rkpq
xp xq xr 6   3   2 1 + − ∇r ∇s Rkpq
+ Rkpqm R
rsm xp xq xr xs + O r (x)5 , 20 45 where, on the rhs, the components of the curvature and its covariant derivatives are evaluated at y. From (23.113) one can show that the determinant

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252

23. HEAT KERNEL FOR STATIC METRICS

of the metric has the expansion (see formula (23.139) below) 1 1 det (gk
(x)) = 1 − δ k
Rkpq
xp xq − δ k
∇r Rkpq
xp xq xr 3 6   2   1 k
1 1 1 1 δ Rkpq
xp xq − Rkpq
xp xq Rkrs
xr xs + 2 3 2 3 3   2 1 + δ k
− ∇r ∇s Rkpq
+ Rkpqm R
rsm xp xq xr xs 20 45   5 + O r (x) , so that (see Lemma 3.4 on p. 210 of [168])

(23.114)

det (gk
(x)) 1 1 = 1 − Rpq xp xq − ∇r Rpq xp xq xr 6  3 1 1 1 ∇r ∇s Rpq + Rtpqu Rtrsu − Rpq Rrs xp xq xr xs − 20 90 18 5

+O r ,

where Rpq is the Ricci tensor. This formula implies Lemma 23.30.  α (x, y) = det (gk
) (x)   1 1 (23.115) = 1 − Rpq (y) xp xq − ∇r Rpq (y) xp xq xr + O r (x)4 , 6 12 and so φ0 has the expansion (which may be differentiated term by term)   φ0 (x, y) = α−1/2 (x, y)   1 1 (23.116) = 1 + Rpq (y) xp xq + ∇r Rpq (y) xp xq xr + O r (x)4 . 12 24 Next we consider the Laplacian of the distance (squared) function r2 = d2 ( · , y). By (23.17), the Laplacian of the distance function is given by11 n − 1 ∂ log α + . ∆x r = r ∂r Therefore  

∆x r2 (x) = 2 |∇r|2 + 2r∆x r (x) (23.117) 11

= 2n + 2r (x)

Alternatively,

 (∆x r) (x) = H (x) =

∂ log ∂r

∂ log α (x) ∂r

  S (x) , det gij

where H is the mean curvature at x of the geodesic sphere S (p, r (x)).

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4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC

253

since |∇r|2 = 1. On the derivative of (23.115) holds  hand,  the space   other 4 3 ∂ in the sense that ∂r O r (x) = O r (x) , so that    ∂ log α ∂ 1 1 4 p q p q r (x) = log 1 − Rpq (y) x x − ∇r Rpq (y) x x x + O r (x) ∂r ∂r 6 12   1 1 Rpq (y) xp xq − ∇r Rpq (y) xp xq xr + O r (x)3 =− 3r (x) 4r (x) p

p

x since ∂x ∂r = r . Hence the Laplacian of the distance squared function has the expansion

Lemma 23.31. (23.118)  

1 2 ∆x r2 (x) = 2n − Rpq (y) xp xq − ∇r Rpq (y) xp xq xr + O r (x)4 . 3 2

We compute the expansion for (∆x φ0 ) (x, y) = ∆x α−1/2 (x, y) for x near y. For any function f we have    ∂ 1 ij ∂f det g g k
∂xj det gk
∂xi   ∂ ∂f = α−1 i αg ij j , ∂x ∂x

∆f = √

so that (23.119)   −1/2  1/2    ∂ α ∂ α ∂ ∂ = −α−1 i g ij . ∆x α−1/2 = α−1 i αg ij j ∂x ∂x ∂x ∂xj We now compute the rhs of (23.119). Taking the derivative of the formula for α1/2 which is analogous to (23.116), we have (23.120)

  ∂ α1/2 1 1 3 q p q R (∇ = − (y) x − R + 2∇ R ) (y) x x + O r (x) . jq j pq p jq ∂xj 6 24 Since the inverse of the metric g −1 has the expansion   1 1 g ij = δij + Ripqj (y) xp xq + ∇r Ripqj (y) xp xq xr + O r (x)4 , 3 6

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254

23. HEAT KERNEL FOR STATIC METRICS

we have (23.121) 1/2

ij ∂ α −g ∂xj 

 3

1 p q = δij + Ripqj (y) x x + O r 3    1 1 3 q p q Rjq (y) x + (∇j Rpq + 2∇p Rjq ) (y) x x + O r (x) × 6 24   1 1 (∇i Rpq + 2∇p Riq ) (y) xp xq + O r (x)3 . = Rir (y) xr + 6 24   3 ∂ ∂ = Thus, by (23.119) and by α−1 ∂x i applied to (23.121) with ∂xi O r (x)   2 O r (x) , we have  1/2    ∂ α ∂ ∆x α−1/2 = −α−1 i g ij ∂x ∂xj   α−1 α−1 Rii + (2∇i Riq + ∇q Rii ) xq + O r (x)2 , (23.122) = 6 12 where the Ricci tensors and their derivatives on the rhs are evaluated at y and where we used ∂ 1 (∇i Rpq + 2∇p Riq ) i (xp xq ) 24 ∂x 1 1 (∇i Riq + 2∇i Riq ) xq + (∇i Rpi + 2∇p Rii ) xp = 24 24 1 (2∇i Riq + ∇q Rii ) xq . = 12 Lemma 23.32. The norm squared of the gradient and the Laplacian of φ0 = α−1/2 are given by (1)   ∂φ0 ∂φ0 2 = O r (x) , (23.123) |∇x φ0 |2 (x) = g ij i ∂x ∂xj (2) (23.124)     R (y) 1 + (∇r R) (y) xr + O r (x)2 . (∆x φ0 ) (x) = ∆x α−1/2 (x) = 6 6 Proof. (1) By (23.120) we have   1 ∂φ0 2 q R = (y) x + O r (x) . jq ∂xj 6

2

(2) This follows from applying to (23.122) the equations α = 1 + O r ,


g ij = δij + O r2 , and g ij ∇i Rjk = 12 ∇k R.

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4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC

255

Now we compute the expansion for φ1 up to first order. From (23.26) we have  1 

−1/2 (x, y) α1/2 ∆x φ0 expy (ρ r (x) V ) , y dρ, φ1 (x, y) = α 0

where V ∈ Ty M is the

unit vector tangent to the unique minimal geodesic from y to x. Let xi be geodesic normal coordinates centered at y and let i i x also denote the i-th coordinate of x. We have x expy (ρ r (x) V ) = ρxi .

By (23.124) and since α1/2 = 1 + O r2 , we have   

R (y) ρ + (∇r R) (y) xr + O ρ2 r (x)2 . expy (ρ r (x) V ) , y = 6 6   Hence, using α−1/2 (x, y) = 1 + O r (x)2 , we have 

α1/2 ∆x φ0

   1 R (y) ρ  2 2 r 2 + (∇r R)(y) x + O ρ r (x) dρ. φ1 (x, y) = 1 + O r (x) 6 6 0 

From this we immediately obtain Lemma 23.33. (23.125)

φ1 (x, y) =

  1 R (y) + (∇r R) (y) xr + O r (x)2 . 6 12

Remark 23.34. We have (23.126)

φ2 (y, y) =

1 1 1 1 ∆R + R2 − |Rc|2 + |Rm|2 , 30 72 180 180

where |Rc|2  g ik g j
Rij Rk
and |Rm|2  g ip g jq g kr g
s Rijk
Rpqrs and the rhs is evaluated at y (see Gilkey [72]). 4.2. An aspect of the heat kernel asymptotics on a Riemannian manifold. Motivated by §9.6 in Perelman [152], we consider an aspect of the heat kernel asymptotics on a fixed Riemannian manifold related to entropy. Differentiating the logarithm of (23.9), i.e., differentiating the equation N   d2 (x, y) n k + log φk t , (23.127) log HN = − log (4πt) − 2 4t k=0

in both time and space, we have d2 (x, y) n ∂ log HN = − + + ∂t 2t 4t2 =−

N k=1  N

kφk tk−1

k=0 φk t

k

d2 (x, y) φ1 n + + + O (t) 2t 4t2 φ0

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256

23. HEAT KERNEL FOR STATIC METRICS

and ∆x log HN = −

∆x





d2 (x, y) 4t

N k=0 +  N

∆x φk

k=0 φk t

tk

k

 2  N   k=0 ∇x φk tk  −  2 N k φ t k k=0

∆x d2 (x, y) ∆x φ0 |∇x φ0 |2 + =− − + O (t) . 4t φ0 φ20 Note that since, by (23.125) with x = y, we have

1 φ1 (y, y) = (∆x φ0 ) (x, y)|x=y = R (y) 6 and φ0 (y, y) = 1, summing the above two equations and evaluating at x = y yields     n 1 ∂ + ∆x log HN (x, y, t) = − + R (y) + O (t) , ∂t t 3 x=y 

  = 0 and ∆x d2 (x, y) x=y = 2n.12 where we also used |∇x φ0 |2 (x, y) x=y

Since (23.128)

 log

N  k=0

 φk tk

= log φ0 +



φ1 t + O t2 , φ0

by (23.127) we have log HN +



d2 (x, y) n φ1 log (4πt) = − + log φ0 + t + O t2 . 2 4t φ0

Thus, in general, for any x, y ∈ M, we have    log H + n log (4πt) n ∂ N 2 + ∆x log HN + log (4πt) + ∂t 2 t

−∆x d2 (x, y) + 4 log φ0 2φ1 + ∆x φ0 |∇x φ0 |2 + (23.129) = − + O (t) . 4t φ0 φ20 In particular, evaluating (23.129) along the diagonal x = y, we have (23.130) 2

12

  log H + n log (4πt) 3 n ∂ N 2  + ∆x log HN + log (4πt) +  ∂t 2 t x=y 1 n = − + R (y) + O (t) . 2t 2

On the other hand, 



∂ − ∆x log HN (x, y, t)

= O (t) . ∂t x=y

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4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC

257

More generally, for x near y and t near 0, we have    log H + n log (4πt) n ∂ N 2 + ∆x log HN + log (4πt) + ∂t 2 t

− 2n − 23 Rpq (y) xp xq + 13 Rpq (y) xp xq = 4t   1 R (y) 1 R (y) r + (∇r R) (y) x + + (∇r R) (y) xr +2 6 12 6 6     r (x)3 + O r (x)2 + O (t) +O t (23.131)

Rpq (y) xp xq 1 1 n + R (y) + + (∇r R) (y) xr 2t  2 4t 3  3   r (x) + O r (x)2 + O (t) . +O t

=−

Recall that for Euclidean space we have   2 −n/2 − |x| −n/2 (4πt) e 4t dx = π Rn

√

2

e−|˜x| d˜ x = 1, Rn

by the change of variables x ˜ = x/ 4t. Differentiating this under the integral sign, we see that    |x|2 |x|2 n − + 2 (4πt)−n/2 e− 4t dx. 0= 2t 4t Rn Hence



|x|2 n |x|2 (4πt)−n/2 e− 4t dx = . 2 Rn 4t Moreover, we also deduce from the same change of variables that (1)    |x|2 O (|x|) (4πt)−n/2 e− 4t dx = O t1/2 (23.132)

Rn

and (2)



 O

(23.133) Rn

|x|3 t

 (4πt)−n/2 e−

|x|2 4t

  dx = O t1/2 .

(3) If A = (Aij ) is a symmetric n × n matrix, then  |x|2 Aij xi xj tr (A) (4πt)−n/2 e− 4t dx = . (23.134) 4t 2 Rn Indeed, after conjugating by a rotation, we may assume that Aei = λi ei , i = 1, . . . , n, where {e1 , . . . , en } is the standard basis of Rn .

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258

23. HEAT KERNEL FOR STATIC METRICS

Then 

2 |x|2 λi xi (4πt)−n/2 e− 4t dx 4t n i=1 R 2 2 n   (xi ) λi xi −1/2 − 4t (4πt) e dxi = 4t R

 |x|2 Aij xi xj (4πt)−n/2 e− 4t dx = 4t n

Rn



i=1

=

n  λi i=1

2

=

tr (A) . 2

Let H be the heat kernel on a closed Riemannian manifold (Mn , g). From (23.132)–(23.134) we may deduce the following. (1) If |ϕ (x, y, t)| ≤ C0 d (x, y), where C0 < ∞, then

(23.135)

   

M

  ϕ (x, y, t) H (x, y, t) dµ (x) ≤ Ct1/2

for t small, where C < ∞ is independent of y. (2) If |ψ (x, y, t)| ≤ C1 d (x, y)3 , where C1 < ∞, then

(23.136)

   

M

  ψ (x, y, t) H (x, y, t) dµ (x) ≤ Ct3/2

for t small, where C < ∞ is independent of y.

(3) If xi are geodesic normal coordinates centered at y and if (Aij ) is a symmetric n × n matrix, then

(23.137)

   

 Aij xi xj tr (A)  H (x, y, t) dµ (x) − ≤ Ct χ (x) 4t 2  M

for t small, where χ : M → R is any cutoff function equal to 1 in a neighborhood of y with support in B (y, inj (g)) and where C < ∞ is independent of y. With this in mind, on a Riemannian manifold, let H = (4πt)−n/2 e−f be as in Theorem 23.16 and let η = η (d (x, y)) be the cutoff function defined

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5. SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS

259

in (23.36). Using (23.127) and (23.128), we then have near t = 0 that13 

    n η log HN + log (4πt) Hdµ + O t3/2 2 M  2     2

d (x, y) φ1 + log φ0 + t + O t η − Hdµ + O t3/2 = 4t φ0 M  2    1 d (x, y) 3 p q + Rpq (y) x x + O r (x) η − Hdµ = 4t 12 M      1 R (y) + O (r (x)) tHdµ + O t3/2 η + 6 M   n 1 = − + tR (y) + O t3/2 . 2 3

In particular, by (23.130),   log H + n log (4πt)  n ∂ N 2  + ∆x log HN + log (4πt) +  ∂t 2 t x=y     ∂ 1 n + − η log HN + log (4πt) Hdµ t ∂t 2 M       1 1 n 1 ∂ 1 n − + tR (y) − tR (y) + O t1/2 = − + R (y) − 2t 2 t 2 3 ∂t 3   1 = − R (y) + O t1/2 . 6 

Remark 23.35. In contrast, we shall see in §4 of the next chapter that, by a result of Perelman, a corresponding quantity for the Ricci flow is o (1).
Problem 23.36. Calculate the asymptotics near t = 0 of − M f Hdµ.

5. Supplementary material: Elementary tools 5.1. Expansion for the determinant of a square matrix. Here we recall an elementary result used in the derivation of the first four terms of (23.114). Lemma 23.37 (Expansion for the determinant). If a square matrix has the expansion (23.138)

13



Mij (s) = δij + sAij + s2 Bij + s3 Cij + s4 Dij + O s5 ,

Note that limx→0 x log x = 0.

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260

then

23. HEAT KERNEL FOR STATIC METRICS

 1 2

1 2 det M (s) = 1 + s · tr (A) + s tr (B) + tr (A) − tr A 2 2  ⎛    ⎞ 2 1 2 1 tr (A) − tr (A) + tr (B) tr (A) 2 6  ⎠ + s3 ⎝ 3 1 + 3 tr (A) − tr (AB) + tr (C)

+ O s4 . 

2

Moreover, if A = 0, then (23.139)

det M (s) = 1 + s2 tr (B) + s3 tr (C)  

1 2

4 1 2 tr (B) − tr B + tr (D) + O s5 . +s 2 2

Proof. Recall that if M = M (s) is a time-dependent invertible matrix, then   d −1 dM det M = det M · tr M , (23.140) ds ds so that if M is given by (23.138), then    dM d  . det M = tr ds s=0 ds Differentiating (23.140), we have (23.141)      d2 2 −1 dM −1 dM −1 dM − tr M M det M = det M · tr M ds2 ds ds ds   2 d M + det M · tr M −1 2 , ds so that if M is given by (23.138), then 

d2  det M = tr2 (A) − tr A2 + 2 tr (B) .  2 ds s=0 We compute (23.142) d3 det M ds3 ⎛

⎞

2 −1 dM



M −1 dM tr M ds − 3 tr M −1 dM tr M −1 dM ds ds ds  ⎜

 −1 dM −1 dM −1 dM ⎟ ⎜ +3 tr M −1 dM tr M −1 d2 M + 2 tr M ds M ds M ds ⎟ 2 = det M ⎜ ⎟, ds ds     ⎠ ⎝ dM d2 M d3 M −1 −1 −1 −3 tr M ds M ds2 + tr M ds3

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5. SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS

261

so that if M is given by (23.138), then      d3  2 2 det M = tr (A) tr (A) − 3 tr (A) + 6 tr (B) ds3 s=0   + 2 tr (A)3 − 6 tr (AB) + 6 tr (C) . Finally, assuming dM ds (0) = A = 0, we have (where the rhs is evaluated at s = 0)  d4  det M ds4 s=0       d2 M d2 M d2 M d2 M = 3 tr M −1 2 tr M −1 2 − 3 tr M −1 2 M −1 2 ds ds ds ds   4 d M + tr M −1 4 ds

2 = 12 tr (B) − 12 tr B 2 + 24 tr (D) . The lemma now follows from

  s2 d2  d  det M + det M det M (s) = det M (0) + s  ds s=0 2 ds2 s=0  

s4 d4  s3 d3  det M + det M + O s5 . +   3 4 6 ds 24 ds s=0

s=0


5.2. Interchanging differentiation and integration. We recall an elementary fact about interchanging differentiation and integration. The following is Theorem 13 on p. 297 of Widder [187]. Lemma 23.38 (Fubini’s theorem on noncompact space-time). Let (Mn , g) be a noncompact Riemannian manifold. If (i) f (x, t) ∈ C 1 (M × [α,
ω]), (ii) the improper integral M f (x, t) dµ (x) converges uniformly in t ∈ [α, ω], then



ω







ω

f (x, t) dµ (x) dt = α

M

f (x, t) dtdµ (x) . M

α

Remark 23.39. By assumption (ii), we have that for every O ∈ M and ε > 0, there exists Rε < ∞ independent of t ∈ [α, ω] such that for every R ∈ [Rε , ∞),       f (x, t) dµ (x) < ε.    M−B(O,R)

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262

23. HEAT KERNEL FOR STATIC METRICS

Proof. By the remark, for every ε > 0 there exists Rε < ∞ independent of t ∈ [α, ω] such that for every R ∈ [Rε , ∞),      ω   f (x, t) dµ (x) dt < ε (ω − α) . (23.143)    α M−B(O,R)
Let F (t)  M f (x, t) dµ (x). Then (23.143) says for every ε > 0 and R ∈ [Rε , ∞),        ω   f (x, t) dµ (x) dt < ε (ω − α) , F (t) −    α B(O,R) so that

 lim

ω

R→∞ α

That is,







 F (t) −

f (x, t) dµ (x) dt = 0. B(O,R)



ω

ω



F (t) dt = lim

R→∞ α

α



f (x, t) dµ (x) dt B(O,R)  ω

f (x, t) dtdµ (x)

= lim

R→∞ B(O,R)   ω

α

f (x, t) dtdµ (x) .

= M

α


The following is Theorem 14 on p. 298 of Widder [187]. Lemma 23.40 (Differentiation under the integral sign). Let (Mn , g) be a noncompact Riemannian manifold. If (i) f (x, t) ∈ C 1 (M × [α,
ω]), (ii) the improper integral M f (x, t) dµ (x) converges uniformly in t ∈ [α, ω],
(iii) the improper integral M ∂f ∂t (x, t) dµ (x) converges uniformly in t ∈ [α, ω], then   ∂f d (x, t) dµ (x) . f (x, t) dµ (x) = dt M M ∂t
¯ Proof. Let ϕ (t)  M ∂f ∂t (x, t) dµ (x) and t ∈ [α, ω]. We compute  t¯   t¯ ∂f (x, t) dµ (x) dt ϕ (t) dt = α α M ∂t   t¯ ∂f (x, t) dtdµ (x) = ∂t  M α ¯ f (x, t) dµ (x) − f (x, α) dµ (x) . = M

M

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6. NOTES AND COMMENTARY

By the fundamental theorem of calculus, we have   ¯ d t d ¯ f (x, t) dµ (x) = ϕ (t) dt dt¯ M dt¯ α = ϕ (t¯)  ∂f (x, t¯) dµ (x) . = M ∂t

263




6. Notes and commentary First we make some general comments regarding the history and literature for the heat kernel on manifolds. The parametrix method employed to prove the existence of a fundamental solution to a linear elliptic or parabolic partial differential equation of second order is due to E. E. Levi. For a thorough discussion of the parabolic case, see Chapter 1 of Friedman [61]. For the heat kernel on a closed Riemannian manifold, this method was adapted by Minakshisundaram and Pleijel [131], [130], which also leads to a geometric understanding of the asymptotics of the heat kernel. We closely follow the presentations in [61], Berger, Gauduchon, and Mazet [13], and Chavel [27]. See also the forthcoming book by Li [118]. For some additional references besides those mentioned above on the parametrix and the existence and asymptotics of the heat kernel, see Berline, Getzler, and Vergne [14] and Gilkey [73]. Second we mention some source material for this chapter. §1. For Lemma 23.9 see p. 209 of Berger, Gauduchon, and Mazet [13] or p. 150 of Chavel [27]. Proposition 23.12 is Lemma E.III.3 on pp. 210–211 of [13]. §2. Equation (23.54) is (4.1) on p. 14 of Friedman [61]. Definition (23.55) is (4.4)–(4.5) on p. 14 of [61]. We refer the reader to § 4 in Chapter 1 of [61] for a proof of the convergence of the series for fundamental solutions of variable coefficient second-order parabolic equations. §3. We follow Friedman [61] in presenting the properties of differentiating a convolution with the parametrix. For Lemma 23.26 see Theorem 3 on pp. 8–9 of [61]. For Lemma 23.27 see Theorem 4 on p. 9 of [61]. §4. Regarding the formulas for the expansion of φk near the diagonal, see Branson, Gilkey, and Vassilevich [17] for much more general formulas for the heat kernel asymptotic coefficients for ‘Laplace-type’ operators.

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http://dx.doi.org/10.1090/surv/163/08

CHAPTER 24

Heat Kernel for Evolving Metrics I guess I’ll never learn ... another page is turned. – From “I’ll Wait” by Van Halen

In this chapter we discuss the existence and asymptotics of the minimal fundamental solution to the heat equation with a potential function and a complete time-dependent metric (the manifold may be either compact or noncompact). Recall that the existence of the fundamental solution to the adjoint heat equation with respect to a solution of the Ricci flow, as well as Perelman’s Harnack-type inequality, was used in the proof of pseudolocality in Chapters 21 and 22. In §1 we discuss the existence of a parametrix for linear heat-type equations associated to a 1-parameter family of Riemannian metrics on a closed manifold. Similarly to the fixed metric case, in §2 we use this parametrix to prove the existence of the heat kernel associated to a 1-parameter family of Riemannian metrics on a closed manifold (we do not discuss all of the details of the proof, which requires relatively minor modifications). In §3 we modify the techniques in the fixed metric case to obtain an asymptotic expansion for the heat kernel associated to an evolving metric (for example, a solution to the Ricci flow). In §4 we discuss aspects of the heat kernel asymptotic expansion related to §9.6 of Perelman [152]. In §5 we consider the existence of heat kernels on noncompact manifolds with evolving metrics as the limit of Dirichlet heat kernels for an exhaustion by an increasing sequence of compact domains. The setup in this chapter is the following. Let g (τ ), τ ∈ [0, T ], be a smooth 1-parameter family of complete metrics on a C ∞ manifold Mn . We write its evolution as ∂ gij  2Rij , (24.1) ∂τ where Rij is a general time-dependent symmetric 2-tensor (the notation Rij is motivated by, but not to be confused with, the special case of the Ricci tensor Rij ). Consider the heat operator with a potential term: (24.2)

L

∂ − ∆τ + Q, ∂τ 265

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266

24. HEAT KERNEL FOR EVOLVING METRICS

where Q : M × [0, T ] → R is a C ∞ function and where ∆τ = ∆g(τ ) denotes the Laplacian with respect to g (τ ). As a special case, we have Rij = Rij

(24.3)

and

Q = R,

in which case g (τ ) satisfies the backward Ricci flow and ∂ − ∆τ + R ∂τ is the adjoint heat operator considered by Perelman. Note that under (24.1) we have ∂ dµ = Rdµ, (24.4) ∂τ where R  g ij Rij . L =
∗ =

1. Heat kernel for a time-dependent metric In this section we state the main result on the existence of the heat kernel for a time-dependent metric on a closed manifold. We then begin its proof by considering estimates for the transplanted heat kernel using the metric at time τ . 1.1. Statement of the existence of the heat kernel for a timedependent metric.

As in (23.30), let R2T = (τ, υ) ∈ R2 : 0 < τ − υ ≤ T . Definition 24.1 (Heat kernel with respect to an evolving metric). Let g (τ ), τ ∈ [0, T ], be a smooth 1-parameter family of complete metrics on a manifold Mn . We say that H : M × M × R2T → R is the fundamental solution for

∂ ∂τ

− ∆x,τ + Q if

C2

in the first two space variables, and C 1 in the (1) H is continuous, last two time variables, (2) letting H (x, τ ; y, υ)  H (x, y, τ, υ), (24.5)     ∂ ∂ − ∆x,τ + Q H( · , · ; y, υ) = 0, + ∆y,υ − Q + R H(x, τ ; · , · ) = 0, ∂τ ∂υ (3) (24.6)

lim H ( · , τ ; y, υ) = δy ,

τ υ

lim H (x, τ ; · , υ) = δx .

υτ

The minimal positive fundamental solution is called the heat kernel for ∂ ∂τ − ∆x,τ + Q. The main result proved in this section and the next section is the following.

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1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC

267

Theorem 24.2 (Existence of the heat kernel on closed (Mn , g (τ ))). In the setup above, assume that M is closed. Then there exists a unique ∂ −∆x,τ +Q. Moreover, H is positive fundamental solution H (x, τ ; y, υ) for ∂τ ∞ and C , and hence H is the heat kernel. 1.2. A good approximation to the heat kernel for a time-dependent metric. In order to prove Theorem 24.2, we first construct a good approximation to the heat kernel on a closed manifold Mn with respect to a time-dependent metric g (τ ), τ ∈ [0, T ]. ˜ : M × M × R2 → (0, ∞) by Define the transplanted heat kernel E T   d2τ (x, y) −n/2 ˜ , exp − (24.7) E (x, y, τ, υ)  (4π (τ − υ)) 4 (τ − υ) where dτ : M × M → [0, ∞) denotes the distance function with respect to g (τ ) for τ ∈ [0, T ]. Recall that Minj(g(τ ))  {(x, y) ∈ M × M : dτ (x, y) < inj (g (τ ))} . For each N ∈ N, with N > n/2, we shall construct a function . Minj(g(τ )) × {τ } × [0, τ ) → R HN : τ ∈(0,T ]

of the form (24.8)

˜ (x, y, τ, υ) HN (x, y, τ, υ)  E

N 

ψk (x, y, τ, υ) (τ − υ)k ,

k=0

where the functions ψk :

.

Minj(g(τ )) × {τ } × [0, τ ) → R

τ ∈(0,T ]

are to be defined. Here, as in (23.8), Minj(g(τ ))  {(x, y) ∈ M × M : dτ (x, y) < inj (g (τ ))} . Note that, unlike in our previous discussions of the heat kernel parametrix, ψk depends on (τ, υ). The reason for why this is assumed is that the recursive odes we shall use to define the ψk depend on (τ, υ). Contrast this with the time independence of the ode (23.23a)–(23.23b). Given τ ∈ [0, T ] and y ∈ M, let rτ (x)  dτ (x, y) .

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268

24. HEAT KERNEL FOR EVOLVING METRICS

/ Analogous to (23.18), we have that in τ ∈(0,T ] Minj(g(τ )) × {τ } × [0, τ ) ⊂ M × M × R2T , 2

  ∂ ˜ ∂ log ατ ∂ E ∂ ∂τ rτ ˜ ˜ (24.9) E= E + ∆x,τ − ∂τ ∂rτ ∂rτ 4 (τ − υ)   τ rτ ∂r ∂ log ατ rτ ∂τ ˜ (24.10) E, + = − 2 (τ − υ) ∂rτ 2 (τ − υ) 9  where ατ  det g S (τ ) rτn−1 and g S is as in (23.13). Note that ατ (y, y) = 1. 2

∂ rτ explicRemark 24.3. Although it may be difficult to calculate ∂τ itly, we shall only need its qualitative properties. In particular, we have an

2 ∂ rτ (x) for x near y. expansion of ∂τ Similarly to (23.10) and (23.11), we shall define the {ψk }N k=0 so that (24.11)

ψ0 (y, y, τ, υ) = 1

and so that the approximate heat kernel HN satisfies the defining equation: ˜ (x, y, τ, υ) Lx,τ (ψN ) (x, y, τ, υ) (24.12) Lx,τ (HN ) (x, y, τ, υ) = (τ − υ)N E / in τ ∈(0,T ] Minj(g(τ )) × {τ } × [0, τ ), where ∂ − ∆x,τ + Q. ∂τ Given τ ∈ [0, T ] and (x, y) ∈ Minj(g(τ )) , let V ∈ Ty M be the unit vector tangent to the unique unit speed minimal geodesic from y to x with respect to g (τ ); call this geodesic (24.13)

Lx,τ 

(24.14)

γτ,V : [0, rτ (x)] → M.

d

γτ,V (s). The unit tangent (radial) vector to γτ,V is ∂r∂τ (s) = ds Similarly to (23.23a)–(23.23b), we define, recursively in k, the {ψk }N k=0 to solve the following first-order linear odes along geodesics γτ,V emanating from y with respect to the first variable in ( · , y, τ, υ):   1 ∂ log ατ 1 ∂rτ ∂ψ0 ψ0 , = − + (24.15) ∂rτ 2 ∂rτ 2 ∂τ with the initial condition (24.11), and for 1 ≤ k ≤ N   rτ ∂ log ατ rτ ∂rτ ∂ψk ψk = −Lx,τ (ψk−1 ) . + +k− (24.16) rτ ∂rτ 2 ∂rτ 2 ∂τ As in the fixed metric case, we have the following. Lemma 24.4 (Recursive odes for ψk ). (1) With the initial data (24.11) and the assumption that the ψk ’s are 2 finite / along the diagonal of M×M×RT , the odes (24.15)–(24.16) on τ ∈(0,T ] Minj(g(τ )) × {τ } × [0, τ ) may be solved recursively in k.

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1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC

˜ (2) If the {ψk }N k=0 satisfy (24.15)–(24.16), then HN = E a solution to (24.12), i.e.,

269

N

k=0 ψk τ

k

is

˜ · Lx,τ (ψN ) . Lx,τ (HN ) = (τ − υ)N E Remark 24.5. Similarly to Remark 23.8, we have that ψk is independent of N ≥ k. Proof. (1) From (24.15)–(24.16) we have ∂  k 1/2 − 1
rτ r α e 2 0 ∂s τ τ

∂rτ ∂τ

(γτ,V (¯s))d¯s ψ



k

= −ατ1/2 rτk−1 e− 2

1


rτ 0

∂rτ ∂τ

(γτ,V (¯s))d¯s L

x,τ

(ψk−1 ) ,

where rτ , ατ , ψk , and Lx,τ (ψk−1 ) are all evaluated at (γτ,V (s) , y) (in particular rτ = rτ (γτ,V (s) , y) = s). Thus, integrating over the interval [0, rτ (x)], we have the recursive formula (24.17) ψk (x, y, τ, υ)


rτ (x)

(γ(s))ds ∂τ = −rτ (x)−k ατ−1/2 (x, y) e 2 0  rτ (x)
1 s ∂rτ × ατ1/2 (γ (s), y) sk−1 e− 2 0 ∂τ (γ(¯s))d¯s Lx,τ (ψk−1 )(γ (s), y, τ, υ)ds, 1

∂rτ

0

where γ  γτ,V is the unit speed minimal geodesic emanating from y to x (compare with (23.25)). Part (1) easily follows. (2) Applying the heat-type operator Lx,τ to (24.8) and using (24.12) and (24.10) yields ˜ x,τ (ψN ) (τ − υ)N EL  N   k ˜ ψk (τ − υ) = Lx,τ E  =

k=0 τ rτ ∂r ∂ log ατ rτ ∂τ − 2 (τ − υ) ∂rτ 2 (τ − υ)

−2

N 

 ˜ E

N 

ψk (τ − υ)k

k=0

0 1 ˜ ∇ψk (τ − υ)k ∇x E,

k=0 N 

˜ +E

k=0

g(τ )

k

˜ Lx,τ (ψk ) (τ − υ) + E

N 

kψk (τ − υ)k−1 .

k=0

˜ collecting like powers of τ − υ, cancelling the two (τ − υ)N Factoring out E, ∂ ˜ = − rτ terms in the above expression, and using ∇x log E 2(τ −υ) ∂rτ , we then

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270

24. HEAT KERNEL FOR EVOLVING METRICS

have that equation (24.12) is equivalent to the following equation: (24.18)



0=

τ rτ ∂r rτ ∂ log ατ ∂τ − 2 ∂rτ 2

+ 

N 



N 

k−1

ψk (τ − υ)

k=0

+

N 

rτ

k=0

Lx,τ (ψk−1 ) (τ − υ)k−1 +

k=1

N 

∂ψk (τ − υ)k−1 ∂rτ

kψk (τ − υ)k−1

k=0

 ∂ψ0 rτ ∂ log ατ ψ0 (τ − υ)−1 = rτ + ψ0 − ∂rτ 2 ∂rτ 2     N τ  rτ ∂r rτ ∂ log ατ ∂ψk k−1 ∂τ + k ψk + (τ − υ) + − rτ ∂rτ 2 ∂rτ 2 τ rτ ∂r ∂τ

k=1

+

N 

(τ − υ)k−1 Lx,τ (ψk−1 ) .

k=1 −1 and If the {ψk }N k=0 solve (24.15)–(24.16), then the coefficients of (τ − υ) k−1 for 1 ≤ k ≤ N in (24.18) are zero. Hence part (2) follows.
(τ − υ)

Remark 24.6. By (24.16) we have (24.19)

  1 . ψk (y, y, τ, υ) = − Lx,τ (ψk−1 ) (x, y, τ, υ) k x=y

Note that, in regards to the quantity the elementary estimate:

∂rτ ∂τ

on the rhs of (24.17), we have

Lemma 24.7 (Elementary bound for changing distances).     ∂rτ   (24.20)  ∂τ (x) ≤ Crτ (x) . Proof. If γ is a minimal geodesic joining y to x at time τ , then under (24.1) we have  ∂ L g(τ ) (γ) = Rτ (γ, ˙ γ) ˙ ds, ∂τ γ where ds is the arc length element of γ (s) with respect to g (τ ) and where Rτ is the symmetric 2-tensor in (24.1) at time τ .1 In particular,  ∂ ∂rτ (x) = inf L g(τ ) (γ) = inf Rτ (γ, ˙ γ) ˙ ds, (24.21) γ ∂τ γ ∂τ γ In the above notation, R simultaneously denotes a symmetric 2-tensor Rij and its trace g ij Rij . Here, as elsewhere, it will be clear from the context which quantity R denotes. 1

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2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 271

where the infimum is taken over all minimal geodesics joining y to x, with respect to g (τ ). Hence, assuming |Rij | ≤ C on M × [0, T ], we have     ∂rτ    ∂τ (x) ≤ C L g(τ ) (γ) = Crτ (x) ,


which is (24.20). 2. Existence of the heat kernel for a time-dependent metric

In this section we sketch how to proceed analogously to §1 and §2 of the previous chapter to complete the proof of Theorem 24.2 on the existence of ∂ − ∆x,τ + Q. In particular, we the heat kernel for the operator Lx,τ = ∂τ discuss a version of the Levi–Minakshisundaram–Pleijel method to derive a parametrix for the heat operator with respect to a time-dependent metric on a closed manifold. We leave it as an exercise for the reader to fill in the details of the proof (consult [61], [69], [70], and [85]). 2.1. Parametrix for the heat operator for a time-dependent metric. / For N > n2 we may extend Lx,τ (HN ) continuously to τ ∈[0,T ] Minj(g(τ )) × / {τ } × [0, τ ] by having it take the value 0 on τ ∈[0,T ] Minj(g(τ )) × {τ } × {τ } 1.3 of Chapter 23). More(this generalizes extending
x HN in subsection n

over, given local coordinates U , xi i=1 , we have for k + 2 < N − n2 ∂t
∂xk (Lx,τ HN ) (x, y, τ, υ) = (τ − υ) in

/

N −(n/2)−k−2


  d2τ (x, y) F k,
(x, y, τ, υ) exp − 4 (τ − υ)



(U × M) ∩ Minj(g(τ )) × {τ } × [0, τ ], where ⎞ ⎛ .

(U × M) ∩ Minj(g(τ )) × {τ } × [0, τ ]⎠ . Fk,
∈ C ∞ ⎝

τ ∈[0,T ]

τ ∈[0,T ]

From this we may obtain the following (compare with Lemma 23.10). Lemma 24.8 (Covariant derivatives of Lx,τ HN ). For any k,  ∈ N ∪ {0}, (24.22)     d2τ (x, y)  
k N −(n/2)−k−2
exp − ∂t ∇x (Lx,τ HN ) (x, y, τ, υ) ≤ C (τ − υ) 4 (τ − υ) / n on τ ∈[0,T ] Minj(g(τ )) × {τ } × [0, τ ]. Moreover, if N > 2 + k + 2, then (24.23) ∂t
∇kx (Lx,τ HN ) (x, y, τ, υ) −n/2

= (4π)

N −(n/2)−k−2


(τ − υ)



d2 (x, y) exp − τ 4 (τ − υ)



: k,
(x, y, τ, υ) , F

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272

24. HEAT KERNEL FOR EVOLVING METRICS

: k,
is a C ∞ covariant k-tensor on / where F τ ∈[0,T ] Minj(g(τ )) × {τ } × [0, τ ]. Exercise 24.9. Prove the above lemma. Now let η˜ : [0, ∞) → [0, 1] be a nonincreasing C ∞ cutoff function with η˜ (s) = 1 for s ≤ 1 and η˜ (s) = 0 for s ≥ 2. Define ρ  14 minτ ∈[0,T ] inj (g (τ )). Given HN as above, define the parametrix PN : M × M × R2T → R by



dτ (x, y) (24.24) PN (x, y, τ, υ)  η˜ ρ Analogous to Definition 23.11 we have

 HN (x, y, τ, υ) .

Definition 24.10 (Parametrix for Lx,τ ). We say that a C ∞ function P : M × M × R2T → R is a parametrix for Lx,τ if (1) the functions Lx,τ P and L∗y,υ P both extend continuously to M × M × R2T , where L∗y,υ 

∂ + ∆y,υ − Q + R, ∂υ

and (2) limτ υ P ( · , τ ; y, υ) = δy and limυτ P (x, τ ; · , υ) = δx , that is, for any function f ∈ C 0 (M),  (24.25a) P (x, τ ; y, υ) f (x) dµ (x) = f (y) , lim τ υ M  (24.25b) P (x, τ ; y, υ) f (y) dµ (y) = f (x) . lim υτ

M

By essentially the same proof as for Proposition 23.12, one can show the following. Proposition 24.11 (Existence of a parametrix for Lx,τ ). If N > n/2, then PN is a parametrix for Lx,τ . Moreover, analogous to the Lemma 23.14, we have Lemma 24.12 (Derivatives of Lx,τ PN ). (24.26) ∂t
∇kx (Lx,τ PN ) (x, y, τ, υ)

d2 (x,y)

τ N− n −k−2
− 5(τ −υ) 2

= (τ − υ)

e

G k,
(x, y, τ, υ) ,

where G k,
is a C ∞ covariant k-tensor on M × M × R2T . In particular (compare with (23.44)), (24.27)

|Lx,τ PN | (x, y, τ, υ) ≤ C0 (τ − υ)N − 2 e n

d2 (x,y)

τ − 5(τ −υ)

for some constant C0 < ∞.2 2 Similarly to as in the previous chapter, one may replace the factor 5 on the rhs of (24.27) by 4 + ε for any ε > 0.

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2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 273

2.2. The parametrix convolution series. Similarly to the previous chapter, we shall construct the

heat kernel by a convolution series. Recall that R2> = (τ, υ) ∈ R2 : τ > υ . Definition 24.13 (Space-time convolution). Given two functions

H, J ∈ C 0 M × M × R2> , their space-time convolution is defined by  τ H (x, τ ; z, σ) J (z, σ; y, υ) dµg(σ) (z) dσ (24.28) (H ∗ J) (x, τ ; y, υ)  υ

M

as long as the integral is well defined. Remark 24.14. Note that when g (σ) ≡ g, by letting u = 0 and by taking H (x, t; z, s) = F (x, z, t − s) and J (z, s; y, u) = G (z, y, s − u) in (24.28), we have  t F (x, z, t − s) G (z, y, s) dµg (z) ds, (H ∗ J) (x, t; y, u)  0

M

which is the same as (23.46). Lemma 24.15. The space-time convolution in (24.28) is associative. Proof. We compute (24.29) ((H ∗ J) ∗ K) (x, τ ; y, υ)  τ (H ∗ J) (x, τ ; z, σ) K (z, σ; y, υ) dµg(σ) (z) dσ = υ M  τ τ H(x, τ ; w, ρ) J(w, ρ; z, σ) K(z, σ; y, υ)dµg(ρ) (w)dµg(σ) (z)dρdσ = υ

σ

M×M

and (H ∗ (J ∗ K)) (x, τ ; y, υ)  τ H (x, τ ; z, σ) (J ∗ K) (z, σ; y, υ) dµg(σ) (z) dσ = υ M  τ σ H(x, τ ; z, σ) J(z, σ; w, ρ) K(w, ρ; y, υ)dµg(ρ) (w)dµg(σ) (z)dρdσ. = υ

υ

M×M

Exchanging the order of integration with respect to ρ and σ (and switching the labels w and z), we obtain (H ∗ (J ∗ K)) (x, τ ; y, υ)  τ τ H(x, τ ; w, σ) J(w, σ; z, ρ) K(z, ρ; y, υ)dµg(ρ) (z)dµg(σ) (w)dσdρ = υ ρ M×M  τ τ H(x, τ ; w, ρ) J(w, ρ; z, σ) K(z, σ; y, υ)dµg(σ) (z)dµg(ρ) (w)dρdσ, = υ

σ

M×M

where in the last line we relabelled ρ and σ; this is the same as (24.29).


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274

24. HEAT KERNEL FOR EVOLVING METRICS

Now define F ∗k  F ∗ F ∗k−1 for k ∈ N (F ∗1  F ); since convolution is associative, we may write F ∗k = F ∗ · · · ∗ F (k-fold product). With the parametrix PN defined by (24.24) and satisfying Proposition 24.11, we now follow §2 of the last chapter to establish the convergence of the associated parametrix convolution series: H  PN + PN ∗

(24.30)

∞ 

(Lx,τ PN )∗k .

k=1

Using the facts that (Lx,τ PN ) ∗

∞ 

(Lx,τ PN )

∗k

=

k=1

and that for any G ∈ (24.31)

C 0 (M

∞ 

(Lx,τ PN )∗k

k=2

× M × [0, ∞))

Lx,τ (PN ∗ G) = (Lx,τ PN ) ∗ G − G,

we find that Lx,τ H = 0. By Lemma 26.4, we also have L∗y,υ H = 0. Toward verifying (24.31), we first generalize to the time-dependent metric case the properties of differentiating a convolution with the parametrix discussed in §3 of the previous chapter. Similarly to Lemma 23.26, we have Lemma

n 24.16 (First space derivatives of a convolution with PN ). Let i U , x i=1 be a local coordinate system on M. If   G ∈ C 0 M × M × R2T , then PN ∗ G is C 1 with respect to the space variables and for x ∈ U , the first space derivatives of PN ∗ G are given by (24.32)  τ ∂PN ∂ (PN ∗ G) (x, τ ; y, υ) = (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) dσ. i i ∂x υ M ∂x Sketch of proof. Define  (24.33) JN (x, τ ; y, υ; σ) 

M

PN (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) ,

so that



τ

(PN ∗ G) (x, τ ; y, υ) =

JN (x, τ ; y, υ; σ) dσ. υ

We have  ∂PN ∂JN (x, τ ; y, υ; σ) = (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) i i ∂x M ∂x and

    ∂PN  ≤ C (τ − σ)−α rτ,x (z)2α−n−1  (x, τ ; z, σ)   ∂xi

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2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 275



for α ∈ (0, 1). Taking α ∈ 12 , 1 , we have         ∂PN  ∂JN   dµg(σ) (z)   (x, τ ; y, υ; σ) ≤ (x, τ ; z, σ) G (z, σ; y, υ)    ∂xi  ∂xi M

≤ C (τ − σ)−α and the improper integral  τ  τ ∂PN ∂JN (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) dσ = (x, τ ; y, υ; σ) dσ i i υ υ ∂x M ∂x on the rhs of (24.32) converges absolutely. Finally, similarly to (23.90), we may show that  τ ∂JN ∂ (PN ∗ G) (x, τ ; y, υ) = (x, τ ; y, υ; σ) dσ i ∂xi υ ∂x exists.
Let Bg(τ ) (x, r) = {y ∈ M : dτ (y, x) < r}. Analogous to Lemma 23.27, we have Lemma 24.17 (Second space derivatives of a convolution with PN ). Under the same hypotheses as Lemma 24.16, for (x, τ ) ∈ U × (0, T ] such that

Bg(τ ) x, 12 inj (g (τ )) ⊂ U , we have PN ∗ G is C 2 with respect to the space variable x and the second space derivatives of PN ∗ G are given by (24.34)  τ ∂ 2 PN ∂ 2 (PN ∗ G) (x, τ ; y, υ) = (x, τ ; z, σ) G(z, σ; y, υ) dµg(σ) (z) dσ, i j ∂xi ∂xj υ M ∂x ∂x where xi are geodesic coordinates centered at x with respect to g (τ ). Sketch of proof. Let JN be as in (24.33). We have ∂ 2 JN (x, τ ; y, υ; σ) ∂xi ∂xj    2 ∂ 2 PN ∂ PN − (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) = ∂xix ∂xjx ∂xiz ∂xjz M  ∂ 2 PN (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) . + j M ∂xiz ∂xz (1) Similarly to (23.102), we have      ∂ 2P d2 τ (x,z) ∂ 2 PN  d2τ (x, z)  − 4(τ N −n −σ) . 2 (τ − σ) − e (x, τ ; z, σ) ≤ C 1 +    ∂xix ∂xjx ∂xiz ∂xjz  τ −σ

(2) Similarly to (23.106), we have for α ∈ 12 , 1    2    ∂ PN −α (x, τ ; z, σ) G (z, σ; y, υ)  dµg(σ) (z) ≤ C (τ − σ) .  j i  M  ∂xz ∂xz

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276

24. HEAT KERNEL FOR EVOLVING METRICS

Now (1) and (2) imply that the integral on the rhs of (24.34) is the sum of two integrals, each of which converges absolutely. Finally, we may finish the proof by showing that (similarly to (23.107))  τ 2  τ ∂JN ∂ JN ∂ (x, τ ; y, υ; σ) dσ = (x, τ ; y, υ; σ) dσ. i j i j ∂x υ ∂x υ ∂x ∂x
Analogous to Lemma 23.29, we have Lemma 24.18 (Time derivative of a convolution with PN ). If G ∈ M × M × R2T , then PN ∗ G is C 1 with respect to the time variable τ and

C0

(24.35) ∂ (PN ∗ G) (x, τ ; y, υ) = G (x, τ ; y, υ) ∂τ  τ ∂PN (x, τ ; z, σ) G(z, σ; y, υ) dµg(σ) (z) dσ. + υ M ∂τ Sketch of proof. We compute the time difference quotient (24.36) (PN ∗ G) (x, τ + h; y, υ) − (PN ∗ G) (x, τ ; y, υ) h  τ +h   τ 1 JN (x, τ + h; y, υ; σ) dσ − JN (x, τ ; y, υ; σ) dσ = h υ υ  1 τ +h JN (x, τ + h; y, υ; σ) dσ = h τ  τ JN (x, τ + h; y, υ; σ) dσ − JN (x, τ ; y, υ; σ) dσ + h υ    τ  ∂ 1 τ +h JN  JN (x, τ + h; y, υ; σ) dσ + (x, τ ; y, υ; σ) dσ, = h τ υ ∂τ τ =τ ∗ where τ ∗ ∈ (τ, τ + h). To establish (24.35) from taking the limit as h  0 of (24.36), one shows that   τ  τ ∂JN  ∂JN (x, τ ; y, υ; σ) dσ (x, τ ; y, υ; σ) dσ = lim  h0 υ ∂τ τ =τ ∗ ∂τ υ and 

τ υ

∂JN (x, τ ; y, υ; σ) dσ ∂τ  τ = υ

M

∂PN (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) dσ. ∂τ

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2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 277

The second formula is obvious. The first formula is true since by (24.27) we have         ∂JN ∂P N    (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z)  ∂τ (x, τ ; y, υ; σ) =  ∂τ  M    (∆x,τ − Q) PN (x, τ ; z, σ) G (z, σ; y, υ) dµg(σ) (z) ≤ M



+ C0

N− n 2

(τ − σ)

M −α

≤ C (t − s)

for α ∈ 2 , 1 . 1

d2 (x,z)

e

τ − 5(τ −σ)

|G (z, σ; y, υ)| dµg(σ) (z)

+C


Continuing with our verification of (24.31), we now estimate (Lx,τ PN )∗k . Recall that by (24.27) we have N− n 2

|Lx,τ PN | (x, y, τ, υ) ≤ C0 (τ − υ)

d2 (x,y)

e

τ − 5(τ −υ)

.

We shall prove by induction that there exist constants Ck < ∞ (defined in (24.39) below) such that on M × M × R2T   c2(k−1) d2 τ (x,y) n   − 5(τ −υ) (24.37) (Lx,τ PN )∗k  (x, y, τ, υ) ≤ Ck (τ − υ)k(N − 2 +1)−1 e for all k ∈ N, where c ∈ (0, 1) is defined in (24.38) below. Assuming (24.37) holds for a given k, we have    ∗k+1   (x, τ ; y, υ) (Lx,τ PN )     = (Lx,τ PN ) ∗ (Lx,τ PN )∗k  (x, τ ; y, υ)   τ    ∗k  Lx,τ PN (x, τ ; z, σ) (Lx,τ PN ) (z, σ; y, υ) dµg(σ) (z) dσ  = 

≤

υ τ



υ

M

C0 (τ − σ)N − 2 e n

M

d2 (x,z)

τ − 5(τ −σ)

−

c2(k−1) d2 σ (z,y)

5(σ−υ) × Ck (σ − υ)k(N − 2 +1)−1 e dµg(σ) (z) dσ  τ  d2 (x,z) c2k d2 τ (z,y) n n − τ − (σ − υ)k(N − 2 +1)−1 e 5(τ −σ) 5(σ−υ) dµg(σ) (z) dσ ≤ C0 Ck (τ − υ)N − 2 n

M

υ

≤

C C 0 nk

(τ − υ)(k+1)( k N − 2 +1

N− n +1 2

)−1 e

c2k d2 (x,y) − 5(ττ−υ)

C˜ Vol (g (τ ))

since υ ≤ σ ≤ τ , (24.38)

dσ (z, y) ≥ c · dτ (x, z) ,

and ˜ g(τ ) dµg(σ) ≤ Cdµ

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24. HEAT KERNEL FOR EVOLVING METRICS

for some constants c > 0 and C˜ < ∞. Thus, defining  k−1 C0k C˜ Vol (g (τ )) (24.39) Ck 

k−1 , (k − 1)! N − n2 + 1 we obtain   c2k d2 (x,y)  ∗k+1  (k+1)(N − n +1)−1 − 5(ττ−υ) 2 e .  (x, τ ; y, υ) ≤ Ck+1 (τ − υ) (Lx,τ PN ) By induction we conclude that for all k ∈ N we have (24.37) with Ck defined by (24.39). The C 0 convergence of the convolution series (24.30) follows since  k−1 n k C ˜ ∞ C (τ − υ)k(N − 2 +1)−1 Vol (g (τ ))  0

k−1 (k − 1)! N − n2 + 1 k=1   n C0 C˜ Vol (g (τ )) (τ − υ)N − 2 +1 N− n = C0 (τ − υ) 2 exp N − n2 + 1 < ∞. Compare the above with Lemma 23.20. Analogous to Lemma 23.24 we have the following. Lemma 24.19 (Covariant derivatives of the convolution series). Given , m ∈ N ∪ {0} and N > n2 + 2 + m, the series of covariant m-tensors ∞ 

  ∗k ∂t
∇m P ) (L x,τ N x

k=1

converges absolutely and uniformly on M × M × [0, T ]. Hence ∞  ∞     ∗k ∗k
m (Lx,τ PN ) ∂t
∇m = ∂t ∇x x (Lx,τ PN ) k=1

k=1

exists and is continuous. This concludes our discussion of the sketch of a proof of the existence of heat-type kernels in the time-dependent metric case. Exercise 24.20. Complete the details of the proof of the existence of a fundamental solution to the heat-type equation Lx,τ u = 0 (defined using an evolving metric). 3. Aspects of the asymptotics of the heat kernel for a time-dependent metric Similarly to as in the last chapter, the proof of Theorem 24.2 yields the following asymptotic expansion.

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3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC

279

Theorem 24.21 (Heat kernel expansion for evolving metrics). On a closed manifold there exists a sequence of functions uj ∈ C ∞ (M × M × R2T ), with (24.40)

uk (x, y, τ, υ) = ψk (x, y, τ, υ)

for all x, y ∈ M such that dτ (x, y) ≤ ρ = (τ, υ) ∈ R2T , such that the function

1 4

minτ ∈[0,T ] inj (g (τ )) and for all

(24.41) wN (x, y, τ, υ)  H (x, y, τ, υ) − (4π(τ − υ))

−n 2

 N  2 dτ (x, y)  exp − (τ − υ)k uk (x, y, τ, υ) 4 (τ − υ) k=0

satisfies

  n wN (x, y, τ, υ) = O (τ − υ)N +1− 2

as τ  υ, uniformly for all x, y ∈ M. Moreover, for any k, ˙ ≥ 0,      
k N +1− n −k−2
2 (24.42) ∂τ ∇x wN  (x, y, τ, υ) = O (τ − υ) as τ  υ, uniformly for all x, y ∈ M. We now discuss, based on the above theorem and the recursive formula (24.17) for ψk , aspects of the heat kernel asymptotic expansion. Assume for simplicity that υ = 0. With this we abbreviate (x, y, τ, 0) as (x, y, τ ). τ We begin by deriving the first few terms of the expansion for ∂r ∂τ (x) for x near y. Let x be a point near y with x = y. As in (24.14), let γ = γτ,V : [0, rτ (x)] → M be the unique unit speed minimal geodesic joining y to x at time

nτ , with unit tangent vector V ∈ Ty M. In geodesic normal coordinates xiτ i=1 with respect to g (τ ) centered at y, we have s xi (24.43) γ (s)i  xiτ (γ (s)) = rτ (x) τ for s ∈ [0, rτ (x)], where for the last term xiτ  xiτ (x). By (24.21) and expanding the 2-tensor Rτ along γ, we have  rτ (x) ∂rτ (x) = Rτ (γ˙ (s) , γ˙ (s)) ds ∂τ 0   rτ (x) 2 xiτ xjτ xkτ (∇k Rij )(y, τ ) + O s ds. Rij (y, τ ) + s = rτ (x) rτ (x) rτ (x) 0 Evaluating this integral, we obtain the following.

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24. HEAT KERNEL FOR EVOLVING METRICS

Lemma 24.22 (Expansion for n i 2 i=1 xτ . Then

∂rτ ∂τ ).

Let xiτ = xiτ (x), so that rτ2 (x) =

∂rτ 1 1 (x) = Rij (y, τ ) xiτ xjτ + (∇k Rij ) (y, τ ) xkτ xiτ xjτ ∂τ rτ (x) 2rτ (x)   + O rτ (x)3 .

(24.44)

Now we compute the first few terms of the expansion for ψ0 . By (24.15), we have   1 ∂r ∂ τ . log ατ1/2 ψ0 = ∂rτ 2 ∂τ Since ψ0 (y, y, τ ) = 1, we have    rτ (x) 1 ∂r τ (γ (s)) ds , ψ0 (x, y, τ ) = ατ−1/2 (x, y) exp 2 0 ∂τ

(24.45)

where γ is the unique unit speed minimal geodesic joining x to y with respect to g (τ ) and where dτ (x, y) < inj (g (τ )). From (23.116), we have (24.46)     1 1 ατ−1/2 (x, y) = 1+ Rpq (y, τ ) xpτ xqτ + ∇r Rpq (y, τ ) xpτ xqτ xrτ +O rτ (x)4 . 12 24 Moreover, from (24.44) we have 

rτ (x)

∂rτ (γ (s)) ds ∂τ 0   rτ (x)  1 1 i j k i j Rij (y, τ ) x (∇k Rij ) (y, τ ) x ˜x ˜ + ˜ x ˜x ˜ ds = rτ (γ (s)) 2rτ (γ (s)) 0  rτ (x)   O rτ (γ (s))3 ds, + 0

n where x ˜i = rτ s(x) xiτ and where xiτ i=1 are the coordinates of x. Making the change of variable u  rτ s(x) , we have (note that rτ (γ (s)) = urτ (x)) (24.47)  rτ (x) 0

∂rτ (γ (s)) ds ∂τ  1   1 2 4 i j k i j du uRij (y, τ ) xτ xτ + u (∇k Rij )(y, τ ) xτ xτ xτ + O rτ (x) = 2 0   1 1 = Rij (y, τ ) xiτ xjτ + (∇k Rij ) (y, τ ) xkτ xiτ xjτ + O rτ (x)4 . 2 6

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3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC

281

Remark 24.23. Below we shall use the following variant of (24.47). Integrating (24.44), we have for s ∈ (0, rτ (x)] 

s 0

 2

s¯ i j Rij (y, τ ) xτ xτ + O s¯ d¯ s rτ (x)2 0 3

s2 i j R (y, τ ) x x + O s , = ij τ τ 2rτ (x)2

∂rτ (γ (¯ s)) d¯ s= ∂τ

(24.48)

s)) = where we used xiτ (γ (¯



s

s¯xiτ rτ (x)

and rτ (γ (¯ s)) = s¯ in the first line.

By applying the formulas (24.46) and (24.47) to (24.45), we have 

  1 1 4 p q p q r ψ0 (x, y, τ ) = 1 + Rpq (y, τ ) xτ xτ + ∇r Rpq (y) xτ xτ xτ + O rτ (x) 12 24   j   i (∇k Rij )(y, τ ) xkτ xiτ xjτ Rij (y, τ ) xτ xτ + + O rτ (x)4 . × 1+ 4 12 Expanding this formula, we obtain the following.3 Lemma 24.24 (Asymptotic expansion for ψ0 ). (24.49)



 1 1 Rij + Rij (y, τ ) xiτ xjτ ψ0 (x, y, τ ) = 1 + 12 4     1 1 ∇k Rij + ∇k Rij (y, τ ) xkτ xiτ xjτ + O rτ (x)4 . + 24 12

Next we consider ψ1 . In view of (24.17) for k = 1, we first compute ∆x,τ ψ0 . In general, consider the expansion in normal coordinates for the Laplacian on a Riemannian manifold (Mn , g)  ∆f = g

ij

∂f ∂ 2f − Γkij k i j ∂x ∂x ∂x



3 Compare with the asymptotic expansion (23.116) of φ0 (x, y) for a fixed Riemannian metric.

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24. HEAT KERNEL FOR EVOLVING METRICS

  of a C 2 function f . Using g ij = δij + O r (x)2 , we compute using (23.113) that   ∂ 1 ∂ 2 g − g + O r (x) g ij Γkij = ii ik ∂xi  2 ∂xk  1 1 ∂ p q p q r − Rkpqi x x − ∇r Rkpqi x x x = ∂xi 3 6     1 1 1 ∂ 2 p q p q r R ∇ x x − R x x x − + O r (x) − ipqi r ipqi 2 ∂xk 3 6   1 ∂ ∂ 1 = − Rkpqi i (xp xq ) + Ripqi k (xp xq ) + O r (x)2 3 ∂x 6 ∂x   2 = Rkq xq + O r (x)2 . 3 Therefore   2 ∂ 2f ∂f ∆f = g ij i j − Rkq xq k + O r (x)2 . ∂x ∂x 3 ∂x Now applying this formula to the Laplacian of (24.49), we have4   1 1 ij Rij + Rij (y, τ ) (∆x,τ ψ0 )(x, y, τ ) = 2g 12 4   1  1 ij k kj i ki j ∇k Rij + ∇k Rij (y, τ ) + 2 g xτ + g xτ + g xτ 24 12   ∂ψ0 2 − Rkq xq k + O rτ (x)2 3 ∂x 1 1 (24.50) = R (y, τ ) + R (y, τ ) 6 2   xi + τ (∇i R + 2 (div R)i + ∇i R) (y, τ ) + O rτ (x)2 , 6   2 0 = O r (x) and where we used Rkq xq ∂ψ τ ∂xk    1 ∂2  1 ∇k Rij + ∇k Rij g pq p q xkτ xiτ xjτ 24 12 ∂xτ ∂xτ   1  1 ij k kj i ki j ∇k Rij + ∇k Rij = g xτ + g xτ + g xτ 12 6 xiτ (∇i R + 2 (div R)i + ∇i R) 6 (here we applied the contracted second Bianchi identity). To warm up, we first consider ψ1 along the diagonal. Evaluating (24.16) for k = 1 at x = y, we have =

ψ1 (y, y, τ ) = −Lx,τ (ψ0 ) (x, y, τ )|x=y . 4

Compare with (23.124) in the static metric case.

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3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC

283

Recall that by assumption (24.11), we have ∂ψ0 (y, y, τ ) = 0. ∂τ Hence, by (24.50), along the diagonal of M × M × (0, T ], ψ1 (y, y, τ ) = (∆x,τ ψ0 ) (x, y, τ )|x=y − Q (y, τ ) 1 1 = R (y, τ ) + R (y, τ ) − Q (y, τ ) . 6 2 Now we compute the first few terms of the expansion for ψ1 (x, y, τ ) for x near y and τ near 0. First note that from (24.46) we may easily deduce  

1 s2 Rpq (y, τ ) xpτ xqτ + O s3 . (24.52) ατ1/2 (γ (s) , y) = 1 − 2 12 rτ (x) (24.51)

Moreover, by (24.17), we have
rτ (x)

(γ(s))ds ∂τ ψ1 (x, y, τ ) = −rτ (x)−1 ατ−1/2 (x, y) e 2 0  rτ (x)
1 s ∂rτ × ατ1/2 (γ (s) , y) e− 2 0 ∂τ (γ(¯s))d¯s Lx,τ (ψ0 ) (γ (s) , y, τ ) ds, 1

∂rτ

0

where Lx,τ = we have

∂ ∂τ

− ∆x,τ + Q. Therefore, by applying (24.52) and (24.48), 

ψ1 (x, y, τ ) = −rτ (x)

rτ (x)

−1

ατ1/2 (γ(s), y) e− 2

1


s 0

∂rτ ∂τ

(γ(¯ s))d¯ s

0

× Lx,τ (ψ0 ) (γ (s) , y, τ ) ds  + O rτ (x)2   rτ (x)  3

1 s2 −1 p q Rpq (y, τ ) xτ xτ + O s 1− = −rτ (x) 12 rτ (x)2 0   3

1 s2 i j Rij (y, τ ) xτ xτ + O s × 1− 4 rτ (x)2 

(24.53)

× Lx,τ (ψ0 ) (γ (s) , y, τ ) ds  + O rτ (x)2 . 

Observe that by differentiating (24.45) we immediately get     1 ∂  rτ (x) ∂r ∂ 1 ∂ψ0 τ −1/2 = log ατ (γ (s)) ds . + (24.54) ψ0 ∂τ ∂τ 2 ∂τ ∂τ 0 Now we expand the factor Lx,τ (ψ0 ) (γ (s) , y, τ ) on the rhs of the equation above. Since by (24.46) we easily deduce  

∂ log ατ−1/2 (γ (s) , y) = O s2 ∂τ

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24. HEAT KERNEL FOR EVOLVING METRICS

and by (24.47) we obtain  s 

∂rτ ∂ (γ (¯ s)) d¯ s = O s2 , ∂τ 0 ∂τ it follows from (24.54) that

∂ψ0 (γ (s) , y, τ ) = O s2 (24.55) ∂τ for s ∈ [0, rτ (x)]. Furthermore, similar to (24.50), we have (24.56) 1 (∆x,τ ψ0 ) (γ (s) , y, τ ) = R (y, τ ) + 6 s xiτ + rτ (x) 6 Using (24.55), (24.56), and Q (γ (s) , τ ) = Q (y, τ ) +

1 R (y, τ ) 2



(∇i R + 2 (div R)i + ∇i R)(y, τ ) + O s2 .

s xiτ ∇i Q (y, τ ) + O s2 , rτ (x)

we compute Lx,τ (ψ0 ) (γ (s) , y, τ )   ∂ψ0 − ∆x,τ ψ0 + Qψ0 (γ (s) , y, τ ) = ∂τ 1 s xiτ 1 (∇i R + 2 (div R)i + ∇i R) (y, τ ) = − R (y, τ ) − R (y, τ ) − 6 2 rτ (x) 6

s xiτ ∇i Q (y, τ ) + O s2 + Q (y, τ ) + rτ (x)   1 1 = − R − R + Q (y, τ ) 6 2

s xiτ (∇i R + 2 (div R)i + ∇i R − 6∇i Q) (y, τ ) + O s2 . − rτ (x) 6 We conclude from (24.53) that ψ1 (x, y, τ ) = −rτ (x)−1 = rτ (x)

−1



rτ (x)

  Lx,τ (ψ0 ) (γ (s) , y, τ ) ds + O rτ (x)2

0 rτ (x) 

 0

  2

1 1 R + R − Q (y, τ ) + O s ds 6 2

 rτ (x) s xiτ (∇i R + 2 (div R)i + ∇i R − 6∇i Q) (y, τ ) ds + rτ (x)−1 rτ (x) 6 0   + O rτ (x)2 . Thus we obtain the following.

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4. CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL

285

Lemma 24.25 (Asymptotic expansion for ψ1 ).

(24.57)

ψ1 (x, y, τ )   1 1 R + R − Q (y, τ ) = 6 2 1 (∇i R + 2 (div R)i + ∇i R − 6∇i Q) (y, τ ) · xiτ (x) + 12  + O rτ (x)2 .

4. Characterizing Ricci flow by the asymptotics of the heat kernel In this section, by modifying the discussion regarding the static metric case in subsection 4.2 of Chapter 23, we consider calculations related to the asymptotic formula in §9.6 of Perelman [152]. There he wrote: “Ricci flow can be characterized among all other evolution equations by the infinitesimal behavior of the fundamental solutions of the conjugate heat equation. Namely, suppose we have a Riemannian metric gij (t) evolving with time according to an equation (gij )t = Aij (t). Then we have the heat ∂ ∂ −  and its conjugate
∗ = − ∂t −  − 12 A, operator
= ∂t

d uv = ((
u) v − u (
∗ v)) . (Here A = g ij Aij .) so that dt n Consider the fundamental solution u = (−4πt)− 2 e−f for
∗ , starting as δ-function at some point (p, 0). Then for general
¯ Aij the function (
f¯ + ft )(q, t), where f¯ = f − f u, is of the order O(1) for (q, t) near (p, 0). The Ricci flow Aij = −2Rij ¯ is characterized by the condition (
f¯ + ft )(q, t) = o(1) ; in fact, it is O(|pq|2 + |t|) in this case.” Let g (τ ), τ ∈ [0, T ], be a smooth family of metrics on a closed manifold satisfying (24.1), i.e., ∂ gij = 2Rij , ∂τ where Rij is some time-dependent symmetric 2-tensor. Given y ∈ M, let Mn

H : M × (0, T ] → (0, ∞) be the corresponding fundamental solution to the adjoint heat equation: (24.58) (24.59)

∂H − ∆g(τ ) H + R ( · , τ ) H = 0, ∂τ lim H ( · , τ ) = δy .
∗ H 

τ →0

Define f : M × (0, T ] → R by (24.60)

H ( · , τ ) = (4πτ )−n/2 e−f ( · ,τ ) .

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24. HEAT KERNEL FOR EVOLVING METRICS

From (24.58) we derive n ∂f − ∆g(τ ) f + |∇f |2 − R + = 0. (24.61) ∂τ 2τ Define f¯ : M × (0, T ] → R by  f H dµg(τ ) . f¯  f − M

Let


∂ − ∂τ

− ∆g(τ ) . We have

   ¯ ∂f d f 1 f − ∆g(τ ) f + + f Hdµg(τ ) − f Hdµg(τ ) .
f¯ + = − τ ∂τ τ dτ τ M M The following is a special case of Perelman’s result.5

Claim 24.26 (Heat kernel asymptotics characterization of Ricci flow). Let H : M × (0, T ] → (0, ∞) be the adjoint heat kernel centered at (y, 0) and let f : M × (0, T ] → R be as in (24.60). If Rij = Rij , i.e., g (τ ) evolves by backward Ricci flow, then   f¯ (x, τ ) = O d2g(0) (x, y) + τ . (24.62)
f¯ (x, τ ) + τ In particular, f¯ (x, τ ) = o (1) (24.63)
f¯ (x, τ ) + τ for x near y and τ near 0, i.e.,   f¯ (x, τ ) ¯ = 0.
f (x, τ ) + lim τ d2g(0) (x,y)+τ →0 Assume, as in the previous section, that υ = 0 and abbreviate (x, y, τ, 0) as (x, y, τ ). A discussion of aspects of the heat kernel asymptotic expansion related to Claim 24.26 shall occupy the rest of this section. Recall from (24.8), corresponding to (Mn , g (τ )), τ ∈ [0, T ], we have (24.64)

˜ (x, y, τ ) HN (x, y, τ ) = E

N 

ψk (x, y, τ ) τ k ,

k=0

˜ is given by (24.7), i.e., where E (24.65)

 −n/2

˜ (x, y, τ )  (4πτ ) E

exp −

d2g(τ ) (x, y) 4τ

 ,

and where the ψk are defined by (24.15)–(24.16). By (24.60), we have    log H + n log (4πτ ) ∂ n f 2 + ∆ log H + log (4πτ ) + .
f − = τ ∂τ 2 τ 5 More generally, one may wish to study the asymptotic expansions of fundamental solutions of heat-type equations coupled to the Ricci flow.

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4. CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL

287

By Theorem 24.21, for τ small and for x and y close, HN is a good approximation to the adjoint heat kernel H. For this reason we shall calculate the asymptotics of the quantity: W  W1 + W2 , where (24.66a) (24.66b)



  ∂ n + ∆x,τ log HN + log (4πτ ) , W1  ∂τ 2 log HN + n2 log (4πτ ) W2  ; τ

our calculation culminates in Lemma 24.28 below. Note that, by (24.44), we have the expansion

  ∂ rτ2 (x) = 2Rij (y, τ ) xiτ xjτ +(∇k Rij ) (y, τ ) xkτ xiτ xjτ +O rτ (x)4 , (24.67) ∂τ i where xτ are the coordinates of x in geodesic normal coordinates with respect to g (τ ) centered at y. From this we may deduce   rτ2 (x) = r02 (x) + 2Rij (y, 0) xiτ xjτ τ + O τ rτ (x)3 + τ 2 rτ (x)2 . Furthermore, by (23.118), we have  

2 ∆x,τ rτ2 (x) = 2n − Rpq (y, τ ) xpτ xqτ + O rτ (x)3 . 3 Hence (24.68)       2

2 ∂ + ∆x,τ rτ (x) = 2n − Rij − 2Rij (y, τ ) xiτ xjτ + O rτ (x)3 . ∂τ 3 The logarithm of (24.64) yields (24.69)

N   rτ2 n ψk τ k . log HN + log (4πτ ) = − + log 2 4τ k=0

Taking its time derivative and Laplacian, we have   2 N k−1 + ∂ψk τ k τ kψ k k=0 rτ ∂τ + N k 4τ k=0 ψk τ

∂ 0 ψ1 + ∂ψ r2 r2 ∂τ = τ2 − ∂τ τ + + O (τ ) 4τ 4τ ψ0

 r2 n ∂  log HN + log (4πτ ) = τ2 − ∂τ 2 4τ

∂ ∂τ

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24. HEAT KERNEL FOR EVOLVING METRICS

and

  n ∆x,τ log HN + log (4πτ ) 2  2  N 2 N k k ∇ ψ τ   x,τ k ∆x,τ rτ k=0 k=0 ∆x,τ ψk τ +  − =−   2 N N k 4τ k k=0 ψk τ k=0 ψk τ

∆x,τ rτ2 ∆x,τ ψ0 |∇x,τ ψ0 |2 + =− − + O (τ ) , 4τ ψ0 ψ02

respectively. Summing the above two formulas yields the following expression for (24.66a): ∂

2

rτ |∇x,τ ψ0 |2 rτ2 ∂τ + ∆x,τ − W1 = 2 − 4τ 4τ ψ02 ∂

ψ1 + ∂τ + ∆x,τ ψ0 + + O (τ ) . ψ0 By (24.69), for (24.66b) we have (24.70)

W2 = −

rτ2 log ψ0 ψ1 + + + O (τ ) . 4τ 2 τ ψ0

Summing these two formulas yields



∂ + ∆x,τ rτ2 + 4 log ψ0 |∇x,τ ψ0 |2 − ∂τ − (24.71) W = 4τ ψ02 ∂

2ψ1 + ∂τ + ∆x,τ ψ0 + O (τ ) . + ψ0 On the other hand, we obtain from (24.68) and (24.49) that  

∂ + ∆x,τ rτ2 + 4 log ψ0 − ∂τ   2 Rij − 2Rij (y, τ ) xiτ xjτ = −2n + 3     1 Rij + Rij (y, τ ) xiτ xjτ + O rτ (x)3 + 3   (24.72) = −2n + (Rij − Rij ) (y, τ ) xiτ xjτ + O rτ (x)3 . We also have from (24.49) that (24.73)

  |∇x,τ ψ0 |2 (x) = O rτ (x)2 .

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4. CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL

289

Applying (24.72) and (24.73) to (24.71) yields (24.74)

2ψ1 + −n (Rij − Rij ) (y, τ ) xiτ xjτ + + W = 2τ 4τ   3   rτ (x) + O τ + rτ (x)2 . +O τ 

Remark 24.27 (About the O rτ (x)2 τ

≤ C, then

rτ (x)3 τ

rτ (x)3 τ





+ ∆x,τ ψ0 ψ0

∂ ∂τ

term). If rτ (x)2 + τ → 0 and

→ 0.

  By applying to (24.74) the formulas ψ0 = 1 + O rτ (x)2 , (24.57) (24.50), and (24.55), we calculate that   1 3 −n + R + R − 2Q (y, τ ) W = 2τ 2 2 i x + τ (∇i R + 2 (div R)i + ∇i R − 3∇i Q) (y, τ ) 3 (Rij − Rij ) (y, τ ) + xiτ xjτ 4τ  3   rτ (x) + O τ + rτ (x)2 . +O τ In particular, if Q = R, then   1 1 −n + R − R (y, τ ) W = 2τ 2 2 i x + τ (∇i R + 2 (div R)i − 2∇i R) (y, τ ) 3 (Rij − Rij ) (y, τ ) + xiτ xjτ 4τ  3   rτ (x) + O τ + rτ (x)2 . +O τ Further specializing, we have the following. Lemma 24.28. If g (τ ) evolves by the backward Ricci flow (i.e., Rij = Rij ) and if Q = R, then the approximate adjoint heat kernel HN in (24.8) satisfies (see (24.66) for the definition of W = W1 + W2 )     rτ (x)3 n + O τ + rτ (x)2 . W =− +O 2τ τ Problem 24.29. Prove Claim 24.26.

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24. HEAT KERNEL FOR EVOLVING METRICS

5. Heat kernel on noncompact manifolds Thus far in this chapter and the previous chapter, we have discussed heat kernels and parametrixes on closed manifolds. In this section we prove the existence of heat kernels on noncompact manifolds via the limit of Dirichlet heat kernels of smooth bounded domains. 5.1. Dirichlet heat kernels for fixed metrics. First recall the following heat kernel existence result for the Dirichlet problem; see Li [116] or Chapter VII of Chavel [27]. Theorem 24.30 (Existence and uniqueness of Dirichlet heat kernel). Let (Mn , g) be a compact smooth Riemannian manifold with nonempty boundary ∂M. Then there exists a unique heat kernel HD (x, y, t) for the Dirichlet problem (also called the Dirichlet heat kernel ). That is, there exists a unique continuous function HD : M × M × (0, ∞) → [0, ∞), which is C ∞ on int (M) × int (M) × (0, ∞), such that ∂ HD (x, y, t) = ∆x HD (x, y, t) in int (M) × int (M) × (0, ∞) , ∂t HD (x, y, t) = 0 on ∂M × int (M) × (0, ∞) , lim HD ( · , y, t) = δy

t0

for y ∈ int (M) ,

where int (M) denotes the interior of M. More specifically, the Dirichlet heat kernel may be obtained as the convergent series ∞  e−λk t ϕk (x) ϕk (y) , HD (x, y, t) = k=1

where ϕk is the k-th Dirichlet eigenfunction and λk is its corresponding eigenvalue, so that ∆ϕk + λk ϕk = 0, ϕk |∂M = 0, {ϕk }∞ k=1 is an orthonormal 2 basis for L (M, g), and 0 < λ1 < λ2 ≤ λ3 ≤ · · · . See the beginning part of [116] for details. 5.2. Dirichlet heat kernels for evolving metrics. Now we consider the time-dependent metric case. Let g (τ ), τ ∈ [0, T ], be a smooth 1-parameter family of Riemannian metrics on a compact manifold Mn with nonempty boundary ∂M. We adopt the same setup as in the ∂ ∂ gij  2Rij and Lx,τ  ∂τ − ∆x,τ + Q, where beginning of this chapter: ∂τ ∞ Q : M × [0, T ] → R is a C function. 4n without boundary and We extend M to a smooth compact manifold M we extend g (τ ), τ ∈ [0, T ], to a smooth 1-parameter family of Riemannian 4 For example, we may do this as follows. metrics g: (τ ), τ ∈ [0, T ], on M. First, extend the differentiable manifold M to a collar past its boundary. Second, extend the metric so that in a collar of the new boundary it is

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

291

isometric to the product of a metric on ∂M with an interval; we may do this in a way which depends smoothly on τ . Third, we double the extension of the differentiable manifold and we double the extended metric to obtain 4 g: (τ )). (M, ::M 4 × [0, T ] → R. Let Now extend Q to a C ∞ function Q : :M 4×M 4 × R2 → R H T : We shall obtain the Dirichlet − ∆x,g(τ ) + Q.   : on (M, g (τ )) by adding to H  a solution to 2

: x,τ  be the heat kernel for L heat kernel for Lx,τ

∂ ∂τ

the Dirichlet problem for the heat equation. We shall prove the following.

M×M×RT

Lemma 24.31 (Dirichlet problem for the heat equation). Given any υ ∈ [0, T ) and any continuous function b : ∂M × (υ, T ] → R with lim b (x, τ ) = 0

τ υ

for all x ∈ ∂M,

there exists a unique C 0 (C ∞ in the interior ) solution u : M × [υ, T ] → R to Lx,τ u = 0 with the boundary conditions u (x, υ) = 0

for x ∈ int (M) ,

u (x, τ ) = b (x, τ )

for x ∈ ∂M and τ ∈ (υ, T ].

By the lemma, given y ∈ int (M) and υ ∈ [0, T ), we may let fy,υ : M × [υ, T ] → R be the solution to Lx,τ fy,υ = 0 with the boundary conditions fy,υ (x, υ) = 0

for x ∈ int (M) ,

: (x, τ ; y, υ) fy,υ (x, τ ) = −H

for x ∈ ∂M and τ ∈ (υ, T ].

We then define H : M × M × R2T → R, by (24.75)

: (x, τ ; y, υ) + fy,υ (x, τ ) . H (x, τ ; y, υ)  H

Given y ∈ int (M) and υ ∈ [0, T ),    : lim max H  (x, τ ; y, υ) = 0 τ υ x∈∂M

and by the maximum principle we have for τ ∈ (υ, T ]    : max |fy,υ | (x, τ ) ≤ C max H  (x, τ ; y, υ) x∈M

x∈∂M

where C < ∞ is a constant depending only on Q and T . Therefore we have the following.

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292

24. HEAT KERNEL FOR EVOLVING METRICS

Theorem 24.32 (Existence of Dirichlet heat kernel). The function H defined by (24.75) is the fundamental solution for Lx,τ satisfying the Dirichlet boundary condition: H (x, τ ; y, υ) = 0 lim H ( · , τ ; y, υ) = δy

τ
υ

on ∂M × int (M) × (0, ∞) , for y ∈ int (M) .

In the remainder of this subsection we give a proof of Lemma 24.31. Without loss of generality we may assume that υ = 0. Given a continuous function ψ : ∂M × [0, T ] → R, define uψ : int (M) × [0, T ] → R by
τ
 ∂H dσ (x, τ ; z, σ) ψ (z, σ) dµg(σ) (z) , (24.76) uψ (x, τ )
− 0 ∂M ∂νz,σ where dµg(σ) denotes the induced volume (n − 1)-form of ∂M and where νz,σ denotes the outward unit normal to ∂M at z, all with respect to g (σ). Given x ∈ int (M), we have    g(σ)   sup H  (x, τ ; z, σ) = 0. lim ∇ τ →0 (z,σ)∈∂M×[0,τ )

Thus we have lim uψ (x, τ ) = uψ (x, 0) = 0

τ →0

for x ∈ int (M)




∂H (x, τ ; z, τ )’) and also (abusing notation with ‘ ∂ν z,τ  
τ
 ∂H dσ Lx,τ (x, τ ; z, σ) ψ (z, σ) dµg(σ) (z) Lx,τ uψ (x, τ ) = − ∂νz,σ 0 ∂M
 ∂H (x, τ ; z, τ ) ψ (z, τ ) dµg(τ ) (z) − ∂M ∂νz,τ = 0,    
∂  = 0 (since Lx,τ and ∂ act on H = L because Lx,τ ∂ν∂ H x,τ ∂νz,σ ∂νz,σ z,σ


(x, τ ; z, τ ) = 0 for x ∈ int (M) distinct factors, they commute) and ∂ν∂ H z,τ and z ∈ ∂M. To understand the boundary values of uψ , we have the following. Lemma 24.33 (Jump relation). For any x0 ∈ ∂M we have (24.77)
τ
 1 ∂H dσ (x0 , τ ; z, σ) ψ (z, σ) dµg(σ) (z) . lim uψ (x, τ ) = ψ (x0 , τ )− x→x0 2 0 ∂M ∂νz,σ Here the limit is taken with x inside any finite closed cone C ⊂ Int(M)∪{x0 }.

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

293

Note that (1) compared to formula (24.76) in the interior, we have the extra ‘jump’ term 12 ψ (x0 , τ ), 

(x0 , τ ; z, σ) has a singularity at (z, σ) = (x0 , τ ), (2) the integrand ∂ν∂ H z,σ (3) for any x0 ∈ ∂M and ε > 0  : ∂H (x, τ ; z, σ) ψ(z, σ) dµg(σ) (z) dσ lim x→x0 (∂M×[0,τ −ε])∪((∂M−B(x ,ε))×[τ −ε,τ )) ∂νz,σ 0  : ∂H (x0 , τ ; z, σ) ψ (z, σ) dµg(σ) (z) dσ, = (∂M×[0,τ −ε])∪((∂M−B(x0 ,ε))×[τ −ε,τ )) ∂νz,σ so that the jump term is due solely to the singularity at (x0 , τ ). Exercise 24.34. Prove Lemma 24.33. Hint: See the notes and commentary at the end of this chapter. By Lemma 24.33, in order to prove Lemma 24.31, it suffices to find a continuous function ψ : ∂M × [0, T ] → R satisfying the equation (24.78)  τ  : ∂H dσ (x0 , τ ; z, σ) ψ (z, σ) dµg(σ) (z) ψ (x0 , τ ) = 2b (x0 , τ ) + 2 0 ∂M ∂νz,σ for all x0 ∈ ∂M and τ ∈ [0, T ], where b : ∂M × (0, T ] → R is continuous and limτ 0 b (x, τ ) = 0 for all x ∈ ∂M. To prove existence, we iterate this equation; namely, consider the sequence {ψk }∞ k=−1 of functions on ∂M × [0, T ] defined recursively by ψ−1 = 0 and (24.79)  τ  : ∂H dσ (x0 , τ ; z, σ) ψk−1 (z, σ) dµg(σ) (z) ψk (x0 , τ )  2b (x0 , τ ) + 2 0 ∂M ∂νz,σ for k ∈ N ∪ {0}. Let (24.80)

Ak (x0 , τ )  ψk (x0 , τ ) − ψk−1 (x0 , τ ) .

Since ψ−1 = 0, we have (24.81)

A0 (x0 , τ ) = ψ0 (x0 , τ ) = 2b (x0 , τ )

and by (24.79) we have the recurrence relation  τ  : ∂H dσ (x0 , τ ; z, σ) Ak−1 (z, σ) dµg(σ) (z) (24.82) Ak (x0 , τ ) = 2 0 ∂M ∂νz,σ for k ∈ N. Define ψ∞ : ∂M × [0, T ] → R by (24.83)

ψ∞ (x0 , τ ) 

∞  k=0

Ak (x0 , τ ) = lim ψk (x0 , τ ) . k→∞

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294

24. HEAT KERNEL FOR EVOLVING METRICS

Claim. The series in (24.83) converges uniformly on ∂M × [0, T ] to a continuous function and we may exchange the sum and integration:  ∞  τ  : ∂H (24.84) dσ (x0 , τ ; z, σ) Ak (z, σ) dµg(σ) (z) ∂M ∂νz,σ k=1 0   τ ∞  : ∂H dσ (x0 , τ ; z, σ) Ak (z, σ) dµg(σ) (z) . = 0 ∂M ∂νz,σ k=1

The claim, (24.83), and summing (24.82) from k = 1 to ∞ imply (24.85)  τ  : ∂H dσ (x0 , τ ; z, σ) ψ∞ (z, σ) dµg(σ) (z) , ψ∞ (x0 , τ ) = 2b (x0 , τ ) + 2 0 ∂M ∂νz,σ i.e., ψ∞ is a solution of (24.78). By (24.76) we conclude  τ  : ∂H dσ (x, τ ; z, σ) ψ∞ (z, σ) dµg(σ) (z) (24.86) uψ∞ (x, τ ) = − 0 ∂M ∂νz,σ is the desired solution to Lemma 24.31. Now we prove the claim. First we rewrite the Ak . For k = 1 we have  τ  : ∂H dσ (x0 , τ ; z, σ) A0 (z, σ) dµg(σ) (z) A1 (x0 , τ ) = 2 ∂νz,σ ∂M 0 τ dσ M1 (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) , =2 0

∂M

where M1 : ∂M × ∂M × R2T → R is defined by (24.87)

M1 (x0 , τ ; z, σ)  2

By induction we can show that  τ  dσ (24.88) Ak (x0 , τ ) = 2 0

: ∂H (x0 , τ ; z, σ) . ∂νz,σ

Mk (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) ,

∂M

where the functions Mk : ∂M × ∂M × R2T → R are defined recursively by (24.89)  τ Mk+1 (x0 , τ ; z, σ)  2 σ

∂M

: ∂H (x0 , τ ; w, ρ) Mk (w, ρ; z, σ) dµg(ρ) (w) dρ ∂νw,ρ

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

295

for k ∈ N and by (24.87). Indeed, assuming (24.88) holds for some k ∈ N, we have  τ : ∂H (x0 , τ ; z, σ) Ak (z, σ) dµg(σ) (z) dσ Ak+1 (x0 , τ ) = 2 0 ∂M ∂νz,σ  τ : ∂H (x0 , τ ; z, σ) dµg(σ) (z) =2 0 ∂M ∂νz,σ  σ Mk (z, σ; w, ρ) b (w, ρ) dµg(ρ) (w) dρdσ. ×2 0

∂M

Switching the order of integration for ρ and σ and switching names for w and z, we have  τ τ  : ∂H (x0 , τ ; w, σ) Mk (w, σ; z, ρ) b (z, ρ) Ak+1 (x0 , τ ) = 4 0 ρ ∂M ∂M ∂νw,σ 

τ



τ



× dµg(σ) (w) dµg(ρ) (z) dσdρ



=4 0



σ

∂M



τ

dσ

=2 0

∂M

: ∂H (x0 , τ ; w, ρ) Mk (w, ρ; z, σ) b (z, σ) ∂νw,ρ × dµg(ρ) (w) dµg(σ) (z) dρdσ

Mk+1 (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) . ∂M

This completes the proof of (24.88). By (24.88), the claimed formula (24.84) is equivalent to the facts that the series  ∞ ∞  τ   Ak (x0 , τ ) = 2 dσ Mk (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) (24.90) k=1

k=1

0

∂M

converges and (24.91) ∞  

τ 0

k=1



τ

= 0



: ∂H (x0 , τ ; z, σ) ∂M ∂νz,σ  σ  dρ Mk (z, σ; w, ρ) b (w, ρ) dµg(ρ) (w) dµg(σ) (z) ×

dσ



0

∂M

: ∂H dσ (x0 , τ ; z, σ) ∂M ∂νz,σ  ∞  σ  dρ Mk (z, σ; w, ρ) b (w, ρ) dµg(ρ) (w) dµg(σ) (z) . × k=1

0

∂M

To see both of these facts, we estimate |Mk |. First note that 2   n dτ (x0 , z) − dτ (x0 ,z)  g(σ) :  e 5(τ −σ) . H  (x0 , τ ; z, σ) ≤ C (τ − σ)− 2 (24.92) ∇ τ −σ Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

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24. HEAT KERNEL FOR EVOLVING METRICS

On the other hand, we shall obtain a better estimate for the normal deriv: on ∂M. ative of H Let α be any number in the interval ( 12 , 1). (1) k = 1. We have for 0 ≤ σ < τ and x0 , z ∈ ∂M with x0 = z    ∂H   :  |M1 | (x0 , τ ; z, σ) = 2   (x0 , τ ; z, σ)  ∂νz,σ   d2 (x ,z)  2 dτ (x0 , z) − τ 0 −n + C e 5(τ −σ) (24.93a) ≤ C (τ − σ) 2 τ −σ −

d2 τ (x0 ,z) 6(τ −σ)

(24.93b)

≤ C (τ − σ)− 2 e

(24.93c)

(x0 , z) , ≤ C# (τ − σ)−α d−n+2α τ

n

where C# < ∞, where we used Lemma 24.35 below to obtain (24.93a), and where the fourth line is true since f (x)  xλ e−x , λ ≥ 0, is bounded for x ≥ 0. Note that for the model case of Euclidean space Rn with heat kernel |x|2

H (x, t) = (4πt)− 2 e− 4t centered at the origin and a hyperplane P perpendicular to a vector ν and passing through the origin, we have n

x, ν ∂H (x, t) = − H (x, t) = 0 for x ∈ P, t > 0. ∂ν 2t This is essentially the reason for the in the estimate (24.93a)    improvement  g(σ) :   ∂ H  H . Formally, we may summarize for  ∂νz,σ  on ∂M as compared to ∇ the above as follows. Lemma 24.35. Let g (τ ), τ ∈ [0, T ], be a smooth family of Riemannian metrics on a closed Riemannian manifold Mn with heat kernel H (x, τ ; y, σ) for the operator Lx,τ . Let N n−1 ⊂ M be a compact smooth hypersurface. Then there exists a constant C < ∞ such that    d2 (x,y)  2  ∂H  τ n  (x, τ ; y, σ) ≤ C (τ − σ)− 2 dτ (x, y) + C e− 5(τ −σ)   ∂νy,σ  τ −σ ≤ C (τ − σ)− 2 e n

d2 (x,y)

τ − 6(τ −σ)

for any x, y ∈ N , 0 ≤ σ < τ ≤ T , and unit normal νy,σ to N at y with respect to g (σ). Exercise 24.36. Prove this lemma using the facts that  ( )  2 dτ (x, y) ∂ dτ (x, y) ∂ = , ∇dτ (y) ∂νy,τ 4 (τ − σ) 2 (τ − σ) ∂νy,τ ≤C

d2τ (x, y) τ −σ

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

297

   ∂ ∂  √  ≤ τ − σ. −  ∂νy,τ ∂νy,σ 

and

Remark 24.37. Also note that by (24.93c) we have for α ∈ ( 12 , 1) that   τ dσ |M1 | (x0 , τ ; z, σ) dµg(σ) (z) ≤ Cτ 1−α 0

∂M

is finite. This implies |A1 | (x0 , τ ) ≤ Cτ 1−α . Remark 24.38. Note that (24.93a) and (24.93b) follow directly from Lemma 24.35. (2) k ≥ 2. Recall that α ∈ ( 12 , 1). We estimate |Mk | by a type of induction. Suppose we have for some k ∈ N an estimate of the form k (x0 , z) |Mk | (x0 , τ ; z, σ) ≤ Ck (τ − σ)−βk d−γ τ

(24.94)

for some Ck < ∞, where βk < 1 and γk ≤ n − 2α. Note that by (24.93c), for k = 1 we have (24.94) with C1 = C# , β1 = α, and γ1 = n − 2α. Let C0 ∈ [1, ∞) be such that C0−1 g (τ1 ) ≤ g (τ2 ) ≤ C0 g (τ1 )

(24.95)

for all τ1 , τ2 ∈ [0, T ]. Recall also that for any α, β < 1,  1  τ (τ − ρ)−α (ρ − σ)−β dρ = (τ − σ)1−α−β (1 − ρ˜)−α ρ˜−β d˜ ρ σ

0

= Cα,β (τ − σ)1−α−β ,

(24.96) where Cα,β 

(24.97)

Γ (1 − α) Γ (1 − β) Γ (2 − α − β)

and Γ is the Gamma function. Thus, by applying (24.93c) and (24.94) to (24.89), we obtain |Mk+1 (x0 , τ ; z, σ)| (24.98a)

   ∂H  :   ≤2   (x0 , τ ; w, ρ) |Mk | (w, ρ; z, σ) dµg(ρ) (w) dρ   ∂ν w,ρ σ ∂M  τ (τ − ρ)−α d−n+2α (x0 , w) ≤ 2C# Ck τ 

τ



σ

∂M

k (w, z) dµg(ρ) (w)dρ × (ρ − σ)−βk d−γ ρ

(24.98b) γ /2

(τ − σ)1−α−βk  k d−n+2α (x0 , w) d−γ (w, z) dµg(τ ) (w) , × τ τ

≤ CCk Cα,βk C0 k

∂M

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24. HEAT KERNEL FOR EVOLVING METRICS

k k where C < ∞ is independent of k, βk , γk and where we used d−γ ≤ C0 k d−γ ρ τ n/2 and dµg(ρ) ≤ C0 dµg(τ ) by (24.95). We break up the region of integration ∂M into three subregions:

R1  w ∈ ∂M : dτ (w, x0 ) ≤ 12 dτ (x0 , z) = Bg(τ ) x0 , 12 dτ (x0 , z) ,

R2  w ∈ ∂M : dτ (w, z) ≤ 12 dτ (x0 , z) = Bg(τ ) z, 12 dτ (x0 , z) ,

R3  w ∈ ∂M : dτ (w, x0 ) ≥ 12 dτ (x0 , z) and dτ (w, z) ≥ 12 dτ (x0 , z) ,

γ /2

where the balls are contained in ∂M. Since dτ (w, z) ≥ 12 dτ (x0 , z) for w ∈ R1 , we have  k d−n+2α (x0 , w) d−γ (w, z) dµg(τ ) (w) τ τ R1  −n+2α k (x0 , z) (x0 , w) dµg(τ ) (w) ≤ 2γk d−γ   dτ τ 1 Bg(τ ) x0 , 2 dτ (x0 ,z)

(24.99)

≤ C2γk dτ2α−γk −1 (x0 , z) ,

where C < ∞ is independent of k, βk , γk (the region of integration is an (n − 1)-dimensional ball). Since dτ (x0 , w) ≥ 12 dτ (x0 , z) for w ∈ R2 , we have  k d−n+2α (x0 , w) d−γ (w, z) dµg(τ ) (w) τ τ R2  −γk (x , z) (w, z) dµg(τ ) (w) ≤ 2n−2α d−n+2α   dτ 0 τ 1 Bg(τ ) z, 2 dτ (x0 ,z)

C d2α−γk −1 (x0 , z) , −γk + n − 1 τ where C < ∞ is independent of k, βk , γk and where we used γk ≤ n − 2α < n − 1. Since dτ (x0 , w) ≥ 13 dτ (w, z) for w ∈ R3 and R3 ⊂ ∂M − R2 , we have  k d−n+2α (x0 , w) d−γ (w, z) dµg(τ ) (w) τ τ R3  n−2α k d−n+2α−γ (w, z) dµg(τ ) (w) ≤3 τ (24.100)

≤

 ≤C

∂M−R2 diam(M,g(τ )) 1 2 dτ (x0 ,z)

r2α−γk −2 dr

diam(M,g(τ )) C r2α−γk −1  1 2α − γk − 1 2 dτ (x0 ,z) γ −2α+1 2 k C 2α−γk −1 d (24.101) ≤ (x0 , z) γk − 2α + 1 τ provided γk > 2α − 1, where we used the fact that ∂M is compact and the volume comparison theorem in ∂M − R2 . =

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

299

Now suppose 2α − 1 < γk ≤ n − 2α.

(24.102)

Regarding the integral in (24.98b), since ∂M = R1 ∪ R2 ∪ R3 and by summing the estimates (24.99), (24.100), and (24.101), we have  k d−n+2α (x0 , w) d−γ (w, z) dµg(τ ) (w) ≤ Cdτ2α−γk −1 (x0 , z) , (24.103) τ τ ∂M

where C < ∞ is independent of k. Thus, assuming (24.94) and (24.102), we have that (24.98b) implies (24.104)

−γk+1

|Mk+1 (x0 , τ ; z, σ)| ≤ Ck+1 (τ − σ)−βk+1 dτ

(x0 , z) ,

where γ /2

Ck+1  CCk Cα,βk C0 k , −βk+1  1 − α − βk , −γk+1  2α − γk − 1. ¯ we have Therefore there exists k¯ ∈ N such that for any 1 ≤ k ≤ k, k (x0 , z) , |Mk | (x0 , τ ; z, σ) ≤ Ck (τ − σ)−βk d−γ τ

(24.105)

≥ 0. where either −βk¯ ≥ 0 or −γk¯ ≥ 1 − 2α, i.e., −γk+1 ¯ Case 1. There exists k0 = k¯ + 1 ∈ N such that −γk0 ≥ 0. Then by (24.104) we have |Mk0 | (x0 , τ ; z, σ) ≤ C (τ − σ)−β , where β = βk0 +1 and C = Ck0 +1 diam (g (τ ))−γk0 +1 . Substituting this in (24.98b), we have |Mk0 +1 (x0 , τ ; z, σ)|   τ −α −β (τ − ρ) (ρ − σ) dρ ≤C σ

≤ CC 

d−n+2α (x0 , w) dµg(τ ) (w) τ ∂M

Γ (1 − α) Γ (1 − β) (τ − σ)1−α−β , Γ (2 − α − β)

where C   sup



x0 ∈∂M ∂M

d−n+2α (x0 , w) dµg(τ ) (w) < ∞. τ

In general, we have for  ∈ N ∪ {0}
Γ
(1 − α) Γ (1 − β) (τ − σ)
(1−α)−β . |Mk0 +
(x0 , τ ; z, σ)| ≤ C C  Γ (1 +  (1 − α) − β)

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300

24. HEAT KERNEL FOR EVOLVING METRICS

Since ∞  
Γ
(1 − α) Γ (1 − β) (τ − σ)
(1−α)−β C Γ (1 +  (1 − α) − β)
=0  
∞ C  Γ (1 − α) (τ − σ)1−α  = Γ (1 − β) (τ − σ)−β Γ (1 − β +  (1 − α))
=0 ∞ (the rhs converges since α < 1), the series
=0 |Mk0 +
(x0 , τ ; z, σ)| converges uniformly on ∂M × ∂M × R2T . Case 2. There exists k1 = k¯ ∈ N such that −βk1 ≥ 0 and −γk < 1 − 2α for all 1 ≤ k ≤ k1 . Then (24.105) implies −γk1

|Mk1 | (x0 , τ ; z, σ) ≤ Ck1 (τ − σ)−βk1 dτ ≤

−γk C˜k1 dτ 1

(x0 , z)

(x0 , z) ,

where C˜k1  Ck1 T −βk1 . Substituting this in (24.98a), we have 2α−1−γk1 (x0 , z) |Mk1 +1 (x0 , τ ; z, σ)| ≤ C˜k1 +1 dτ

for some C˜k1 +1 < ∞. Iterating this, we have
(2α−1)−γk1 (x0 , z) |Mk1 +
(x0 , τ ; z, σ)| ≤ C˜k1 +
dτ

for all  ∈ N ∪ {0}. Hence, by taking  sufficiently large so that  (2α − 1) − γk1 ≥ 0, this brings us back to Case 1. This completes the proof of the claim (i.e., (24.84)) and hence Lemma 24.31.  Now we make a couple of observations. Since ∞
=0 |Mk+
(x0 , τ ; z, σ)| converges uniformly on ∂M × ∂M × R2T , by (24.91) we have  ∞  τ  dσ Mk (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) k=1

0

∂M





τ

dσ

= 0

∂M

M∞ (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) ,

where M∞ (x0 , τ ; z, σ) 

∞ 

Mk (x0 , τ ; z, σ) .

k=1

Hence, using (24.88), we see that ψ∞ (defined by (24.83)) may be written as ∞  Ak (x0 , τ ) ψ∞ (x0 , τ ) − 2b (x0 , τ ) = k=1





τ

dσ

=2 0

∂M

M∞ (x0 , τ ; z, σ) b (z, σ) dµg(σ) (z) .

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5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

301

Since M∞ (x0 , τ ; z, σ) − M1 (x0 , τ ; z, σ) =

∞ 

Mk+1 (x0 , τ ; z, σ)

k=1 ∞  τ 



=2

k=1 σ τ 

∂M

 =2

σ

∂M

: ∂H (x0 , τ ; w, ρ) Mk (w, ρ; z, σ) dµg(ρ) (w) dρ ∂νw,ρ

: ∂H (x0 , τ ; w, ρ) M∞ (w, ρ; z, σ) dµg(ρ) (w) dρ, ∂νw,ρ

we have that M∞ satisfies the equation M∞ (x0 , τ ; z, σ) : ∂H (x0 , τ ; z, σ) ∂νz,σ  τ : ∂H (x0 , τ ; w, ρ) M∞ (w, ρ; z, σ) dµg(ρ) (w) dρ. +2 σ ∂M ∂νw,ρ

=2

Finally, by (24.86) we note that  τ  ∞  : ∂H dσ (x, τ ; z, σ) Ak (z, σ) dµg(σ) (z) uψ∞ (x, τ ) = − 0 ∂M ∂νz,σ k=0  ∞  τ  : ∂H dσ (x, τ ; z, σ) Ak−1 (z, σ) dµg(σ) (z) =− ∂M ∂νz,σ k=1 0  ∞  τ  dσ Mk (x, τ ; z, σ) b (z, σ) dµg(σ) (z) =− k=1 τ

 =−

0

∂M



dσ 0

∂M

M∞ (x, τ ; z, σ) b (z, σ) dµg(σ) (z) ;

here we equated (24.82) and (24.88) to obtain the third equality. 5.3. Heat kernels on noncompact manifolds. The existence of the heat kernel on noncompact manifolds is given by the following; see Theorem 4 and its proof on pp. 188–191 of [27]. Theorem 24.39 (Existence and uniqueness of heat kernel on noncompact manifolds — fixed metric). Let (Mn , g) be a (not necessarily complete) noncompact Riemannian manifold. There exists a unique minimal positive fundamental solution HM (x, y, t) of the heat equation (also called the heat kernel ). Moreover, HM (y, x, t) is C ∞ and symmetric in x and y. Now we consider the time-dependent metric case.

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302

24. HEAT KERNEL FOR EVOLVING METRICS

Theorem 24.40 (Existence and uniqueness of heat kernel on noncompact manifolds — time-dependent metrics). Let Mn be a noncompact manifold and let g (τ ), τ ∈ [0, T ], be a smooth family of Riemannian metrics on M. If Q is uniformly bounded, then there exists a unique C ∞ minimal positive fundamental solution HM (x, τ ; y, υ) for the heat-type operator ∂ − ∆x,τ + Q. Lx,τ = ∂τ The idea of the proof is to take an exhaustion {Ωi }i∈N of M by smooth domains with compact closure such that Ωi ⊂ Ωi+1 . Let HΩi (x, τ ; y, υ) denote the Dirichlet heat kernel of Ωi , g|Ωi , which exists by Theorem 24.32. By the maximum principle we have 0 < HΩi ≤ HΩi+1

(24.106)

on Ωi × Ωi × R2T .

Therefore HM (x, τ ; y, υ)  lim HΩi (x, τ ; y, υ) ∈ (0, ∞]

(24.107)

i→∞

exists for any (x, τ ; y, υ) ∈ M × M × R2T (note that HΩi is defined at any (x, τ ; y, υ) for i sufficiently large). Note that the positivity of HM follows from (24.106). We shall show that HM is finite, C ∞ , and a fundamental solution. One way to see that HM is finite is to apply the methods used in the proof of Lemma 22.9. Lemma 24.41. Given y ∈ Ωi , we have  HΩi (x, τ ; y, υ) dµg(τ ) (x) ≤ C (T ) < ∞ Ωi

for τ ∈ (0, T ]. Proof. We compute  d HΩi (x, τ ; y, υ) dµg(τ ) (x) dτ Ωi  ∂HΩi (x, τ ; y, υ) dµg(τ ) (x) = Ωi ∂τ  (∆x,τ − Q) HΩi (x, τ ; y, υ) dµg(τ ) (x) = Ω   i ∂HΩi (x, τ ; y, υ) dµg(τ ) (x) − QHΩi (x, τ ; y, υ) dµg(τ ) (x) = Ωi ∂ντ,i Ωi  QHΩi (x, τ ; y, υ) dµg(τ ) (x) ≤− Ωi  Q HΩi (x, τ ; y, υ) dµg(τ ) (x) , ≤− inf M×M×R2T

Ωi

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6. NOTES AND COMMENTARY

303

where ντ,i is the outward unit normal to ∂Ωi with respect to g (τ ); here we used

∂HΩi ∂ντ,i

≤ 0.6 The lemma follows since  lim HΩi (x, τ ; y, υ) dµg(τ ) (x) = 1. τ υ

Ωi


With Lemma 24.41, we may apply the parabolic mean value inequality (i.e., Theorem 25.2, which is a local result) to obtain a uniform upper bound for HΩi on any compact subset of M × M × R2T (in a bounded subset of M and away from υ = τ ). We conclude that HM (x, τ ; y, υ) is finite on all of M × M × R2T . By the Bernstein local derivative estimates for heat-type equations, we obtain that HM is C ∞ . To show that HM is a fundamental solution, we may apply the method in S. Zhang [196]. Finally, HM is the minimal positive fundamental solution since for any positive fundamental solution H  we have H  ≥ HΩi for all i. 6. Notes and commentary The heat kernel with time-dependent coefficients has been treated in [14], [61], by Garofalo and Lanconelli [69], [70], and by one of the authors [85]. There is a work by Molchanov [132] using probabilistic methods regarding the asymptotics of heat kernels with respect to time-dependent metrics (we would like to thank Peter Topping for bringing this reference to our attention). §1. The existence of the heat kernel associated to a 1-parameter family of Riemannian metrics on a closed manifold is due to one of the authors [85] and is related to [69]. §5. In subsection 5.2 we follow Chapter VII of Chavel [27]. For another approach to Lemma 24.31, see Theorems 7 and 9 in Chapter 3 of Friedman [61]. For Lemma 24.33, see §5.2 of Friedman [61] or, for a special case, see Theorem 1 on pp. 159–160 of Chavel [27]. To solve Exercise 24.34, one needs to adapt the arguments in [61].

6

In fact,

∂HΩi ∂ντ,i

< 0 by the Hopf boundary point lemma.

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http://dx.doi.org/10.1090/surv/163/09

CHAPTER 25

Estimates of the Heat Equation for Evolving Metrics You talk about things that nobody cares. – From “Sweet Emotion” by Aerosmith

In this chapter we discuss estimates for heat-type equations with respect to evolving Riemannian metrics. In particular, we consider mean value-type inequalities and differential Harnack estimates. Generally, the proofs of the former are based on integral estimates and the proofs of the latter rely on applications of the maximum principle to gradient-type quantities. In §1 we present the mean value inequality for subsolutions to heat-type equations using Moser iteration. In §2 we discuss the Li–Yau differential Harnack estimate for positive solutions to heat-type equations. 1. Mean value inequality for solutions of heat-type equations with respect to evolving metrics For a second-order elliptic or parabolic equation, the mean value inequality bounds a subsolution in a smaller ball in terms of its integral in a larger ball. This fundamental inequality, which we now consider for solutions of heat-type equations with respect to evolving metrics, is useful for obtaining upper estimates for the heat kernel (see §2 in Chapter 26). 1.1. Statement of the parabolic mean value inequality for an evolving metric. First, in the fixed metric case, we may formalize the parabolic mean value property as follows. Recall B (p, r) = {x ∈ M : d (x, p) < r}. Definition 25.1. We say that a complete Riemannian manifold (Mn , g) satisfies the parabolic mean value inequality up to time T ∈ (0, ∞] with constant C < ∞ if for any (p, t) ∈ M × [0, √T ) and any nonnegative subsolution f to the heat equation on P = B p, t × [0, t], i.e.,

we have

∂f ≤ ∆f, ∂t  C f dµ dt, f (p, t) ≤ Vol (P ) P 305

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306

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

where the volume is measured with respect to the metric g + dt2 . The time-dependent setup is as follows. Let g˜ be a complete C ∞ metric on a smooth manifold Mn and suppose that Ω is a compact domain in M with C ∞ boundary ∂Ω. (The boundary ∂Ω may be empty. For example, if M is closed, we may take Ω = M.) Let g (τ ), τ ∈ [0, T ], where T ∈ (0, ∞), be a C ∞ family of C ∞ metrics on Ω such that the initial metric satisfies the bounds (25.1)

C0−1 g˜ ≤ g (0) ≤ C0 g˜

in Ω for some constant C0 ∈ [1, ∞). Sometimes we shall take g˜ = g (0), in which case C0 = 1. Define the time-dependent symmetric 2-tensor Rij by ∂ gij  2Rij . ∂τ

(25.2) Let (25.3)

R (x, τ )  g ij (x, τ ) Rij (x, τ ) .

A special case we shall later consider is where Rij = Rij is the Ricci tensor (so that g (τ ) is a solution to the backward Ricci flow). Let (25.4)

Λ  sup |Rij (x, τ )|g(τ ) < ∞. Ω×[0,T ]

By (25.1) and (25.2), we have (25.5)

C˜0−1 g˜ ≤ g (τ ) ≤ C˜0 g˜

in Ω × [0, T ], where C˜0  C0 e2ΛT < ∞. Let Q : Ω × [0, T ] → R be a C ∞ function and let u : Ω × [0, T ] → R+ be a positive subsolution to ∂u ≤ ∆g(τ ) u − Q u. ∂τ When Rij = Rij , we are most interested in solutions to the adjoint heat equation ∂u ∂τ = ∆g(τ ) u − R u. Define the parabolic cylinder

(25.7) Pg˜ x, τ, r, −r2  Bg˜ (x, r) × [τ − r2 , τ ]

(25.6)

based at the point (x, τ ) with radius r. The following is a version of Moser’s parabolic mean value inequality for heat-type equations with respect to evolving metrics. Theorem 25.2 (Parabolic mean value inequality). In the above setup, suppose that we have the Ricci curvature lower bound Rc (˜ g ) ≥ −K in Ω, where K ≥ 0, and let u : Ω × [0, T ] → R+

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1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS

307

be a positive subsolution to (25.6). If (x0 , τ0 ) ∈ Ω × (0, T ] and r0 > 0 are such that   Pg˜ x0 , τ0 , 4r0 , − (2r0 )2 ⊂ Ω × [0, T ] , (25.8) then1 (25.9)

√

C1 eC2 τ0 +C3 Kr0 u≤ 2 sup r0 Volg˜ (Bg˜ (x0 , r0 )) Pg˜ (x0 ,τ0 ,r0 ,−r02 )

 Pg˜ (x0 ,τ0 ,2r0 ,−(2r0 )2 )

u(x, τ ) dµg˜(x) dτ,

where (1) C1 depends only on C0 , n, T , and Λ, (2) C2 depends only on supΩ×[0,T ] |Q| and supΩ×[0,T ] |R (x, τ )|, (3) C3 depends only on n. The proof of this theorem shall occupy the rest of this section. The idea is to use Moser iteration to bound higher and higher Lp -norms of the subsolution u of (25.6) on ‘slightly’ smaller and smaller parabolic cylinders based at the same point. Starting with an L1 bound on a parabolic cylinder, by applying the Sobolev inequality to a sequence of reverse Poincar´e-type inequalities obtained by integration by parts against the equation, we shall recursively derive an L∞ bound (on the parabolic cylinder based at the same point with half the radius) as the limit of uniform Lp bounds as p → ∞. We now proceed with the details. 1.2. Proof of the parabolic mean value inequality via Moser iteration. To simplify our notation, let ∆ = ∆g(τ ) , | · | = | · |g(τ ) , and dµ = dµg(τ ) . Given our subsolution u : Ω × [0, T ] → R+ to (25.6), define (25.10)

v  e−Aτ u,

where A ≥ 0. By (25.6), we have ∂v − ∆v + (Q + A) v ≤ 0. ∂τ We shall choose A at least as large so that (25.11)

Q + A ≥ 0.

Step 1. Reverse Poincar´e-type inequality. We have for any real number p ∈ [1, ∞) (25.12) 1

∂ (v p ) − ∆ (v p ) + p (Q + A) v p ≤ −p (p − 1) v p−2 |∇v|2 ≤ 0 ∂τ

Note that the denominator on the rhs of (25.9) is    r02 Volg˜ (Bg˜ (x0 , r0 )) = Volg˜+dτ 2 Pg˜ x0 , τ0 , r0 , −r02 .

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308

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

on Ω × [0, T ]. We shall integrate this inequality after localizing it (to enable integration by parts). Let 0 ≤ τ1 < τ2 ≤ T and let ψ : Ω × [τ1 , τ2 ] → [0, 1]

(25.13)

be a cutoff function with support contained in D × [τ1 , τ2 ], where D ⊂ Ω is a compact regular (C 1 ) subdomain. Assume that ψ (x, τ1 ) ≡ 0

(25.14)

for all x ∈ Ω. We shall discuss the choices of D and construction of the corresponding ψ and bounds for their first derivatives later (see (25.31)). Multiplying the heat inequality (25.12) by ψ 2 v p and integrating by parts in space and time, we have (25.15)    τ2  2p

2 1 ∂ p p 2p v ψ − v ∆ (v ) + p (Q + A) v dµ dτ 0≥ 2 ∂τ τ1 D     τ2   1 ∂ψ 2p 1 2 2p 2 2p v − ψ v R dµ dτ + ψ v dµ (τ2 ) −ψ = ∂τ 2 2 τ1 D D  τ2    |∇ (ψv p )|2 − v 2p |∇ψ|2 + p (Q + A) ψ 2 v 2p dµ dτ + τ1

D

∂ dµ = R dµ and since since the volume form evolves by ∂τ    2 p p p 2 ψ v ∆ (v ) dµ = |∇ (ψv )| dµ − v 2p |∇ψ|2 dµ. − D

D

D

Note that, in retrospect, we were able to throw away the good gradient term on the rhs of (25.12) since, in (25.15), integrating by parts on the term involving −∆ (v p ) yields a comparable good term. Now choose A ∈ [0, ∞), depending only on Q and supΩ×[0,T ] R (in particular, A is independent of p and D), so that 1 (25.16) Q+A− R≥0 2p on D × [0, T ] for all p ≥ 1. Applying this to (25.15), we have    τ2  1 p 2 2 2p (25.17) |∇ (ψv )| dµ dτ + ψ v dµ (τ2 ) 0≥ 2 τ1 D D    τ2  ∂ψ + |∇ψ|2 v 2p dµ dτ. ψ − ∂τ τ1 D Thus, given 0 ≤ τ1 < τ2 ≤ T , if ψ : Ω × [τ1 , τ2 ] → [0, 1] in (25.13) with ψ ( · , τ1 ) ≡ 0 is defined so that it further satisfies the inequality ∂ψ + |∇ψ|2g(τ ) ≤ L (25.18) ψ ∂τ for some constant L ∈ [0, ∞), then since both terms on the rhs of the first line of (25.17) are nonnegative, we have

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1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS

Lemma 25.3.   τ2  |∇ (ψv p )|2g(τ ) dµg(τ ) dτ ≤ L (25.19) τ1

and

D







τ2

ψ v dµg(τ ) (τ2 ) ≤ 2L 2 2p

(25.20) D

τ2 τ1

309

 v 2p dµg(τ ) dτ D



τ1

v 2p dµg(τ ) dτ, D

where supp (ψ) ⊂ D × [τ1 , τ2 ] and D ⊂ Ω. Inequality (25.19) is sometimes
called a ‘reverse
2 Poincar´e-type inequal2 ity’ since it is roughly of the form |∇f | ≤ C f . Since (25.5) implies −n/2 n/2 dµg˜ ≤ dµg(τ ) ≤ C˜0 dµg˜ C˜0 1/2 and |α|g˜ ≤ C˜0 |α|g(τ ) for any 1-form α, inequality (25.19) implies (using g˜ instead of g (τ ))  τ2   τ2  n+2 p 2 2 ˜ |∇ (ψv )|g˜ dµg˜ dτ ≤ C0 |∇ (ψv p )|2g(τ ) dµg(τ ) dτ τ1 D τ1 D  τ2  (25.21) v 2p dµg˜ dτ. ≤ C˜0n+1 L τ1

Similarly, (25.20) implies    ψ 2 v 2p dµg˜ (τ2 ) ≤ 2C˜0n L (25.22) D

D

τ2 τ1

 v 2p dµg˜ dτ. D

Lp -norms

of v by lower Lp -norms. To make Step 2. Bounding higher use of the reverse Poincar´e-type inequality (25.21), we now recall the following version of the L2 -Sobolev inequality (see [165]). Proposition 25.4 (L2 -Sobolev inequality in a ball). There exists a constant CS < ∞ depending only on n such that for any complete Riemannian manifold (Mn , g˜), where n ≥ 3, if K ≥ 0, x0 ∈ M, and r ∈ (0, ∞) are such that Rcg˜ ≥ −K in Bg˜ (x0 , 2r) , then 

 n−2 |f |

2n n−2

n

dµg˜

Bg˜ (x0 ,r)

≤e for any

C∞

√ CS (1+ Kr )

−2 Volg˜ n





Bg˜ (x0 , r) Bg˜ (x0 ,r)

 r2 |∇f |2g˜ + f 2 dµg˜

function f with compact support in Bg˜ (x0 , r).

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310

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

Remark 25.5. If n = 2, then under the assumptions of the above lemma we have for any real number p > 2  p−2

 |f |

2p p−2

p

dµg˜

Bg˜ (x0 ,r)

≤e

√ CS (p)(1+ Kr )

−2 Volg˜ p





Bg˜ (x0 , r) Bg˜ (x0 ,r)

 r2 |∇f |2g˜ + f 2 dµg˜,

where CS (p) < ∞ depends only on p. Now given x0 ∈ int (Ω), we take D = Bg˜ (x0 , r), where r ∈ (0, 2r0 ], where r0 satisfies (25.8). We then have supp (ψ) ⊂ Bg˜ (x0 , r) × [τ1 , τ2 ] . Assume from now on that n ≥ 3 (we leave the n = 2 case as an exercise). In view of (25.19), we apply the above L2 -Sobolev inequality to the function ψv p , which is possible since Rcg˜ ≥ −K in Ω. We obtain for any p ∈ [1, ∞) and τ ∈ [τ1 , τ2 ],  n−2  |ψv | p

2n n−2

n

dµg˜

(τ )

Bg˜ (x0 ,r)

≤e

√ CS (1+ Kr )

−2 Volg˜ n





Bg˜ (x0 , r) Bg˜ (x0 ,r)



r2 |∇ (ψv p )|2g˜ + ψ 2 v 2p dµg˜ (τ ) .

On the other hand, by H¨older’s inequality, since n−2 n +   2(n+2) 4 |ψv p | n dµg˜ = |ψv p |2 |ψv p | n dµg˜ Bg˜ (x0 ,r)

Bg˜ (x0 ,r)

 ≤

|ψv p |

2n n−2

τ1

|ψv p |

2(n+2) n

τ2

≤



|ψv p |

2n n−2

 ×

τ2 τ1



Bg˜ (x0 ,r)

2

n

n

|ψv p |2 dµg˜ .

dµg˜ Bg˜ (x0 ,r)

 n−2  n

2

n

|ψv p |2 dµg˜

dµg˜

τ1

≤

Bg˜ (x0 ,r) √ −2 eCS (1+ Kr) Volg˜ n

= 1, we have

dµg˜ dτ

Bg˜ (x0 ,r)



2 n

n−2

Bg˜ (x0 ,r)

Thus (25.23)  τ2 



dτ

Bg˜ (x0 ,r)

Bg˜ (x0 , r)

  r2 |∇ (ψv p )|2g˜ + ψ 2 v 2p dµg˜



2

n

ψ 2 v 2p dµg˜ Bg˜ (x0 ,r)

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dτ.

1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS

311

Let



Pr  Pg˜ x0 , τ0 , r, −r2 , where Pg˜ is defined in (25.7). For the cutoff function ψ in (25.13) we take τ1 = τ0 − r2 , τ2 = τ ∈ τ0 − r2 , τ0 , and D = Bg˜ (x0 , r). Let 0 < r < r. We further require that ψ satisfies ψ ≡ 1 on Pr .

(25.24)

For such ψ, by (25.22) and supp (ψ) ⊂ Pr , we have    τ  2 2p n ˜ ψ v dµg˜ (τ ) ≤ 2C L 0

Bg˜ (x0 ,r)

τ0



−r 2

≤ 2C˜0n L

(25.25)

v 2p dµg˜ dτ

Bg˜ (x0 ,r)

v 2p dµg˜ dτ Pr

  for τ ∈ τ0 − r2 , τ0 , where L is as in (25.18). Substituting the reverse Poincar´e-type inequality (25.21), with τ1 = τ0 − r2 and now τ2 = τ0 , and (25.25) into (25.23), we obtain (25.26)  τ0 

|ψv p |

τ0 −r 2

≤e

Bg˜ (x0 ,r) √ CS (1+ Kr )

2(n+2) n

dµg˜ dτ

−2

Volg˜ n Bg˜ (x0 , r) 2    n n 2p v dµg˜ dτ × 2C˜0 L Pr



τ0 τ0 −r 2

 Bg˜ (x0 ,r)

 r2 C˜0n+1 L + ψ 2 v 2p dµg˜dτ.

Next, by (25.26) while using ψ ≡ 1 in Pr and ψ 2 ≤ 1 in Pr , we have  n+2 v n 2p dµg˜ dτ Pr

 |ψv p |

≤

2(n+2) n

dµg˜ dτ

Pr

  √ −2 ≤ eCS (1+ Kr) Volg˜ n Bg˜ (x0 , r) r2 C˜0n+1 L + 1  2  τ0    n n 2p ˜ v dµg˜ dτ v 2p dµg˜ dτ × 2C 0 L Pr

 

v 2p dµg˜ dτ

=M

τ0 −r 2

 n+2

Bg˜ (x0 ,r)

n

,

Pr

where (25.27)

M  eCS (1+

√ Kr )

  2 −2 n Volg˜ n Bg˜ (x0 , r) r2 C˜0n+1 L + 1 2C˜0n L ,

where Rc (˜ g ) ≥ −K in Ω, C˜0 is as in (25.5), C is given by Proposition 25.4, and L is as in (25.18).

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312

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

Summarizing, we have proved for 0 < r < r ≤ 2r0 that (25.28)

v

n

n+2 L n 2p (Pr )

1

≤ M n+2 2p v L2p (Pr ) ,

where the Lp -norms are with respect to the product measure dµg˜ dτ , i.e., for a space-time region P and a function f defined on P ,   1 q q f dµg˜ dτ .

f Lq (P )  P

n+2

That is, for the subsolution v, we have estimated the higher L n 2p -norm in the smaller parabolic cylinder Pr in terms of the lower L2p -norm in the larger parabolic cylinder Pr . Step 3. Bounding the L∞ -norm by the L2 -norm. We now iterate this estimate to obtain an L∞ -norm estimate. In particular, when considering the L2p -norm in (25.28), for some p ≥ 1, we shall choose r, r , and ψ to depend on p. For i ∈ N define (in view of r < r in (25.28))   1 (25.29) ri  r0 1 + i−1 2 (so that ri is decreasing in i, r1 = 2r0 , and limi→∞ ri = r0 ) and, motivated by (25.28), define   n + 2 i−1 pi  n (so that p1 = 1 and limi→∞ pi = ∞). Corresponding to ri (and pi ), we shall now construct space-time cutoff functions ψi : Ω × [0, T ] → [0, 1] ∞ for all i ∈ N. Define a C function φi : [0, ∞) → [0, 1] so that  1 on [0, ri+1 ] , φi = 0 on [ri , ∞) and

2i+1 ≤ φi ≤ 0 r0 = r20i ). Define a C ∞ function ηi : [0, T ] → [0, 1] so that    2 ,T , 1 on τ0 − ri+1 ηi = 0 on [0, τ0 − ri2 ] −

(note that ri − ri+1 (25.30) and

0 ≤ ηi ≤ 2 (note that ri2 − ri+1 ≥

(25.31)

r02 ). 2i−1

2i r02

Let

ψi (x, τ )  φi (dg˜ (x, x0 )) · ηi (τ ) .

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1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS

313

Note that 0≤

(25.32)

2i ∂ψi ≤ 2 ∂τ r0

and

|∇ψi |g˜ ≤

2i+1 . r0

Here ∇ψi is a C ∞ vector field on the complement of the cut locus of x0 (the cut locus is a closed set with measure zero). We then have supp (ψi ) ⊂ Pri ,

ψi ≡ 1 on Pri+1 .

Moreover, by (25.32) and ψi ≤ 1, we have ∂ψi + |∇ψi |2g˜ ≤ 4i+2 r0−2  Li . ∂τ Now we may take ψ = ψi in Step 2. Corresponding to (25.27), define (25.34) √     2 1 2 ˜ n+1 i+2 eCS (1+ Kri ) n i+2 −2 n ˜ 1 + i−1 C0 4 + 1 2 C 0 4 r0 . Mi  2/n 2 Vol Bg˜ (x0 , ri ) (25.33)

ψi

g˜

By (25.28), with r = ri , r = ri+1 , p = pi , and M = Mi , we have (note that pi+1 = n+2 n pi ) 1 n n+2 2pi

≤ Mi i+1 )

v L2pi (Pr ) , i ¯ (x0 , 4r0 ) ⊂ Ω. where we used the fact that ri ≤ 2r0 and (25.8) imply B Hence, for any integer j ≥ 2 j−1  n 1 ; n+2 2pi Mi

v L2 (P2r ) .

v L2pj (Pr ) ≤ 0 j

v L2pi+1 (Pr

i=1

Taking the limit as j → ∞, we have ∞  n 1 ; n+2 2pi Mi

v L2 (P2r )

v L∞ (Pr ) ≤ 0 0 i=1

≤ C˜ v L2 (P2r ) , 0

(25.35) where √   ∞ ; eCS (1+ Kri ) 2/n

i=1

Volg˜ Bg˜(x0 , ri ) √



1+

1 2i−1

−i

2



 2 n C˜0n+1 4i+2 + 1 2C˜0n 4i+2 r0−2 

−i

∞ n+2 1 ∞ n+2 1 ≤ eCS (1+2 Kr0 ) 2 i=1 ( n ) (Volg˜ Bg˜ (x0 , r0 ))− n i=1 ( n ) −i ∞    2  12 ( n+2 ; n ) n × 1 + C˜0n+1 4i+3 2C˜0n 4i+2 r0−2

i=1

˜  C.

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1 n n+2 2p

i

314

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

Now since (25.36)

∞ n+2 −i i=1

n

= n2 , we have √

n C˜ = eCS (1+2 Kr0 ) 4 (Volg˜ Bg˜ (x0 , r0 ))− 2 −i ∞    2  12 ( n+2 ; n ) n+1 i+3 n i+2 −2 n ˜ ˜ 2 C 0 4 r0 . 1 + C0 4 × 1

i=1

Since C˜0n+1 44 ≥ 1, we have  2 

n+2 − 4 n+2 n ≤ 2 n C˜0n+3 4i+3 n r0 n , 1 + C˜0n+1 4i+3 2C˜0n 4i+2 r0−2 which in turn implies C˜ ≤ eCS (1+2

√ Kr0 ) n 4

(Volg˜ Bg˜ (x0 , r0 ))− 2 2 1

n+2 4

(n+3)n 4

C˜0

r0−1 Cn ,

where  ( ∞  ; 2 i+3 n+2 n 4 Cn  1

(25.37)

n+2 −i n

)

=2

∞

( n+2 n )

i=1 (i+3)

−i+1

0, iterating this inequality for suitable choices of r and r between r0 and 2r0 , we obtain the in the L2 mean value inequality (here we need to keep track of the constants  −k < k β iterations of the previous inequality; in regards to this, we use ∞ k=1 ∞ for β ∈ (1, ∞))

v L∞ (Pr ) ≤ C v L2 (P2r ) . 0

0

1/2

1/2

An improvement, based on combining this with v L2 ≤ v L1 v L∞ and an elementary iteration, yields the L1 mean value inequality

v L∞ (Pr ) ≤ C v L1 (P2r ) . 0 0 2. Li–Yau differential Harnack estimate for positive solutions of heat-type equations with respect to evolving metrics For a heat-type equation, a differential Harnack estimate of Li–Yau type yields a space-time gradient estimate for a positive solution, which when integrated compares the solution at different points in space and time. It is one of the main ingredients in obtaining lower bounds for heat kernels (see subsection 2.2 in Chapter 26). In this section we shall prove differential Harnack estimates, with respect to evolving metrics, by adjustments of Li and Yau’s proof of Theorem 1.2 and Theorem 1.3 in [121]. We state the estimates and some consequences in subsections 2.1 and 2.2 and we begin the proofs in subsection 2.3. In the case of the Ricci flow the hypotheses one assumes are local; this is related to the cutoff function in the maximum principle argument (i.e., localization) and uses Perelman’s changing distance estimate. Before being immersed in the more technical exposition below, the reader may wish to see the exposition of the Li–Yau inequality in the ‘cleaner’ case of the heat equation with respect to a static metric on a closed manifold with Rc ≥ 0; see Theorem 10.1 in [45]. 2.1. Statement of the differential Harnack estimate and some consequences. Let g (τ ), τ ∈ [0, T ], be a smooth family of complete Riemannian metrics on a manifold Mn . As in the previous section, define ∂ gij  2Rij , ∂τ which is a time-dependent symmetric 2-tensor on M defined for τ ∈ [0, T ], and again let R  g ij Rij . Let (25.49)

Q : M × [0, T ] → R

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318

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

(‘potential function’ for the heat-type equation below) be a C ∞ function satisfying   −2g ij ∇i Rjk + ∇k R + 4 |∇Q| ≤ C  and ∆Q ≤ C  (25.50) 3  on M × [0, T ] for some constant C < ∞. Suppose that |sect (g (0))| ≤ K on M and suppose Rc ≥ −K

(25.51)

and

|Rij | ≤ Λ

on M × [0, T ] for some finite and nonnegative constants K and Λ. We have the following result. Theorem 25.8 (Harnack estimate of Li–Yau-type for evolving metrics). Let g (τ ), τ ∈ [0, T ], be a family of Riemannian metrics on Mn as above and let Q be as above. If u : Bg(0) (p, 2 R) × [0, T ] → R+ , where R ≥ 1, is a positive solution to ∂u = ∆g(τ ) u − Qu, ∂τ then for any ε ∈ (0, 2/3) we have the gradient estimate   1 C5 C6 n ∂ log u 2 − (1 − ε) |∇ log u| + Q ≥ − + + 2 + C7 (25.53) ∂τ 2 − 3ε τ R R

(25.52)

in Bg(0) (p, R −Cn,K ) × (0, T ], where C5 depends only on n, K, Λ, T , and supM×[0,T ] |∇i Rjk |, where C6 depends only on n, K, Λ, T , and ε, where C7 depends only on n, K, Λ, ε, and C  , and where Cn,K ∈ (1, ∞) depends only on n and K. In particular, taking R → ∞, we have   1 n ∂ log u − (1 − ε) |∇ log u|2 + Q ≥ − + C7 (25.54) ∂τ 2 − 3ε τ in all of M × (0, T ]. In the case of the Ricci flow, we have the following. Theorem 25.9 (Harnack estimate of Li–Yau-type for Ricci flow). Let ∂ gij = −2Rij on g (τ ), τ ∈ [0, T ], be a complete solution of the Ricci flow ∂τ n a manifold M and let Q : M × [0, T ] → R be a C ∞ function. Suppose there are constants K, C  , and a radius R ≥ max{1, √1K } such that in Bg(τ ) (p, 2 R) we have 2 (25.55)

|Rij | ≤ K,

4 |∇Q| ≤ C  , 3

and

∆Q ≤ C 

Note that in the cases where Rij = cRij , the inequality to the first inequality in (25.50). 2

4 3

|∇Q| ≤ C  is equivalent

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

319

for each τ ∈ [0, T ]. If .

u:

Bg(τ ) (p, 2 R) × {τ } → R+

τ ∈[0,T ]

is a positive solution to ∂u = ∆g(τ ) u − Qu, ∂τ then for each τ ∈ [0, T ] we have in Bg(τ ) (p, R) n ∂ log u − (1 − ε) |∇ log u|2 + Q ≥ − (25.56) ∂τ 2 − 3ε



 1 C8 C9 + + 2 + C10 , τ R R

where C8 depends only on n and K, where C9 depends only on n and ε, and where C10 depends only on n, K, ε, and C  . Remark 25.10. A difference in the above two theorems is that, in the latter case, the bounds in the hypothesis are assumed only on a ball instead of on the whole manifold. Problem 25.11. Prove a Li–Yau-type differential Harnack estimate for positive solutions of heat-type equations coupled to the backward Ricci flow using only local bounds on the curvature and potential function. 2.2. Some consequences — classical-type Harnack estimates. A standard integration along paths of the gradient estimate (25.54) yields the following ‘classical-type’ Harnack inequality, which compares the solution at different points in space-time. Corollary 25.12 (Classical-type Harnack inequality for evolving metrics). Assume that g˜ is a complete metric on Mn such that for each τ ∈ [0, T ], (25.57)

C˜0−1 g˜ ≤ g (τ ) ≤ C˜0 g˜

on M, where C˜0 < ∞. Then under the assumptions of Theorem 25.8 we have that for any x1 , x2 ∈ M and 0 < τ1 < τ2 ≤ T, u(x2 , τ2 ) ≥ e−C11 (τ2 −τ1 ) (25.58) u(x1 , τ1 )



τ2 τ1

−

n 2−3ε



2 C˜0 dg˜ (x1 , x2 ) exp − 4(1 − ε) τ2 − τ1

where C11 = C7 + supM×[τ1 ,τ2 ] Q.

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 ,

320

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

Proof. Let γ : [τ1 , τ2 ] → M be a smooth path joining x1 to x2 . By the fundamental theorem of calculus, u(x2 , τ2 ) log u(x1 , τ1 )  τ2 d (log u (γ (τ ) , τ )) dτ = τ1 dτ  ( )   τ 2  dγ ∂ log u (γ (τ ) , τ ) + (∇ log u) (γ (τ ) , τ ) , (τ ) dτ. = ∂τ dτ τ1 g(τ ) Applying (25.54), we have u(x2 , τ2 ) u(x1 , τ1 ) )  (  τ2  dγ 2 (1 − ε) |∇ log u|g(τ ) (γ (τ ) , τ ) + (∇ log u) (γ (τ ) , τ ) , dτ ≥ dτ g(τ ) τ1   τ2  n − Q (γ (τ ) , τ ) − C7 dτ − + (2 − 3ε) τ τ1 2    τ2    dγ τ2 n 1  (τ ) log dτ − − C11 (τ2 − τ1 ) , ≥−   4(1 − ε) dτ 2 − 3ε τ1

log

τ1

g(τ )

where C11 = C7 + supM×[τ1 ,τ2 ] Q. Exponentiating, we obtain for any path γ  − n Λ(γ) 2−3ε u(x2 , τ2 ) − −C11 (τ2 −τ1 ) τ2 ≥e e 4(1−ε) , u(x1 , τ1 ) τ1 where 2  τ2    dγ   (τ ) dτ. Λ (γ) =   dτ τ1 g(τ ) Now let γ be a constant speed minimal geodesic with respect to g˜, so that     dγ  (τ ) ≡ dg˜ (x1 , x2 ) .   dτ τ2 − τ1 g˜ Then, using (25.57), we obtain 2  τ2    dγ d2 (x , x ) ˜  dτ = C˜0 g˜ 1 2 .  (τ ) Λ (γ) ≤ C0   dτ τ2 − τ1 τ1 g˜


The corollary follows.

In the case of the Ricci flow, by the same argument as in the proof of Corollary 25.12, we have the following. Corollary 25.13 (Classical-type Harnack inequality for Ricci flow). Let g (τ ), τ ∈ [0, T ], be a complete solution of the Ricci flow as in Theorem 25.9 and such that for each τ ∈ [0, T ], C˜ −1 g˜ ≤ g (τ ) ≤ C˜0 g˜ 0

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

321

on Bg(τ ) (p, R), where g˜ is some complete metric on M and C˜0 < ∞. Suppose that x1 , x2 ∈ M are such that there exists a minimal geodesic γ joining x1 and x2 with respect to (M, g˜) with γ (τ ) ∈ Bg(τ ) (p, R) for each τ ∈ [0, T ]. Let . Bg(τ ) (p, 2 R) × {τ } . P= τ ∈[0,T ]

Then for a positive solution u : P → R+ to (25.52) and 0 < τ1 < τ2 ≤ T we have    − n ˜0 d2g˜ (x1 , x2 ) 2−3ε C τ u(x2 , τ2 ) 2 ≥ e−C12 (τ2 −τ1 ) exp − , u(x1 , τ1 ) τ1 4(1 − ε) τ2 − τ1 where C12 =

C8 R

+

C9 R2

+ C10 + supP Q.

Exercise 25.14. Prove the above corollary. As a special case of Corollary 25.12, (25.59)   n ˜ C √

0 τ2 2−3ε C11 (τ2 −τ1 )+ 4(1−ε) u(x2 , τ2 ) for x2 ∈ Bg˜ x1 , τ2 − τ1 . u(x1 , τ1 ) ≤ e τ1 Integrating this with respect to x2 over the ball, we have (25.60)   n  τ2 2−3ε C12 √ u(x, τ2 )dµg˜ (x) , u(x1 , τ1 ) ≤ √ Volg˜ Bg˜ (x1 , τ2 − τ1 ) τ1 Bg˜ (x1 , τ2 −τ1 ) ˜ C 0 √ C (τ −τ )+ e 11 2 1 4(1−ε) . Since x2 ∈ Bg˜ (x1 , τ2 − τ1 ) is equivalent to where C12 =√ x1 ∈ Bg˜ (x2 , τ2 − τ1 ), integrating (25.59) with respect to x1 over a ball, we obtain (25.61)   n  −1 τ1 2−3ε C12 √ u(x, τ1 )dµg˜ (x) . u(x2 , τ2 ) ≥ √ Volg˜ Bg˜ (x2 , τ2 − τ1 ) τ2 Bg˜ (x2 , τ2 −τ1 )

The above inequalities (25.60) and (25.61) may be considered as mean value-type inequalities. Compare them with the mean value inequality of §1 of this chapter, proved by Moser iteration, where the integral on the rhs of the inequality is over a parabolic cylinder instead of a spatial ball at a given time. Remark 25.15. In view of Perelman’s differential Harnack estimate for the adjoint heat kernel, we are interested in obtaining bounds for this adjoint heat kernel under local hypotheses. We shall obtain upper and lower bounds for the heat kernel using the Li–Yau inequality and other estimates and then use symmetry (see Lemma 26.3) to obtain upper and lower bounds for the adjoint heat kernel. One also has gradient estimates as discussed in §2 of Chapter 16 and §4 of Appendix E both in Part II.

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322

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

2.3. The Harnack quantity. We proceed to give the proofs of Theorem 25.8 and Theorem 25.9. Let u be a positive solution to the heat-type equation (25.52) and let (25.62)

L  log u.

Note that (25.52) implies ∂L = ∆L + |∇L|2 − Q. ∂τ Here and below, the inner products and norms are with respect to g (τ ). Given ε ∈ R, define the gradient-type (Harnack) quantity ∂L − (1 − ε) |∇L|2 + Q = ∆L + ε |∇L|2 . (25.64) P  ∂τ (Later, we shall choose ε positive and small enough.) (25.63)

Remark 25.16. (1) In the model Euclidean case, where u is the fundamental solution to the heat equation on Rn (Rij = 0) and ε = 0, we have that   n |x − y|2 =− τ P = τ ∆L = τ ∆x − 4τ 2 is constant. (2) For the heat equation on a static manifold with Rc ≥ 0 one considn . ers the quantity P with ε = 0 and shows that P ≥ − 2τ   We wish to compute the evolution of P . Since ∆ = g ij ∂i ∂j − Γkij ∂k , by (25.49) and Lemma 3.2 in Volume One, we have

∂ ∆g(τ ) = −2Rij ∇i ∇j − g ij (∇i Rj
+ ∇j Ri
− ∇
Rij ) g k
∇k (25.65) ∂τ

= −2Rij ∇i ∇j + −2g ij ∇i Rjk + ∇k R · ∇k as differential operators acting on functions. Recall that in the special case where Rij = cRij , c ∈ R, is a constant multiple of the Ricci tensor, by the contracted second Bianchi identity, the evolution of the Laplacian simplifies to

∂ ∆g(τ ) = −2cRij ∇i ∇j . (25.66) ∂τ 2.4. Evolution of the Harnack quantity. We have the following equation (compare with (E.39) and (E.40) in Part II). Lemma 25.17 (Evolution equation for the Harnack quantity). If u is a positive solution to the heat-type equation (25.52), where the time-dependent metric g (τ ) satisfies (25.49), then for any ε ∈ R the Harnack quantity P = ∆L + ε |∇L|2 ,

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

323

where L = log u, satisfies the heat-type equation (25.67)

∂P = ∆P + 2 ∇L, ∇P  + 2 (1 − ε) |∇∇L|2 − 2Rij ∇i ∇j L ∂τ + 2 ((1 − ε) Rij − εRij ) ∇i L∇j L

+ −2g ij ∇i Rjk + ∇k R − 2ε∇k Q · ∇k L − ∆Q.

Proof. Taking the time derivative of (25.64) while using (25.63) and (25.65), we compute     (  ) ∂ ∂L ∂L ∂P = ∆ L+∆ + 2ε ∇L, ∇ ∂τ ∂τ ∂τ ∂τ − 2εRij ∇i L∇j L

(25.68) = −2Rij ∇i ∇j L + −2g ij ∇i Rjk + ∇k R · ∇k L   + ∆P + ∆ (1 − ε) |∇L|2 − Q + 2ε ∇L, ∇P  1 0  + 2ε ∇L, ∇ (1 − ε) |∇L|2 − Q − 2εRij ∇i L∇j L. Applying the standard Bochner-type formula ∆ |∇L|2 = 2 |∇∇L|2 + 2 ∇L, ∇ (∆L) + 2 Rc (∇L, ∇L) and the identity

1 0 2 (1 − ε) ∇L, ∇ (∆L) + 2ε (1 − ε) ∇L, ∇ |∇L|2 = 2 (1 − ε) ∇L, ∇P 

(which follows from ∇P = ∇∆L + ε∇ |∇L|2 ) to (25.68), we obtain (25.67) by rearranging and combining terms.
If we assume a lower bound on the Ricci curvature and bounds on Rij and parts of its first two covariant derivatives, then we obtain the following. Corollary 25.18 (Evolution inequality for the Harnack quantity). Suppose that we have the bounds (25.69)

Rc ≥ −K,   −2g ij ∇i Rjk + ∇k R + 4 |∇Q| ≤ C  , 3

|Rij | ≤ Λ, ∆Q ≤ C 

on M × [0, T ] for some nonnegative constants K, Λ, and C  . If ε ∈ (0, 2/3), then the Harnack quantity P satisfies the following heat-type inequality (25.70)

2 − 3ε ∂P ≥ ∆P + 2 ∇L, ∇P  + (∆L)2 − C1 |∇L|2 − C2 , ∂τ n

where (25.71)



C1  2 (1 − ε) K + Λε + C 

and

C2 

Λ2 + 2C  . ε

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25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

Proof. Applying the assumed bounds in (25.69) to the evolution equation (25.67), we obtain3 ∂P ≥ ∆P + 2 ∇L, ∇P  + 2 (1 − ε) |∇∇L|2 − 2Λ |∇∇L| ∂τ − 2 ((1 − ε) K + Λε) |∇L|2 − C  |∇L| − C  . 2

b to both the first Then applying the Peter–Paul inequality ax2 − bx ≥ − 4a and second derivatives of L, we have

Λ2 ∂P ≥ ∆P + 2 ∇L, ∇P  + (2 − 3ε) |∇∇L|2 − ∂τ ε

− 2 (1 − ε) K + Λε + C  |∇L|2 − 2C  . Finally, applying |∇∇L|2 ≥

1 n

(∆L)2 to this, we obtain the corollary.




Note that we are trying to obtain a lower bound for P , so that the term (∆L)2 on the rhs of (25.70), which is similar to P 2 , is a ‘good’ term.

2−3ε n

2.5. Localizing the Harnack calculation. To apply the maximum principle to the evolution inequality for P when M is noncompact, we need to ‘localize’ the above calculation of the evolution of P , i.e., multiply P by a cutoff function so that it has compact support. First we make some calculations using a general cutoff function and then we choose a suitable cutoff function. Let R ∈ [1, ∞) and let φ : M × [0, T ] → [0, 1] be a C 2 function with (25.72)

supp (φ) ⊂ Bg(0) (p, 2 R) × [0, T ] .

Multiplying the evolution inequality (25.70) for P by τ φ (the factor τ is motivated by Remark 25.16), since φ is independent of time, we have at any

3

Note that since ε ∈ (0, 2/3),






4
ij ij 
−2g ∇i Rjk + ∇k R − 2ε∇k q
≤
−2g ∇i Rjk + ∇k R
+ |∇q| ≤ C . 3

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

325

point where φ = 0 ∂P ∂φ φP 1 ∂ (τ φP ) = φ + P+ τ ∂τ ∂τ ∂τ τ 

(25.73)

2 − 3ε (∆L)2 − C1 |∇L|2 − C2 ≥ φ ∆P + 2 ∇L, ∇P  + n φP ∂φ P+ + ∂τ  ( ) τ ∇φ ∂φ − ∆φ P − 2 , ∇ (φP ) = ∆ (φP ) + ∂τ φ



|∇φ|2 P + 2 ∇L, ∇ (φP ) − 2P ∇L, ∇φ φ φP 2 − 3ε φ (∆L)2 − C1 φ |∇L|2 − C2 φ + , + n τ

+2

where C1 , C2 ≥ 0 are as in (25.71). We now apply the weak maximum principle to our calculation. By (25.72), the support of τ φ (x) P (x, τ ) is contained in B(p, 2 R) × [0, T ]. Let τ¯ ∈ (0, T ] and suppose that τ φP is negative somewhere in B(p, 2 R) × [0, τ¯]. (Otherwise we have the estimate τ φP ≥ 0 in M × [0, τ¯].) Then there exists a point (x0 , τ0 ) ∈ B(p, 2 R) × (0, τ¯] at which τ φP attains a negative minimum. We shall obtain a lower bound for τ φP at this negative minimum. At (x0 , τ0 ) we have φ = 0 and ∇(φP ) = 0,

(25.74)

∆(φP ) ≥ 0,

and

∂ (τ φP ) ≤ 0. ∂τ

The rest of the calculations in this subsection occur at (x0 , τ0 ). Substituting these inequalities into (25.73), we have   |∇φ|2 2 − 3ε ∂φ − ∆φ P + 2 P − 2P ∇L, ∇φ + φ (∆L)2 (25.75) 0 ≥ ∂τ φ n φP . − C1 φ |∇L|2 − C2 φ + τ0 Since by (25.64) (25.76)

 2 (∆L)2 = P − ε |∇L|2 = P 2 − 2εP |∇L|2 + ε2 |∇L|4

and since for any δ > 0 (25.77)

−2P ∇L, ∇φ ≥ δφ |∇L|2 P + δ −1

|∇φ|2 P φ

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326

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

wherever P < 0, we obtain from (25.75) that (25.78) 2 − 3ε 2 φP n     2

φ 2 (2 − 3ε) ∂φ 2 −1 |∇φ| − ∆φ + 2 + δ + ε − δ φ |∇L| P − + ∂τ φ τ0 n   2 − 3ε 2 4 2 ε |∇L| − C1 |∇L| − C2 . +φ n

0≥

ε, which is positive since ε ∈ (0, 2/3).4 Suppose Now we take δ = 2(2−3ε) n that the first two terms on the second line of (25.78) satisfy   n |∇φ|2 ∂φ − ∆φ + 2 + ≤ C3 , (25.79) ∂τ 2 (2 − 3ε) ε φ where C3 ∈ [0, ∞) is to be determined after we choose φ in (25.85) below. Then we have at (x0 , τ0 ), where τ φP attains a negative minimum,     φ nC12 2 − 3ε (φP )2 + C3 + φP − φ2 C2 + , (25.80) 0≥ n τ0 (2 − 3ε) ε2 where we multiplied the whole equation by φ and where we also used a |∇L|4 − C1 |∇L|2 ≥ −

C12 a

for a > 0. We have a quadratic inequality in φP , which we use to bound φP . Note that if there is a number x ∈ R satisfying an inequality of the form ax2 + bx + c ≤ 0, where a > 0, b ≥ 0, and c < 0, then b2 − 4ac > 0 and we have the lower bound √ √ b + −ac −b − b2 − 4ac ≥− . x≥ 2a a Hence   !   2 n nC1 τ0 C3 + φ + τ0 φ 2−3ε C2 + (2−3ε)ε . τ0 φP ≥ − 2 n 2 − 3ε Furthermore, since τ0 ≤ τ¯ ≤ T and 0 ≤ φ ≤ 1, we have (25.81)  !  n 1 + τ¯C3 + τ¯ 2−3ε C2 + τ0 (φP ) (x0 , τ0 ) ≥ − n 2 − 3ε

nC12

 

(2−3ε)ε2

˜  −C.

2(2−3ε) More generally, we may

take δ ∈ (0, n ε] since then on the second line of (25.78) the term − 2(2−3ε) ε − δ φ |∇L|2 P is nonnegative. n 4

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

327

Since (x0 , τ0 ) is a point where τ φ (x) P (x, τ ) attains a negative minimum, we have the estimate (25.82) τ φ (x) P (x, τ ) ≥ −C˜ in all of M × [0, τ¯]. 2.6. The form of the cutoff function. From (25.79) we see that the remaining issue is to obtain a cutoff function φ : M × [0, T ] → [0, 1] such that both Define a C ∞ function

∂φ ∂τ − ∆φ

and

|∇φ|2 φ

have upper bounds.

ψ : [0, ∞) → [0, 1]

(25.83)

so that (1) ψ (r) = 1 for r ∈ [0, 1] and ψ (r) = 0 for r ∈ [2, ∞), (2) ψ  ≤ 0,

(25.84)

|ψ  |2 ¯ ≤ C, ψ

and

   ¯ ψ  ≤ C,

where C¯ ∈ (0, ∞) is a universal constant. Let f : M × [0, T ] → R+ be a ‘distance-like’ function to be defined as in §4 of Chapter 26 of this volume. Given any R ∈ [1, ∞), we now assume that φ has the form   f (x, τ ) . (25.85) φ (x, τ )  ψ R Note that ψ  (f / R) ∇f ∇φ = R and ψ  (f / R) ψ  (f / R) ∇∇f + ∇f ⊗ ∇f. (25.86) ∇∇φ = R R2 Let F 

f (x,τ ) R ,

(ψ  (F ))2 |∇φ|2 |∇f |2 = 2 φ R ψ (F ) C¯ ≤ 2 |∇f |2 R

(25.87) and

(25.88)

so that φ = ψ (F ). We calculate

  ψ  (F ) ∂f ψ  (F ) ∂φ − ∆φ = − ∆f − |∇f |2 ∂τ R ∂τ R2    ¯ ∂f C¯ Cφ max 0, − + ∆f + 2 |∇f |2 ≤ R ∂τ R

since ψ  ≤ 0.

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328

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS |∇φ|2 φ

To obtain upper bounds for 2

and

∂φ ∂τ

− ∆φ, evidently it suffices

− ∂f ∂τ

+ ∆f . Combining (25.87) and to obtain upper bounds for |∇f | and (25.88), we have   n |∇φ|2 ∂φ − ∆φ + 2 + ∂τ 2 (2 − 3ε) ε φ      ¯ ¯ ∂f n C Cφ max 0, − + ∆f + 3 + (25.89) ≤ |∇f |2 . R ∂τ 2 (2 − 3ε) ε R2 Note that the differential Harnack estimate of Theorem 25.8 shall follow from (25.82), which in turn relies on showing (25.79). In view of (25.89), the desired estimate (25.79) shall follow from upper bounds for |∇f |2 and − ∂f ∂τ + ∆f ; we prove this in the next subsection. There are a few ways to try to approach the upper bounds for |∇f |2 and − ∂f ∂τ + ∆f : (1) Define f ( · , τ ) to be the distance function to p at time 0 (then φ is independent of time). (2) Define f ( · , τ ) to be the distance function to p at time τ . (3) Define f ( · , τ ) to be a smoothing of the distance function at time 0, so that one has Hessian bounds, which in turn controls the timedependent Laplacian ∆ of f . An issue with (1) is that the consideration of controlling ∆ = ∆g(τ ) f ∂ (∆f ) yields a term which is −Rij ∇i ∇j f (see (25.65)); by ∆g(0) f and ∂τ however it is difficult to obtain two-sided bounds for the Hessian of the distance function. We do not consider this method. An issue with (3) is that one construction of such a function f (see §4 of Chapter 26 in this volume) appears to use global bounds (on all of M) for the curvature whereas we wish to use only curvature bounds in a ball. In the next subsection we consider this method for the case of a general family of metrics. An issue with (2) is to estimate the heat operator of the time-dependent distance function. In view of Perelman’s changing distances estimate (Theorem 18.7), in the subsection after the next, we shall consider this method for the case of a solution to the Ricci flow. 2.7. Completing the proof of the Li–Yau inequality in the case of a general family of metrics. Recall that in the hypothesis of Theorem 25.8 we have assumed the bound |sect (g (0))| ≤ K on M. By Proposition 26.49 below, we may define f : M → R in (25.85) to be a C ∞ function such that (1) (distance-like) (25.90)

dg(0) (x, p) + 1 ≤ f (x) ≤ dg(0) (x, p) + Cn,K

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2. LI–YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS

329

and (2) (uniform bounds on its first and second covariant derivatives with respect to g (0))     (x) ≤ Cn,K (25.91) |∇f |g(0) (x) ≤ Cn,K and ∇g(0) ∇f  g(0)

for all x ∈ M, where Cn,K ∈ (1, ∞) depends only on n and K. Since

∂ ∂τ gij

= 2Rij and since (25.51) implies

−Λgij (x, τ ) ≤ Rij (x, τ ) ≤ Λgij (x, τ )

on M × [0, T ] ,

we have (25.92)

e−2ΛT g (x, 0) ≤ g (x, τ ) ≤ e2ΛT g (x, 0)

and (25.93)

e−ΛT dg(0) (x, p) ≤ dg(τ ) (x, p) ≤ eΛT dg(0) (x, p)

for all (x, τ ) ∈ M × [0, T ]. Hence (25.90) implies e−ΛT dg(τ ) (x, p) + 1 ≤ f (x) ≤ eΛT dg(τ ) (x, p) + Cn,K . The bounds (25.91) and (25.92) also imply that on M × [0, T ] (25.94) and (25.95)

|∇f |g(τ ) (x) ≤ eΛT Cn,K    1   √ ∆g(τ ) f  ≤ ∇g(τ ) ∇f  (x) ≤ C4 , n g(τ )

where C4 < ∞ depends only on n, K, Λ, T , and supM×[0,T ] |∇i Rjk |. Here we used (25.91),   g(τ ) g(0) ∇i ∇j f − ∇i ∇j f = Γkij (0) − Γkij (τ ) ∇k f, and

   τ¯ d      k k k Γij (τ ) dτ  τ ) =  Γij (0) − Γij (¯ 0 dτ  τ¯    g(τ )  ≤3 ∇i Rjk  (τ ) dτ. 0

Note that since f is independent of time, (25.95) implies    √  ∂f  −  ∂τ + ∆f  ≤ nC4 . By (25.89) now we have   n |∇φ|2 ∂φ − ∆φ + 2 + ∂τ 2 (2 − 3ε) ε φ   ¯  ¯ √ C 2ΛT 2 n Cφ (25.96) nC4 + 3 + e Cn,K  C3 ≤ R 2 (2 − 3ε) ε R2

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330

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

since ∂f ∂τ = 0 and by (25.95) and (25.94). By our definition of C3 in (25.96), we finally obtain (25.79). By the lower bound for f in (25.90), the cutoff function φ defined by (25.85) satisfies supp (φ) ⊂ Bg(0) (p, 2 R −1) × [0, T ] . On the other hand, by the upper bound for f in (25.90), we have (25.97)

φ≡1

in Bg(0) (p, R −Cn,K ) × [0, T ] .

Hence, by (25.82) we have C˜ τ for (x, τ ) ∈ Bg(0) (p, R −Cn,K ) × (0, τ¯] and for any τ¯ ∈ (0, T ]. In particular, given any (x, τ ) ∈ Bg(0) (p, R −Cn,K ) × (0, T ], by taking τ¯ = τ , we obtain from (25.98) and (25.81),   !   1 n nC12 2−3ε + C3 + C2 + (2−3ε)ε2 . (25.99) P (x, τ ) ≥ − n 2 − 3ε τ (25.98)

P (x, τ ) ≥ −

Hence, using (25.71), we obtain n P (x, τ ) ≥ − 2 − 3ε



 1 C5 C6 + + 2 + C7 , τ R R

where the dependences of C5 , C6 , and C7 are exactly as in the statement of Theorem 25.8. Hence we obtain (25.53); this completes the proof of Theorem 25.8. 2.8. The case of the Ricci flow. We now complete the proof of Theorem 25.9, where Rij = −Rij . Suppose for each τ ∈ [0, T ] we have Rc ( · , τ ) ≤ (n − 1)K in Bg(τ ) (p, 2 R) for some constants K ≥ 0 and R ≥ √1K . Let f (x, τ ) = dg(τ ) (x, p) . Then |∇f |2 = 1 and by (18.8) we have   √ 5 ∂f − ∆f (x, τ ) ≥ − (n − 1) K ∂τ 3 for all x ∈ M − Bg(τ ) (p, √1K ). By (25.89) we now have   n |∇φ|2 ∂φ − ∆φ + 2 + ∂τ 2 (2 − 3ε) ε φ   ¯  ¯ 5 √ C n Cφ (n − 1) K + 3 + (25.100)  C˜3 ≤ R 3 2 (2 − 3ε) ε R2

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3. NOTES AND COMMENTARY

331

in x ∈ M − Bg(τ ) (p, √1K ) for each τ ∈ [0, T ]. Since R ≥ √1K and for each τ ∈ [0, T ] we have φ ≡ 1 in Bg(τ ) (p, R), the estimate (25.100) actually holds on all of M × [0, T ]. Similarly to (25.99), we have   !   2 1 n nC 1 + C˜3 + 2−3ε C2 + (2−3ε)ε P (x, τ ) ≥ − 2 n 2 − 3ε τ for x ∈ Bg(τ ) (p, R) and τ ∈ (0, T ]. Therefore   1 C8 C9 n + + 2 + C10 , P (x, τ ) ≥ − 2 − 3ε τ R R where the dependences of C8 , C9 , and C10 are as in the statement of Theorem 25.9. This completes the proof of Theorem 25.9.
We remark that there is the standard issue of the distance function f = dg(τ ) ( · , p) being only Lipschitz in Bg(τ ) (p, 2 R) and C ∞ a.e. in Bg(τ ) (p, 2 R). There are two ways to address this issue: use a barrier function for f or use Calabi’s trick. (1) Barrier function. Let (x0 , τ0 ) ∈ B(p, 2 R) × (0, τ¯] be a point at which τ φP attains a negative minimum. Then, in a neighborhood U of this point, P < 0. Now the function fˆ defined using lengths of paths as in subsection 1 of Chapter 17 is an upper barrier for f in a neighborhood V ⊂ U of x0 , ˆ i.e., fˆ ≥  fˆ in V and f (x0 ) = f (x0 ). Sinceψ is decreasing, this implies that f f is a lower barrier for φ = ψ R in V. Since P < 0 in V, we φˆ  ψ R have ˆ τ φP ≤ τ φP in V. ˆ attains a Since τ φP attains a minimum at (x0 , τ0 ), we conclude that τ φP local minimum at (x0 , τ0 ). We then obtain the corresponding estimate for P (x0 , τ0 ) with the bounds for f and its derivatives replaced by the bounds for fˆ and its derivatives. However these bounds are by definition the same. (2) Calabi’s trick. We leave this as an exercise; see also subsection 1.13 of Appendix A in Part I. 3. Notes and commentary For excellent modern references on geometric analysis and the heat equation, see Schoen and Yau [168] and the forthcoming book by Peter Li [118]. §1. The parabolic mean value inequality is intimately related to the volume doubling property (or weak volume growth condition), Sobolev inequality (in the form of Proposition 25.4), and the upper bound for the heat kernel; see Li and Wang [120], especially Corollary 2.3 and the discussion in the introduction therein. Note that the method of Moser iteration is standard and there are other works using this technique, especially in the fixed metric case. See the

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332

25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

original papers [135] and [134] for both the parabolic and elliptic versions of Moser iteration. For Theorem 25.2 see Lemma 3.1 in Chau, Tam, and Yu [26], which is based on §5 of Zhang [195]. For Proposition 25.4, see Theorem 3.1 of Saloff-Coste [165]. See Gallot [65] for earlier work and see Aubin [8], Hebey [96], and Hebey and Vaugon [98] for sharp forms. §2. Originally, for linear second-order parabolic operators in divergence form on Euclidean space, Harnack estimates have been proved by Moser iteration (see Moser [135]). On Riemannian manifolds, Li and Yau [121] proved a Harnack estimate by integrating a gradient estimate along spacetime paths. In Saloff-Coste [165], partly based on the observation that qualitatively, Sobolev constants on Riemannian manifolds depend only on their rough isometry class, i.e., if g ≤ C g˜, then the Sobolev constants of g and g˜ differ at most by a factor depending only on C and n, a Harnack estimate on Riemannian manifolds was proved by Moser iteration. In Chapters 15 and 16 of Part II we discussed Hamilton’s matrix Harnack estimate for the Ricci flow and Perelman’s differential Harnack estimate for the adjoint heat equation coupled to the Ricci flow, respectively. For Theorem 25.8 see Li and Yau [121] and Lemma 4.1 of Chau, Tam, and Yu [26].

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http://dx.doi.org/10.1090/surv/163/10

CHAPTER 26

Bounds for the Heat Kernel for Evolving Metrics Only love can bring the rain. – From “Love, Reign o’er Me” by The Who

In this chapter we discuss applications of the a priori estimates of the previous chapter to obtain bounds for the heat kernel. In §1 we discuss some basic general properties of solutions of the heat equation. In §2 we discuss the application of the mean value inequality and the Li–Yau differential Harnack estimate, discussed in the previous chapter, to obtain upper and lower bounds for heat kernels on evolving complete Riemannian manifolds. In §3 we present the space-time mean value property (MVP). In §4 we discuss Tam’s work, which is related to the earlier works of others, on the existence of distance-like functions with uniformly bounded gradients and Hessians on complete noncompact manifolds with bounded curvature. 1. Heat kernel for an evolving metric In this section we discuss general properties of the heat equation with respect to an evolving metric and consider the adjoint heat equation and its associated heat kernel. In particular we consider Duhamel’s principle, the boundedness of the L1 -norm, and the semigroup property. 1.1. The heat equation and its adjoint. Let g (τ ), τ ∈ [0, T ], be a smooth family of Riemannian metrics on a manifold Mn and define ∂ gij  2Rij , (26.1) ∂τ where Rij is a time-dependent symmetric 2-tensor. The application we shall later make is to the case where Rij = Rij is the Ricci tensor. Let (26.2)

R  g ij Rij ,

so that (26.3) where dµg(τ )

∂ dµ = Rdµg(τ ) , ∂τ g(τ ) denotes the volume form of g (τ ). 333

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334

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

We consider the heat(-type) equation Lx,τ u 

(26.4)

∂u − ∆g(τ ) u + Qu = 0, ∂τ

where Q : M × [0, T ] → R is a C ∞ function and where ∆g(τ ) denotes the Laplacian corresponding to g (τ ), and we consider the corresponding adjoint heat equation (26.5)

L∗x,τ u  −

∂u − ∆g(τ ) u + (Q − R) u = 0. ∂τ

Let H (x, τ ; y, υ), where x, y ∈ M and 0 ≤ υ < τ ≤ T , be the heat kernel for L on a closed manifold, that is, (26.6)   ∂H (x, τ ; y, υ) − (∆x,τ H) (x, τ ; y, υ) + Q (x, τ ) H (x, τ ; y, υ) = 0, ∂τ (26.7) lim H ( · , τ ; y, υ) = δy . τ υ

Equation (26.6) says that, with respect to the first pair of variables x and τ , H satisfies Lu = 0. Equation (26.7) says that for any φ ∈ Cc0 (M)  H ( · , τ ; y, υ) φ (x) dµg(τ ) (x) = φ (y) . (26.8) lim τ υ

M

Define the adjoint heat kernel H ∗ (x, τ ; y, υ), where x, y ∈ M and 0 ≤ τ < υ ≤ T , for L∗ by   ∂H ∗ + ∆x,τ H ∗ (x, τ ; y, υ) + (R − Q) (x, τ ) H ∗ (x, τ ; y, υ) = 0, (26.9) ∂τ (26.10) lim H ∗ ( · , τ ; y, υ) = δy . τ υ

1.2. Elementary properties of the heat kernel and adjoint heat kernel on a closed manifold. The following is a space-time version of Green’s second identity for the operator L and its adjoint L∗ .1 Lemma 26.1 (Duhamel’s principle on a closed manifold). Let g (τ ), τ ∈ [0, T ], be a time-dependent Riemannian metric on a closed manifold Mn . If The space version of Green’s identity for the operator ∆ says that if Ω ⊂ M  second  ¯ , then is a C 1 domain and if F, G ∈ C 2 Ω     ∂F ∂G G −F dσ, (G∆F − F ∆G) dµ = ∂ν ∂ν Ω ∂Ω 1

where ν is the unit outward normal to ∂Ω; see (2.11) on p. 17 of Gilbarg and Trudinger [71].

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1. HEAT KERNEL FOR AN EVOLVING METRIC

335

A and B are functions on M × [0, T ] which are both C 2 in space and C 1 in time, then for any 0 ≤ τ1 < τ2 ≤ T we have (26.11)  τ2







(Lx,τ A) (x, τ ) B (x, τ ) − A (x, τ ) L∗x,τ B (x, τ ) dµg(τ ) (x) M  A (x, τ2 ) B (x, τ2 ) dµg(τ2 ) (x) − A (x, τ1 ) B (x, τ1 ) dµg(τ1 ) (x) .

dτ τ1



= M

M

Proof. The lhs of (26.11) is equal to      τ2   ∂B ∂A (x, τ ) B (x, τ ) + A (x, τ ) (x, τ ) dµg(τ ) (x) dτ ∂τ ∂τ τ1 M  τ2 





dτ − ∆g(τ ) A (x, τ ) B (x, τ ) + A (x, τ ) ∆g(τ ) B (x, τ ) dµg(τ ) (x) + τ M  1τ2  dτ A (x, τ ) R (x, τ ) B (x, τ ) dµg(τ ) (x) + τ1 M    τ2 d A (x, τ ) B (x, τ ) dµg(τ ) (x) dτ = τ1 dτ M   A (x, τ2 ) B (x, τ2 ) dµg(τ2 ) (x) − A (x, τ1 ) B (x, τ1 ) dµg(τ1 ) (x) , = M

M

where we used (26.3) and the divergence theorem, i.e., 



− ∆g(τ ) A B + A ∆g(τ ) B dµg(τ ) = 0. M


As a special case of (26.11), we see that if a and b are C 2 functions with compact support in M × (0, T ), then 

T





(26.12) 0

M

T

b Lx,τ a dµg(τ ) dτ = 0

 M

a L∗x,τ b dµg(τ ) dτ.

So, indeed, the operator L∗x,τ is the (formal) adjoint of Lx,τ . Exercise 26.2 (Duhamel’s principle on a compact manifold with boundary). Show that if M is compact and has nonempty boundary ∂M, then (26.11) still holds provided both A = 0 and B = 0 on ∂M × [0, T ]. Let H and H ∗ be the heat kernel and adjoint heat kernel in (26.6) and (26.9), respectively. Given 0 ≤ ρ < υ ≤ T , for τ ∈ (ρ, υ) we take A (x, τ ) = H (x, τ ; z, ρ)

and

B (x, τ ) = H ∗ (x, τ ; y, υ)

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

in (26.11). Since (Lx H) (x, τ ; z, ρ) = 0 and (L∗x H ∗ ) (x, τ ; y, υ) = 0, we have for ρ < τ1 < τ2 < υ  τ2  dτ (Lx,τ H) (x, τ ; z, ρ) H ∗ (x, τ ; y, υ) dµg(τ ) (x) 0= τ1 M  τ2 

dτ H (x, τ ; z, ρ) L∗x,τ H ∗ (x, τ ; y, υ) dµg(τ ) (x) − M  τ1 (26.13) H (x, τ2 ; z, ρ) H ∗ (x, τ2 ; y, υ) dµg(τ2 ) (x) = M  H (x, τ1 ; z, ρ) H ∗ (x, τ1 ; y, υ) dµg(τ1 ) (x) . − M

We then take the limits as τ1  ρ and τ2 υ in (26.13), while using (26.7) and (26.10), to obtain Lemma 26.3 (Symmetry between heat and adjoint heat kernels on a closed manifold). For any y, z ∈ M and 0 ≤ ρ < υ ≤ T we have H (y, υ; z, ρ) = H ∗ (z, ρ; y, υ) .

(26.14)

We can now prove the following. Lemma 26.4 (For second set of space-time variables H is the heat kernel for L∗ ).   ∂H + ∆y,υ H (x, τ ; y, υ) + (R − Q) (y, υ) H (x, τ ; y, υ) = 0, (26.15) ∂υ (26.16) lim H (x, τ ; · , υ) = δx . υτ

That is, with respect to the second pair of variables y and υ, H is the fundamental solution to L∗ u = 0. Proof. Substituting (26.14) into (26.9)–(26.10), we obtain (26.15)– (26.16).
Note that using (26.6) and (26.3), we have for any y ∈ M and 0 ≤ υ < τ ≤ T that2   d H (x, τ ; y, υ) dµg(τ ) (x) dτ M    ∂H (x, τ ; y, υ) + R (x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) = ∂τ M  (26.17) ((R (x, τ ) − Q (x, τ )) H (x, τ ; y, υ)) dµg(τ ) (x) . = M

Assuming that (26.18)

sup |R − Q|  C1 < ∞,

M×[0,T ]

2 An elementary justification of the interchange of the time derivative and the space integral is given by Lemma 23.40.

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we have      d   H (x, τ ; y, υ) dµg(τ ) (x)  ≤ C1 H (x, τ ; y, υ) dµg(τ ) (x) .  dτ M M On the other hand, by taking φ ≡ 1 in (26.8), we have  H (x, τ ; y, υ) dµg(τ ) (x) = 1. lim τ υ

M

Hence we have Lemma 26.5 (L1 -norm of heat kernel on a closed manifold is bounded). If M is closed, then the heat kernel for Lx,τ satisfies  H (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) (26.19) e−C1 (τ −υ) ≤ M

for any y ∈ M and 0 ≤ υ < τ ≤ T . If Q = R, then C1 = 0, so that (26.19) yields the following. Corollary 26.6 (L1 -norm of heat kernel on a closed manifold is preserved when Q = R). If M is closed and Q = R, then  H (x, τ ; y, υ) dµg(τ ) (x) ≡ 1 (26.20) M

for any y ∈ M and 0 ≤ υ < τ ≤ T . On the other hand, by Lemma 26.4 we have     d   H (x, τ ; y, υ) dµ (y) g(υ)   dυ M       ∂H (x, τ ; y, υ) + R (y, υ) H (x, τ ; y, υ) dµg(υ) (y) =  ∂υ   M    (−∆y,υ H (x, τ ; y, υ) + Q (y, υ) H (x, τ ; y, υ)) dµg(υ) (y) = M      H (x, τ ; y, υ) dµg(υ) (y) , ≤ sup |Q|  M

so that Lemma 26.7 (L1 -norm of heat kernel using the second space variables). If M is closed, then the heat kernel for L satisfies  −C2 (τ −υ) ≤ H (x, τ ; y, υ) dµg(υ) (y) ≤ eC2 (τ −υ) , (26.21) e M

where C2  supM×[0,T ] |Q|.

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338

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Given y, z ∈ M and 0 ≤ υ < ρ ≤ T , using (26.6) and (26.15), we compute for τ ∈ (υ, ρ) (26.22)   d H (z, ρ; x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) dτ M    ∂H ∂H (z, ρ; x, τ ) H (x, τ ; y, υ) + H (z, ρ; x, τ ) (x, τ ; y, υ) dµg(τ ) (x) = ∂τ ∂τ M  H (z, ρ; x, τ ) H (x, τ ; y, υ) R (x, τ ) dµg(τ ) (x) + M (∆x,τ H) (z, ρ; x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) =− M  H (z, ρ; x, τ ) (∆x,τ H) (x, τ ; y, υ) dµg(τ ) (x) + M

= 0, where we integrated by parts to obtain the last equality. Thus we have the semigroup property of the heat kernel: Lemma 26.8 (Semigroup property on a closed manifold). Let H be the heat kernel for L. Then for y, z ∈ M and 0 ≤ υ < τ < ρ ≤ T  H (z, ρ; x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) = H (z, ρ; y, υ) . (26.23) M

Proof. Using (26.22) and (26.7), we compute  H (z, ρ; x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) M  H (z, ρ; x, σ) H (x, σ; y, υ) dµg(σ) (x) = lim συ

M

= H (z, ρ; y, υ) .
1.3. Elementary properties of the Dirichlet heat kernel on manifolds with boundary. Let g (τ ), τ ∈ [0, T ], be a smooth 1-parameter family of Riemannian metrics on a compact manifold Mn with nonempty boundary ∂M. Let HD (x, τ ; y, υ) denote the Dirichlet heat kernel for Lx,τ . We have the following symmetry property between HD and the adjoint Dirichlet heat kernel ∗ for L∗ . HD x,τ Lemma 26.9. For any x, y ∈ M and 0 ≤ υ < τ ≤ T we have (26.24)

∗ (y, υ; x, τ ) . HD (x, τ ; y, υ) = HD

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In particular, (26.25)   ∂HD + ∆y,υ HD (x, τ ; y, υ) + (R − Q) (y, υ) HD (x, τ ; y, υ) = 0, ∂υ (26.26) lim HD (x, τ ; · , υ) = δx . υτ

Proof. The proof of Lemma 26.3 holds for the Dirichlet heat kernel since the integration by parts in (26.13) is still valid.
Similarly to Lemma 26.5, we have Lemma 26.10 (L1 -norm of Dirichlet heat kernel is bounded). The Dirichlet heat kernel HD for Lx,τ , on a compact manifold (Mn , g (τ )) with nonempty boundary, satisfies: (1)  HD (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) (26.27) M

for any y ∈ int (M) and 0 ≤ υ < τ ≤ T , where C1 is as in (26.18). (2)



(26.28) M

HD (x, τ ; y, υ) dµg(τ ) (x) ≥ 1 − C (τ − υ)

for any y ∈ int (M) and τ ∈ [0, T ], where C < ∞ depends only on (26.29) (26.30)

Λ r

sup |Rij (x, τ )|g(τ ) < ∞,

M×[0,T ]

1 min 4



 min dg(τ ) (y, ∂M) , min inj g(τ ) (y) ,

τ ∈[0,T ]

τ ∈[0,T ]

C¯ in (25.84), C1 in (26.18), and maxM×[0,T ] |Rc|. Proof. (1) We compute  d HD (x, τ ; y, υ) dµg(τ ) (x) dτ M    ∂HD (x, τ ; y, υ) + R (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) = ∂τ M (∆x,τ HD ) (x, τ ; y, υ) dµg(τ ) (x) = M  (R − Q) (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) + M  νx (HD ) (x, τ ; y, υ) dµg(τ ) (x) ≤ ∂M  HD (x, τ ; y, υ) dµg(τ ) (x) , + C1 M

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where ν denotes the unit outward normal vector field to M on ∂M, where νx denotes the directional derivative in the direction ν with respect to the x variable, and where C1 is as in (26.18). Since HD (x, τ ; y, υ) = 0 for x ∈ ∂M and HD (x, τ ; y, υ) ≥ 0 for x ∈ M, we have νx (HD ) ≤ 0. Hence   d HD (x, τ ; y, υ) dµg(τ ) (x) ≤ C1 HD (x, τ ; y, υ) dµg(τ ) (x) , dτ M M which implies

 M

HD (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) .

(2) Let y ∈ int (M) and let r be as in (26.30). Let ψ : [0, ∞) → [0, 1] be as in (25.83) and define the cutoff function ϕ : M × [0, T ] → R+ by

 ϕ (x, τ )  ψ

Note that ϕ is C ∞ and (26.31)

supp (ϕ) ⊂

.

dg(τ ) (x, y) r

 .

¯g(τ ) (y, 2 r) × {τ } ⊂ int (M) × [0, T ] . B

τ ∈[0,T ]

¯g(τ ) (y, r) and τ ∈ [0, T ]. We also have ϕ (x, τ ) = 1 for x ∈ B Using Lx,τ HD = 0 and (26.3), we compute  d ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) dτ M  (26.32) = ϕ (x, τ ) (∆x,τ HD ) (x, τ ; y, υ) dµg(τ ) (x) M    ∂ϕ (x, τ ) + (R − Q) ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) . + ∂τ M Since

∂ ∂τ gij

we have

(26.33)

= 2Rij and Λ = supM×[0,T ] |Rij (x, τ )|g(τ ) < ∞ implies    ∂  dg(τ )  ≤ Λdg(τ ) ,   ∂τ   1 ∂ ∂ϕ  dg(τ ) (x, y) (x, τ ) = ψ d (x, y) ∂τ r r ∂τ g(τ )   Λdg(τ ) (x, y) ≥ − C¯ ϕ (x, τ ) r  ¯ ≥ −2 CΛ,

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√ where C¯ ∈ (0, ∞) is as in (25.84) and where we used ϕ ≤ 1 and (26.31). By applying (26.33) and (26.18) to (26.32) and integrating by parts, we obtain  d ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) dτ M     ¯ HD (x, τ ; y, υ) dµg(τ ) (x) (26.34) ∆x,τ ϕ (x, τ ) − 2 CΛ ≥ M  ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) , − C1 M

where C1 is as in (26.18). Since there exists C  ∈ (0, ∞) depending only on n, r, and maxM×[0,T ] |Rc| such that ∆x,τ ϕ (x, τ ) ≥ −C  , by applying (26.27) to (26.34), we obtain  d ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) dτ M      C (τ −υ) 1 ¯ − C1 ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) . e ≥ − C + 2 CΛ M

Now (26.28) follows from integrating this in time and from the fact that  ϕ (x, τ ) HD (x, τ ; y, υ) dµg(τ ) (x) = 1. lim τ υ

M


Exercise 26.11. Derive another lower bound for  HD (x, τ ; y, υ) dµg(τ ) (x) M

using the method in the derivation of (22.41) in the proof of Lemma 22.9. Finally, we note that the semigroup property holds for the Dirichlet heat kernel. Lemma 26.12. The semigroup property holds for the Dirichlet heat kernel. Proof. Using Exercise 26.2, the lemma follows from the same technique as when the compact manifold has no boundary.
1.4. Elementary properties of the heat kernel on a noncompact manifold. As before, let g (τ ) be an evolving complete Riemannian metric on a ∂ gij = 2Rij for τ ∈ [0, T ] and let noncompact manifold Mn satisfying ∂τ H (x, τ ; y, υ), where x, y ∈ M and 0 ≤ υ < τ ≤ T , be the heat kernel for

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Lx,τ . By definition, H is the minimal positive solution to (26.6)–(26.7), i.e.,  (26.35a) (26.35b)

 ∂H − ∆x,τ H + Q (x, τ ) H (x, τ ; y, υ) = 0, ∂τ lim H ( · , τ ; y, υ) = δy . τ υ

We have the following symmetry property between H and the adjoint heat kernel H ∗ for L∗x,τ . Lemma 26.13. For any x, y ∈ M and 0 ≤ υ < τ ≤ T we have H (x, τ ; y, υ) = H ∗ (y, υ; x, τ ) . In particular,   ∂H + ∆y,υ H (x, τ ; y, υ) + (R − Q) (y, υ) H (x, τ ; y, υ) = 0, (26.36a) ∂υ (26.36b) lim H (x, τ ; ·, υ) = δx . υτ

Proof. This is true because of the construction of the heat kernel on a noncompact manifold as the limit of a sequence of Dirichlet heat kernels (see §5 of Chapter 24) and the fact that Lemma 26.3 holds for Dirichlet heat kernels.
We have the following L1 estimate for the heat kernel in the noncompact case. Lemma 26.14 (L1 -norm of heat kernel is bounded — noncompact case). Let (Mn , g (τ )), τ ∈ [0, T ], be an evolving complete noncompact Riemannian manifold such that supM |sect (g (0))|, supM×[0,T ] |Rij |, supM×[0,T ] |∇i Rjk | are finite. Then for the minimal positive fundamental solution H to (26.4) we have  −C1 (τ −υ) ≤ H (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) (26.37) e M

for any y ∈ M and 0 ≤ υ < τ ≤ T , where C1 

sup |R − Q| < ∞.

M×[0,T ]

Since C1 = 0 when Q = R, we immediately obtain the following. Corollary 26.15 (L1 -norm of heat kernel is preserved when Q = R — noncompact case). Under the hypotheses of Lemma 26.14, now with Q = R, we have  H (x, τ ; y, υ) dµτ (x) ≡ 1 (26.38) M

for any y ∈ M and 0 ≤ υ < τ ≤ T .

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Proof of Lemma 26.14. (1) Upper bound. Let {Ωi }i∈N be an exhaustion of M as above and let HΩi (x, τ ; y, υ) be the Dirichlet heat kernel of (Ωi , g (τ )) as in (24.107), i.e., lim HΩi (x, τ ; y, υ) = H (x, τ ; y, υ) .

i→∞

By the upper bound (26.27), with ‘M = Ωi ’, we have  HΩi (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) Ωi

for any y ∈ int (Ωi ) and 0 ≤ υ < τ ≤ T . Since HΩi converges to H, we conclude  H (x, τ ; y, υ) dµg(τ ) (x) ≤ eC1 (τ −υ) (26.39) M

for any y ∈ M and 0 ≤ υ < τ ≤ T . (2) Lower bound. We prove a lower bound for the integral on the lhs of (26.39). Let φ be the cutoff function defined in (25.85). Since supM |sect (g (0))|, supM×[0,T ] |Rij |, and supM×[0,T ] |∇i Rjk | are finite, by (25.86), (25.94), and (25.95), we have  √   √  √ C C φ Cn   + 2 ≤ (26.40) ∆g(τ ) φ ≤ n |∇∇φ|g(τ ) ≤ n R R R for some constant Cn < ∞ independent of R, where we assume R ≥ 1. We compute for any τ ∈ (0, T )     d  φ (x) H (x, τ ; y, υ) dµg(τ ) (x)  dτ M       ∂H  (x, τ ; y, υ) + R (x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) φ (x) = ∂τ  M    φ (x) (∆x,τ H) (x, τ ; y, υ) dµg(τ ) (x) ≤ M      φ (x) (R − Q) (x, τ ) H (x, τ ; y, υ) dµg(τ ) (x) +   M   H (x, τ ; y, υ) (∆x,τ φ) (x) dµg(τ ) (x) ≤  M  φ (x) H (x, τ ; y, υ) dµg(τ ) (x) . + C1 M

Applying (26.40) and (26.39) to this, we obtain  d φ (x) H (x, τ ; y, υ) dµg(τ ) (x) dτ M  Cn C1 (τ −υ) − C1 φ (x) H (x, τ ; y, υ) dµg(τ ) (x) . ≥− e R M

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Integrating this, we find that for any τ1 < τ2  C1 (τ2 −υ) e φ (x) H (x, τ2 ; y, υ) dµg(τ2 ) (x) M  φ (x) H (x, τ1 ; y, υ) dµg(τ1 ) (x) − eC1 (τ1 −υ) M     τ2 d C1 (τ −υ) e φ (x) H (x, τ ; y, υ) dµg(τ ) (x) dτ = τ1 dτ M  Cn τ2 2C1 (τ −υ) e dτ. ≥− R τ1 Taking the limit as R → ∞, we obtain   −C1 (τ2 −τ1 ) H (x, τ2 ; y, υ) dµg(τ2 ) (x) ≥ e M

Since limτ1 υ




MH



M

H (x, τ1 ; y, υ) dµg(τ1 ) (x) .

(x, τ1 ; y, υ) dµg(τ1 ) (x) = 1, we conclude

M

H (x, τ2 ; y, υ) dµg(τ2 ) (x) ≥ e−C1 (τ2 −υ) .


The semigroup property for closed manifolds may be extended to the complete case. Lemma 26.16 (Semigroup property — noncompact case). Let (Mn, g(τ )), τ ∈ [0, T ], be a complete noncompact evolving manifold. The minimal positive fundamental solution H to Lx,τ u = 0 satisfies  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµg(ρ) (z) (26.41) H (x, τ ; y, υ) = M

for any x, y ∈ M and 0 ≤ υ < ρ < τ ≤ T . Proof. As in the discussion following Theorem 24.40, let {Ωi }∞ i=1 be an exhaustion of M by smooth domains with compact closure suchthat Ωi ⊂ Ωi+1 and let HΩi (x, τ ; y, υ) denote the Dirichlet heat kernel of Ωi , g|Ωi . By Lemma 26.12, the semigroup property holds for the Dirichlet heat kernels HΩi , that is, for any x, y ∈ Ωi and 0 ≤ υ < ρ < τ ≤ T we have  HΩi (x, τ ; z, ρ) HΩi (z, ρ; y, υ) dµg(ρ) (z) HΩi (x, τ ; y, υ) = Ωi  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµg(ρ) (z) ≤ M

for all i since HΩi ≤ H. Taking the limit as i → ∞ yields  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµg(ρ) (z) . H (x, τ ; y, υ) ≤ M

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On the other hand, fix a compact domain K ⊂ M. We have  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµg(ρ) (z) K  HΩi (x, τ ; z, ρ) HΩi (z, ρ; y, υ) dµg(ρ) (z) = lim i→∞ K

≤ lim HΩi (x, τ ; y, υ) i→∞

= H (x, τ ; y, υ) . Since this is true for any K, we conclude  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµg(ρ) (z) ≤ H (x, τ ; y, υ) . M


2. Upper and lower bounds of the heat kernel for an evolving metric The mean value inequality, the differential Harnack estimate, the boundedness of L1 -norm, and the semigroup property may be used to derive bounds on the heat kernel. 2.1. Upper bounds of the heat kernel for an evolving metric. We first obtain an upper bound for the heat kernel H (x, τ ; y, υ) in terms of volumes of balls which is not sensitive to the distance between x and y. Then we prove that H decays exponential quadratically in the space variables in a weighted L2 sense. Finally we demonstrate that H decays pointwise exponential quadratically. 2.1.1. A rough upper bound for the heat kernel. The following result is obtained by combining the mean value inequality
(at a suitable spatial scale) with the above lemma on H dµ ≤ C. Lemma 26.17 (Rough upper bound for H). Let (Mn , g (τ )), τ ∈ [0, T ], g˜, C˜0 , C1 , C2 , and C3 all be as in Theorem 25.2. Let H (x, τ ; y, υ) be the minimal positive fundamental solution to (26.4). Then for all x, y ∈ M and 0≤υ 0 and Pg˜ x, τ, 2 , − 4 

   u (z, σ) dµg˜ (z) dσ √ Pg˜ x,τ, τ2−υ ,− τ −υ 4

τ

≤

 u (z, σ) dµg˜ (z) dσ

M

υ n/2



≤ C˜0

τ

eC1 (σ−υ) dσ,

υ

where for the last inequality we used the fact that for σ ∈ [υ, τ ]   n/2 ˜ u (z, σ) dµg˜ (z) ≤ C0 H (z, σ; y, υ) dµg(σ) (z) M

≤

M n/2 C C˜0 e 1 (σ−υ)

by Lemma 26.14. Hence (26.44) implies H (x, τ ; y, υ) = u (x, τ ) C

√

KT

3 n/2 4C˜0 C1 eC2 T + 2   ≤ √ Volg˜ Bg˜ x, τ2−υ


τ υ

eC1 (σ−υ) dσ τ −υ

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since τ − υ ≤ T . Thus H (x, τ ; y, υ) ≤

C  √ , Volg˜ Bg˜ x, τ2−υ

where C depends only on n, T , K, C˜0 , C1 , C2 , and C3 . We leave it as an exercise for the reader to prove the similar inequality H (x, τ ; y, υ) ≤

C  √ . Volg˜ Bg˜ y, τ2−υ


Exercise 26.19. Under the setup of Lemma 26.17, let Ω ⊂ M be a smooth compact domain and let HΩ (x, τ ; y, υ) denote the Dirichlet heat kernel for L.  √  (1) Show that for x, y ∈ Ω and 0 ≤ υ < τ ≤ T such that Bg˜ x, τ2−υ ⊂ Ω we have C  √ , (26.45) HΩ (x, τ ; y, υ) ≤ Volg˜ Bg˜ x, τ2−υ where C < ∞ has the same dependence as in Lemma 26.17 (and is independent of Ω).  √  (2) Similarly, show that if Bg˜ y, τ2−υ ⊂ Ω, then (26.46)

HΩ (x, τ ; y, υ) ≤

C  √ . τ −υ Volg˜ Bg˜ y, 2

2.1.2. Average exponential quadratic decay for the heat kernel. We shall improve our pointwise bound so that we obtain exponential quadratic decay in the space variables. In preparation for the next result, we give the following. Definition 26.20. We say that a continuous, strictly increasing function f : (0, T ] → (0, ∞) is (γ, A)-regular, where γ > 1 and A ≥ 1, if f (τ2 ) f (τ1 ) ≤A f (τ1 /γ) f (τ2 /γ) for all 0 < τ1 ≤ τ2 < T . The following result says that if the L2 -norm of a solution has a reasonable time-dependent bound, then an exponential quadratically weighted L2 -norm of the solution has a corresponding time-dependent bound. This integral bound is a precursor to pointwise bounds. Let int (Ω) denote the interior of Ω.

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348

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Lemma 26.21 (Exponential quadratically weighted L2 -estimate in terms of L2 -estimate). Let (Mn , g (τ )) be as in (26.1). Let Ω ⊂ M be a smooth compact domain and let K be a compact set with K ⊂ int (Ω). There exists C = C (γ, sup |Q| , sup |R|) ∈ (0, ∞) and D = D (γ, T, sup |Rij |) with the following property. Let u : Ω × [0, T ] → R be a solution to ∂u = ∆g(τ ) u − Q u (26.47) ∂τ with Dirichlet boundary values u|∂Ω×[0,T ] = 0 and supp (u ( · , 0)) ⊂ K in the sense that for x ∈ Ω−K we have limτ 0 u (x, τ ) = 0 locally uniformly. If  1 u2 (x, τ ) dµg(τ ) (x) ≤ (26.48) f (τ ) Ω for τ ∈ (0, T ], where f is (γ, A)-regular, then  d2 (x,K) g(0) 4AeCτ 2 u (x, τ ) e Dτ dµg(τ ) (x) ≤ (26.49) f (τ /γ) Ω for τ ∈ (0, T ]. Remark 26.22. An important special case is when K is a point and u is a Dirichlet heat kernel centered at that point. Proof. Step 1. Weighted L2 estimate. Let C0 > 0 be such that   1 Q− R ≥0 (26.50) C0 + inf 2 Ω×[0,T ] and let

v (x, τ )  e−C0 τ u (x, τ ) .

Then (26.47) yields (26.51)

∂v = ∆v − (Q + C0 ) v ∂τ

and (26.48) yields  1 e−2C0 τ  . v 2 (x, τ ) dµg(τ ) (x) ≤ (26.52) ˜ f (τ ) f (τ ) Ω Since f is (γ, A)-regular, we have that f˜ (τ ) = e2C0 τ f (τ ) is also (γ, A)regular since f˜ is strictly increasing and −1 f (τ ) f˜ (τ ) , = e2C0 τ (1−γ ) ˜ f (τ /γ) f (τ /γ) −1 whereas e2C0 τ (1−γ ) is nondecreasing in τ .

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

349

Given R > 0, let



NR (K)  x ∈ M : dg(0) (x, K) ≤ R

denote the R-neighborhood of K. We shall assume that R is sufficiently small so that NR (K) ⊂ int (Ω). Define ρR : Ω → [0, R] by  R −dg(0) (x, K) if x ∈ NR (K) , ρR (x)  0 if x ∈ Ω − NR (K) . Since |∇ρR |2g(0) ≤ 1 in Ω and g (τ ) ≥ C˜0−1 g (0) for τ ∈ [0, T ], where C˜0  e2T sup|Rij | < ∞, we have |∇ρR |2g(τ ) ≤ C˜0

(26.53) on Ω × [0, T ]. Given σ > 0, define

ξ : Ω × (0, min {T, σ}) → R by ξ (x, τ )  −

ρ2R (x) . 2C˜0 (σ − τ )

We have that ξ is a Lipschitz function with 1 ∂ξ + |∇ξ|2g(τ ) ∂τ 2 ρ2R (x) |∇ρR |2g(τ ) =− + 2C˜0 (σ − τ )2 2C˜02 (σ − τ )2 ≤0 ρ2R (x)

(26.54)

by (26.53). Using (26.54), (26.51), and (26.50), we compute that     ∂ξ ∂v d + v2 + v 2 R eξ dµg(τ ) v 2 eξ dµg(τ ) = 2v dτ Ω ∂τ ∂τ Ω      1 1 2 2 ≤ 2v∆v − v |∇ξ| − 2 Q + C0 − R v 2 eξ dµg(τ ) 2 2 Ω   1 ∂v (26.55) ≤− |2∇v + v∇ξ| eξ dµg(τ ) + 2v eξ dµg(τ ) 2 Ω ∂ν ∂Ω ≤0 ∂v ≤ 0 on ∂Ω × [0, T ]). since v = 0 on ∂Ω × [0, T ] (all we need is v ∂ν

Step 2. Upper estimate for IR (τ ). Let  v 2 (x, τ ) dµg(τ ) (x) . (26.56) IR (τ )  Ω−NR (K)

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350

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Now fix 0 < τ1 < τ2 < T and 0 < R1 < R2 such that NR2 (K) ⊂ int (Ω). Given T˜ ∈ (τ2 , T ) and δ > 0, let σ = T˜ + δ. By the monotonicity formula (26.55) we have ⎛ ⎞  2 (x) ρ  R2  ⎠ dµg(τ1 ) (x) v 2 (x, τ1 ) exp ⎝− ˜ Ω 2C0 T˜ + δ − τ1 ⎛ ⎞  2 (x) ρ  R2  ⎠ dµg(τ2 ) (x) (26.57) v 2 (x, τ2 ) exp ⎝− ≥ Ω 2C˜0 T˜ + δ − τ2 ≥ IR2 (τ2 ) since ρR2 (x) = 0 for x ∈ Ω−NR2 (K). On the other hand, the lhs of (26.57) is ⎛ ⎞  2 (x) ρ  R2  ⎠ dµg(τ1 ) (x) v 2 (x, τ1 ) exp ⎝− ˜ NR1 (K) 2C0 T˜ + δ − τ1 ⎛ ⎞  2 (x) ρ  R2  ⎠ dµg(τ1 ) (x) v 2 (x, τ1 ) exp ⎝− + Ω−NR1 (K) 2C˜0 T˜ + δ − τ1 ⎛ ⎞  2 (R2 − R1 )   ⎠ dµg(τ1 ) (x) + IR1 (τ1 ) v 2 (x, τ1 ) exp ⎝− ≤ ˜ ˜ NR1 (K) 2C0 T + δ − τ1  since dg(0) (x, K) ≤ R1 for x ∈ NR1 (K) and since exp

ρ2 (x) − 2C˜ RT˜2+δ−τ 0( 1)

for all x ∈ Ω. We conclude that IR2 (τ2 ) − IR1 (τ1 ) ⎛

⎞

 (R2 − R1 ) ⎠   v 2 (x, τ1 ) dµg(τ1 ) (x) ˜ ˜ N (K) R1 2C0 T + δ − τ1 ⎞ ⎛ 1 (R2 − R1 )2  ⎠ ≤ exp ⎝− f˜ (τ1 ) 2C˜0 T˜ + δ − τ1 ≤ exp ⎝−

2

by (26.52). Now we take T˜  τ2 and δ  0 to obtain   (R2 − R1 )2 1 exp − . (26.58) IR2 (τ2 ) − IR1 (τ1 ) ≤ 2C˜0 (τ2 − τ1 ) f˜ (τ1 ) Now given τ and R, for i ∈ N ∪ {0} define τ τi  i , γ

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 ≤1

2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

351

where γ is as above with f being (γ, A)-regular, and define   1 1 + R, Ri  2 i+2 so that both τi and Ri are decreasing in i and τ0 = τ, R0 = R, We have

τi → 0, Ri →

 lim IRi (τi ) = lim

i→∞

i→∞ Ω−N (K) Ri

1 R as i → ∞. 2

e−2C0 τi u2 (x, τi ) dµg(τ ) (x)

=0 since limτ 0 u (x, τ ) = 0 locally uniformly for x ∈ Ω − K (which implies uniformly for x ∈ Ω − N 1 R (K)). Therefore 2

IR (τ ) = ≤

∞  i=0 ∞  i=0

IRi (τi ) − IRi+1 (τi+1 )



  (Ri − Ri+1 )2 1 exp − 2C˜0 (τi − τi+1 ) f˜ (τi+1 )

by (26.58). Since f˜ is (γ, A)-regular, we have i+1 

i 1 ; f˜ τ /γ j 1 f˜ (τ0 ) = ≤ A . f˜ (τi+1 ) f˜ (τ ) j=0 f˜ (τ /γ j+1 ) f˜ (τ ) f˜ (τ1 ) 1

By this, Ri − Ri+1 =

1 (i+2)(i+3)

R≥

R , (i+3)2

and τi − τi+1 =

γ−1 τ, γ i+1

we obtain

    ∞ γ i+1 R2 1  f˜ (τ0 ) − . exp (i + 1) log A IR (τ ) ≤ 2C˜0 (i + 3)4 (γ − 1) τ f˜ (τ ) i=0 f˜ (τ1 ) Let B

γ i+1 . i∈N∪{0} (i + 3)4 (i + 2) (γ − 1) inf

Since γ > 1, we have B > 0. Then (26.59)

2      −BR ∞ B R2 e 2C˜0 τ  f˜ (τ0 ) − . exp (i + 1) log A IR (τ ) ≤ 2C˜0 τ f˜ (τ ) i=0 f˜ (τ1 )

We estimate the quantity on the rhs of (26.59) as follows.

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352

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

 ˜  0) Case 1. log A ff˜(τ − (τ ) 1

B R2 ˜0 τ 2C

≤ − log 2. Then (26.59) implies

IR (τ ) ≤ =  ˜  0) − Case 2. log A ff˜(τ (τ1 ) our assumption, we have

B R2 ˜0 τ 2C

e

− B˜R

2

2 C0 τ

f˜ (τ ) 1 f˜ (τ )

e

∞ 

2−i−1

i=0 − B˜R

2

2 C0 τ

.

> − log 2. Then by (26.56), (26.52), and



IR (τ ) ≤

v 2 (x, τ ) dµg(τ ) (x) Ω

1 ˜ f (τ ) 2 2A −BR e 2C˜0 τ . < f˜ (τ /γ)

≤

In either case we have  v 2 (x, τ ) dµg(τ ) (x) = IR (τ ) ≤ (26.60) Ω−NR (K)

using

1 f˜(τ )


0 to be chosen sufficiently large later  d2 (x,K) g(0) v 2 (x, τ ) e Dτ dµg(τ ) (x) Ω

 v 2 (x, τ ) e

=

d2 (x,K) g(0) Dτ

K0

+

∞   i=1

First note that

v 2 (x, τ ) e

d2 (x,K) g(0) Dτ

Ki



dµg(τ ) (x) dµg(τ ) (x) .

d2 (x,K) g(0)

v 2 (x, τ ) e Dτ dµg(τ ) (x) K0  R2 v 2 (x, τ ) dµg(τ ) (x) ≤ e Dτ K0

≤

1 f˜ (τ )

R2

e Dτ

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

353

by (26.52). Second, we compute  d2 (x,K) g(0) 2 v (x, τ ) e Dτ dµg(τ ) (x) Ki  4 i R2 v 2 (x, τ ) dµg(τ ) (x) ≤ e Dτ Ω−int(N2i−1 R (K)) since Ki ⊂ Ω − int (N2i−1 R (K)) and dg(0) (x, K) ≤ 2i R for x ∈ Ki . Now    4i−1 R2 2A 2 v (x, τ ) dµg(τ ) (x) ≤ exp −B 2C˜0 τ f˜ (τ /γ) Ω−int(N2i−1 R (K)) by (26.60). Combining all of the above estimates, we obtain  d2 (x,K) g(0) 2 v (x, τ ) e Dτ dµg(τ ) (x) Ω

  ∞ R2 2A  4i R2 4i−1 R2 1 Dτ Dτ e + e exp −B ≤ 2C˜0 τ f˜ (τ ) f˜ (τ /γ) i=1 =

1 f˜ (τ )

e

R2 Dτ

+

2A

∞ 

f˜ (τ /γ)

i=1

e

R2 Dτ

  4i 1− BD ˜ 8 C0

.

R2

Taking R so that e Dτ = 2, we conclude    ∞ d2 (x,K) g(0) 2A  4i 1− 8BD 2 2 ˜ C 0 . + v (x, τ ) e Dτ dµg(τ ) (x) ≤ 2 f˜ (τ ) f˜ (τ /γ) i=1 Ω Now choose D =

˜0 16C B ,

so that



v 2 (x, τ ) e

d2 (x,K) g(0) Dτ

Ω

dµg(τ ) (x) ∞

≤

2A  −4i 2 + 2 f˜ (τ ) f˜ (τ /γ) i=1

8+A 4f˜ (τ /γ) 4A ≤ f˜ (τ /γ) ≤

∞

−4i

≤ 18 , f˜ (τ /γ) ≤ f˜ (τ ), and A ≥ 1. Thus we have  d2 (x,K) g(0) 4AeCτ , u2 (x, τ ) e Dτ dµg(τ ) (x) ≤ f (τ /γ) Ω

where C  2C0 1 − γ −1 . This completes the proof of Lemma 26.21. since

i=1 2

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354

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

The following says that, in an integral sense, the heat kernel decays exponential quadratically fast in space; this result follows from Lemma 26.17 and Lemma 26.21. Lemma 26.23 (Heat kernel has average exponential quadratic decay). Let (Mn , g (τ )), τ ∈ [0, T ], be a complete noncompact evolving Riemannian manifold and let H be the minimal positive fundamental solution to (26.4). Suppose that g (0) is such that Rcg(0) ≥ −K. Then there exist constants C = C (n, T, K, sup |Rij |) < ∞ and D = D (T, sup |Rij |) < ∞ such that (26.61)   2  dg(0) (x, y) C  √ , dµg(τ ) (x) ≤ H 2 (x, τ ; y, υ) exp D (τ − υ) M Volg(0) Bg(0) y, τ4−υ (26.62)   2  dg(0) (x, y) C 2  √ . dµg(υ) (y) ≤ H (x, τ ; y, υ) exp D (τ − υ) M Volg(0) Bg(0) x, τ4−υ Proof. As in the discussion following Theorem 24.40, let {Ωi }∞ i=1 be an exhaustion of M and let HΩi (x, τ ; y, υ) denote the corresponding Dirichlet heat kernels. By Lemma 26.17, we have HΩi (x, τ ; y, υ) ≤ H (x, τ ; y, υ) ≤ so that

C

 √ , Volg(0) Bg(0) y, τ2−υ

 Ωi

HΩ2 i (x, τ ; y, υ) dµg(τ ) (x) ≤

C

 √  Volg(0) Bg(0) y, τ2−υ

 HΩi (x, τ ; y, υ) dµg(τ ) (x) Ωi

CeC1 T  √  Volg(0) Bg(0) y, τ2−υ

since Ωi HΩi dµg(τ ) ≤ M H dµg(τ ) ≤ eC1 (τ −υ) ≤ eC1 T by (26.37). Now given y ∈ M and υ ∈ [0, T ), define Gi : Ωi × (0, T − υ] → [0, ∞) by (26.63)

≤

Gi (x, σ)  HΩi (x, υ + σ; y, υ) for i large enough. Define f : (0, ∞) → (0, ∞) by  √  σ 1 , Volg(0) Bg(0) y, f (σ)  C T 1 Ce 2 so that

 G2i (x, σ) dµg(υ+σ) (x) ≤ Ωi

1 . f (σ)

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

Let Vol−

K n−1

355

B (r) denote the volume of a ball of radius r in the simply-

K , connected n-dimensional space form with constant sectional curvature − n−1 where K ≥ 0 (so that Rc ≡ −K). We claim that f is (γ, A)-regular with γ = 4 and √  Vol− K B 2T n−1 √ . A= Vol− K B 4T n−1

Indeed, since Rcg(0) ≥ −K, we have by the Bishop–Gromov volume comparison theorem that √   √  σ1 σ1 Vol B K Vol B y, g(0) g(0) − n−1 2 2 f (σ2 ) f (σ1 ) √  ≤ A ≤ A  √ ≤ = . σ σ 1 f (σ1 /4) f (σ2 /4) Vol B Vol K B y, 1 g(0)

g(0)

4

− n−1

4

Hence, by Lemma 26.21 with Ω = Ωi , u = Gi , and K = {y}, we have  d2 (x,y) g(0) 4AeCσ 2 . Gi (x, σ) e Dσ dµg(υ+σ) (x) ≤ (26.64) f (σ/4) Ωi Taking the limit as i → ∞ (one may first integrate over compact regions), we have  d2 (x,y) g(0) 4AeC(τ −υ) H 2 (x, τ ; y, υ) e D(τ −υ) dµg(τ ) (x) ≤ f ((τ − υ) /4) M √ 

4Ce(C+C1 )T ≤

Vol−

K n−1

B

Vol−

K n−1

B

T 2

√  T 4

 √  . Volg(0) Bg(0) y, τ4−υ

This proves (26.61). Similarly, one may prove (26.62); we leave this as an exercise for the reader.
Exercise 26.24. Justify that Lemma 26.21 can be applied to prove (26.64) even though the fundamental solution Gi (x, σ) is a δ-function distribution centered at y at time σ = 0. 2.1.3. Pointwise upper bound of the heat kernel for an evolving metric. One may combine the estimates in Lemma 26.23 with the semigroup property to prove the following upper estimate for the heat kernel, which extends an estimate of Li and Yau to the case of a time-dependent metric (see Theorem 5.1 in [26]; compare with Corollary 3.1 of Li and Yau [121]). Theorem 26.25 (Upper bound of the heat kernel for an evolving metric). There exists a constant C3 < ∞ depending only on n, T , K, and sup |Rij |

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356

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

and there exists a constant C4 < ∞ depending only on T and sup |Rij | such that the minimal positive fundamental solution H to (26.4) satisfies3   d2g˜ (x,y) C3 exp − C4 (τ −υ) (26.65) H (x, τ ; y, υ) ≤  "   "  1/2 1/2 τ −υ Volg˜ Bg˜ x, τ −υ B · Vol y, g˜ g˜ 2 2 for any x, y ∈ M and 0 ≤ υ < τ ≤ T , where g˜ is as in (25.5). Proof. Let D ∈ (0, ∞) be as in Lemma 26.23. By the inequality d2g˜ (x, z) τ −ρ

+

d2g˜ (z, y) ρ−υ

≥

d2g˜ (x, y) (dg˜ (x, z) + dg˜ (z, y))2 ≥ (τ − ρ) + (ρ − υ) τ −υ

for x, y, z ∈ M and υ < ρ < τ and by the semigroup property (26.41), we have H (x, τ ; y, υ)  H (x, τ ; z, ρ) H (z, ρ; y, υ) dµρ (z) = M

d2 (x,y)

≤e ≤e

g ˜ − 2D(τ −υ)



d2 g ˜ (x,z)

M

d2 g ˜ (x,y) − 2D(τ −υ)

×



2

H 2 (x, τ ; z, ρ) e D(τ −ρ) dµg(ρ) (z) d2 g ˜ (z,y)

M

1

d2 g ˜ (x,z)

M



d2 g ˜ (z,y)

H (x, τ ; z, ρ) e 2D(τ −ρ) · H (z, ρ; y, υ) e 2D(ρ−υ) dµg(ρ) (z)

1

H 2 (z, ρ; y, υ) e D(ρ−υ) dµg(ρ) (z)

2

,

where the last inequality follows from H¨ older’s inequality. Substituting the exponentially weighted integral bounds (26.61) and (26.62) into this, we obtain   d2g˜ (x,y) C exp − 2D(τ −υ) , H (x, τ ; y, υ) ≤ √ √ 1/2 1/2 Volg˜ Bg˜ (x, τ − ρ) Volg˜ Bg˜ (y, ρ − υ) where C depends on n, T , K, and sup |Rij |. The theorem now follows from
taking ρ = τ +υ 2 . The following qualitative improvement of the previous theorem is Corollary 5.2 in [26]. Corollary 26.26. The heat kernel H for L satisfies ⎫ ⎧ ⎪ ⎪ 2 ⎬ ⎨ dg ˜ (x,y) 1 1 − D(τ −υ) min H (x, τ ; y, υ) ≤ C e  " ,  "  . ⎪ ⎭ ⎩ Volg˜Bg˜ x, τ −υ Volg˜ Bg˜ y, τ −υ ⎪ 2 2 3

In fact, C4 = 2D, where D is as in Lemma 26.23.

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

357

Proof. We only prove the inequality d2 (x,y)

Ce

g ˜ − D(τ −υ)

 "  Volg˜ Bg˜ x, τ −υ 2 " ˜  dg˜ (x, y). and leave the other one as an exercise. Let σ  τ −υ 2 and r H (x, τ ; y, υ) ≤

(1) If σ ≥ r˜, then we may invoke Lemma 26.17 to obtain H (x, τ ; y, υ) ≤

C Volg˜ Bg˜ (x, σ) 1

2

Ce 2C5 − r˜ e 2C5 σ2 ≤ Volg˜ Bg˜ (x, σ) for any constant C5 > 0. (2) Suppose σ < r˜. Recall that (26.65) says   2 C3 exp − 2Cr˜4 σ2 . (26.66) H (x, τ ; y, υ) ≤ 1/2 1/2 Volg˜ Bg˜ (x, σ) · Volg˜ Bg˜ (y, σ) Since Bg˜ (x, σ) ⊂ Bg˜ (y, r˜ + σ), regarding the rhs of (26.66), we have −1 Vol−1 g˜ Bg˜ (y, σ) ≤ Volg˜ Bg˜ (x, σ)

≤ Vol−1 g˜ Bg˜ (x, σ)

Volg˜ Bg˜ (y, r˜ + σ) Volg˜ Bg˜ (y, σ) Vol− K B (˜ r + σ) n−1

Vol−

K n−1

B (σ)

,

where we used the Bishop–Gromov relative volume comparison theorem and where Vol− K B (r) denotes the volume of a ball of radius r in the simplyn−1

K . Thus connected space form with constant sectional curvature − n−1 1   2 2 r + σ) C3 exp − 2Cr˜4 σ2 Vol− K B (˜ n−1 . (26.67) H (x, τ ; y, υ) ≤ 1 Volg˜ Bg˜ (x, σ) Vol−2 K B (σ) n−1

 "  For all σ, r˜ > 0 such that σ ≤ max r˜, T2 , we have  Vol K B (˜  r + σ) − n−1 r˜2 exp − 2C4 σ 2 Vol− K B (σ) n−1  Vol K B (2˜  r) 2 − n−1 r˜ ≤ exp − 2C4 σ 2 Vol− K B (σ) n−1

(26.68)

≤ C4 ,

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358

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where C4 depends only on n, K, and T . Thus (26.67) implies   1 2 C3 C42 exp − 4Cr˜4 σ2 . H (x, τ ; y, υ) ≤ Volg˜ Bg˜ (x, σ) The corollary follows from taking D  C5  2C4 .




Exercise 26.27. Verify that (26.68) is true. 2.2. Lower bound of the heat kernel for an evolving metric. We now derive a lower bound for the heat kernel with respect to an evolving metric. We begin with the following, which is Lemma 5.5 in [26]. Lemma 26.28 (L1 -norm of H concentrates at y as τ − υ → 0). We have the following. (1)  H (x, τ ; y, υ) dµg(τ ) (x) → 0 (26.69) √ M−Bg˜ (y,A τ −υ ) as A → ∞ uniformly in 0 ≤ υ < τ ≤ T . (2) (26.70)

 H √ M−Bg˜ (x,A τ −υ )

(x, τ ; y, υ) dµg(υ) (y) → 0

as A → ∞ uniformly in 0 ≤ υ < τ ≤ T . Proof. (1) Using d2 (x,y)

H (x, τ ; y, υ) ≤

Ce

g ˜ − D(τ −υ)

 "  Volg˜ Bg˜ y, τ −υ 2

from Corollary 26.26, we compute  H (x, τ ; y, υ) dµg(τ ) (x) √ M−Bg˜ (y,A τ −υ )  d2 g ˜ (x,y) C − D(τ −υ) dµ e ≤  "  g(τ ) (x) √ τ −υ M−Bg˜ (y,A τ −υ ) Volg˜ Bg˜ y, 2

 n/2 d2 g ˜ (x,y) C C˜0 − D(τ −υ) dµ (x) e ≤  "  g˜ √ τ −υ M−B y,A τ −υ ( ) g ˜ Volg˜ Bg˜ y, 2  ∞ n/2 2 d (Volg˜ Bg˜ (y, ρ)) C C˜0 − ρ dρ. e D(τ −υ) =  "  √ dρ τ −υ A τ −υ Volg˜ Bg˜ y, 2

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

359

Integrating by parts and throwing away a negative boundary term, we have (26.71)  H √ M−Bg˜ (y,A τ −υ ) n/2 ≤ C C˜0



(x, τ ; y, υ) dµg(τ ) (x)

∞ √ A τ −υ

2 Volg˜ Bg˜ (y, ρ) d − ρ  "  e D(τ −υ) dρ Volg˜ Bg˜ y, τ −υ 2



ρ2 D (τ − υ)

 dρ.

√ Now by the volume comparison theorem, for ρ ∈ [ τ − υ, ∞), Vol− K (ρ) Volg˜ Bg˜ (y, ρ) n−1  " ≤ " , τ −υ τ −υ Volg˜ Bg˜ y, Vol K − 2 2 n−1

so that

 H √ M−Bg˜ (y,A τ −υ ) n/2 ≤ 2C C˜0



(x, τ ; y, υ) dµg(τ ) (x) Vol−

∞ √ A σ

Vol−

(ρ) ρ2 ρ  σ e− Dσ dρ, Dσ 2

K n−1

K n−1

where σ  τ − υ. It is not difficult to see that, as a function of A, the rhs of (26.71) tends to 0 as A → ∞ uniformly in σ ∈ (0, T ]. Basically, the exponential quadratic decaying term beats the exponential linear growing term from the volume comparison. (2) We leave this as an exercise.




From combining Lemma 26.28 with Lemma 26.14, we immediately obtain Corollary 26.29. There exist constants A < ∞ and c > 0 such that  H (x, τ ; y, υ) dµg(τ ) (x) ≥ c (26.72) √ Bg˜ (y,A τ −υ ) and (26.73)

 H √ Bg˜ (x,A τ −υ )

(x, τ ; y, υ) dµg(υ) (y) ≥ c.

We have the following lower bound along the diagonal (see Lemma 5.6 in [26]). Lemma 26.30 (Lower bound for the heat kernel along the diagonal). There exists a constant c > 0 such that for any y ∈ M and 0 ≤ υ < τ ≤ T c

. √ (26.74) H (y, τ ; y, υ) ≥ Volg˜ Bg˜ y, τ − υ

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360

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Proof. Let u (x, τ )  H (x, τ ; y, υ) . By the Li–Yau inequality (25.58) with ε = 12 , we have u(y, τ ) H (y, τ ; y, υ) τ +υ

= u(x, τ +υ H x, 2 ; y, υ 2 )



 ˜0 d2g˜ (x, y) C ( ) exp − ≥e τ +υ 2 τ − τ +υ 2 2   −2n  2 d (x, y) C11 2τ g˜ exp −C˜0 = e− 2 (τ −υ) τ +υ τ −υ   d2g˜ (x, y) C11 T , ≥ e− 2 2−2n exp −C˜0 τ −υ −C11 τ − τ +υ 2

(26.75)



τ

−2n

for A and c as in where C11 < ∞ is as in Corollary 25.12. Therefore, √ Corollary 26.29 (without loss of generality, assume A ≥ 2), we have H (y, τ ; y, υ) ≥

e−

C11 T 2

2−2n "   Volg˜ Bg˜ y, A τ −υ 2 e−

C11 T 2

˜ A2 C 0

2−2n e− 2 " ≥   Volg˜ Bg˜ y, A τ −υ 2 ce−

C11 T 2

 Bg˜ y,A





τ −υ 2

˜  e −C 0

d2 g ˜ (x,y) τ −υ

  τ +υ ; y, υ dµg˜ (x) H x, 2

 τ +υ  H x, ; y, υ dµg˜ (x)

2 Bg˜ y,A τ −υ 

2

˜ A2 C 0

2−2n e− 2 " ≥   Volg˜ Bg˜ y, A τ −υ 2 ≥

const

, √ Volg˜ Bg˜ y, τ − υ

where we used (26.72) and the"volume comparison theorem to change the " τ −υ T
radius of the ball (note that A 2 ≤A 2 ). The following is Proposition 5.1 and Corollary 5.3 in [26]. Theorem 26.31 (Lower bound for the heat kernel). There exists a positive constant c1 depending only on n, T , and the bounds on |Rij |, |Rij |, |∇k Rij |, and |∆R| and there exists a positive constant c2 depending only on T and the bound on |Rij | such that (26.76)  d2 (x,y) 1 1 − c g˜(τ −υ)

,

e 2 √ √ . H (x, τ ; y, υ) ≥ c1 min Volg˜Bg˜ x, τ − υ Volg˜ Bg˜ y, τ − υ

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2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC

361

Hence d2 (x,y)

(26.77)

H (x, τ ; y, υ) ≥

1/2

Volg˜ Bg˜



c1 e

− c g˜(τ −υ)

2 .

. √ √ 1/2 x, τ − υ Volg˜ Bg˜ y, τ − υ

Proof. Again let u (x, τ )  H (x, τ ; y, υ). (1) By the Li–Yau inequality (25.58) with ε = 12 , we have (essentially we have switched x and y in the first variable of (26.75)) H (x, τ ; y, υ) u(x, τ ) τ +υ

= u(y, τ +υ H y, 2 ; y, υ 2 ) ≥e

C T − 11 2



2−2n exp −C˜0

d2g˜ (x, y) τ −υ

 .

Then applying the lower bound (26.74) along the diagonal, we obtain   C T d2g˜ (x,y) − 11 −2n ˜ exp −C0 τ −υ ce 2 2 H (x, τ ; y, υ) ≥  "  Volg˜ Bg˜ y, τ −υ 2   2 dg˜ (x,y) ˜ const exp −C0 τ −υ

. √ ≥ Volg˜ Bg˜ y, τ − υ The estimate with x and y switched on the rhs is proved analogously. (2) Finally, (26.76) implies H (x, τ ; y, υ)2 ≥ c21 e

2d2 g ˜ (x,y)

−c

2 (τ −υ)

1

, √ √ Volg˜ Bg˜ x, τ − υ Volg˜ Bg˜ y, τ − υ


which is (26.77).

2.3. Double integral upper bound of the heat kernel for a fixed metric. The proof of the following special case of a result of Davies [52] uses techniques similar to that in Theorem 26.25. Theorem 26.32 (Davies’ double integral upper bound for the heat kernel). If (Mn , g) is a complete Riemannian manifold and if H (x, y, t) is its heat kernel, then for any two bounded subsets U1 and U2 of M we have   d2 (U1 ,U2 ) 1 1 H (x, y, t) dµ (x) dµ (y) ≤ Vol 2 (U1 ) Vol 2 (U2 ) e− 4t , (26.78) U1

U2

where d (U1 , U2 ) = inf x∈U1 ,

y∈U2

d (x, y) is the distance between U1 and U2 .

Proof. Recall from Lemma 26.13 that H (x, y, t) = H (y, x, t). Let  H (x, y, t) dµ (y) ui (x, t)  Ui

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362

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

for i = 1, 2. Using the semigroup property (26.41), which implies      t t H z, y, dµ (z) , H x, z, H (x, y, t) = 2 2 M we compute for t > 0   H (x, y, t) dµ (x) dµ (y) U1 U2        t t H z, y, dµ (x) dµ (y) dµ (z) H x, z, = 2 2 M U1 U2      t t (26.79) u2 z, dµ (z) . u1 z, = 2 2 M Now let (26.80)

ξi (x, t) 

d2 (x, Ui ) 2t

for i = 1, 2 (note that ξi = 0 in Ui ), where d (x, Ui )  inf y∈Ui d (x, y) denotes the distance from x to Ui . Note that by the triangle inequality, d ( · , Ui ) is a Lipschitz function with Lipschitz constant 1, so that (26.81)

|∇d ( · , Ui )| ≤ 1

wherever d ( · , Ui ) is differentiable, which is a.e. on M by Rademacher’s theorem. Since for any x ∈ M     1 d2 (x, U1 ) + d2 (x, U2 ) t t 1 ξ1 x, + ξ2 x, = 2 2 2 2 2t d2 (U1 , U2 ) , ≥ 4t formula (26.79) implies   d2 (U1 ,U2 ) H (x, y, t) dµ (x) dµ (y) e 4t U1 U2      1 t 1 t t t ξ z, ( ) 1 2 u e2 e 2 ξ2 (z, 2 ) dµ (z) u1 z, ≤ 2 z, 2 2 M  1    1    2 2 t t 2 ξ1 2 ξ2 (26.82) · . u1 e dµ u2 e dµ ≤ 2 2 M M One calculates from (26.80) and (26.81) that d2 (x, Ui ) 1 ∂ξi =− ≤ − |∇ξi |2 . ∂t 2t2 2

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

363

By the monotonicity formula (26.55) in the proof of Lemma 26.21, we have      t 2 ξi 2 ξi ≤ lim ui e dµ ui e dµ (s) s→0 2 M M  χ2Ui dµ = M

= Vol (Ui ) for t > 0, where χUi denotes the characteristic function of Ui . We conclude from this and (26.82) that   d2 (U1 ,U2 ) 1 1 H (x, y, t) dµ (x) dµ (y) ≤ Vol 2 (U1 ) Vol 2 (U2 ) , e 4t U1

U2

which is (26.78).




Remark 26.33. Davies’ result may be used to give an alternative proof of a pointwise upper bound for the heat kernel on complete Riemannian manifolds satisfying the parabolic mean value inequality (see Theorem 1.2 in Li and Wang [120]). Problem 26.34. Is there a ‘Ricci flow’ version of Davies’ estimate? 3. Heat balls and the space-time mean value property The heat equation may be considered as an ‘averaging process’. This idea is especially important in physics. A beautiful illustration of this averaging process is the space-time mean value property (MVP) for solutions to the heat equation. We consider first the classical case of Euclidean space and then Riemannian manifolds. 3.1. Space-time mean value property on Euclidean space. We consider the elliptic case of the Laplace equation and the parabolic case of the heat equation. 3.1.1. Euclidean mean value property for the Laplace equation. In Euclidean space En , the mean value property for solutions of the Laplace equation is given by the following. Let ωn denote the volume of the unit n-ball and let B(O, r) denote the ball of radius r centered at a point O ∈ Rn . Theorem 26.35 (MVP for Laplace equation on En ). If u is a C 2 solution to ∆u = 0 in a domain Ω ⊂ Rn , then  1 u(x)dµE (x) (26.83) u(O) = ωn rn B(O,r) for any O ∈ Ω and r > 0 such that B(O, r) ⊂ Ω.

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364

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Proof. Consider the function of r defined by the rhs of (26.83), which is the average of u on the ball of radius r,  1 u(x)dµE (x) . (26.84) φO (r)  ωn rn B(O,r) Note that lim φO (r) = u(O)

r→0

since u is continuous. We compute (26.85)   n 1 ∂ φO (r) = − u(x)dµE (x) + u(x)dσr (x) , ∂r ωn rn+1 B(O,r) ωn rn ∂B(O,r) where dσr is the induced volume form on ∂B(O, r). On the other hand,4     2n u(x)dµE (x) = ∆ |x|2 u(x)dµE (x) B(O,r) B(O,r)    ∇ |x|2 · ∇u(x)dµE (x) =− B(O,r)  2 |x| u(x)dσr (x) + ∂B(O,r)    (26.86) |x|2 − r2 ∆u(x)dµE (x) = B(O,r)  u(x)dσr (x) . + 2r ∂B(O,r)

Since ∆u = 0, by combining (26.85) and (26.86), we conclude that ∂ φO (r) = 0. ∂r

(26.87) Since φO is constant, we have

φO (r) = lim φO (ρ) = u(O). ρ→0


Note that the boundary of the ball that appears in (26.84), i.e., ∂B(O, r), is a level set of the fundamental solution of the Laplace equation centered at O. Note that by (26.83) and (26.86) we also have  1 u(x)dσr (x) , (26.88) u(O) = nωn rn−1 ∂B(O,r) where the rhs is the average of u on the sphere of radius r. 4

One may think of this as a Pohozaev-type identity; note that x = conformal Killing vector field.

1 ∇ |x|2 2

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is a

3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

365

Exercise 26.36 (Mean value inequality when Rc ≥ 0). Show that if is a complete Riemannian manifold with Rc ≥ 0 and if f : M → (−∞, 0] is a nonpositive C 2 function with ∆f ≥ 0, then for any O ∈ M and 0 < r < inj (O)  1 f dµ. (26.89) f (O) ≤ ωn rn B(O,r)

Hint: By the Laplacian comparison theorem, we have ∆ ρ2 ≤ 2n, where ρ (x)  d (x, O). Defining  1 f (x)dµ (x) , ψO (r)  ωn rn B(O,r) (Mn , g)

∂ ψO (r) ≥ 0 (note that ρ (x)2 − r2 ≤ 0 for x ∈ B(O, r)). (See we find that ∂r for example Proposition 1.142 in [45] for another proof.)

Remark 26.37. By taking f ≡ −1 in (26.89), we have Vol B (O, r) ≤ ωn rn , which, of course, is a special case of the Bishop–Gromov volume comparison theorem. 3.1.2. Space-time Euclidean mean value property for the heat equation. Analogous to the mean value property for harmonic functions there is a space-time mean value property for solutions to the heat equation on Euclidean space, which we now present. First we discuss the heat kernel and the associated heat balls. Recall that the Euclidean heat kernel is defined by HE (x, t; y, s)  (4π (t − s))− 2 e n

|x−y|2

− 4(t−s)

for x, y ∈ Rn and −∞ < s < t < ∞. Given r ∈ (0, ∞), we define the heat ball of radius r based at (x, t) as the following superlevel set of the heat kernel   1 n Er (x, t)  (y, s) ∈ R × (−∞, t) : HE (x, t; y, s) > n , r which is an open convex subset of Rn × (−∞, t). Note that the point (x, t) is actually at the zenith (or top) of the heat ball Er (x, t); the nadir (or r2 ). The boundaries of the heat bottom) of the heat ball is the point (x, t − 4π balls are the level sets of the fundamental solution of the heat equation. Let (26.90)

ψr (x, t; y, s)  log(rn HE ) = n log r −

|x − y|2 n log (4π (t − s)) − . 2 4 (t − s)

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Another useful way to describe the heat ball is Er (x, t) = {(y, s) ∈ Rn × (−∞, t) : ψr (x, t; y, s) > 0} . Hence (26.91)

∂Er (x, t) = {(y, s) : ψr (x, t; y, s) = 0} .

Each time slice of Er (x, t) is a round ball. In particular, given s ∈ (−∞, t), (26.92)

Er,s (x, t)  {y ∈ Rn : (y, s) ∈ Er (x, t)} = {y ∈ Rn : |x − y| < Φt−s (r)} , !

where

  n 4 (t − s) n log r − log (4π (t − s)) 2 is the radius of the time s slice of Er (x, t). Fixing r, note that the supremum r2 . Hence the maximal radius of a time of Φt−s (r) occurs when t − s = 4πe slice of Er (x, t) is '   2  n r r2 n log r − log . sup Φt−s (r) = πe 2 e s∈[0,t) Φt−s (r) 

Now we are ready to discuss the space-time mean value property. Let u : Rn × [0, t) → R be a C 2 solution to the heat equation. Define  1 |x − y|2 u(y, s) dµE (y) ds, (26.93) φx,t (r)  n 4r (t − s)2 Er (x,t) where dµE denotes the Euclidean measure on Rn . Exercise 26.38. Show that  |x − y|2 1 dµE (y) ds = 1. (26.94) 2 4rn Er (x,t) (t − s) Hint: See p. 409 of [185]. Note that (26.94) implies that φx,t (r) is a weighted average of u on the heat ball of radius r based at (x, t). Since limr→0 Er (x, t) = {(x, t)} in the Gromov–Hausdorff sense and since u is continuous, by (26.94) we obtain (26.95)

u (x, t) = lim φx,t (r) . r→0

With these preliminaries, we may state the following mean value property for heat balls. Theorem 26.39 (Space-time mean value property for the heat equation — Euclidean space). If u : Rn × [0, t] → R is a classical solution to the heat r2 > 0, then equation and if t ≥ 4π  |x − y|2 1 u(y, s) dµE (y) ds. (26.96) u (x, t) = n 4r (t − s)2 Er (x,t)

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

Note that the condition t ≥

r2 4π

367

is equivalent to

Er (x, t) ⊂ Rn × [0, ∞). Theorem 26.39 follows from (26.95) and the following (compare this with (26.87) in the elliptic case). √ Lemma 26.40. For any (x, t) ∈ Rn × (0, ∞) and r < 4πt, ∂ φx,t (r) = 0. ∂r

(26.97)

The proof of Lemma 26.40 uses integration by parts. Since we shall discuss its generalization to the heat equation on Riemannian manifolds in the next subsection, we omit its proof. Note that since   |x − y|2 ∂ − ∆y log HE = |∇y log HE |2 = , − ∂s 4 (t − s)2 we may rewrite the mean value property (26.96), in terms of integrating against the heat kernel, as (26.98) u (x, t) =

1 rn

(26.99) 1 = n r

 Er (x,t)

u(y, s) |∇y log HE |2 (x, t; y, s) dµE (y) ds 

 u(y, s) Er (x,t)

  ∂ − − ∆y log HE (x, t; y, s) dµE (y) ds. ∂s

This observation is useful in formulating its generalization to Riemannian manifolds, where one does not have an explicit formula for the heat kernel. Note also that in (26.99) we may replace log HE by ψr since they differ by a constant (by (26.90)), i.e., (26.100)    ∂ 1 u(y, s) − − ∆y ψr (x, t; y, s) dµE (y) ds. u (x, t) = n r ∂s Er (x,t) We now derive the mean value property for heat spheres. Since ψr = 0 on ∂Er (x, t), replacing log HE by ψr and integrating by parts on (26.100) in the time and space directions separately, we have    ∂u 1 − ∆y u (y, s)ψr (x, t; y, s) dµE (y) ds u (x, t) = n r ∂s Er (x,t)  t  1 u(y, s) ∇y ψr , ν (x, t; y, s) dσs (y) ds, − n r t− r2 ∂Er,s (x,t) 4π

where Er,s (x, t) is the time slice of Er (x, t) defined by (26.92), ν is the outward unit normal to its boundary ∂Er,s (x, t), and dσs is the induced

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

(n − 1)-dimensional measure on ∂Er,s (x, t). Since y−x x−y and ν = , ∇y ψr = 2 (t − s) |y − x| we have

|x − y| = −|∇ψr | 2 (t − s) on ∂Er,s (x, t). Combining this with the fact that u is a solution to the heat equation, we have  t  1 u(y, s)|∇ψr | dσs (y) ds. (26.101) u (x, t) = n r t− r2 ∂Er,s (x,t) 4π   r Note that by (26.91), a normal vector to ∂Er (x, t) is ∇ψr , ∂ψ ∂t . Thus !    r 2 |∇ψr |2 +  ∂ψ ∂t  dσs (y) ds. dσ(y, s) = |∇ψr | ∇y ψr , ν = −

Therefore (26.101) implies  |∇ψr |2 1 u(y, s) ! (26.102) u (x, t) = n   (y, s) dσ(y, s), r ∂Er (x,t)  ∂ψr 2 2 |∇ψr | +  ∂t  where dσ is the induced n-dimensional measure of ∂Er (x, t) (compare with (26.117) in the more general case of a Riemannian manifold).5 3.2. Mean value property on Riemannian manifolds. Let (Mn , g) be a complete Riemannian manifold and let H(x0 , y, τ ) denote the heat kernel centered at (x0 , 0). Let τ (t)  t0 − t, so that H(x0 , y, τ (t)) is the adjoint heat kernel centered at (x0 , t0 ), i.e., ∂ H(x0 , y, τ (t)) = −∆y H(x0 , y, τ (t)), ∂t lim H(x0 , y, τ (t)) = δx0 .

tt0

Define the corresponding Riemannian heat ball of radius r based at (x0 , t0 ) by

(26.103) Er (x0 , t0 )  (y, t) : H(x0 , y, τ ) > r−n ⊂ M × (−∞, t0 ) . 5

To see the equivalence of (26.102) and (26.117), note that 

|∇H|2 |∇ log H|2 −n
∂H
2 = r 

H
2 |∇H|2 +
∂t
|∇ log H|2 +
∂ log ∂t

since H = r−n on ∂Er .

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

369

Its time slices are given by Erτ1 (x0 , t0 )  Er (x0 , t0 ) ∩ {τ (t) = τ1 } ⊂ M × {t0 − τ1 } for τ1 > 0. Equivalently, define Er,t1 (x0 , t0 )  Er (x0 , t0 ) ∩ {t = t1 } ⊂ M × {t1 } , so that Er,t1 (x0 , t0 ) = Ert0 −t1 (x0 , t0 ) for t1 < t0 . In order to ensure that the integral (26.106) below, which appears in the statement of the mean value property, is well defined, we shall make some additional assumptions on the heat kernel H(x, y, τ ) which guarantee that the time slices Erτ are compact. In particular, we assume that along the diagonal we have a bound of the form (26.104)

H(y, y, τ ) ≤

C , f (τ )

(y, τ ) ∈ M × (0, ∞) ,

for some constant C < ∞ and some positive, continuous, strictly increasing function f : (0, ∞) → (0, ∞) which is (γ, A)-regular in the sense of Definition 26.20 with T = ∞. Under assumption (26.104) on the heat kernel along the diagonal, by a general result of Davies and Grigor yan (see Theorem 1.1 of [76]), there exist constants δ, D, C1 ∈ (0, ∞) such that the heat kernel has the upper bound  2  d (x, y) C1 exp − (26.105) H(x, y, τ ) ≤ f (δτ ) Dτ for all x, y ∈ M and τ ∈ (0, ∞).6 Using this estimate, we shall prove the following, which is a generalization of (26.98). Theorem 26.41 (Space-time mean value property for the heat equation — manifold version). Let (Mn , g) be a complete Riemannian manifold with heat kernel H(x, y, τ ) satisfying assumption (26.104). If u : M × [0, T ] → R is a classical solution to the heat equation ∂u ∂t − ∆u = 0, then for any fixed t0 ∈ (0, T ] there exists r¯ > 0 such that for any r ∈ (0, r¯) and x0 ∈ M, the heat ball Er (x0 , t0 ) lies inside a compact subset of M × (0, T ] and  1 u(y, t)|∇ log H|2 (x0 , y, τ (t)) dµ (y) dt, (26.106) u(x0 , t0 ) = n r Er (x0 ,t0 )

where τ (t) = t0 − t. 6

In fact, for any D > 4 there exists δ > 0 and C1 < ∞ such that (26.105) holds.

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Proof. (1) The integral is finite. Let (x0 , t0 ) ∈ M × (0, T ]. By (26.103) and (26.105), we have    C1 r n τ 2 × {t0 − τ } Er (x0 , t0 ) ⊂ y ∈ M : d (x0 , y) < Dτ log f (δτ ) for τ ∈ (0, t0 ] and r ∈ (0, ∞). In particular, for each such τ and r, the heat ball slice Erτ (x0 , t0 ) is bounded. Define r¯ ∈ (0, ∞) by C1 r¯n = f (δt0 ).   ¯n 1r = 0, so that Er¯t0 (x0 , t0 ) is empty. Moreover, by the conThen log fC(δt 0) tinuity of f , for any r ∈ (0, r¯) there exists τr < t0 such that Erτ (x0 , t0 ) is empty for τ > τr . Since |∇ log H|2 (x0 , y, τ ) is bounded for τ away from 0, the integral in (26.106) is finite over any subdomain of Erτ (x0 , t0 ) where τ is bounded away from 0. To obtain the finiteness of the integral in (26.106) when τ is close to zero, we note the following gradient estimate due to Hamilton [91], which is related in spirit to the Li–Yau differential Harnack estimate [121]   B C 2 (26.107) |∇ log H| ≤ log τ τ n/2 H (see Corollary E.34 in Part II). Now by the asymptotics of H(x0 , y, τ ) as τ → 0, we have that the integral on the right-hand side of (26.106) is finite for τ close to zero. (Exercise: Prove this.)7 (2) MVP via a conservation law. Let Er  Er (x0 , t0 ) and let Er,t1  Er ∩ {t = t1 }. Given r > 0, as in (26.90) we let (26.108)

ψr (y, τ ) = log H(x0 , y, τ ) + n log r.

Again, the heat ball may be expressed as Er = {(y, t) : ψr (y, τ (t)) > 0}. It is easy to check that ψr (y, τ (t)) satisfies ∂ψr + ∆ψr = −|∇ψr |2 . (26.109) ∂t Given any smooth function u(x, t), define  1 u(y, t)|∇ψr |2 (y, τ (t)) dµ dt. (26.110) I (r)  n r Er

Having discussed a sufficient condition for the integral I to be well defined, d I. i.e., when H satisfies (26.104), we may proceed to compute dr Without loss of generality, we may change the time interval to [−T, 0] and assume that t0 = 0, so that τ (t) = −t. Let (26.111) 7

J (r)  rn I (r) .

See Theorem 10 and Remark 11 in §3 of [57].

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

371

Noticing that the set Er,t is empty for t sufficiently close to −T , by the co-area formula (see Evans and Gariepy [59] for example), we have 



0

u(y, t)|∇ψr |2 dµ dt

J (r) = 

−∞ 0

Er,t



∞



 u(y, t)|∇ψ| dσ

= −∞

−n log r

 dc

dt,

ψ(y,τ )=c

where ψ(y, τ )  log H(x0 , y, τ ) = ψr (y, τ ) − n log r and dσ is the induced volume element on ψ(y, τ ) = c. Hence, for almost every r, using the fact that the outward unit normal ∇ψr , we have ν of ∂Ert is given by − |∇ψ r|   n u(y, t)|∇ψr | dσ dt −∞ r ∂Er,t    n 0 u(y, t)∇ψr , ν dσ dt =− r −∞ ∂Er,t    n 0 (∇u, ∇ψr  + u∆ψr ) dµ dt =− r −∞ Er,t    n 0 (ψr ∆u − u∆ψr ) dµ dt, = r −∞ Er,t

d J= dr

(26.112)



0

where we have used the divergence theorem and where we have also used the fact that ψr = 0 on ∂Er,t .8 Now we may apply (26.109) to (26.112) and

8

Note that div (u∇ψr ) = ∇u, ∇ψr  + u∆ψr , div (ψr ∇u) = ∇ψr , ∇u + ψr ∆u.

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26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

deduce n d J= dr r



0







∂ψr + |∇ψr |2 ψr ∆u + u ∂t −∞ Er,t    0  ∂ψr ∂u n +u dµ dt ψr = r −∞ Er,t ∂t ∂t   n 0 u|∇ψr |2 dµ dt + r −∞ Er,t     ∂u n 0 dµ dt ψr ∆u − + r −∞ Er,t ∂t     n 0 ∂u n dµ dt. ψr ∆u − = J+ r r −∞ Er,t ∂t

(26.113)





dµ dt

Note that, since ψr = 0 on ∂Er,t , we have 

 Er,t

so that

∂ψr ∂u +u ψr ∂t ∂t 

0 −∞



 dµ = Er,t



 Er,t

d ∂ (ψr u) dµ = ∂t dt

∂ψr ∂u +u ψr ∂t ∂t



 ψr udµ , Er,t

 dµ dt = 0.

Therefore, from (26.113) we have, for almost every r, that      J n ∂u d dµ dt. ψr ∆u − = n+1 (26.114) dr rn r ∂t Er

In the special case where u(x, t) is a solution to the heat equation, we have d I ≡ 0. dr Note that 0, the sequence of (incomplete) Riemannian for any δ >

manifolds B(x0 , δ), x0 , i2 g i∈N converges as i → ∞, in the C ∞ pointed Cheeger–Gromov sense, to (Rn , 0, gcan ), where gcan denotes the standard Euclidean metric. Moreover, from the scaling property r 

, x0 , g I r, x0 , c2 g = I c and the heat kernel asymptotics we can show that lim I(r, x0 , g) = u(x0 , t0 ).

r→0

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

373

Hence, from the absolute continuity of the function I(r), we have  1 u(y, t)|∇ log H|2 (x0 , y, τ ) dµ dt u(x0 , t0 ) = n r Er ⎞ ⎛     r n ⎝ ∂u − ∆u ψs dµ dt⎠ ds. (26.115) + n+1 ∂t 0 s Es

The theorem follows from this equality.




A consequence of the mean value property for harmonic functions is the strong maximum principle (see p. 27 in Evans [58]). Similarly, a consequence of the mean value property for solutions of the heat equation is the strong maximum principle for the heat equation. In particular, tracing through the proof of the theorem, one sees that if u is a subsolution of the heat equation, then formula (26.106) holds with ‘=’ replaced by ‘≤’. Moreover, if u(x, t) is a subsolution to the heat equation and if u achieves its maximum on M × (0, T ] at some point (x0 , t0 ), then one may infer from (26.106) that u(x, t) ≡ u(x0 , t0 ) for all (x, t) with t < t0 . 3.3. Mean value property using heat spheres. In this subsection we present the mean value theorem for the heat equation on Riemannian manifolds in terms of integrals on heat spheres by adapting an argument of Fabes and Garofalo [60]. ˜ = M × (−∞, 0] Without loss of generality, assume that t0 = 0. Let M be equipped with the Riemannian product metric g˜(x, t) =

n 

gij (x)dxi ⊗ dxj + dt2 ,

i,j=1

where t is the global (time) coordinate on (−∞, 0]. Again let H(x0 , y, τ ) denote the heat kernel centered at (x0 , 0), where τ  −t. Applying the divergence theorem to the space-time vector field given by ∂ + u∇H − H∇u ∂t ˜ we obtain on a bounded (space-time) domain D in M, uH

(26.116)         ∂H ∂u ∂u − ∆u H dµ dt = − ∆u H + u + ∆H dµ dt ∂t ∂t ∂t D D    ∂ div g˜ u H + u∇H − H∇u dµ dt = ∂t D )  ( ∂ σ, u H + u∇H − H∇u, ν˜ d˜ = ∂t ∂D g˜

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374

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where ν˜ is the outward unit normal to ∂D and where d˜ σ is the volume element of ∂D, both with respect to g˜. Recall that Er = {(y, τ ) : H(x0 , y, τ ) ≥ r−n }. We define Drs  {(y, τ (t)) ∈ Er : t < s} and two portions of its boundary P1s  {(y, τ (t)) : H(x0 , y, τ (t)) = r−n and t < s} and P2s  {(y, τ (t)) ∈ Drs : t = s}. Applying (26.116) to Drs , we have    ∂u − ∆u H dµdt 0= ∂t Drs )  (  ∂ σ uH dµ + uH + u∇H − H∇u, ν˜ d˜ = ∂t P2s P1s )   (  ∂ −n σ+ uH dµ + r u∇H, ν˜ d˜ σ. u − ∇u, ν˜ d˜ = ∂t P2s P1s P1s Letting s → 0, we then have  uH dµ u(x0 , 0) = lim s→0 P s 2

)  ∂ σ− u∇H, ν˜ d˜ σ u − ∇u, ν˜ d˜ = −r ∂t P10 P10     |∇H|2 ∂ − ∆ u dµ dt + u" σ. = −r−n  ∂H 2 d˜ ∂t Dr0 P10 2   |∇H| + ∂t −n



(

Summing together the above equalities, we have the following mean value property for solutions to the heat equation. Theorem 26.42 (MVP using heat spheres). Let (Mn , g) be a Riemannian manifold such that the heat kernel H(x0 , y, τ ) satisfies (26.104). If u : M × [T1 , 0] → R is a solution to the heat equation, then  |∇H|2 u (y, t) " σ (y, t) , (26.117) u(x0 , 0) =  ∂H 2 (x0 , y, τ (t)) d˜ ∂Er 2   |∇H| + ∂t where we note that ∂Er = P10 . Exercise 26.43. Give another proof of Theorem 26.41 by integrating (26.117) and applying the co-area formula as in [60].

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3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY

375

3.4. Mean value property under Ricci flow. The mean value property can also be extended, as an inequality, to the case where the Riemannian metric is evolving by the Ricci flow for supersolutions to the associated heat equation under the assumption that the scalar curvature of the evolving metric is nonnegative. For this we invoke the concept of the reduced distance function (see Chapter 7 of Part I). Let (Mn , g(t)), t ∈ [0, T ], be a solution to the Ricci flow. Let (y, τ ) denote the reduced distance function, defined with respect to some basepoint (x0 , t0 ) ∈ M × (0, T ] and where τ  t0 − t. Let v(y, τ )  (4πτ )−n/2 e−
(y,τ ) .

(26.118)

For any r > 0, one defines the heat ball using the reduced distance function by Er  {(y, t) : v(y, τ (t)) ≥ r−n and t ≤ t0 }.

(26.119)

Using standard estimates for the reduced distance, one can check that Er is compact for r sufficiently small. Analogous to (26.90) and (26.108), let ψr (y, t)  log v(y, τ (t)) + n log r and for any smooth function u(x, t) let 

|∇ψr |2 + ψr R u dµg(t) dt (26.120) P (r)  Er

(compare with (26.98) and (26.110)). Note that ψr ≥ 0 in Er . The following monotonicity formula was proved in [57]. Theorem 26.44. Let (Mn , g(t)), t ∈ [0, T ], be a complete solution to the Ricci flow and let P (r) I(r)  n . r Then for any smooth function u and 0 < r1 < r2 , (26.121) I(r2 ) − I(r1 )         r2 n u ∂ ∂ + ∆ − R v + ψr − ∆ u dµg(t) dt dr. =− n+1 v ∂t ∂t r1 r Er Recall that Perelman proved that (see (7.91) in Part I) n ∂ − ∆ + |∇|2 − R + ≥ 0, ∂τ 2τ that is,



   ∂ n ∂ 2 +∆−R v = − ∆ + |∇| − R + v ≥ 0. ∂t ∂τ 2τ

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376

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Hence, if u ≥ 0, then



I(r2 ) − I(r1 ) ≤ −

r2 r1

n





rn+1

ψr Er

 ∂ − ∆ u dµg(t) dt dr. ∂t

As an immediate consequence, we have the following. Corollary 26.45. Let (Mn , g(t)) be a complete solution to the Ricci flow. If u is a nonnegative supersolution to the heat equation, then I(r2 ) ≤ I(r1 )

(26.122)

for 0 < r1 < r2 . Moreover, for r > 0 1 u(x0 , t0 ) ≥ I(r) = n r

(26.123)





|∇ψr |2 + ψr R u dµg(t) dt.

Er

Note that inequality (26.123) follows from taking r1 → 0 in (26.122) since limr1 →0 I(r1 ) = u(x0 , t0 ). Corollary 26.46. Let (Mn , g(t)) be a complete solution to the Ricci flow. If w(x, t) is a supersolution to the heat equation which attains its minimum over M × [0, t0 ] at a point (x0 , t0 ) and if the scalar curvature is nonnegative, then w(x, t) ≡ w(x0 , t0 ) for all x ∈ M and t ≤ t0 . Proof. Applying (26.123) to u(x, t)  w(x, t) − w(x0 , t0 ), we obtain for all r > 0  

2 |∇ψr | w dµg(t) dt ≤ |∇ψr |2 + ψr R w dµg(t) dt ≤ 0. Er

Er




This implies u(x, t) ≡ 0 on Er for all r > 0.

3.5. Strong maximum principle for weak solutions. For simplicity we shall only state the result below for weak supersolutions. This result is also known as a weak Harnack inequality in the literature. Let (Mn , g) be a Riemannian manifold and let O ∈ M be such that the following two properties hold. (1) For some R, P > 0 and κ > 1, we have for any C ∞ function f : B(O, κr) → R, where r ∈ (0, R], that  (26.124)

 |f − favg |2 dµ ≤ P r2

B(O,r)




|∇f |2 dµ, B(O,κr)

where favg  B(O,r) f dµ. (2) There exists D < ∞ such that for any r ∈ (0, R], 1 Vol B(O,r)

(26.125)

Vol B(O, 2r) ≤ D Vol B(O, r).

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4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS

377

Given r > 0, t ∈ (−∞, ∞), and δ ∈ (0, 1), we define the following parabolic regions in M × (−∞, ∞):   3+δ 2 3−δ 2 r ,t − r , Q−  B(O, δr) × t − 4 4   3−δ 2 r , Q−  B(O, δr) × t − r2 , t − 4   1+δ 2 r ,t . Q+  B(O, δr) × t − 4 We have the following. Theorem 26.47 (Weak Harnack inequality). Let (Mn , g) be a Riemannian manifold satisfying the properties above. Then there exist p0 > 0 and A < ∞ such that for any weak supersolution to the heat equation u with u(x, t) > 0 a.e. in B(O, r) × (t − r2 , t), where 0 < r < R, we have 1   p0 1 p0 u dµ ≤ A inf u. (26.126) Q+ Vol(Q− ) Q− We refer the reader to the original articles by Moser [135] and SaloffCoste [165] for the proof, which uses Moser iteration (compare with §1 of this chapter). There is an excellent exposition in the corresponding elliptic case by Li [115]. Moreover, the result above also holds for uniformly parabolic operators in divergence form. An important consequence of Theorem 26.47 is the Harnack estimate for the heat equation. Theorem 26.48 (Harnack estimate). Under the same assumptions as in Theorem 26.47, there exists A < ∞ such that for any weak solution u to the heat equation with u(x, t) > 0 a.e. in B(O, r) × (t − r2 , t), 0 < r < R, we have (26.127)

sup u ≤ A inf u. Q−

Q+

We remark that for classical solutions to the heat equation on a manifold with Rc ≥ 0, the Li–Yau differential Harnack estimate implies a sharp form of the Harnack estimate above. 4. Distance-like functions on complete noncompact manifolds In this section we discuss the existence of distance-like functions with uniform bounds on all higher covariant derivatives on complete noncompact manifolds with bounded curvature. Such functions are useful in constructing barrier functions used in applying the maximum principle on noncompact manifolds (see for example §2 of Chapter 12 in Part II).9 The following is Theorem 1 in Tam [177]. 9

We also used such a function in §2 of the present chapter.

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378

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Proposition 26.49 (Distance-like functions with bounds on derivatives). Given n ∈ N − {1} and K ∈ (0, ∞), there exists a constant Cn,K ∈ (1, ∞), depending only on n and K, such that if (Mn , g) is a complete noncompact Riemannian manifold with sectional curvature |sect (g)| ≤ K and if O ∈ M, then there exists a C ∞ function f : M → R with the following bounds on M: (1) (distance-like) r (x) + 1 ≤ f (x) ≤ r (x) + Cn,K ,

(26.128)

where r (x)  d (x, O), and (2) (uniform bounds on first two derivatives) |∇f | (x) ≤ Cn,K

(26.129)

and

|∇∇f | (x) ≤ Cn,K .

Remark 26.50. There exist constants Cn,K,m ∈ (0, ∞), depending only n on n,  and m, such that if we further assume that (M , g) satisfies  k K, ∇ Rm ≤ K for 0 ≤ k ≤ m, where m ∈ N, then we have    k  (26.130) ∇ f  (x) ≤ Cn,K,m for 1 ≤ k ≤ m + 2. We now give the proof of this proposition, closely following Tam’s paper [177]. The idea is to start with a distance-like function with uniformly bounded gradient and to evolve this function by the heat equation. The evolved function retains the original properties of the initial function and obtains the additional property of having uniformly bounded Hessian (and even bounded higher derivatives if bounds on the derivatives of the curvature are assumed). By Proposition 2.1 in Greene and Wu [75], given (Mn , g) and O ∈ M, there exists a C ∞ function u : M → R with (26.131)

|u (x) − r (x)| ≤ 1 and

|∇u| (x) ≤ 2

on M (see also Lemma 12.30 in Part II). Let H : M × M × (0, ∞) → (0, ∞) be the heat kernel (minimal positive fundamental solution of the heat equation) of (M, g). Then the function f : M × (0, ∞) → R defined by  H (x, y, t) u (y) dµ (y) (26.132) f (x, t)  M

is a solution to the heat equation with limt→0 f (x, t) = u (x) uniformly in x ∈ M. Remark 26.51. The uniform convergence may be seen from the following estimates which we shall prove below: (26.134), (26.139), and the bound corresponding to (26.141), where the integration is over M − B (x, δ) for any δ ∈ (0, 1], together with Remark 26.52.

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4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS

379

Step 1. We shall show that there exists a constant C1 < ∞ depending only on n and K such that |f (x, t) − u (x)| ≤ C1

(26.133)

for all x ∈ M and t ∈ (0, 1]. By (26.131), this implies |f (x, t) − r (x)| ≤ C1 + 1 for all x ∈ M and t ∈ (0, 1]. By adding the constant C1 + 2 to f (which does not affect the derivatives of f ), we obtain (26.128) for any Cn,K ≥ 2C1 + 3.
Using (26.132), M H (x, y, t) dµ (y) = 1, and |∇u| ≤ 2, we compute  H (x, y, t) |u (y) − u (x)| dµ (y) |f (x, t) − u (x)| ≤ M  (26.134) H (x, y, t) d (x, y) dµ (y) . ≤2 M

(Note that the rhs of (26.134) is independent of the choice of O.) Clearly,   H (x, y, t) d (x, y) dµ (y) ≤ H (x, y, t) dµ (y) = 1. (26.135) B(x,1)

B(x,1)

On the other hand, since Rc ≥ − (n − 1) K, by Corollary 3.1 of Li and Yau [121] we have10    √  √ d2 (x, y) −1/2 −1/2 H(x, y, t) ≤ C1 Vol B x, t Vol B y, t exp C2 Kt − 5t for all x, y ∈ M and t ∈ (0, ∞), where C1 < ∞ is an absolute constant and where C2 < ∞ depends only on n. In particular, if t ∈ (0, 1], then (26.136)   2  √  √ d (x, y) −1/2 −1/2 , B x, t Vol B y, t exp − H (x, y, t) ≤ C3 Vol 5t depends only where C3 < ∞ √

√on n and K.

Since B x, t ⊂ B y, t + d (x, y) , we have for t ∈ (0, 1],  √   √ Vol B x, t ≤ Vol B y, t + d (x, y)  √   √ (26.137) t + d (x, y) ≤ Vol B y, t C4 t−n/2 exp C5 for some C4 < ∞ and C5 < ∞ depending only on n and K, where we used the following result. Since sect (g) ≥ −K, the Bishop–Gromov volume comparison theorem implies √ √ VolK B( t + d (x, y)) Vol B(y, t + d (x, y)) √ √ ≤ Vol B(y, t) VolK B( t)  √  (26.138) t + d (x, y) , ≤ C4 t−n/2 exp C5 10

Compare with the analogous upper estimate (26.65) for the time-dependent metric

case.

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380

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where VolK B(s) denotes the volume of the ball of radius s in the simplyn of constant sectional curvature connected complete Riemannian manifold SK 11 K. Applying (26.137) to (26.136), we have     2  √ 1 d (x, y) −n/4 −1 C5 d (x, y) exp − Vol B x, t exp H (x, y, t) ≤ C6 t 2 5t   2  √ d (x, y) (26.139) ≤ C7 t−n/4 Vol−1 B x, t exp − 6t for t ∈ (0, 1] and d (x, y) ≥ 1 and where C6 < ∞ and C7 < ∞ depend only on n and K. Now, using this upper bound for the heat kernel, we compute (26.140)  2

H (x, y, t) d (x, y) dµ (y)

M−B(x,1)

 d2 (x, y) dµ (y) Vol d (x, y) exp − ≤ 2C7 t 6t M−B(x,1)  2  √  ∞ s −n/4 −1 Area ∂B (x, s) ds Vol B x, t s exp − = 2C7 t 6t 1  2  √  ∞ s −n/4 −1 Area ∂B (x, s) ds, Vol B x, t exp − ≤ C8 t 8t 1 −n/4

−1



√ B x, t





where C8 < ∞ depends only on n and K. d Vol B (x, s), so that by integratOn the other hand, Area ∂B (x, s) = ds ing by parts in (26.140) (and using the Bishop–Gromov volume comparison theorem to control the growth of Area ∂B (x, s) and to see that the boundary term at s = ∞ vanishes), we obtain for all x ∈ M and t ∈ (0, 1],  H (x, y, t) d (x, y) dµ (y) 2 M−B(x,1)

≤ C8 t−n/4



∞ 1

 2 s Vol B (x, s) s √ ds exp − 4t 8t Vol B x, t

since the boundary term at s = 1 is equal to   Vol B (x, 1) 1 √ ≤ 0. −C8 t−n/4 exp − 8t Vol B x, t √ Since t ≤ 1, we have VolK B( t) ≥ c1 tn/2 for some c1 > 0 depending only on n and K. Note also that for any r ∈ (0, ∞) we have 11

VolK B(r) ≤ C5 eC5 r , where C5 < ∞ depends only on n and K.

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4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS

381

Applying the relative volume comparison theorem, which says   s VolK B(s) Vol B (x, s) √ ≤ √ ≤ exp C9 √ Vol B x, t VolK B( t) t for some C9 < ∞ depending only on n and K (this holds since s ≥ 1 ≥ t), we have  H (x, y, t) d (x, y) dµ (y) 2 M−B(x,1)

≤ C8 t

−n/4



∞ 1

  2 s s s √ ds exp − + C9 4t 8t t

≤ C10 ,

(26.141)

where C10 < ∞ depends only on n and K. Remark 26.52. If we assume that t ∈ (0, ε], then the corresponding bound C10 tends to zero as ε → 0. Hence, by applying this and (26.135) to (26.134), we obtain |f (x, t) − u (x)| ≤ C11 , where C11 < ∞ depends only on n and K. This proves (26.133) and hence completes the proof of Step 1. Step 2. There exists Cn,K,1 ∈ [1, ∞) depending only on n and K such that |∇f | (x, t) ≤ Cn,K,1

(26.142)

for all x ∈ M and t ∈ (0, 1]. First observe that  

∂ − ∆ f 2 = −2 |∇f |2 . (26.143) ∂t We have (see Exercise 2.20 in [45]) ∂ |∇f |2 = ∆ |∇f |2 − 2 |∇∇f |2 − 2 Rc (∇f, ∇f ) ∂t ≤ ∆ |∇f |2 + 2 (n − 1) K |∇f |2 . Hence



(26.144)

  ∂ − ∆ e−2(n−1)Kt |∇f |2 ≤ 0. ∂t

We would like to apply the maximum principle to the above equation; the issue is the noncompactness of M. Note that |∇f | (x, 0) = |∇u| (x) ≤ 2. We claim that there exists a constant a > 0 such that  1 2 e−ar (x) |∇f |2 (x, t) dµ (x) dt < ∞. (26.145) 0

M

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382

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Once we have (26.145), we may apply the maximum principle of Karp and Li (see the proof of Theorem 1 in [106] or Theorem 1.2 in [144])12 to obtain (26.142). Now we turn to the proof of (26.145). Given R ∈ [1, ∞), let η : M → [0, 1] be a C ∞ cutoff function with13 η (x) = 1 if d (x, O) ≤ R, η (x) = 0 if d (x, O) ≥ 2R, |∇η|2 /η (x) ≤ C/R2

if d (x, O) < 2R,

where C < ∞ is a universal constant. Using (26.143) and integrating by parts, we have 1



e−ar (x) η (x) |∇f |2 (x, t) dµ (x) dt M    

∂ 1 1 −ar 2 (x) − ∆ f 2 (x, t) dµ (x) dt e η (x) =− 2 0 M ∂t  1 0   1 2 f (x, t) ∇ e−ar (x) η (x) , ∇f (x, t) dµ (x) dt, ≤C−

0

2

0

B(O,2R)

where we used 1 2





η (x) f 2 (x, 0) dµ (x) M 

1 2 e−ar (x) η (x) f 2 (x, 1) dµ (x) − 2  M 1 2 e−ar (x) η (x) (r (x) + 1)2 dµ (x) ≤ 2 M ≤ C. e−ar

2 (x)

Since 1

 − 0

≤

0   1 2 f (x, t) ∇ e−ar (x) η (x) , ∇f (x, t) dµ (x) dt B(O,2R)  1

1 2

0

1 + 2

12 13

e−ar

B(O,2R)



1

0

B(O,2R)

2 (x)

η (x) |∇f |2 (x, t) dµ (x) dt

2 2 ear (x)   −ar2 (x)  η (x)  f 2 (x, t) dµ (x) dt, ∇ e η (x)

See also Theorem 12.22 on pp. 153–154 in Part II. We only need that η is Lipschitz.

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4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS

we obtain  1 0

e−ar

2 (x)

η (x) |∇f |2 (x, t) dµ (x) dt

B(O,2R)

1



≤ 2C + 0

383

B(O,2R)

2 2 ear (x)   −ar2 (x)  η (x)  f 2 (x, t) dµ (x) dt. ∇ e η (x)

Now since f (x, t) ≤ r (x) + C and   2   2 2   2 ∇ e−ar (x) η (x)  ≤ 2e−2ar (x) 4a2 r2 (x) η 2 (x) + |∇η| (x) (note that |∇r| ≤ 1 a.e.), we have  1 2 2 ear (x)   −ar2 (x)  η (x)  f 2 (x, t) dµ (x) ∇ e 0 B(O,2R) η (x)    1 2 |∇η| (x) 2 −ar 2 (x) 2 dµ (x) , 2 (r (x) + C) e 4a2 r (x) η (x) + ≤ η (x) 0 B(O,2R) which, for a sufficiently large depending only on n and K, is bounded independent of the R ∈ [1, ∞) used in the definition of η (where we used the volume comparison theorem). Step 3. There exist constants Cn,K,2 < ∞ depending only on n and K such that |∇∇f | (x, 1) ≤ Cn,K,2 for all x ∈ M. Given any point x ∈ M, we have that   √  √  ˆ 0, ρ/ K → B x, ρ/ K (26.146) expx : B is a local diffeomorphism, where ρ > 0 is a universal constant and where √ √  ˆ B 0, ρ/ K ⊂ Tx M is the ball of radius ρ/ K centered at 0 with respect to the inner product g (x). We consider the pulled-back metric (26.147)  √  ˆ 0, ρ/ K . Define on B

gˆ  (expx )∗ g  √  ˆ 0, ρ/ K → R fˆ : B

by fˆ (V, t)  f (expx (V ) , t) − u (x) . Note that from Step 1,   ˆ  f (0, t)   = |f (x, t) − u (x)| ≤ C1

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384

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

 √  ˆ 0, ρ/ K × (0, 1]. By the gradient estimate (26.142), we have in B   ρ ˆ  f (y, t) ≤ C1 + √ Cn,K,1 K √ for all t ∈ (0, 1] and x, y ∈ M with d (x, y) < ρ/ K. By the diffeomorphism invariance of the norm of the Hessian, we have     |Hess g f |g (x, t) = Hess gˆfˆ (0, t) . gˆ

Now we have



and (26.148)  √  ˆ 0, ρ/ K × (0, 1]. in B

 ∂ − ∆gˆ fˆ = 0 ∂t

  ρ  ˆ f  ≤ C1 + √ Cn,K,1 K

n Recall that a local coordinate system xi i=1 is called harmonic if i

∆ x = 0 for all i, i.e., g jk Γijk = 0. √ Since |sect (ˆ g )| ≤ K and injgˆ (0) ≥ ρ/ K, by Jost and Karcher [102] √ ∈ (0, ρ/ K], depending only (cf. DeTurck and Kazdan [53]), there exist ρ 0 i n ˆ on n and K, and harmonic coordinates yˆ i=1 on B (0, ρ0 ) such that

ˆ gij C 1,1/2 ≤ C12

(26.149) and

−1 δij ≤ gˆij ≤ C12 δij C12

(26.150)

for C12 < ∞ depending only on n and K, where gˆij    some constant ∂ ∂ gˆ ∂ yˆi , ∂ yˆj . With respect to these coordinates, the heat equation is   n ˆ  1  ∂ ∂ f ∂ fˆ = |ˆ g |ˆ g ij j ∂t ∂ yˆ |ˆ g | i,j=1 ∂ yˆi   n   ∂ fˆ 2 fˆ  ∂ 1 ∂ |ˆ g |ˆ g ij (26.151) gˆij i j +  , = ∂ yˆ ∂ yˆ ∂ yˆj |ˆ g | ∂ yˆi i,j=1

where |ˆ g |  det (ˆ gij ). Hence, by (26.149), (26.150), (26.148), and the interior Schauder estimates (see Chapter 3 of Friedman [61] or §3 of Chapter IV on pp. 51–61 of Lieberman [122]), we have ? ? ? ˆ? ?f ? 2,α ≤ C13 C

ˆ (0, ρ0 /2) × [1/2, 1] for some constant C13 < ∞ depending only on n and in B K.

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4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS

385

ˆ (0, ρ0 /2) with respect to In particular, we have f (·, 1) C 2,α ≤ C13 in B harmonic coordinates. Since   ∂2 1 k
∂ ∂ ∂ k ∂ k gˆj
+ j gˆi
−
gˆij , ∇i ∇j = i j − Γij k and Γij = gˆ ∂ yˆ ∂ yˆ ∂ yˆ 2 ∂ yˆi ∂ yˆ ∂ yˆ by (26.149) and (26.150) we have (corresponding to yˆ = 0) |∇∇f | (x, 1) ≤ C14 for all x ∈ M, where C14 < ∞ depends only on n and K. This completes the proof of the proposition.
n Now we give a proof of Remark 26.50. Suppose that (M , g) satisfies     k ∇ Rm ≤ K for 0 ≤ k ≤ m, where m ∈ N. Let ρ be  as in (26.146). By Theorem 6 in Hebey and Herzlich 14 k  [97], since ∇ Rm ≤ K for 0 ≤ k ≤ m, there exists a constant C < ∞ √ depending only on n, K, m, and ρ0 ∈ (0, ρ/ K] such that for any x ∈ M (with respect to harmonic coordinates yˆi centered at x)     β m+1     ∂ ∂     (26.152)  ∂ yˆi1 · · · ∂ yˆiβ gˆpq  ,  ∂ yˆi1 · · · ∂ yˆim+1 gˆpq  ≤ C 0,α ˆ (0, ρ0 /2) for all 0 ≤ β ≤ m + 1. in B Using this estimate, we can apply the high-order interior estimate of parabolic Schauder theory to (26.151) and we have ? ? ? ˆ? (26.153) ?f ? m+2,α ≤ Cn,K,
C

ˆ (0, ρ0 /4) × in B However, this estimate is only a local estimate with respect to partial derivatives in harmonic  β  coordinates. To get an estimate ∇ f  of f , we use induction. By the for all higher covariant derivatives   [ 34 , 1].

formula Γkij = 12 gˆk
∇ i1 · · · ∇ iβ f − we obtain

∂ gˆ ∂ yˆi j


+

∂ gˆ ∂ yˆj i


−

∂ gˆ ∂ yˆ ij

and by

  ∂β f k−1 k−2 k−2 = Γ∗∇ f +(∂Γ)∗∇ f +· · ·+ ∂ Γ ∗∇f, ∂ yˆi1 · · · ∂ yˆiβ    β  ∇ f  (x, 1) ≤ C

for 1 ≤ β ≤ m + 2

using estimates (26.152) and (26.153). Since x is arbitrary and C is independent of x, estimate (26.130) is proved. Remark 26.53. By Corollary 4.12 of Hamilton [93] (or Proposition 4.32 in Part I), we have bounds on the partial derivatives, with respect to geodesic coordinates, of gˆpq up to m-th order. On the other hand, by DeTurck and Kazdan [53], harmonic coordinates have maximal regularity. 14 In [97] only bounds on the Ricci tensor and its covariant derivatives are assumed. See also [95].

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386

26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

5. Notes and commentary §1. The existence and positivity of the heat kernel for a time-dependent metric was considered by one of the authors [85] (see also [61] and [26]). For the convenience of the reader we give a proof of this existence in §1 of Chapter 23. For Lemma 26.4 see Theorem 15 on pp. 28–29 of Friedman [61]. For Lemma 26.1 see (8.4) and (8.5) on p. 27 of [61]. For Corollary 26.15 see for example Lemma 5.1 of [26]. §2. For Lemma 26.17 see Lemma 5.2 in [26]. For Definition 26.20 see §2 of Grigor yan [76]. For Lemma 26.21 see Theorem 2.1 in Grigor yan [76] and Lemma 2.1 in Chau, Tam, and Yu [26]. Compare also with Lemma 5.1 of Li [117] and Proposition 10.68 in Part II. For Lemma 26.23 see Lemma 5.3 of [26]. Our exposition of the proof of Theorem 26.32 follows Lemma 5.3 of Li [117]; note that if λ1 (M, g) > 0, then one can obtain a better estimate. §3. For the mean value property for harmonic functions, see pp. 25–26 of Evans [58]. The space-time mean value property for solutions to the heat equation on Euclidean space is due to Pini [158], Fulks [64], and Watson [185]. A nice exposition of this property is given in Theorem 3 of §2.3.2 in Evans [58]. For the mean curvature flow, see Ecker [55]; for the Ricci flow, see Ecker, Topping, and two of the authors [57]. By the heat kernel upper estimates of Carlen, Kusuoka, and Stroock [24], Carron [25], Cheng, Li, and Yau [38], Davies [51], Li and Yau [121], Grigor yan [76], Nash [137], Varopoulos [183], and others, the class of Riemannian manifolds with backward heat kernel satisfying condition (26.104) includes the following classes: (1) manifolds whose Ricci curvatures are bounded from below, (2) manifolds with an appropriate Sobolev, Nash, logarithmic Sobolev, or Faber–Krahn-type inequality, (3) manifolds with an isoperimetric inequality, (4) manifolds with bounded geometry. For (26.107), due to Hamilton [91], see also Malliavin and Stroock [126], Theorem 10 in §3 of Ecker, Topping, and two of the authors [57], and §2 of Chapter 16 and §4 of Appendix E both in Part II. The local monotonicity formula in Theorem 26.44 has additional consequences, including an improved Harnack estimate. The interested reader may consult Garofalo and Lanconelli [70] for results in this direction. §4. The work of Tam [177] is related to the earlier work of Shi [173] and Lin and Wang [123] and uses some techniques of Greene and Wu [75], Cheng and Yau [39], Jost and Karcher [102], Schoen and Yau [168], Karp and Li [106], Li and Yau [121], and Tam and one of the authors [143] and [144].

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http://dx.doi.org/10.1090/surv/163/11

APPENDIX G

Elementary Aspects of Metric Geometry I’m still at it, After-mathematics. – From “Still D.R.E.” by Dr. Dre featuring Snoop Dogg

When studying a class of objects, even if one is only interested in the properties of objects in this class, it is often useful to enlarge the class. In metric geometry, enlarging the class of Riemannian manifolds, we shall consider the notions of metric spaces, length spaces, and Aleksandrov spaces. One property which we usually wish the enlarged class of geometric objects to have is that of (pre)compactness. In addition, we wish to study the geometric properties of objects in the enlarged class. In this appendix, we discuss enlargements of the class of smooth manifolds and their properties. In contrast, most of the other topics in this book series may be classified as parts of geometric analysis (i.e., analysis on smooth manifolds). With this in mind, we remark that many of Perelman’s (ingenious) arguments bring the synthesis of traditional geometric analysis and metric/comparison geometry to a higher level. As in other areas of mathematics, one may anticipate a further broadening and deepening of this synthesis. We provide two quotations which reflect some motivations for the inclusion of the metric geometry in this appendix. On p. xiii of [18] it is written: ‘... it was a common belief that “geometry of manifolds” basically boiled down to “analysis on manifolds”. ... It is now understood that a tremendous part of geometry essentially belongs to metric geometry, ...’ Cheeger and Grove have written (see p. vi of [34]): ‘Thus, it seems that distinctions such as “metric geometry” versus “geometric analysis” are to some extent artificial and if pressed too far, are genuinely destructive. To reiterate, increasingly, the solution of specific geometric problems requires a mixture of synthetic, analytic and topological arguments ... the work of Perelman (on the program originated by Hamilton) being just one, albeit spectacular, example. This circumstance can only make the subject more interesting.’ Metric geometry is now an integral part of the study of Ricci flow. Here we only provide a cursory introduction to this subject. Excellent reference 387 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

388

G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

books on metric and related Riemannian geometry are Gromov’s [78] and Burago, Burago, and Ivanov’s [18]. We shall refer to these books freely throughout this appendix. In §1 we discuss length spaces, the Gromov–Hausdorff distance and convergence, the Gromov precompactness theorem (I), and the tangent and asymptotic cones. In §2 we recall Aleksandrov spaces and their basic properties and associated notions, the Gromov precompactness theorem (II), the Aleksandrov space splitting theorem, and the existence of the tangent cone. 1. Metric spaces and length spaces In this section we recall some basic facts about metric and length spaces. This serves two purposes: (1) One of Perelman’s many contributions to Ricci flow is the introduction of a space-time length-type geometry (see [152] or the expository Chapter 7 in Part I). We hope that the discussion of background material on metric and length spaces may facilitate the reader’s study of the foundations of Perelman’s theory. (2) Metric and length spaces prepare us for the study of Aleksandrov spaces, which are natural spaces for discussing compactness theorems for Riemannian manifolds in the absence of injectivity radius bounds. 1.1. (Quasi-)metric and (quasi-)length spaces. In this subsection we present a few basic notions related to metric spaces and length spaces. At its end, we also indicate a relevance of metric and length spaces to Ricci flow. 1.1.1. Metric-type spaces. Recall that a metric d on a set X is a nonnegative symmetric function defined on the Cartesian product X ×X which vanishes only on the diagonal and satisfies the triangle inequality, i.e., d (x, z) ≤ d (x, y) + d (y, z) for all x, y, z ∈ X. We also refer to d as the distance function. A metric space is the pair (X, d) of a topological space and a metric, where the topology on X is the coarsest topology for which B (x, r) is an open set for all x ∈ X and r > 0. A metric space is said to be complete if every Cauchy sequence converges. If S is a subset of a metric space (X, dX ), it has a subspace metric dS simply defined by dS (x, y)  dX (x, y) for x, y ∈ S.

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1. METRIC SPACES AND LENGTH SPACES

389

Remark G.1 (Scaling metric spaces). If (X, d) is a metric space, then so is (X, αd), where α > 0 and (αd) (x, y)  α · d (x, y). Two important aspects of metrics are the behaviors of αd as α → 0 (where large scales become ‘visible’) and as α → ∞ (where small scales become visible).   More generally, we say that a pair X, d˜ , where X is a set and where d˜ : X × X → R ∪ {∞} is a map, is a quasi-metric space1 if the triangle inequality holds, i.e., for every x, y, z ∈ X (G.1) d˜(x, z) ≤ d˜(x, y) + d˜(y, z) . In particular, d˜ may be neither nonnegative nor symmetric and d˜ may be infinite.   Remark G.2. Let X, T, d˜ be a triple where X is a set, where T ⊂ X × X is a transitive relation on X,2 and where d˜ : T → R is a (finite) function such that for every (x, y) , (y, z) ∈ T (G.2)

d˜(x, z) ≤ d˜(x, y) + d˜(y, z)

(by transitivity, (x, z) ∈ T ). If we extend d˜ to a map from X ×X to R∪{∞} ˜ ∞ on (X × X) − T , then the triangle inequality still holds. by defining  d=  That is, X, d˜ is a quasi-metric space.   If for a quasi-metric space X, d˜ , we also have (1) (distance is nonnegative and finite) 0 ≤ d˜ < ∞, (2) (distance from a point to itself is zero) d˜(x, x) = 0 for all x ∈ X, and (3) (distance is symmetric) d˜(x, y) = d˜(y, x) for all x, y ∈ X,   then d˜ is called a pseudo-metric3 and X, d˜ is called a pseudo-metric space.   In general, given a pseudo-metric space X, d˜ , we may define the equivalence relation ∼ on X by x ∼ y if and only if d˜(x, y) = 0. Let X   X/ ∼ be the quotient space. Define d : X  × X  → [0, ∞) by (G.3)

d ([x] , [y])  d˜(x, y) .

Then d is a well-defined metric on X  .4 We call (X  , d ) the  (quotient) metric space induced by the pseudo-metric space X, d˜ . 1 A more standard use of the term ‘quasi-metric space’ is for a metric space without the axiom of symmetry. 2 A relation R on X is a subset of X × X. It is transitive if (x, y) , (y, z) ∈ R implies (x, z) ∈ R. 3 We also use the terminology of quasi-distance and pseudo-distance. 4 See Proposition 1.1.5 on p. 2 of [18]; there the proof is left as an exercise.

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390

G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

If (X, dX ) and (Y, dY ) are metric spaces, then the product metric on X × Y is defined by5 " dX×Y ((x1 , y1 ) , (x2 , y2 )) = dX (x1 , x2 )2 + dY (y1 , y2 )2 . We call the metric space (X × Y, dX×Y ) the metric product of (X, dX ) and (Y, dY ). A map f : (X, dX ) → (Y, dY ) between metric spaces is an isometry if it is bijective and distance-preserving, i.e., dY (f (x1 ) , f (x2 )) = dX (x1 , x2 ) for all x1 , x2 ∈ X. In this case we say that (X, dX ) and (Y, dY ) are isometric. An injective distance-preserving map is called an isometric embedding. Given a metric space (Y, dY ) and a map f : X → Y , the pullback of the metric dY by f is defined by (f ∗ dY ) (x1 , x2 )  dY (f (x1 ) , f (x2 )) for all x1 , x2 ∈ X. Exercise G.3 (Pullback metric). Show that (X, f ∗ dY ) is a pseudometric space. Moreover, if f is injective, then (X, f ∗ dY ) is a metric space. 1.1.2. Length-type spaces. There are different ways to define length spaces; here we present one of them. We say that a quadruple (X, I, A, L), where X is a topological space, I ⊂ R is an interval, A is a set of continuous paths γ : [a, b] → X where [a, b] ⊂ I, and L : A → R ∪ {∞} , is a quasi-length space if A is closed under restrictions, concatenations, and linear reparametrizations and L is additive:6 



L (α  β) = L (α) + L (β)

and L γ|[a,τ ] depends continuously on τ . Here, if α : [a, b] → X and β : [b, c] → X satisfy α (b) = β (b), where [a, c] ⊂ I, then  α (u) if u ∈ [a, b] , (α  β) (u)  β (u) if u ∈ [b, c] denotes the concatenation of α and β. The set A is called the class of admissible paths and an element of it is called an admissible path. If γ ∈ A is such that L (γ) < ∞, then we say that γ is rectifiable. Given a Hausdorff quasi-length space (X, I, A, L), we may define the associated quasi-metric (or associated quasi-distance) by dL (x, y)  inf {L (γ) : γ ∈ A with γ (a) = x and γ (b) = y} 5 6

The notion of product clearly extends to pseudo-metric spaces. By convention, a + ∞ = ∞ + a = ∞ for a ∈ R.

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1. METRIC SPACES AND LENGTH SPACES

391

(see Exercise G.6 below). If there are no paths in A between x and y, then the distance is defined to be ∞. Note that if L : A → R is finite-valued, then dL (x, y) < ∞ for any (x, y) ∈ TL , where TL  {(x, y) : ∃γ ∈ A with γ (a) = x and γ (b) = y} . Definition G.4 (Length space). If the function L of a quasi-length space (X, I, A, L) is nonnegative, invariant under reparametrizations, and agrees with the topology of X, i.e., for every x ∈ X and neighborhood U of x     /U inf L γ|[a,b] : γ ∈ A with γ (a) = x and γ (b) ∈

> 0,

then we say that (X, I, A, L) is a length space. We call the function L a length structure; see §2.1 of [18] for more details. When I and A are understood, we denote a length space simply by (X, L). Let (X, I, A, L) be a length space. We say that a path α : I → X contained in A, where I ⊂ R is an interval, is parametrized by arc length if for all [c, d] ⊂ I we have   (G.4) L α|[c,d] = d − c. Lemma G.5. If β : I → X is a rectifiable path in a length space, i.e., if L (β) < ∞, then there exists a continuous nondecreasing function φ : I → [0, L (β)] and a rectifiable path β˜ parametrized by arc length such that β = β˜ ◦ φ (see Proposition 2.5.9 of [18]). Let (X, L) be a length space. A path γ : [a, b] → X is a shortest path if for every path β joining γ (a) and γ (b) we have L (β) ≥ L (γ). Note that if γ : [a, b] → X is a shortest path and if [c, d] ⊂ [a, b], then γ|[c,d] is also a shortest path. A path γ : [a, b] → X is a geodesic if for every τ ∈ [a, b] there exists ε > 0 such that γ|[τ −ε,τ +ε]∩[a,b] is a shortest path. A length space (X, L) is complete if every two points x, y ∈ X can be joined by a shortest path, i.e., there exists γ ∈ A such that L (γ) = dL (x, y) . Exercise G.6 (Metric induced by a length space). Given a quasi-length space (X, I, A, L), show that dL satisfies the triangle inequality so that (X, dL ) is a quasi-metric space. Moreover, if (X, I, A, L) is a length space, then (X, dL ) is a metric space (where dL is allowed to take the value ∞). This is Exercise 2.1.2 of [18]. If a metric d is the associated metric of a length structure, then we say that d is an intrinsic metric. If the length structure is complete, then we say that d is a strictly intrinsic metric.

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1.1.3. Relations between metric spaces and length spaces. Given a (quasi-)metric space (X, d) and an interval I ⊂ R, we let A denote the set of continuous paths γ : [a, b] → X, where [a, b] ⊂ I, and we define Ld : A → R ∪ {∞} by  (G.5) Ld (γ) = sup ki=1 d (γ (τi−1 ) , γ (τi )) , T

where T is a partition a = τ0 ≤ τ1 ≤ · · · ≤ τk = b of [a, b]. We call Ld the (quasi-)length structure induced by d.7 Exercise G.7 (Length induced by metric space). Show that (X, I, A, Ld ) is a quasi-length space. Moreover, show that if (X, d) is a metric space, then (X, I, A, Ld ) is a length space (see pp. 34–35 of [18]). Note that, by definition, for a length structure induced by a metric, all continuous paths are admissible (but certainly not necessarily rectifiable). Moreover, length structures induced by a metric are lower semi-continuous (see Theorem 2.3.4(iv) of [18]). Given a metric space (X, d), we have the induced intrinsic metric defined by (G.6)

d  dLd ;

that is, the intrinsic metric is the metric associated to the length induced by the original metric. We have the following two facts. 1. If d is an intrinsic metric, then dLd = d (see Proposition 2.3.12 on p. 37 of [18]). 2. Given a lower semi-continuous length structure L (with respect to pointwise convergence), we have LdL = L (see Theorem 2.4.3 of [18]). Given two length spaces (X, LX ) and (Y, LY ), we define the product length space as follows. Let dLX and dLY be the associated metrics of LX and LY , respectively. The product length structure LX×Y on X × Y is the length structure induced by the product metric dLX × dLY (see p. 88 of [18]). Recall that a Riemannian metric defines at each point an infinitesimal measure of length and angle. Integrating the infinitesimal measure of length along paths, we obtain a length structure. Example G.8 (Riemannian manifold). Let (Mn , g) be a connected Riemannian manifold. Given a piecewise C 1 path γ : [a, b] → M, its length is  Given a partition T = {τi }ki=0 of [a, b], let Ld (γ; T )
ki=1 d (γ (τi−1 ) , γ (τi )). Note  1  k1  2  k2 that if T1 = τi i=0 and T2 = τi i=0 are partitions of [a, b], then the partition T1 ∪ T2 satisfies Ld (γ; T1 ∪ T2 ) ≥ max {Ld (γ; T1 ) , Ld (γ; T2 )} . 7

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1. METRIC SPACES AND LENGTH SPACES

given by

393

  b   dγ   L (γ)  Lg (γ)   du (u) du. a g

Let A denote the space of piecewise C 1 paths in M; then (M, R, A, L) is a length space. The distance function is defined by d (x, y) = inf L (γ) , γ

where the infimum is taken over all piecewise C 1 paths γ joining x to y. Note that the infimum is realized by at least one geodesic joining x and y provided g is complete. We then have that (M, d) is a metric space. 1.1.4. Relevance to the Ricci flow. A more sophisticated example related to length-type spaces, fundamental to the study of the Ricci flow, is the reduced distance; see §7 of Perelman [152] and Chapter 7 of Part I. Note that the L-length may be negative and is not an example of a length function. Another relevance of length spaces to Ricci flow (and more generally, Riemannian geometry) is via Gromov’s compactness theorem, which we recall in the next subsection. Aleksandrov spaces (see the next section for a definition), as limits of Riemannian manifolds, naturally occur in Perelman’s work on singularity analysis in Ricci flow [152]. 1.2. Gromov–Hausdorff distance and Gromov’s precompactness theorem. In this subsection we review the definition of Gromov–Hausdorff (GH) distance between metric spaces and Gromov’s precompactness theorem for Riemannian manifolds with Ricci curvature bounded from below in the collection of metric spaces. 1.2.1. Gromov–Hausdorff distance between metric spaces. Given a metric space (Z, dZ ), the (closed) ε-neighborhood of a subset S is defined by Nε (S)  {z ∈ Z : dZ (z, S) ≤ ε} . Given two subsets A and B of Z, the Hausdorff distance dZ H (A, B) between them is defined to be the infimum over all ε > 0 such that A is contained in the ε-neighborhood of B and B is contained in the ε-neighborhood of A, that is, dZ H (A, B)  inf {ε > 0 : A ⊂ Nε (B) and B ⊂ Nε (A)} . Z If there is no such ε, then we define dZ H (A, B)  ∞. Note that dH is a metric on the set of compact subsets of Z. (See §3.3 of Gromov [78].) Using the Hausdorff distance, we may define the Gromov–Hausdorff distance dGH ((X, dX ) , (Y, dY )) between two metric spaces (X, dX ) and

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(Y, dY ) as the infimum, over all metric spaces (Z, dZ ) and isometric embeddings f : X → Z and g : Y → Z, of the Hausdorff distance in Z between f (X) and g (Y ), that is, (G.7)

dGH ((X, dX ) , (Y, dY ))  inf dZ H (f (X) , g (Y )) , Z,f,g

where the infimum is taken over all Z, f, g as above. The Gromov–Hausdorff distance is a (finite) metric on the set of all compact metric spaces. In particular, we have that dGH is nonnegative, symmetric, and satisfies the following properties: (1) If (X, dX ) and (Y, dY ) are compact metric spaces such that dGH ((X, dX ) , (Y, dY )) = 0, then (X, dX ) and (Y, dY ) are isometric. (2) (Triangle inequality) dGH ((X, dX ) , (Z, dZ )) ≤ dGH ((X, dX ) , (Y, dY )) + dGH ((Y, dY ) , (Z, dZ )) . Remark G.9. For the proof of (1) see Proposition 3.6 of [78] or Theorem 7.3.30 of [18]. For the proof of (2) see Proposition 7.3.16 of [18] (or Exercise 7.3.26 in [18]). To give a more intrinsic and computable way to determine the Gromov– Hausdorff distance, we need a notion to measure how much a map distorts distances. The distortion of a map f : (X, dX ) → (Y, dY ) is defined by dis f 

sup |dX (x1 , x2 ) − dY (f (x1 ) , f (x2 ))|

x1 ,x2 ∈X

(see also Definition 7.1.4 of [18]). Clearly dis f = 0 if and only if f is distance preserving (i.e., an isometric embedding). Definition G.10 (ε-net). A subset S of a metric space X is a called an ε-net if Nε (S) = X, i.e., every point of X is within distance ε of S. We say that a map f : (X, dX ) → (Y, dY ) is an ε-isometry (or εHausdorff approximation) if dis f ≤ ε and f (X) in an ε-net in Y , i.e., Nε (f (X)) = Y . Remark G.11. The map f in the definition above need not be continuous. A relation between the Gromov–Hausdorff distance and the distortion of maps, exact up to factors of 2, is given by the following (see Corollary 7.3.28 of [18]). Lemma G.12 (Gromov–Hausdorff distance and ε-isometries). Let (X, dX) and (Y, dY ) be metric spaces and let ε > 0. (1) If dGH ((X, dX ) , (Y, dY )) < ε, then there exist 2ε-isometries f : (X, dX ) → (Y, dY ) and g : (Y, dY ) → (X, dX ).

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1. METRIC SPACES AND LENGTH SPACES

395

(2) If f : (X, dX ) → (Y, dY ) is an ε-isometry, then dGH ((X, dX ) , (Y, dY )) < 2ε. Remark G.13. There is an exact relation between the Gromov–Hausdorff distance and the distortion of correspondences (see Theorem 7.3.25 of [18]; the definition of ‘correspondence’ is on p. 256 of [18]). We say that a sequence of metric spaces {(Xi , dXi )}i∈N converges in the Gromov–Hausdorff distance to a metric space (Y, dY ) if lim dGH ((Xi , dXi ) , (Y, dY )) = 0.

i→∞

1.2.2. Gromov–Hausdorff distance between pointed metric spaces. When the metric spaces under discussion may be unbounded, we consider pointed metric spaces obtained by adding a basepoint to the metric space and we modify the definition of distance between them.8 A (pointed) map f : (X, x0 ) → (Y, y0 ) is a map f : X → Y with f (x0 ) = y0 . The following is Definition 1.6 in [63]. Definition G.14 (Approximate pointed isometries). Let (X, dX , x0 ) and (Y, dY , y0 ) be two pointed metric spaces. A map f : (X, x0 ) → (Y, y0 ) is called an ε-approximate pointed isometry (or ε-pointed Hausdorff approximation) if (1)





, (G.8) BY y0 , ε−1 ⊂ Nε f BX x0 , ε−1 (2) (G.9)

|dY (f (x1 ) , f (x2 )) − dX (x1 , x2 )| < ε

for all x1 , x2 ∈ BX x0 , ε−1 .

The pointed Gromov–Hausdorff distance dpt GH ((X, dX , x0 ) , (Y, dY , y0 )) between two pointed metric spaces is defined to be the infimum over all ε > 0 such that there exist ε-approximate pointed isometries from (X, x0 ) ).9

to (Y, y0 ) and from (Y, y0 ) to (X, x0−1 Note that for any x ∈ BX x0 , ε we have |dY (f (x) , y0 ) − dX (x, x0 )| < ε. This implies that



f BX x0 , ε−1 ⊂ BY y0 , ε−1 + ε .

8

We are interested in pointed spaces because solutions to the Ricci flow on noncompact manifolds arise in the singularity theory of solutions of the Ricci flow on closed manifolds. 9 Note that when both (X, dX ) and (Y, dY , y0 ) have bounded diameter, the pointed Gromov–Hausdorff distance does not necessarily agree with the Gromov–Hausdorff distance.

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−1 + ε ⊂ N −1 From (G.8) we have B , ε , ε y f B x ; hence by 0 2ε 0 Y X



−1 −1 → BY y0 , ε + ε is a 2ε-isometry. (G.9), f : BX x0 , ε The above discussion implies (using Lemma G.12) Lemma G.15 (Pointed Gromov–Hausdorff distance and ε-isometries). Let (X, dX , x0 ) and (Y, dY , y0 ) be two pointed metric spaces. If dpt GH ((X, dX , x0 ) , (Y, dY , y0 )) < ε, then there exists a 2ε-isometry







f : BX x0 , ε−1 , dX → BY y0 , ε−1 + ε , dY with f (x0 ) = y0 and there exists an 2ε-isometry







g : BY y0 , ε−1 , dY → BX x0 , ε−1 + ε , dX with g (y0 ) = x0 . A sequence {(Xi , dXi , xi )}i∈N is said to converge in the pointed Gromov–Hausdorff distance to (Y, dY , y0 ) if lim dpt ((Xi , dXi , xi ) , (Y, dY , y0 )) i→∞ GH

= 0.

1.2.3. Gromov’s precompactness theorem. Definition G.16 (Precompact topological space). A subset S in a topological space X is (sequentially) precompact if, for every sequence of points in S, there exists a subsequence which converges to a point in X. The Gromov precompactness theorem says the following (see Theorem 5.3 of [78]). Given K ∈ R and n ≥ 2, let Mpt n,K be the collection of pointed complete Riemannian manifolds (Mn , g, O) with Rc ≥ (n − 1) K. Theorem G.17 (Gromov precompactness theorem, I). The subcollection in the collection of pointed metric spaces, is precompact with respect to the pointed Gromov–Hausdorff distance. Mpt n,K ,

One of the ideas of the proof of the Gromov precompactness theorem is to use the Bishop–Gromov volume comparison theorem. Another is the following general sufficient condition for compactness (see Theorem 7.4.15 of [18]). Theorem G.18 (General criterion for precompactness). Suppose that a collection M of compact metric spaces is uniformly totally bounded, that is, (1) there exists D < ∞ such that for every X ∈ M we have the diameter bound diam X ≤ D, (2) for every ε > 0 there exists N (ε) ∈ N such that for every X ∈ M there exists an ε-net S ⊂ X consisting of at most N (ε) points.

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1. METRIC SPACES AND LENGTH SPACES

397

Then M is precompact in the collection of metric spaces with respect to the Gromov–Hausdorff distance. The pointed version of this is the following (see Theorem 8.1.10 of [18]). Theorem G.19 (Criterion for precompactness — pointed version). Suppose that a collection Mpt of pointed metric spaces has the property that for every ε > 0 and ρ > 0 there exists N (ε, ρ) ∈ N such that for every (X, x0 ) ∈ Mpt , there exists an ε-net S ⊂ B (x0 , ρ) of B (x0 , ρ) ⊂ X consisting of at most N (ε, ρ) points. Then Mpt is precompact in the collection of pointed metric spaces with respect to the pointed Gromov–Hausdorff distance. As an application of Gromov–Hausdorff distance, in the next subsection we discuss another important notion associated with metric spaces. 1.3. Tangent cone and asymptotic cone of a metric space. We shall discuss the tangent cones and asymptotic cones of a special class of metric spaces. Definition G.20. We say that a metric space is boundedly compact if every closed and bounded subset is compact. 1.3.1. Tangent cone of a boundedly compact metric space. The tangent cone of a boundedly compact metric space (X, d) at a point p ∈ X is defined as the pointed Gromov–Hausdorff limit (G.10)

(Tp X, dp , 0p )  lim (X, αk d, p) , αk →∞

provided this limit exists for any sequence {αk } → ∞ and is independent of {αk } (up to isometry).10 The tangent cone, which is a metric space, reflects the infinitesimal geometry at a point. The tangent cone of an n-dimensional Riemannian manifold at any point p∈M is isometric Euclidean space En .11 Given

and a local coordinate toi

containing p, we have U , αk2 g, p ⊂ M, αk2 g, p is isometsystem U , x ric to



i j

˜ (U ) , gij αk−1 x ˜ d˜ x d˜ x , x (p) , x (U ) , αk2 gij (x) dxi dxj , x (p) = x where x ˜i  αk xi and we abused notation. The rhs limits to (Tp M, g (p) , 0p ), which is isometric to En . The reader may wish to consider the following. Question G.21 (Tangent cone of a subset). Suppose that (X, d) is a boundedly compact metric space and suppose that S ⊂ X is a boundedly

compact subset. Let p ∈ S be such that both tangent cones Tp S, dTp S and 10 Later we abuse notation and call any sequential limit ‘a tangent cone’, where the limits may be different for different sequences;   see Exercise G.24.   11 Note that in normal coordinates xi we have gij (x) = δij + O |x|2 , that is, |gij (x) − δij | ≤ C |x|2 for some constant C < ∞.

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Tp X, dTp X exist. Does there exist a (natural ) embedding ι : Tp S → Tp X such that ι∗ dTp X = dTp S ? Question G.22 (Existence of tangent cone of locally convex subset). Let (Mn , g) be a Riemannian manifold. If S is a locally convex subset of M (see subsection 2.1.2 in Appendix H below for the definition) and if p ∈ S, then does the tangent cone Tp S exist? Exercise G.23 (Invariant of the tangent cone under rescalings). Show that for any c > 0, (Tp X, cdp , 0p ) is isometric to (Tp X, dp , 0p ). Exercise G.24 (Nonuniqueness of tangent-type cone limits). Give an example of a connected compact metric space (X, d) and a point p ∈ X such that there exist sequences αi → ∞ and βi → ∞ such that the Gromov– Hausdorff limits lim (X, αi d, p)

αi →∞

and

lim (X, βi d, p)

βi →∞

both exist but are not isometric. 1.3.2. Asymptotic cone of a boundedly compact metric space. The Gromov–Hausdorff asymptotic cone (or tangent cone at ∞) of a boundedly compact metric space (X, d) is the pointed Gromov– Hausdorff limit (G.11)

(AX, dAX , 0)  lim (X, ωi d, p) , ωi →0

provided this limit exists and is independent (up to isometry) of the sequence {ωi }. Justifying the notation, the Gromov–Hausdorff asymptotic cone is independent of the choice of p (see Proposition 8.2.8 of [18]).12 Therefore, by its definition, the asymptotic cone is unique when it exists. Note that the asymptotic cone reflects the geometry at infinity of X. Exercise G.25 (Invariance of the asymptotic cone under rescalings). Show that for any c > 0, (AX, cdAX , 0) is isometric to (AX, dAX , 0). Example G.26 (Examples of asymptotic cones which are half-lines). (1) The asymptotic cone of a curvature pinching set in a vector space of algebraic curvature operators, as in §7 of Chapter 11 in Part II, is a ray. (2) The asymptotic cone of a complete noncompact Riemannian manifold with aperture equal to zero is a half-line.13 Example G.27. Note that, for hyperbolic space Hn , n ≥ 2, the Gromov– Hausdorff asymptotic cone does not exist; intuitively, the reason for this is that the hyperbolic metric at a point ‘grows too fast’ in terms of the distance of the point to the origin (see Exercise 8.2.13 and the paragraph after That is, if the limit in (G.11) exists for some p ∈ X, then the limit exists for all p ∈ X and it is independent of the choice of p. 13 See §18 of [92] for the definition of aperture. 12

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399

it in [18]). However, in general, for simply-connected complete noncompact manifolds with nonpositive curvature, there is the notion of the ideal boundary or sphere at infinity (see Ballmann, Gromov, and Schroeder [9]). If (Mn , g) is a complete noncompact Riemannian manifold with nonnegative sectional curvature, then the asymptotic cone exists (see §6 of Appendix I). On the other hand, there exist complete noncompact Riemannian manifolds with positive Ricci curvature which do not have asymptotic cones (see Perelman’s [150]). More elementarily, we have the following analogue of Exercise G.24. Exercise G.28. Give an example of a connected noncompact metric subspace (X, d) of E2 and a point p ∈ X such that there exist sequences αi → 0 and βi → 0 such that the Gromov–Hausdorff limits lim (X, αi d, p)

i→∞

and

lim (X, βi d, p)

i→∞

both exist but are not isometric. 1.3.3. Euclidean metric cone. Let X be a topological space. The topological cone Cone (X) over X is the quotient of X × [0, ∞) where the subset X × {0} is identified to a point, n−1 ∼ called the vertex. For example, Cone S = Rn (homeomorphic) for n−1 denotes the unit (n − 1)-sphere. Note that Cone (X) is n ∈ N, where S homeomorphic to a topological n-manifold if and only if X is homeomorphic to S n−1 (in which case Cone (X) ∼ = Rn ). Given x ∈ X and r ∈ [0, ∞), let [(x, r)] denote the equivalence class in Cone (X) of (x, r). Definition G.29 (Euclidean metric cone). The Euclidean metric cone of a metric space (X, d) with diam (X, d) ≤ π is Cone (X) with the metric " (G.12) dCone(X) ([(x1 , r1 )] , [(x2 , r2 )]) = r12 + r22 − 2r1 r2 cos (d (x1 , x2 )) defined for all x1 , x2 ∈ X and r1 , r2 ∈ [0, ∞). We call dCone(X) the cone metric. For the definition of the Euclidean metric cone and its properties when we have diam (X, d) > π, see p. 93 ff. in [18]. Observe that for any c ∈ (0, ∞) dCone(X) ([(x1 , cr1 )] , [(x2 , cr2 )]) = cdCone(X) ([(x1 , r1 )] , [(x2 , r2 )]) . Hence the map c : Cone (X) → Cone (X) defined by c ([(x, r)]) = [(x, cr)] is a homothety, i.e., for any y, z ∈ Cone (X) we have dCone(X) (c (y) , c (z)) = cdCone(X) (y, z).

Exercise G.30. Show that indeed Cone (X) , dCone(X) is a metric space. See the proof of Proposition 3.6.13 in [18]. The following remark gives one motivation for the definition in (G.12).

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Remark G.31. (1) Note that the distance between two points x and y in the sphere S n−1 of radius 1 is the same as the angle x0y, where 0 is the origin of Rn . (2) When X = S n−1 , the metric (G.12) on

Cone S n−1 ∼ = Rn , where the isomorphism is given by polar coordinates, is the same as the Euclidean metric (by the law of cosines). In the remainder of this subsection we give another motivation for the definition of dCone(X) from the Riemannian metric tensor viewpoint. Let x1 , x2 ∈ S n−1 and let α  d (x1 , x2 ). Consider the Euclidean planar triangle with side-angle-side equal to r1 -α-r2 and vertices with polar coordinates (r1 , 0), (0, 0), and (r2 , α). Let β be the angle of the triangle at the vertex (r1 , 0) and let A be the straight line segment in the plane from (r1 , 0) to (r2 , α). For u ∈ [0, α], consider the line segment B emanating from (0, 0) with  angle u and endpoint on the line A. By the law of cosines, L (A) = r12 + r22 − 2r1 r2 cos α. By the law of sines, we have (G.13)

sin α sin β , = 2 r2 r1 + r22 − 2r1 r2 cos α sin (π − β − u) sin (β + u) sin β = = . L (B) r1 r1

Hence (G.14)

L (B) =

r1 sin β , sin β cos u + cos β sin u

where sin β and cos β can be determined from (G.13). Now let (Mn , g) be a Riemannian manifold and define the metric gCone  r2 g + dr2 on M × (0, ∞). Given two points x1 , x2 ∈ M, let γ : [0, d (x1 , x2 )] → M be a unit speed minimal geodesic joining x1 to x2 . Given r1 , r2 ∈ (0, ∞), join the two points (x1 , r1 ) , (x2 , r2 ) ∈ M × (0, ∞) by paths γ¯ : [0, d (x, y)] → M × (0, ∞) of the form γ¯ (u) = (γ (u) , r (u)) , where r : [0, d (x1 , x2 )] → (0, ∞) satisfies r (0) = r1 and r (d (x1 , x2 )) = r2 . The length of γ¯ with respect to gCone is given by '  2  d(x1 ,x2 ) dr r2 + du. L (¯ γ) = du 0

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If we have a variation δr = s, then (G.15)   2 −1/2   d(x1 ,x2 )  dr dr ds 2 du rs + r + δ L (¯ γ) = du du du 0   2 −3/2   2  d(x1 ,x2 )  2r dr dr d = + r2 rsdu, r2 + −r 2 + 2 du du du 0 where we integrated by parts and used the boundary conditions s (0) = s (d (x1 , x2 )) = 0. For any a, b ∈ R, the function 1 (G.16) r (u) = a cos u + b sin u is a solution to the Euler–Lagrange equation of (G.15):  2 dr d2 r + r2 = 0. −r 2 + 2 du du Note that the form of (G.16) matches with (G.14). Exercise G.32 (Riemannian cone is the metric cone). Show that if (Mn , g) is a Riemannian manifold, then the distance function dgCone of the Riemannian metric gCone = r2 g + dr2 on M × (0, ∞) ⊂ Cone (M) is the same as the cone metric dCone(M) on Cone (M) restricted to M × (0, ∞). It is a straightforward calculation that, for n ≥ 2, (Mn , g) has sectional curvature bounded from below by 1 if and only if (M × (0, ∞) , gCone ) has sectional curvature bounded from below by 0. This result has been generalized to complete length spaces (see Theorem 4.2.3 on p. 13 of [19]). 2. Aleksandrov spaces with curvature bounded from below To define the curvature in the traditional way (as the Riemann curvature tensor), a Riemannian metric must be C 2 . However, the subcollection of complete C k Riemannian manifolds, for any k ≥ 2, even with both curvature and diameter bounds, is not precompact in the collection of complete C 2 Riemannian manifolds. This lack of precompactness is a motivation to enlarge the collection of complete Riemannian manifolds to geometric spaces with ‘less regularity’, in particular, including at least all the limiting metric spaces. Note that one of the main issues in the failure of compactness is that the injectivity radii of a sequence of such manifolds may not be bounded from below by a positive constant. In understanding the limiting behavior of sequences and families of Riemannian metrics, important notions are that of Gromov–Hausdorff convergence and compactness. As we have seen, these notions are defined for spaces with less regularity and provide a framework to discuss Gromov’s precompactness theorem. Moreover, one of the main techniques to prove precompactness (Theorem G.17) is the volume comparison theorem. In

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Ricci flow, spaces of less regularity arise when we take limits of smooth solutions to the Ricci flow without a uniform lower bound on the injectivity radius, i.e., when collapse occurs. A natural collection of spaces with less regularity, well-suited for the study of Ricci flow and more generally Riemannian geometry, are Aleksandrov spaces with curvature bounded from below. In this section, only for the sake of the convenience of the reader and borrowing especially from [18], we recall some basic definitions and results on Aleksandrov spaces. 2.1. Motivation. Returning to the (pre)compactness of a class of Riemannian manifolds, it turns out that for a complete length space which is a Gromov–Hausdorff limit of a sequence of Riemannian manifolds with a uniform lower or upper sectional curvature bound, although its curvature may not be well defined, the notion of a lower or upper bound of curvature may still make sense.14 A reason for this is the Toponogov comparison theorem for complete Riemannian manifolds with a lower sectional curvature bound, which we now recall. Theorem G.33 (Toponogov comparison theorem). Let (Mn , g) be a complete Riemannian manifold with sectional curvatures bounded below by k ∈ R. (1) Triangle version. Let ∆ be a triangle in M with vertices (p, q, r) and sides qr, rp, pq which are geodesics15 and whose lengths satisfy the triangle inequality: L (qr) ≤ L (rp)+L (pq) ,

L (rp) ≤ L (qr)+L (pq) ,

L (pq) ≤ L (qr)+L (rp) .

Let rpq, pqr, √ qrp ∈ [0, π] denote the interior angles and assume that L (pq) ≤ π/ k if k > 0. If the geodesics qr and rp are minimal, then there exists a geodesic trian˜ = (˜ gle ∆ p, q˜, r˜) in the complete simply-connected surface of constant Gauss curvature k with the same side lengths





L q˜r˜ = L (qr) , L r˜p˜ = L (rp) , L p˜q˜ = L (pq) such that rpq ≥ ˜ r p˜q˜, pqr ≥ ˜ pq˜r˜. ˜ the k-comparison triangle of ∆. We call ∆ 14

We shall primarily be interested in spaces with a lower bound for the curvature. Such a triangle is not necessarily a ‘geodesic triangle’, which is defined for the more general setting of Aleksandrov spaces in subsection 2.2 below. 15

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(2) Hinge version. Let ∠ be a hinge16 in M with vertices (p, q, r), sides qr and rp which are geodesics, and interior angle qrp ∈ [0, π]. Suppose that √ qr is minimal and that L (rp) ≤ π/ k if k > 0. If ∠ is a geodesic hinge with vertices (p , q  , r ) in the complete simplyconnected surface of constant Gauss curvature k with the same side lengths and same angle, then

d (p, q) ≤ d p , q  . (3) Opposite side version. For any geodesic triangle pqr in M and point s ∈ rp, we have d (q, s) ≥ d (˜ q , s˜) , where ˜ pq˜r˜ is the k-comparison triangle of pqr and s˜ ∈ r˜p˜ is the point with d (˜ p, s˜) = d (p, s). Remark G.34. There are other versions of the Toponogov comparison theorem. For example, there is the Toponogov monotonicity principle; see Lemma G.38 below. 2.2. Aleksandrov spaces with curvature bounded from below. With the above motivations, we make the following definitions. A (geodesic) triangle in a complete length space (X, L) consists of three points p, q, r ∈ X and three shortest paths qr, rp, pq joining these points. The points p, q, r are called the vertices of the triangle and the paths qr, rp, pq are called the sides of the triangle. We denote this triangle by pqr. Since (X, L) is complete, given any three points in X, there exists a triangle with these three points as its vertices. A pair of geodesic segments with a common vertex is called a hinge. Given a triangle pqr in a complete length space (X, L), with √ L (qr) + L (rp) + L (pq) < 2π/ k if k > 0, there exists a triangle ˜ pq˜r˜ in the complete simply-connected surface M2k of constant Gauss curvature k with the same side lengths. This triangle is unique up to isometry. As in the Riemannian case, we call the triangle ˜ pq˜r˜ the k-comparison triangle of pqr. There are several equivalent definitions of Aleksandrov space with curvature bounded from below; we give the following. The advantage of this definition is that it only uses length. Definition G.35 (Aleksandrov space—local version). Given k ∈ R, a complete locally compact17 length space (X, L) (with strictly intrinsic metric dL ) is an Aleksandrov space of curvature ≥ k if for every x ∈ X 16

See subsection 2.2 below for the definition of hinge in an Aleksandrov space. Local compactness is not assumed in [18] and [159], whereas it is assumed in [19], [175], and [156]. 17

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there exists an open neighborhood U of x such that for any triangle pqr contained in U and point s ∈ rp we have q , s˜) , dL (q, s) ≥ d (˜ where ˜ pq˜r˜ is the k-comparison triangle of pqr and s˜ ∈ r˜p˜ is the point with d (˜ p, s˜) = dL (p, s). In this definition, if we drop the local compactness assumption, we then call X a complete length space with curvature bounded from below. If k = 0, then we say that (X, L) is an Aleksandrov space of nonnegative curvature. By Theorem G.33(3), if (Mn , g) is a complete Riemannian manifold with sectional curvatures bounded below by k ∈ R, then (Mn , Lg ) is an Aleksandrov space of curvature ≥ k. The global version of the definition of Aleksandrov space is as follows. Definition G.36 (Aleksandrov space—global version). Given k ∈ R, we say that a complete locally compact length space (X, L) has curvature ≥ k in the large if for any triangle pqr contained in X and point s ∈ rp we have q , s˜) dL (q, s) ≥ d (˜ whenever the k-comparison triangle ˜ pq˜r˜ of pqr exists and s˜ ∈ r˜p˜ satisfies d (˜ p, s˜) = dL (p, s). The local and global versions of the definition of Aleksandrov space are the same (see Theorem 10.3.1 of [18]). Theorem G.37 (Toponogov’s globalization theorem). If (X, L) is an Aleksandrov space of curvature ≥ k, then (X, L) has curvature ≥ k in the large. The following notions are used in Aleksandrov space theory. Let (X, d) be a metric space. Given x, y, z ∈ X distinct, the Euclidean comparison angle of (x, y, z) at y is (G.17)

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

There is an equivalent definition of Aleksandrov spaces with curvature bounded from below given by using comparison angles (see Definition 2.3 of [19]). If (X, d) is a length space and if α and β are paths emanating from p ∈ X, i.e., α, β : [0, ε] → X for some ε > 0 and α (0) = β (0) = p, then the angle between α and β is defined by (G.18)

˜ (u) p β (v) , p (α, β)  lim α u,v→0+

provided the limit exists.

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We say that a path γ emanating from p has a direction at p if its angle with itself exists and is zero, i.e., p (γ, γ) = 0.18 Lemma G.38 (Comparison angle monotonicity). Let (X, d) be an Aleksandrov space with curvature bounded from below. Suppose α is a shortest path between α (0) and α (ε) and suppose β is a shortest path between β (0) and β (ε). Then ˜ (u) p β (v) α is a nonincreasing function of u and v. For a proof of the nonnegative curvature case, see Proposition 4.3.2 of [18] or, for a proof of the general case, see §2.6(B) of [19]. In particular, the angle p (α, β) exists. Proposition G.39 (Existence of angle between shortest paths). If X is an Aleksandrov space of curvature ≥ k and if α and β are shortest paths emanating from a common point p, then p (α, β) is well defined. In particular, any shortest path emanating from a point has a direction. ˜ (u) p β (v) is that a Another consequence of the monotonicity of α shortest path does not bifurcate in an Aleksandrov space with curvature bounded from below. That is, let α (t) and β (t), t ∈ [0, b], be two (unit speed) minimal paths; if there is a δ > 0 such that α (t) = β (t) for t ∈ [0, δ], then α (t) = β (t) for all t ∈ [0, b]. 2.3. Some basic notions and properties of Aleksandrov spaces with curvature bounded from below. Given d ∈ [0, ∞), there is the standard notion of d-dimensional Hausdorff measure µd (X) of a metric space X (see p. 19 of [18] for the definition). The Hausdorff dimension of a metric space X is defined to be the unique number dimHaus (X) ∈ [0, ∞] such that µd (X) = 0 for all d > dimHaus (X) and µd (X) = ∞ for all d < dimHaus (X). By Theorem 1.7.16 of [18], such a number always exists. For the proof of the following result, see Theorems 10.8.1 and 10.8.2 of [18].19 Theorem G.40 (Aleksandrov spaces have integer dimension). The Hausdorff dimension of an Aleksandrov space X with curvature bounded from below is a finite nonnegative integer. A basic tool used to study the local topological structure of Aleksandrov spaces is strainers, which we now define. Note that if p (γ, γ) is well defined, then it must be equal to zero (by taking u = v in the limit on the rhs of (G.18)). 19 Since X is locally compact by definition, we have that dimHaus (X) is not equal to ∞. 18

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Definition G.41 (Strained points and strainers). Let X be an Aleksandrov space with curvature bounded from below. Given m ∈ N and ε > 0, a point p ∈ X is an (m, ε)-strained point if there exist m pairs of points {(ai , bi )}m i=1 in X such that ˜ i , p bi > π − ε, a ˜ i p bj > π − 10ε, a 2

˜ i p aj > π − 10ε, a 2 π ˜ bi p bj > − 10ε, 2

where i = j ∈ {1, 2, . . . , m}. The collection {(ai , bi )}m i=1 is called an (m, ε)strainer for p. The following result, which is Theorem 10.8.3 of [18], is proved using strainers. Theorem G.42 (Regularity of Aleksandrov spaces). Let X be a complete length space with curvature bounded from below. If dimH X = n < ∞, then X is locally compact and an open dense subset of X is homeomorphic to an n-manifold. Besides the notion of Hausdorff dimension given above, there are other definitions of dimension (rough dimension, covering dimension, and strainer number) on Aleksandrov spaces with curvature bounded from below, but they are more or less equivalent (see Proposition 155 of [159]). A key property of Aleksandrov spaces is the following (see Proposition 10.7.1 of [18]); we do not discuss the pointed version. Theorem G.43 (Closedness under Gromov–Hausdorff limit). If a metric space (X, d) is the Gromov–Hausdorff limit (pointed or not) of a sequence of Aleksandrov spaces of curvature ≥ k, then (X, Ld ) is a complete length space of curvature ≥ k. Let M (n, k, D) denote the space of Aleksandrov spaces of curvature ≥ k with Hausdorff dimension ≤ n and diameter ≤ D. The following is Theorem 10.7.2 of [18]. Theorem G.44 (Gromov compactness theorem, II). For any sequence in M (n, k, D) there exists a subsequence which converges in the Gromov– Hausdorff topology to an element of M (n, k, D). Since Riemannian manifolds with sectional curvatures bounded from below are Aleksandrov spaces of curvature bounded from below, the above theorem implies that any sequence of n-dimensional Riemannian manifolds with sectional curvature ≥ k and diameter ≤ D subconverges in M (n, k, D). Hence Aleksandrov spaces of curvature bounded from below are nice generalizations of Riemannian manifolds with sectional curvatures bounded from below. There are other nice aspects of this generalization: some of the theorems and tools in Riemannian geometry can be generalized to Aleksandrov spaces. The following splitting theorem of Toponogov and of Milka [128] is an example (see Theorem 10.5.1 of [18]).

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A line in a length space X is a (unit speed) path γ : R → X such that γ|[a,b] is a shortest path between γ (a) and γ (b) for any a, b ∈ R. Theorem G.45 (Aleksandrov space splitting theorem). If X is an Aleksandrov space of nonnegative curvature which contains a line, then X is isometric to the metric product of an Aleksandrov space of nonnegative curvature with R. This splitting theorem generalizes the splitting theorem for Riemannian manifolds with nonnegative sectional curvature. However, for Riemannian manifolds, one only needs to assume that the Ricci curvatures are nonnegative to prove a splitting theorem. As far as we know, the corresponding generalization has not been achieved yet; related to this is the problem of finding a good notion of nonnegative Ricci curvature on complete length spaces. Note also that the notion of Busemann function, used in the proof of the Riemannian version of the splitting theorem, can be generalized to Aleksandrov spaces and is used in the proof of the above theorem. We shall review Busemann functions on Riemannian manifolds in §1 of Appendix I. The following is Theorem 10.4.1 of [18], which generalizes the corresponding theorem for Riemannian manifolds. Theorem G.46 (Bonnet–Myers-type diameter bound). If X is an Aleksandrov space of curvature ≥ k, where k > 0, then we have √ diam (X) ≤ π/ k. Remark G.47 (Volume comparison for Aleksandrov spaces). Another basic technique of comparison geometry, namely the Bishop–Gromov volume comparison theorem for Riemannian manifolds, also extends to Aleksandrov spaces of curvature ≥ k (see §§10.6.2–10.6.3 of [18]). In the next subsection we list some other important tools used in the study of Aleksandrov spaces. We end this subsection by stating a wellknown open problem which asks how large the closure of the subcollection of Riemannian manifolds with sectional curvature bounded from below in the collection of Aleksandrov spaces of curvature bounded from below is (see [108]). Problem G.48 (Aleksandrov spaces as Riemannian limits). Is every finite n-dimensional Aleksandrov space of curvature ≥ k isometric to the limit of a sequence of complete Riemannian manifolds (whose dimensions may be higher than n) with sectional curvatures ≥ k  for some k  ∈ R? Apparently, the consensus is that the answer to this question should be ‘no’. 2.4. Tools for Aleksandrov spaces with curvature bounded from below. In this subsection we give a brief account of a few tools used in the study of Aleksandrov spaces.

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2.4.1. The tangent cone of Aleksandrov spaces with curvature bounded from below. Let X be an Aleksandrov space of curvature ≥ k, where k ∈ R, and let p ∈ X. On the space Sp of shortest paths emanating from p, the (welldefined) angle p at p defines a pseudo-metric. Let Σp denote the space of equivalence classes of shortest paths emanating from p, where two shortest

paths are equivalent if the angle between them at p is zero. Then Σp , p , where p is the quotient metric, is a metric space. The (metric)

of directions Σp is defined to be the metric space space completion of Σp , p . We may take the Euclidean metric cone Cone (Σp ) over Σp . Recall that we also have the notion of tangent cone defined in (G.10). A point p ∈ X is a regular point if Σp is isometric to S n−1 (1). Otherwise, we say that p is a singular point. We have the following (see Theorem 10.9.3, Corollary 10.9.5, and Corollary 10.9.6, all in [18]). Theorem G.49 (Existence of the tangent cone). If X is an n-dimensional Aleksandrov space of curvature ≥ k and p ∈ X, then we have the following: (1) The tangent cone Tp X at p exists20 and is an n-dimensional Aleksandrov space of nonnegative curvature. (2) Tp X is isometric to Cone (Σp ).21 (3) When n ≥ 3, the space of directions Σp is an (n − 1)-dimensional compact 22 Aleksandrov space of curvature ≥ 1; hence, by Theorem G.46, Σp has diameter ≤ π (if n = 1, then the space of directions is either one or two points; if n = 2, then diam (Σp ) ≤ π). (4) If the space of directions Σp is isometric to the unit sphere S n−1 (1), then there exists a neighborhood of p homeomorphic to an open set in Rn (see p. 73 of [175]). With the above theorem, we may define the exponential map on an Aleksandrov space with curvature bounded from below (see p. 68 of [175]). Definition G.50 (Exponential map and cut locus). Let X be an ndimensional Aleksandrov space of curvature ≥ k and let p ∈ X. (i) We define Ωp ⊂ Cone (Σp ) to consist of the set of equivalence classes [(v, t)] ∈ (Σp × R≥0 ) / ∼ such that there exists a geodesic γ from p to some point x such that γ is in the direction of v and t = L (γ). (ii) We define the exponential map expp : Ωp → X 20

By definition the tangent cone is unique if it exists. That is, the tangent cone Tp X at p is isometric to the metric cone over the Aleksandrov space Σp of curvature ≥ 1 (see part (3)). 22 See p. 69 of [175]. 21

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by expp (v, t)  x. (iii) The cut locus C (p) of p is the set of x ∈ X such that any minimal geodesic px is not properly contained in another minimal geodesic starting from p. We define Cp ⊂ Ωp to consist of those equivalence classes [(v, t)] with the property that there exists x ∈ C (p) such that v is in the direction of some px and t = d (p, x). Note that given a geodesic γ : [a, b] → X in an Aleksandrov space, it is not always possible to extend γ beyond γ (b), i.e., X may not be geodesically complete. One application of the notion of the space of directions Σp is in defining the boundary of an Aleksandrov space. Definition G.51 (Stratification). Let X be a topological space and let N {Xi }N i=1 be a collection of subsets of X. Then {Xi }i=1 is called a stratification of X into topological manifolds if (1) Xi are disjoint topological manifolds without boundary and N .

Xi = X,

i=1

(2) dim X1 > dim X2 > · · · > dim XN , / (3) the set Xk+  N i=k Xi is closed in X for each k = 1, . . . , N . We call the Xi strata of X. The following is part of Theorem 10.10.1 in [18]. Theorem G.52 (Aleksandrov spaces admit stratifications). If X is an n-dimensional Aleksandrov space of curvature ≥ k, then X admits a stratification into topological manifolds. We can inductively define the boundary of an Aleksandrov space of curvature ≥ k. The boundary of a 1-dimensional Aleksandrov space is defined to be its topological boundary. Now suppose that the boundary has been defined for Aleksandrov spaces of curvature bounded from below and dimension ≤ n−1. We define the boundary of an n-dimensional Aleksandrov space X of curvature bounded from below as the set of points p ∈ X where the space of directions Σp has nonempty boundary (here we used Theorem G.49(3)). 2.4.2. The distance function and semi-concave functions on Aleksandrov spaces with curvature bounded from below. Given any metric space X, the most natural continuous function is the distance function d (·, p) to a given point p ∈ X. This function plays a prominent role in comparison geometry and geometric analysis. It is well known that even if the underlying space X is a smooth manifold, the distance

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function is not necessarily smooth. In fact, in this case, d (·, p) is a Lipschitz function.23 On Riemannian manifolds with sectional curvature bounded from below, the Hessian of d (·, p) is bounded from above at smooth points. This suggests that one study Lipschitz functions, in particular, functions with an upper Hessian bound in the support sense, on Aleksandrov spaces. Although these functions do not have derivatives everywhere, they behave nicely. When we study such functions on an Aleksandrov space with curvature bounded from below, another complication arises: the underlying topological space is not a smooth manifold. The following lemma generalizes the corresponding result in Riemannian geometry regarding ∇d (the Gauss lemma). Lemma G.53 (Derivative of distance). Let X be an Aleksandrov space of curvature ≥ k and let p, q ∈ X. Let α be a minimal geodesic from q to p and let β be a unit speed minimal geodesic starting from q. Suppose that p is not in the cut locus C (q). Then the following right derivative exists and is given by  d (p, β (∆t)) − d (p, β (0)) d  = − cos q (α, β) . d (p, β (t))  lim  + + dt t=0 ∆t ∆t→0 For a proof of the above lemma, see Corollary 62 of [159] or Remark 4.5.12 in [18], which contain information about the case where q ∈ C (q). Now we give a definition of semi-concave functions, which correspond to functions with a local upper Hessian bound. Definition G.54 (λ-concave function). Let X be an Aleksandrov space of curvature ≥ k without boundary, let Ω ⊂ X be an open subset, and let λ ∈ R. We say that a locally Lipschitz function f : Ω → R is λ-concave if, for any unit speed geodesic γ : (a, b) → Ω, the function 1 s −→ f ◦ γ (s) − λs2 2 is concave on (a, b). A function f : Ω → R is called semi-concave if for every x ∈ Ω there exist a neighborhood U of x and λ ∈ R such that f |U is λ-concave. Note that a C 2 function on a Riemannian manifold is semi-concave. Example G.55. Let f1 (x) = |x|α , where α ∈ (0, 2). Then, for any λ ∈ R, the function s → f1 ◦ γ (s) − 12 λs2 is not concave in any neighborhood of 0. Thus, for α ∈ [1, 2), the function f1 (x), which is Lipschitz for such α, is not semi-concave. We say that a locally Lipschitz function f : X → R is λ-concave at a point x ∈ M if for every ε > 0, the function 1 y −→ f (y) − (λ + ε) d (y, x)2 2 23

See subsection 1.1 for the definition of Lipschitz function.

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is concave in some neighborhood of x. 2.4.3. Quasi-geodesics. Geodesics play a crucial role in Riemannian geometry. For Aleksandrov spaces, the following generalization is useful. Definition G.56 (Quasi-geodesic). Let X be an Aleksandrov space of curvature ≥ k without boundary. A path γ in X is called a quasi-geodesic if for λ ∈ R and any λ-concave function f the function f ◦ γ is λ-concave. Note that a geodesic is a quasi-geodesic. Quasi-geodesics have nice properties such as being closed under limits and such that for any tangent vector V there exists a quasi-geodesic emanating from its base and pointing in the direction of V . There are several other useful analytical tools in the study of Aleksandrov spaces: the directional derivative, the differential, and the gradient curve of a function on an Aleksandrov space. In Appendix H we shall discuss some of these notions and ideas in a simpler setting: closed locally convex subsets in Riemannian manifolds. Another useful notion is that of extremal set, which can be used to give a refinement of the notion of stratification of Aleksandrov spaces. Also note that some of the results in Morse theory have been generalized by Perelman to Aleksandrov spaces [149]. For all of these results, the reader may find more information either from the surveys by Plaut [159] and Petrunin [156] or from the original papers mentioned in the notes and commentary at the end of this appendix. 2.5. Advanced results on Aleksandrov spaces with curvature bounded from below. The following is Perelman’s stability theorem (see Theorem 10.10.5 of [18] for the statement), whose proof is partly based on the so-called deformation theorem of Siebenmann [174]. Theorem G.57 (Perelman’s stability theorem). Let X be a compact n-dimensional Aleksandrov space of curvature ≥ k, where k ∈ R. There exists ε > 0 such that if Y is a compact n-dimensional Aleksandrov space of curvature ≥ k with dGH (X, Y ) < ε, then Y is homeomorphic to X. Remark G.58. A recent detailed expository account of this theorem has been given by V. Kapovitch [103]. As an immediate consequence, we have the following. Corollary G.59 (Topology of small metric balls). Let X be as in the theorem above. For every p ∈ X there exists ε > 0 such that B (p, ε) is homeomorphic to the tangent cone Tp X. The following result of Otsu and Shioya [146] tells us about the differentiable structures of Aleksandrov spaces. (Perelman also studied the Riemannian structure on a finite-dimensional Aleksandrov space; see [151].)

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Theorem G.60 (Regularity of Aleksandrov spaces). Let X be an ndimensional Aleksandrov space of curvature ≥ k. Then we have the following: (1) There exists a set X0 ⊂ X such that X − X0 has n-dimensional Hausdorff measure 0 and contains the set SX of singular points of X. (2) The metric on X is induced by a C 1/2 -Riemannian metric g on X0 . (3) The Riemannian metric g extends continuously to X − SX . (4) The space X has an almost everywhere approximately twice differentiable structure in the sense of Stolz (see [146] for the definition of such a structure; related to this is Definition H.7 below ). A geodesically complete Aleksandrov space of curvature ≥ k is isometric to a C 1,α Riemannian metric on a smooth manifold (this is a result of Nikolaev, with earlier work by Berestovskii; see Theorem 210 of [159]). Remark G.61. An important topic, which we do not discuss here, is the behavior of the notions mentioned above under both Gromov–Hausdorff convergence and collapsing of Aleksandrov spaces. 3. Notes and commentary Advanced (and indispensable) references on metric geometry are the original papers by Burago, Gromov, and Perelman [19] and Perelman [147]. Additional reference and survey articles include Grove [80], Shiohama [175], Plaut [159], and Petrunin [156]. For a classic book on comparison geometry for Riemannian manifolds, see Cheeger and Ebin [30]; more recent aspects of comparison geometry are contained in the collection of papers edited by Grove and Petersen [82]. There are several excellent references for Aleksandrov spaces with curvature bounded from below including the aforementioned [18], [19], and Berestovskii and Nikolaev [12]. See also Fukaya [62], [159], and [156]. We urge the reader to consult the aforementioned references.

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http://dx.doi.org/10.1090/surv/163/12

APPENDIX H

Convex Functions on Riemannian Manifolds But not too many horns can make that sound. – From “Sultans of Swing” by Dire Straits

In this appendix we provide a detailed discussion of convex analysis on Euclidean spaces and on locally convex subsets in Riemannian manifolds, which comprise a ‘baby version’ of the corresponding study of semi-concave functions on Aleksandrov spaces (see for example Petrunin [156] and the references therein). The discussion of convex functions leads to a proof of Theorem I.24 in the next appendix regarding the existence of distancenonincreasing maps between the level sets of Busemann functions in complete noncompact manifolds with nonnegative sectional curvature. In §1 we review aspects of convex analysis on Euclidean space related to the differentiability (e.g., Lipschitz) properties of convex functions. In §2 we discuss the properties of convex sets and functions on Riemannian manifolds, including generalizations of some of the results of the previous section. In §3 we discuss the generalized gradient of a convex function on a Riemannian manifold. In §4 we discuss integral curves of concave functions. 1. Elementary aspects of convex analysis on Euclidean space Convex analysis enters the Ricci flow from two perspectives. First of all, it is useful in the study of the maximum principle for systems, where invariant subsets of vector bundles for solutions of heat-type equations are fiberwise convex (see Chapter 10 of Part II). Secondly, totally convex subsets are fundamental in the study of Riemannian manifolds with nonnegative sectional curvature (see §1 of the next appendix). In part as a warm-up, in this section we give a brief review of some elementary aspects of convex analysis on Euclidean space. In the next section we shall discuss convex analysis on certain subsets of Riemannian manifolds. 1.1. Lipschitz functions. Let (X, dX ) and (Y, dY ) be metric spaces. A map f : X → Y is Lipschitz if there exists a constant C < ∞ such that dY (f (x1 ) , f (x2 )) ≤ CdX (x1 , x2 ) 413 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

for all x1 , x2 ∈ X. The Lipschitz constant (or dilatation) of f is the infimum of all such C: Lip (f )  inf {C : ∀x1 , x2 ∈ X, dY (f (x1 ) , f (x2 )) ≤ CdX (x1 , x2 )} . A map f : X → Y is locally Lipschitz if for every x ∈ X there exists a neighborhood U of x such that f |U is Lipschitz. The dilatation of f at x is defined to be dil x (f )  inf {Lip ( f |U ) : U is a neighborhood of x} . Problem H.1. Under what conditions do we have the equality Lip (f ) = sup dil x (f ) ? x∈X

Exercise 1.4.5 of Burago, Burago, and Ivanov [18] says that this equality is true if X = R or X = S 1 , where S 1 is endowed with the intrinsic (arc length) metric. A function f on a differentiable manifold Mn is called locally Lipschitz if it is locally Lipschitz with respect to some Riemannian metric g on M. Note that the property of a function being locally Lipschitz is independent of the metric. Hence, locally, a locally Lipschitz function on a manifold is in essence a Lipschitz function on some Euclidean ball. Recall that given a Riemannian manifold (Mn , g), there exists an associated Riemannian measure µ. A set S ⊂ M has measure zero if µ (S) = 0. The property of a set having measure zero is independent of the metric g. We say that a property on a differentiable manifold holds almost everywhere (a.e.) if the property holds on the complement of a set with measure zero with respect to some Riemannian measure. Locally Lipschitz functions on differentiable manifolds have the following property (see Theorem 2 in §3.1.2 on p. 81 of Evans and Gariepy [59] or Theorem 3 on p. 250 of Stein [176]). Proposition H.2 (Rademacher). If Mn is a differentiable manifold and if f : M → R is a locally Lipschitz function, then f is differentiable almost everywhere. In particular, the distance function of a Riemannian manifold, which has Lipschitz constant 1, is differentiable a.e. (alternatively, the distance function to a point p is C ∞ outside the cut locus of p, which has measure zero). When the manifold M is 1-dimensional, a Lipschitz function can be expressed as the integral of its derivative. Lemma H.3 (Fundamental theorem of calculus). If f is a Lipschitz function defined on an interval [a, b], then for any x ∈ [a, b],  x f  (s) ds, f (x) = f (a) + a

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1. ELEMENTARY ASPECTS OF CONVEX ANALYSIS ON EUCLIDEAN SPACE 415

where f  is the derivative function defined a.e. on [a, b]. (See Corollary 15 on p. 110 of Royden [162] or Theorem 7.20 of Rudin [163].) Also note that, for Lipschitz functions on manifolds, integration by parts holds (see Lemma 7.113 in Part I, for example). 1.2. Convex functions on Euclidean spaces and their differentiability. It is standard in the convexity literature to consider functions which take the value ∞. In particular, given a locally convex set C ⊂ Rn ,1 a function f : C → R ∪ {∞} is said to be convex if for any line segment xy ⊂ C we have for all s ∈ (0, 1), f (sx + (1 − s) y) ≤ sf (x) + (1 − s) f (y) (with the operation + and relation ≤ on R ∪ {∞} defined in the obvious ways). The function f being convex is denoted by f ∈ Conv (C). Remark H.4. Given a convex set C ⊂ Rn and a convex function f : C → R, we may define f¯ : Rn → R by  f (x) if x ∈ C, ¯ f (x) = ∞ if x ∈ Rn − C. It is then easy to see that f¯ ∈ Conv (Rn ). If a convex function f is defined on a convex set in Euclidean space, then it is a standard result that f is locally Lipschitz in the interior of the set (see Theorem 1(i) on p. 236 of [59] or Lemma 3.1.1 on p. 102 of [99]). We shall give a proof of this result for f defined on Riemannian manifolds (see Lemma H.21). Recall that the interior of a set Σ is denoted by int (Σ). Proposition H.5 (Convex functions are locally Lipschitz). Any finitevalued convex function f defined over a locally convex set C ⊂ Rn is locally Lipschitz on int (C). Hence (by Rademacher’s theorem) f is differentiable almost everywhere on int (C). Concerning the convergence of convex functions, we have the following general result (see Theorem 3.1.4 of [99]). Lemma H.6 (On convergence of sequences). If fk : Rn → R ∪ {∞} are convex functions converging pointwise to a function f∞ : Rn → R ∪ {∞}, then f∞ is convex and fk converges uniformly to f∞ on compact sets. Regarding the differentiability of functions, the following is equivalent to Definition 7.115 in Part I. 1

Locally convex sets in Riemannian manifolds are defined in subsection 2.1.2 below.

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Definition H.7. Let U ⊂ Rn be an open set. A continuous function f : U → R is twice differentiable in the sense of Stolz at x ∈ U if there exists an n-vector W and a symmetric n × n matrix M such that   f (x + y) = f (x) + W · y + M (y, y) + o |y|2 for x + y ∈ U . Recall from Lemma 7.117 in Part I (Aleksandrov’s theorem) that a convex function on a Euclidean ball is twice differentiable in the sense of Stolz almost everywhere. Although there are good reasons for allowing convex functions to take the value ∞, throughout most of this book, we shall assume that convex functions are finite valued. 1.3. A property about convex sets in Euclidean space. Let K ⊂ Rn be a closed convex set. Define the nearest point projection map πK : Rn → K by πK (x) being the unique closest point in K to x. Clearly πK ◦ πK = πK and πK (x) = x if and only if x ∈ K. The following essentially says that a closed convex set is on one side of its support planes. Lemma H.8. For any x ∈ Rn and z ∈ K we have x − πK (x) , z − πK (x) ≤ 0. Proof. Geometrically this is obvious, noting that the vector x − πK (x) is normal to a support plane. An analytic proof of the lemma may be found for example in the proof of Theorem 3.1.1 on p. 47 in [99]. See also Lemma 10.36 in Part II.
Proposition H.9 (Monotonicity of the nearest point projection map). Let K ⊂ Rn be a closed convex set. For any x, y ∈ Rn we have (1) (πK is distance nonincreasing) |πK (x) − πK (y)| ≤ |x − y| ,

(H.1)

(2) (πK is monotone increasing)2 πK (x) − πK (y) , x − y ≥ 0.

(H.2)

Proof. By Lemma H.8 we have for any x, y ∈ Rn x − πK (x) , πK (y) − πK (x) ≤ 0, y − πK (y) , πK (x) − πK (y) ≤ 0, so that by summing we have y − x − πK (y) + πK (x) , πK (x) − πK (y) ≤ 0. 2

Quoting p. 48 of [99].

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 417

Hence, for any x, y ∈ Rn , (H.3)

|πK (x) − πK (y)|2 ≤ πK (x) − πK (y) , x − y .

Part (2) immediately follows from this and also part (1) follows from this and the Cauchy–Schwarz inequality: πK (x) − πK (y) , x − y ≤ |πK (x) − πK (y)| |x − y| .
A simple consequence of (H.1) is the following Busemann–Feller theorem (compare with Theorem H.59). Recall that the length of a path is defined by (G.5). Corollary H.10 (Projection is length nonincreasing). Let K ⊂ Rn be a closed convex set and let γ be a rectifiable path in Rn . Then the projected path πK ◦ γ has length less than or equal to the length of γ. In the next section we develop tools in order to generalize this result to convex sets in Riemannian manifolds. 2. Connected locally convex subsets in Riemannian manifolds In this section we study connected closed locally convex subsets in Riemannian manifolds and the convex functions defined on them. These sets are useful in the study of complete noncompact manifolds with nonnegative sectional curvature.3 Our main goal is to present some tools so that later we may give a proof of Sharafutdinov’s retraction theorem (see Theorem H.59 below). As we shall see (Proposition H.17), connected closed locally convex subsets in Riemannian manifolds are Aleksandrov spaces (see also p. 821 of Plaut [159]). Hence the concepts and results given in this section may help motivate the corresponding concepts and results for Aleksandrov spaces, as we alluded to in the introduction to this appendix. One may view the presentation below as an introduction to some of the techniques used in studying Aleksandrov spaces, albeit in a much simpler form. In this section (Mn , g) shall denote a connected complete Riemannian manifold. Many of the results in this section are from Sharafutdinov [171]. Caveat: In §2–§4, we sometimes employ ad hoc techniques in the proofs of the results contained therein; these proofs may not be the most efficient proofs and these results are not the most general. The reader is encouraged to dig into the literature to learn the related material. 2.1. Locally convex subsets in Riemannian manifolds. In this subsection we discuss locally convex subsets in Riemannian manifolds and their relation to Aleksandrov spaces. 3

For example, sublevel sets of Busemann functions in complete noncompact manifolds with nonnegative sectional curvature satisfy a stronger condition than local convexity (see Proposition I.15).

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

2.1.1. Interior tangent cone of a subset. Let S be a subset in a Riemannian manifold (Mn , g) and let p ∈ S. We define the interior tangent cone by   there exists a δ > 0 such that ˜ . (H.4) Tp S  V ∈ Tp M : expp (sV ) ∈ int (S) for all s ∈ (0, δ] Remark H.11. The above definition of the interior tangent cone T˜p S is one of a few, nearly equivalent, definitions of the tangent cone at p of the set int (S) ∪ {p}. Of course, T˜p S may be empty even when int (S) is top-dimensional, i.e., even when int (S) = ∅. For example, take √ √ (H.5) S  (x, y) : x ≥ 0 and x ≤ y ≤ 2 x ⊂ R2 and take p = (0, 0). Then T˜p S = ∅ whereas Tp S = {(0, y) : y ≥ 0}, where Tp S denotes the tangent cone defined in (G.10). We have the following properties of the interior tangent cone, which follow directly from definition (H.4): (1) If p ∈ int (S), then T˜p S = Tp M. Note that if p ∈ S\ int (S), then 0 ∈ / T˜p S. (2) The set T˜p S has a cone structure, i.e., for any V ∈ T˜p S and r > 0 we have rV ∈ T˜p S. Remark H.12. Assuming that the tangent cone Tp S exists and assuming that Tp S naturally embeds in Tp M (see Question G.21), we expect that T˜p S ⊂ Tp S. When S is convex,4 we have the following. Lemma H.13 (Openness of T˜p S). If S is convex, then T˜p S is an open subset of Tp M. Proof. (1) If p ∈ int (S), then T˜p S = Tp M and we are done. (2) Suppose p ∈ ∂S. If V ∈ T˜p S, then V = 0 and there exists δ ∈ (0, inj (p)) such that   V ∈ int (S) . expp δ |V | Then there exists an open neighborhood U of δ |VV | in Tp M such that   expp (U ) ⊂ int (S) and U ⊂ B 0, inj (p) . 4

See subsection 2.1.2 for the definition of convex set.

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 419

Since S is convex, we have

K  expp (sW ) : W ∈ U , s ∈ (0, 1] ⊂ S. Since K is open, we actually have K ⊂ int (S). Hence U ⊂ T˜p S and we
conclude that T˜p S is open. Note that if S is not convex, then T˜p S need not be open. For example, consider    S  (x, y) : y ≥ |x| ⊂ R2 and p = (0, 0). Then T˜p S = {(0, y) : y > 0} , which is not open in R2 . 2.1.2. Definition of locally convex sets on Riemannian manifolds. Recall that a subset Σ of a Riemannian manifold (Mn , g) is said to be convex if, for all x, y ∈ Σ, every minimal geodesic γ in M joining x and y is contained in Σ. For example: (1) a subset K ⊂ Rn is convex if and only if for all x, y ∈ K, the line segment {(1 − t) x + ty : t ∈ [0, 1]} is contained in K,

(2) the set x1 , . . . , xn+1 ∈ S n (1) ⊂ Rn+1 : xn+1 ≥ c , where c ≥ 0, is convex in the unit sphere S n (1). Note that the intersection of convex sets is convex. A subset C ⊂ Mn is said to be locally convex if for all x ∈ C there exists an open neighborhood U of x such that C ∩ U is convex. For example, the following sets are locally convex but not convex: (1) the disjoint union of two round balls in Rn , (2) a ball of radius r ∈ (1/4, 1/2) in the unit torus T n  Rn /Zn . Note that a convex set is necessarily connected but a locally convex set may not be connected. Since sufficiently small balls are convex in a Riemannian manifold, an equivalent definition for C to be locally convex is that for all x ∈ C there exists ε > 0 depending on x such that C ∩ B (x, ε) is convex. Since the notion of interior tangent cone is local, Lemma H.13 implies that if a set is locally convex, then an interior tangent cone is open. Remark H.14. For some other variants on the definition of convexity, see subsection 1.2 in the next appendix. 2.1.3. Existence of minimal geodesics in locally convex sets. If C is a connected closed locally convex subset of a Riemannian manifold (Mn , g), then C is a topological manifold with boundary ∂C (see pp. 417– 419, including Theorem 1.6, of Cheeger and Gromoll [32]).

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Lemma H.15 (Existence of minimal geodesics in C between points). If C is a connected closed locally convex subset of a complete Riemannian manifold (Mn , g), then for every x, y ∈ C there exists a smooth geodesic (of (M, g)) α : [0, 1] → C with α (0) = x, α (1) = y, and L (α) = dC (x, y)  inf L (β) , β

where the infimum is taken over all paths β : [0, 1] → C with β (0) = x and β (1) = y. Proof. We prove the lemma in three steps. Step 1 (Existence of shorter piecewise C ∞ geodesic paths). For every x, y ∈ C and any path β˜ : [0, 1] → C with β˜ (0) = x and β˜ (1) = y, we claim that there exists a piecewise C ∞ geodesic path β : [0, 1] → C with β (0) = x, β (1) = y, and

  L (β) ≤ L β˜ .

To see this claim, we observe that, since β˜ is continuous, there is an r0 ∈ ¯ (x, r0 ) ∩ C there ¯ (x, r0 ). For every z ∈ B (0, ∞) such that β˜ ([0, 1]) ⊂ B   / exists εz > 0 such that C ∩ B (z, εz ) is convex. Since s∈[0,1] B β˜ (s) , εβ(s) ˜ is an open cover of the compact set β˜ ([0, 1]), it is easy to see that there is a partition s0 = 0 < s1 < · · · < sm = 1 such that   sk ) , εβ(˜ β˜ ([sk−1 , sk ]) ⊂ B β˜ (˜ ˜ sk ) for some s˜k ∈ (sk−1 , sk ), k = 1, . . . , m. We define β|[sk−1 ,sk ] to be the minimal geodesic joining β˜ (sk−1 ) and β˜ (sk ). It is easy to verify that the broken geodesic β has all the properties in the claim. Clearly we also have L (β) < ∞ and dC (x, y) < ∞.   be a length Step 2 (Existence of minimizing path α). Let β˜i i∈N

minimizing sequence for inf β L (β) = dC (x, y). For each i ∈ N, let βi : [0, 1] → C be a piecewise C ∞ path constructed from β˜i as in Step 1. If necessary, we reparametrize so that each βi has constant speed. Then the piecewise C ∞ paths {βi }i∈N form a length minimizing sequence lim L (βi ) = inf L (β) = dC (x, y) .

i→∞

β

As typical in the calculus of variations, we shall use the direct method to prove that a subsequence of {βi }i∈N converges to a path α : [0, 1] → C with

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 421

α (0) = x, α (1) = y, and L (α) = dC (x, y) . Note that we may assume that ¯ (x, dC (x, y) + 1) ∩ C βi ([0, 1]) ⊂ B ¯ (x, dC (x, y) + 1)∩C for all i (the set on the rhs is compact). For every z ∈ B there exists εz > 0 such that C ∩B (z, εz ) is convex. Consider the (relatively) open cover

¯ (x, dC (x, y) + 1) ∩ C B (z, εz ) ∩ B ¯ z∈B(x,dC (x,y)+1)∩C

¯ (x, dC (x, y) + 1) ∩ C. By Lebesgue’s number of the compact metric space B 5 ¯ (x, dC (x, y) + 1) ∩ C, lemma , there exists δ > 0 such that for every z ∈ B

 B (z, 2δ) ⊂ B z , εz  ¯ (x, dC (x, y) + 1) ∩ C. Hence C ∩ B (z, 2δ) is convex for every for some z  ∈ B ¯ z ∈ B (x, dC (x, y) + 1) ∩ C. Since the lengths of the constant speed paths {βi } have a uniform upper bound, there exists m ∈ N such that, for all 1 ≤ k ≤ m and for all i ∈ N, ¯ (x, dC (x, y) + 1) ∩ C such that there exists zk,i ∈ B 3 2 k−1 k , ⊂ B (zk,i , δ) . βi m m Moreover, there exists a subsequence of i such that for any given k, (1) the sequence zk,i converges to some point zk,∞ as i → ∞ and (2) d (zk,i , zk,∞ ) < δ for all k and i. We then have 3 2 k−1 k , ⊂ B (zk,∞ , 2δ) ∩ C βi m m ∞  is unifor all k and i. Since, for each k, the sequence βi |[ k−1 , k ] m

m

i=1

formly bounded (in the ball B (zk,∞ , 2δ)) and is equicontinuous because of its constant speed assumption, it follows from the Arzela–Ascoli  theorem and a careful choice of subsequence ij that a subsequence of βij [ k−1 , k ] will m

converge in C 0 to a C 0 path α|[ k−1 , k ] m

m

m

¯ (zk,∞ , 2δ) ∩ C in B

for each k = 1, . . . , m. We now have constructed a limit C 0 path α : [0, 1] → C. 5

Recall that Lebesgue’s number lemma says that if (X, d) is a compact metric space and if C is an open cover of X, then there exists δ > 0 such that every ball of radius ≤ δ is contained in a member of C.

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Step 3 (Regularity of the path α). This follows from showing that α satisfies L (α) = dC (x, y). Given any partition {s
}N
=0 of [0, 1], we have N 



d βij (s
−1 ) , βij (s
) ≤ L βij .
=1

Taking the limit ij → ∞, we obtain N 
=1

d (α (s
−1 ) , α (s
)) ≤ lim L (βi ) , i→∞

and hence L (α) ≤ dC (x, y) . Since α is a curve in C joining x and y, we conclude L (α) = dC (x, y) . Given any s ∈ (0, 1), there exists an open neighborhood U of α (s) in M such that C ∩ U is convex. In particular, for σ1 < σ2 sufficiently small, we have α (s + σi ) ∈ C ∩ U for i = 1, 2. Since α has minimal length among paths in C and since C ∩ U is convex, we have   L α|[s+σ1 ,s+σ2 ] = dC (α (s + σ1 ) , α (s + σ2 )) = d (α (s + σ1 ) , α (s + σ2 )) . That is, α is locally length minimizing and hence α is a geodesic of (M, g).
Remark H.16. Alternatively, in the above proof, one should be able to minimize length in a space of ‘piecewise-short’ broken geodesics a` la Milnor [129]. The following proposition gives a connection between connected closed locally convex subsets and Aleksandrov spaces. Proposition H.17 (Locally convex subsets of Riemannian manifolds). Let (Mn , g) be a complete Riemannian manifold with sectional curvature ≥ k. Let C be a connected closed locally convex subset in M. Then, as a length subspace, C is an Aleksandrov space (with possibly nonempty boundary) with curvature ≥ k. Proof. Lemma H.15 implies that C is a complete locally compact length space. Note that, from the definition of convex set and the Toponogov comparison theorem for M, we know that C satisfies the local version of the definition of Aleksandrov space with curvature ≥ k.


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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 423

2.1.4. The infinitesimal structure of locally convex subsets. The following lemma indicates the infinitesimal convexity of locally convex sets. Lemma H.18. Let C ⊂ M be a locally convex set, let p ∈ C, and let V1 , V2 ∈ Tp M be two unit vectors. (i) Suppose there exists ε0 > 0 such that expp (sVi ) ∈ int (C) for 0 < s ≤ ε0 and i = 1, 2. Then there exists ε1 > 0, depending on M, C, V1 , and V2 , such that expp (s ((1 − t) V1 + tV2 )) ∈ int (C) for 0 < s ≤ ε1 and t ∈ [0, 1]. (ii) Suppose there exists ε0 > 0 such that expp (sV1 ) ∈ C

and

expp (sV2 ) ∈ int (C)

for 0 < s ≤ ε0 . Given any t ∈ (0, 1], there exists ε2 > 0, depending on M, C, V1 , V2 , and t, such that expp (s ((1 − t) V1 + tV2 )) ∈ int (C) for 0 < s ≤ ε2 . Proof. (i) Let U be an open neighborhood of p such that C ∩ U is convex. Without loss of generality, we may assume that inj (p) ¯ (p, ε0 ) ⊂ U . and B ε0 < 2 ¯ (p, ε0 ) ⊂ U , there exists From expp (ε0 Vi ) ∈ int (C), for i = 1, 2, and from B an open neighborhood Ui ⊂ U of expp (ε0 Vi ) such that Ui ⊂ Ui ⊂ int (C) . It follows from the convexity of C ∩ U that for i = 1, 2 there exists an open neighborhood Wi of Vi in the unit sphere   (H.6) Spn−1  W ∈ Tp M : |W |g(p) = 1 such that expp (ε0 W ) ∈ Ui

and

expp (sW ) ∈ U ∩ int (C)

for all W ∈ Wi and s ∈ (0, ε0 ]. Let ι : (Tp M, g (p)) → En be an isometry and let ViE and WiE be the images in En of Vi and Wi under ι, respectively. Weconsider the pointed   space 

−2 +  blow-ups U , p, λ g , whose limit as λ → 0 is Tp M, 0, g (p) = En , 0 .

Under the blow-up, the limit of the set expp (sW ) : W ∈ Wi and s ∈ (0, ε0 ] is the open cone   sW E : W E ∈ WiE and s > 0 ⊂ En .

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Hence, for any t ∈ [0, 1], the unit vector

(1−t)V1E +tV2E |(1−t)V1E +tV2E |

is an interior point of

the convex set ΩE  C E ∩ BE (0, 2), where C E ⊂ En is the minimal convex set (cone) which contains (i.e., is the convex hull of) both     and sW E : W E ∈ W2E and s ≥ 0 . sW E : W E ∈ W1E and s ≥ 0 Note that under emanating from p of   0 in Tp M, 0, g (p)

the blow-up (i.e., as λ → 0+ ), the geodesics

pointed U , p, λ−2 g approach (straight) rays emanating from

in any C k -norm and on any compact subset. For λ

small enough, the image of ΩE in U , p, λ−2 g is very close to some minimal convex subset of C ∩ U which contains both

expp (sW ) : W ∈ W1 and s ∈ (0, 2λ] and

expp (sW ) : W ∈ W2 and s ∈ (0, 2λ] .



(1−t)V1 +tV2 lies in the interior of It follows that |(1−t)V 1 +tV2 |g(p)

W ∈ Spn−1 : expp (sW ) ∈ C ∩ U for all s ∈ (0, 2λ] .

Hence there exists ε1 > 0 such that expp (ε1 ((1 − t) V1 + tV2 )) is an interior point of C ∩ U . From this we may easily derive (i). (ii) As in (i), we replace ε0 by a smaller positive number if necessary so that there exists an open neighborhood U2 ⊂ U of expp (ε0 V2 ) such that U2 ⊂ U2 ⊂ int (C) . Let W2 be an open neighborhood of V2 in the unit sphere Spn−1 as defined in (i). By the convexity of C, for any ε ∈ (0, ε0 ), the set  ⎫ ⎧  ⎪ ⎪ 0 < s < sW and expexpp (εV1 ) (sZ) is a ⎬ ⎨ [0,sW ] expexpp (εV1 ) (sZ) : minimal geodesic joining exp (εV ) and a point 1 p ⎪

⎪ ⎭ ⎩ in the set expp (rW ) : W ∈ W2 and r ∈ (0, ε0 ) is a cone. By the same blow-up argument as in (i), when ε is small enough (say ε ≤ ε2 for some ε2 ∈ (0, ε0 )), expp (ε ((1 − t) V1 + tV2 )) is an interior point in the cone above. Hence expp (s ((1 − t) V1 + tV2 )) ∈ int (C) for 0 < s ≤ ε2 .




As a simple consequence of Lemma H.18, we have the following. Corollary H.19 (Interior tangent cone of a locally convex set). Let C ⊂ M be a locally convex set. Then T˜p C defined in (H.4) is a convex cone. 2.2. Convex functions on connected locally convex sets. We recall some facts about convex functions on locally convex sets in Riemannian manifolds and their directional derivatives.

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 425

2.2.1. Definition of convex functions. Let C ⊂ M be a connected locally convex set. We say that a function f : C → R is convex if for any x, y ∈ C and any constant speed geodesic γ : [a, b] → C (the geodesic is contained in C) joining x to y, the composite function f ◦ γ : [a, b] → R is convex, that is, for any u1 , u2 ∈ [a, b] and t ∈ [0, 1] we have6 f ◦ γ ((1 − t) u1 + tu2 ) ≤ (1 − t) f ◦ γ (u1 ) + tf ◦ γ (u2 ) . We say that f : C → R is concave if −f is convex. The above definition of the convexity (concavity) of a function makes sense in an Aleksandrov space without boundary (in the definition, one simply replaces ‘constant speed geodesic’ by ‘unit speed shortest path’).7 Let φ : I → R, where I ⊂ R is an interval, be a convex function. For all dφ s0 , s1 ∈ I with s1 > s0 , the right derivative ds + (s0 ) exists and φ (s1 ) − φ (s0 ) dφ (s0 ) ≤ ds+ s1 − s0 since the rhs is a monotonically nondecreasing function of s1 for fixed s0 . dφ Moreover, the function s → ds + (s) is also monotonically nondecreasing. Conversely to the first statement above, if a Lipschitz function φ : I → R satisfies (H.7) for all s0 , s1 ∈ I with s1 > s0 in I, then φ is concave. (H.7)

Remark H.20. Let f : C → R be a convex function. Note that if γ : I → C is a unit speed geodesic, then d s −→ + (f ◦ γ) (s) ds is a monotonically nondecreasing function. 2.2.2. Convex functions are locally Lipschitz. We prove that convex functions on connected locally convex sets in Riemannian manifolds are locally Lipschitz in the interior of the set, which is the analogue of Proposition H.5. Lemma H.21 (Convex functions are locally Lipschitz). Let C ⊂ (Mn , g) be a connected locally convex set. If f : C → R is a convex function, then f is locally Lipschitz in int (C). Proof. We divide the proof into three steps. Step 1. f is locally bounded from above in int (C). For any p ∈ int (C), let r ∈ (0, inj (p) /4) be such that (1) B (p, 2r) is both convex and contained in C, (2) inj (q) > 2r for q ∈ B (p, r), and 6 7

This definition is opposite to the definition of convexity used by Sharafutdinov. In the literature, λ-concave with λ = 0 is the same as concave.

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

1 (3) (B (p, 2r) , g) is 1000n 2 -close to the Euclidean ball (B (0, 2r) , gEuc ) 2 in the C -Cheeger–Gromov topology. Choose an orthonormal basis {e1 , . . . , en } of Tp M. By assumption, for each i = 1, . . . , n, s −→ f ◦ expp (sei )

is a convex function of s ∈ (−2r, 2r). It is well known that f ◦ expp (sei ) is a continuous function of s and that f ◦ expp (sei ) is bounded from above by

max f ◦ expp (−rei ) , f ◦ expp (rei ) on the interval [−r, r]. Let k ∈ N with 2 ≤ k ≤ n. Given a sequence 1 ≤ i1 < · · · < ik ≤ n and a sequence of k signs ±, . . . , ±, we define the (k − 1)-simplex ∆±,...,± i1 ,...,ik ⊂ B (p, 2r) inductively as follows. We take the sequence of signs to be −, . . . , − as an example. Define the 1-simplex ∆−,− ik−1 ,ik

to be the minimal geodesic from expp −reik−1 to expp (−reik ). We then define the 2-simplex ∆−,−,− ik−2 ,ik−1 ,ik to be the smooth surface (with piecewise spanned by

smooth boundary)−,− all minimal geodesics from expp −reik−2 to some point in ∆ik−1 ,ik , and so forth. At the last stage, we obtain the (k − 1)-simplex ∆−,...,− i1 ,...,ik . In general, taking k = n, we have 2n smooth (n − 1)-simplices ∆±,...,± 1,...,n corresponding to the different choices of sign.

From the convexity of s → f expq (sV ) and from the construction of ∆±,...,± 1,...,n , it is easy to see that f |∆±,...,± is bounded by the largest value of f on the vertices of

∆±,...,± 1,...,n ,

1,...,n

i.e.,

f |∆±,...,± 1,...,n

≤ max f ◦expp (−re1 ),f ◦expp (re1 ) , . . . , f ◦expp (−ren ),f ◦expp (ren ) . Now we define an ‘n-body’ ∆⊂C consisting of minimal geodesics from p to some point in any one of ∆±,...,± 1,...,n . Again, by the convexity of f , we have f |∆ is bounded by

max f (p),f ◦expp (−re1 ),f ◦expp (re1 ) , . . . , f ◦expp (−ren ),f ◦expp (ren ) .

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 427

Geometrically it is clear that there exists r0 ∈ (0, r) such that8 ¯ (p, r0 ) ⊂ int (∆) . B ¯ (p, r0 ); Hence we have proved that f is locally bounded from above on B Step 1 is finished. Step 2. f is locally bounded from below in int (C). For any p ∈ int (C), ¯ (p, r0 ) is bounded from above by a let r0 be as in Step 1 so that f on B positive constant C0 . Now we shall show that f is bounded from below on ¯ (p, r0 /2), let ¯ (p, r0 /2). For any point q ∈ B B s −→ expp (sV ) , with |V |g(p) = 1, be the minimal geodesic from p to q, defined for 0 ≤ s ≤ sq . If f (q) ≥ 0, then we are done. On the other hand, if f (q) ≤ 0, then we have



sq r0 f expp (sq V ) + f expp (−r0 V ) f (p) ≤ r0 + sq r0 + sq 2 ≤ f (q) + C0 . 3

¯ (p, r0 /2). Hence f is bounded from below by min 3 (f (p) − C0 ) , 0 on B 2

Step 3. f is locally Lipschitz in int (C). For any p ∈ int (C), let r0 be as ¯ (p, r0 /2) is bounded by a constant C1 . in Steps 1 and 2 such that |f | on B ¯ (p, r0 /4) with f (q1 ) ≤ f (q2 ), let γ (s) be the minimal For any q1 and q2 in B unit speed geodesic joining q1 = γ (0) and q2 = γ (s0 ). We extend γ further, ¯ (p, r0 /2) at some point qˆ2  γ (s1 ). from the end q2 , until it intersects ∂ B It is clear that s0 < s1 , s0 ≤ r0 /2, and s1 ≥ r0 /4. Using the convexity of f ◦ γ (s), we compute   s0 s0 f (γ (0)) + f ◦ γ (s1 ) , f (q2 ) = f ◦ γ (s0 ) ≤ 1 − s1 s1 f (ˆ q2 ) − f (q1 ) 8C1 f (q2 ) − f (q1 ) ≤ ≤ . s0 s1 r0 ¯ (p, r0 /4) with Lipschitz constant Hence we proved that f is Lipschitz in B 8C1 r0 . As a simple consequence of its locally Lipschitz property, f is continuous in int (C).
2.2.3. Directional derivatives of convex functions. Given a subset S ⊂ M, recall that its interior tangent cone, as defined in (H.4), is denoted by T˜p S. 8

Here we used the condition that (B (p, 2r) , g) is (B (2r) , gEuc ) in the C 2 -Cheeger–Gromov topology.

1 -close 1000n2

to the Euclidean ball

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428

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Definition H.22. Given a function f : S → R and V ∈ T˜p S, we define the directional derivative of f in the direction V at p by

f expp (sV ) − f (p) , (H.8) (DV f ) (p)  lim s s→0+ provided the limit exists. Let V be a cone in some finite-dimensional real vector space. A function h : V → R is positively homogenous of degree 1 if for every V ∈ V and λ ∈ (0, ∞) we have h (λV ) = λh (V ) . For example, if | · | is a norm on the vector space, then h (V )  |V | is positively homogenous of degree 1. Clearly we have Lemma H.23. For any V ∈ T˜p S and r > 0, we have DrV f = rDV f, i.e., the function Df : V → DV f is positively homogenous of degree 1. Let C ⊂ M be a connected locally convex set and let f : C → R be a convex function. For each p ∈ C and s > 0 we define the difference quotient function9

f expp (sV ) − f (p) (H.9) Js : V → s   for any V ∈ dom (Js )  V ∈ T˜p C : expp (sV ) ∈ int (C) . Note that for 0 < s1 ≤ s2 , we have dom (Js2 ) ⊂ dom (Js1 ). Clearly (H.10)

(DV f ) (p) = lim Js (V ) . s→0+

Remark H.24. Since expp (sV ) = p + sV on Euclidean space En , we can rewrite the above difference quotient for C ⊂ En as Js : V →

f (p + sV ) − f (p) . s

The difference quotient and directional derivative have the following properties. Lemma H.25 (Directional derivatives of convex functions). Let C ⊂ (Mn , g) be a connected locally convex set in a connected complete Riemannian manifold and let f : C → R be a convex function. 9 Of course, for any function defined in a neighborhood of a point p in a Riemannian manifold, the difference quotient at p is well defined for s sufficiently small.

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 429

(i) For p ∈ C and 0 < s1 ≤ s2 , we have (H.11)

Js1 (V ) ≤ Js2 (V ) for all V ∈ dom (Js2 ). Hence the directional derivative function on T˜p C, V → DV f (p) ∈ [−∞, ∞),

as defined in (H.8), exists for all p ∈ C. (ii) If f is locally Lipschitz at p ∈ C in the sense that there exist ε > 0 and L ∈ [0, ∞) such that |f (q) − f (p)| ≤ Ld (q, p) for q ∈ B (p, ε) ∩ C, then for any s > 0 with |sV | < ε we have (H.12)

|Js (V )| ≤ L |V | for all V ∈ dom (Js ) and we have

(H.13)

|DV f (p)| ≤ L |V |

for any V ∈ T˜p C. (iii) For any V ∈ T˜p C, if −V ∈ T˜p C, then (H.14)

(H.15)

− (D−V f ) (p) ≤ (DV f ) (p) . In particular, if (DV f ) (p) < 0, then (D−V f ) (p) > 0. Hence, if p ∈ int (C), then we have

max (DV f ) (p) : V ∈ Spn−1 ≥ 0, where Spn−1 ⊂ Tp M denotes the sphere of unit vectors.

Proof. (i) Let t = ss12 ∈ (0, 1]. Inequality (H.11) follows from

f expp (s1 V ) − f (p) s 1   f expp (1 − t) 0 + t (s2 V ) − f (p) = s   1

(1 − t) f expp 0 + tf expp (s2 V ) − f (p) ≤ s1

f expp (s2 V ) − f (p) = , s2 where we used that f is convex. From this monotonicity formula and (H.10) we conclude the existence of DV f (p). Note that √ for p ∈ ∂C, the value of DV f (p) could be −∞. For example, f (x) = − x has directional derivative −∞ at 0 ∈ [0, 1]  C. However if p ∈ int (C), then DV f (p) is finite for all V ∈ T˜p C.

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430

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(ii) We compute for s > 0 and V ∈ dom (Js ) that when |sV | < ε,    f exp (sV ) − f (p)  Ld exp (sV ) , p

  p p ≤ L |V | . |Js (V )| =  ≤   s s (iii) Since f is convex, for s > 0 small enough,

1

1 f (p) ≤ f expp (sV ) + f expp (−sV ) 2 2 and we have



f expp (sV ) − f (p) f expp (−sV ) − f (p) ≤ . − s s We conclude that − (D−V f ) (p) ≤ (DV f ) (p) and (DV f ) (p) ∈ (−∞, ∞).




Furthermore, we also have the following. Lemma H.26 (Convexity and upper semi-continuity of directional derivative). Let f be a convex function on a connected locally convex set C in (Mn , g). (i) If (M, g) = En , then the difference quotient Js is convex in V . Hence DV f (p) is a convex function of V on T˜p C. (ii) If f is Lipschitz in some neighborhood of p, then DV f (p) is a convex function of V on T˜p C. (iii) The function . ˜ Df : p∈C T p C → R, defined by Df (p, V )  DV f (p) , is upper semi-continuous. Proof. (i) For V1 , V2 ∈ T˜p C and t ∈ [0, 1], we compute on En that for s > 0 small enough, Js (tV1 + (1 − t) V2 ) f (p + s (tV1 + (1 − t) V2 )) − f (p) s f (t (p + sV1 ) + (1 − t) (p + sV2 )) − f (p) = s tf (p + sV1 ) + (1 − t) f (p + sV2 ) − f (p) ≤ s = tJs (V1 ) + (1 − t) Js (V2 ) . =

The convexity of DV f (p) in V follows from taking the limit, as s → 0+ , of the above inequality. (ii) Let γs (u), u ∈ [0, 1], be the minimal geodesic joining expp (sV1 ) and expp (sV2 ). Given t ∈ (0, 1), let γs (us (t)) be the point on γs (u)|[0,1] which

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2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 431

is closest to expp (s (tV1 + (1 − t) V2 )). From the pointed Cheeger–Gromov  

convergence of M, λ−2 g, p to Tp M, g (p) , 0 as λ → 0+ , we have as s → 0+ , us (t) − (1 − t) = O (s) ,

d γs (us (t)) , expp (s (tV1 + (1 − t) V2 )) = o (s) . We compute, using the convexity of f ◦ γs (u), that Js (tV1 + (1 − t) V2 )

f expp (s (tV1 + (1 − t) V2 )) − f (p) = s

f (γs (us (t))) − f (p) f expp (s (tV1 + (1 − t) V2 )) − f (γs (us (t))) + = s s (1 − us (t)) f (γs (0)) + us (t) f (γs (1)) − f (p) ≤ s

Ld expp (s (tV1 + (1 − t) V2 )) , γs (us (t)) + s t (f (γs (0)) − f (p)) + (1 − t) (f (γs (1)) − f (p)) ≤ s ((1 − t) − us (t)) (f (γs (0)) − f (γs (1))) + o (1) + s





t f expp (sV1 ) − f (p) + (1 − t) f expp (sV2 ) − f (p) + o (1) . = s Taking s → 0+ , this implies that DtV1 +(1−t)V2 f (p) ≤ tDV1 f (p) + (1 − t) DV2 f (p) . / (iii) Suppose that (pi , Vi ) ∈ p∈C T˜p C converges to . T˜p C. (p∞ , V∞ ) ∈ p∈C

For any given ε > 0, there exists s0 > 0 small enough such that     f exp (s V ) − f (p ) 0 ∞ ∞   p∞ − (DV∞ f ) (p∞ ) ≤ ε.    s0 Since f is a continuous function, there exists i0 (depending on s0 ) such that for i ≥ i0 



 f expp (s0 Vi ) − f expp (s0 V∞ )  ≤ εs0 , ∞ i |f (pi ) − f (p∞ )| ≤ εs0 . Hence we obtain     f exp (s V ) − f (p ) 0 i i   pi − (DV∞ f ) (p∞ ) ≤ 3ε.    s0

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

From Lemma H.25(i), we have for i ≥ i0

f exppi (s0 Vi ) − f (pi ) (DVi f ) (pi ) ≤ ≤ (DV∞ f ) (p∞ ) + 3ε. s0 This proves the upper semi-continuity of Df .




A function φ : Rn → R is said to be (finite) sublinear if φ is convex and positively homogeneous of degree 1. We have the following. Lemma H.27 (Sublinearity of directional derivative function). Let C ⊂ M be a connected locally convex set and let f : C → R be a convex function. (i) If for some p ∈ C and for some ε > 0 and L < ∞ we have |f (q1 ) − f (q2 )| ≤ L d (q1 , q2 ) for all q1 , q2 ∈ B (p, ε) ∩ C, then |DW f (p) − DV f (p)| ≤ L |W − V | for all V, W ∈ T˜p C. Hence, given p ∈ int (C), the directional derivative function V −→ DV f (p) is Lipschitz continuous. (ii) The function V → Df (p) (V ) is sublinear for p ∈ int (C). More precisely, if f is Lipschitz in some neighborhood of p ∈ C, then the function V → Df (p) (V ) is sublinear. Proof. (i) We compute |DW f (p) − DV f (p)|  



  f exp (sW ) − f (p) f exp (sV ) − f (p)   p p − lim =  lim   s→0+ s s s→0+  



 f exp (sW ) − f exp (sV )    p p = lim    s s→0+ 

Ld expp (sW ) , expp (sV ) ≤ lim s s→0+ = L |W − V | . (ii) The sublinearity of DV f (x) in V follows from Lemma H.23 and Lemma H.26(ii). The lemma is proved.
The following is an easy consequence of Lemma H.25(iii); for the definition of the directional derivative of a concave function, see (H.31) below. Exercise H.28 (Directional derivative of concave functions). Show that if f : C → R is concave, where C ⊂ M is a connected locally convex set, and if p ∈ int (C), then (DV f ) (p) exists for all V ∈ Tp M. Moreover, V → (DV f ) (p) is a concave function which is positively homogenous of degree 1 and − (D−V f ) (p) ≥ (DV f ) (p)

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3. GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

433

for all V ∈ Tp M. It follows that

min (DV f ) (p) : V ∈ Spn−1 ≤ 0. 3. Generalized gradients of convex functions on Riemannian manifolds If a function f is differentiable at a point p ∈ M, then we may define its gradient ∇f (p), which is a tangent vector at p. In this section we discuss the notion and properties of generalized gradients for locally Lipschitz convex functions on Riemannian manifolds. In this section (Mn , g) shall denote a connected complete Riemannian manifold. 3.1. Definition of generalized gradient. Let C ⊂ M be a connected locally convex set and let f : C → R be a convex function. Given p ∈ C, we assume that f is Lipschitz in some neighborhood of p. By Lemma H.27(i), DV f (p) exists and is a Lipschitz function of V on T˜p C and we can extend Df (p) to a Lipschitz function on its closure T˜p C. We make the following definition of a generalized gradient of f at p. Definition H.29 (Generalized gradient). If p ∈ C and if a convex function f : C → R is Lipschitz in some neighborhood of p, then there exists a unit vector W ∈ T˜p C ∩ Spn−1 such that   (H.16) (DW f ) (p) = min (DV f ) (p) : V ∈ T˜p C ∩ Spn−1  (Dmin f ) (p) . For any such W , the vector (H.17)

∇f (p) = |(DW f ) (p)| W ∈ T˜p C

is called a generalized gradient of f at p. Note that W is a direction of steepest descent and (H.18)

|∇f (p)| = |(Dmin f ) (p)| .

Hence a generalized gradient is a direction of steepest descent times the absolute value of the rate of steepest descent. The generalized gradient has the following elementary properties. Lemma H.30. Suppose f : C → R is a convex function which is Lipschitz in a neighborhood of a point p ∈ C. (i) If f is differentiable at p, then ∇f (p) is equal to the negative of the (standard ) gradient of f . (ii) Since W in (H.16) is not necessarily unique, neither is ∇f (p).

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(iii) Let W be as in (H.16). If (Dmin f ) (p) ≤ 0, then

(H.19) D∇f (p) f (p) = − |∇f (p)|2 ,

(H.20)

whereas if (Dmin f ) (p) ≥ 0, then

D∇f (p) f (p) = |∇f (p)|2 .

(iv) If for some ε > 0 and L < ∞ we have |f (x) − f (p)| ≤ Ld (x, p) for all x ∈ B (p, ε), then |∇f (p)| ≤ L. Proof. (i) This follows directly from definition (H.17). (ii) Consider the convex Lipschitz function f : Rn → R defined by f (x) = |x| at p = 0. Then W and ∇f (0) both may be any unit vector in S0n−1 . (iii) We compute



D∇f (p) f (p) = D|(DW f )(p)|W f (p) = |(DW f ) (p)| (DW f ) (p) = sign ((Dmin f ) (p)) · |∇f (p)|2 . (iv) This follows from Lemma H.25(ii).




If f is a convex function defined on a connected locally convex subset C of M, then we say that p ∈ C is a critical point of f if (Dmin f ) (p) ≥ 0. Lemma H.31 (Interior critical points of convex functions are local minima). Let f : C → R be a convex function. If p ∈ int (C) and (Dmin f ) (p) ≥ 0, then f attains a local minimum at p. Proof. Since f is convex, for V ∈ Spn−1 and s ≥ 0, we have that

f expp (sV ) is a convex function of s. Hence for s ∈ [0, s¯]

f expp (sV ) ≥ f (p) + s (DV f ) (p)

sV ) ∈ C. The lemma follows from provided expp (¯ (DV f ) (p) ≥ (Dmin f ) (p) ≥ 0 for all V ∈ Spn−1 .




More generally, for a convex function f : C → R, if p ∈ int (C), then there is a neighborhood U of p in C ∩ B (p, inj (p)) such that   f (p) (H.21) f (x) ≥ f (p) + Dexp−1 p (x) for x ∈ U . Now we give a definition.

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Definition H.32 (Convex functions (zero at the boundary) — CX0 (C)). Let C ⊂ M be a closed connected locally convex set with nonempty boundary and interior. We denote by CX0 (C) the collection of locally Lipschitz convex functions f : C → R satisfying f |∂C = 0. The derivatives of such functions have the following properties. Lemma H.33 (Convexity of Df (p) for f ∈ CX0 (C)). Let f ∈ CX0 (C) and let p ∈ ∂C. (i) Suppose Vi ∈ T˜p C ∩ Spn−1 is a sequence which converges to a point V∞ in the boundary of T˜p C ∩ Spn−1 in Spn−1 . Then (DVi f ) (p) → 0. Hence the Lipschitz extension Df (p) : T˜p C → R satisfies Df (p) (V )  DV f (p) = 0

on T˜p C − T˜p C.

(ii) The function Df (p) : T˜p C → R is convex. Hence by Proposition H.5, Df (p) is locally Lipschitz. Proof. (i) Since Vi is in the interior tangent cone T˜p C, by (H.4) and the definition of directional derivative, there is a sequence s∗i → 0+ such that for all s ∈ (0, s∗i ), (H.22) and

expp (sVi ) ∈ int (C)     f exp (sV ) − f (p) i   p − (DVi f ) (p) ≤ 2−i .    s

Since V∞ is in the boundary of T˜p C ∩ Spn−1 in Spn−1 , there exist si ∈ (0, s∗i ) such that (H.23)

/ int (C) . expp (si V∞ ) ∈

Consider the shortest path on Spn−1 joining Vi to V∞ . By (H.22) and (H.23), there is a point Vi on this path such that

expp si Vi ∈ ∂C. Let Lp be the Lipschitz constant of f on some

neighborhood of p in C. Since  f ∈ CX0 (C), we have f (p) = f expp (si Vi ) = 0. Using this, we compute    f exp (s V ) − f (p)    p i i |DVi f (p)| ≤   + 2−i   si  

 f exp (s V ) − f exp (s V  )  i i i   p p i =  + 2−i   si

d expp (si Vi ) , expp (si Vi ) ≤ Lp + 2−i . si

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By the Toponogov comparison theorem (the sectional curvature is bounded from below on compact subsets of M), there is a constant C0 < ∞ such that for |V | , |W | ≤ 1 we have

d expp (V ) , expp (W ) ≤ C0 |V − W | . Hence C0 |si Vi − si Vi | + 2−i si   = Lp Vi − Vi  + 2−i → 0 as i → ∞.

|DVi f (p)| ≤ Lp

(ii) Since T˜p C is convex and hence connected, we may apply Lemma H.26(ii) to obtain that Df (p) : T˜p C → R is convex. That its Lipschitz extension Df (p) : T˜p C → R is convex follows from a simple limiting argument. The lemma is proved.
The following is a simple nonsmooth example. Example H.34 (Square in the plane). Let C  [−1, 1] × [−1, 1] ⊂ R2 , which is a compact convex subset. Define f : C → R by f (x, y) = −d ((x, y), ∂C) . Then (H.24)

f (x, y) = max {|x| , |y|} − 1

and f ∈ CX0 (C), i.e., f is convex and f |∂C = 0. Let D  {(x, y) ∈ C : |x| = |y|} . Then f is smooth in C − D and only Lipschitz continuous on D. The above example has the following generalized gradient. Exercise H.35. Show that (or at least convince yourself that) in Example H.34: (1) (a) If x > |y|, then ∇f (x, y) = (−1, 0). (b) If x < − |y|, then ∇f (x, y) = (1, 0). (c) If y > |x|, then ∇f (x, y) = (0, −1). (d) If y < − |x|, then ∇f (x, y) = (0, 1). (2)

(a) If x = y > 0, then ∇f (x, y) = − 12 , − 12 . (b) If x = y < 0, then ∇f (x, y) = 12 , 12 .

(c) If x = −y > 0, then ∇f (x, y) = − 12 , 12 . (d) If x = −y < 0, then ∇f (x, y) = 12 , − 12 .

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Note that ∇f (x, y) is not continuous in (x, y) (it takes eight discrete values). Furthermore, |∇f | is also not continuous: |∇f | = 1 if |x| = |y|, whereas |∇f | = √12 if |x| = |y|. For functions in CX0 (C), the generalized gradient is unique. Lemma H.36 (Uniqueness of the generalized gradient). Let f ∈ CX0 (C). Then (i) For any p ∈ C, T˜p C is nonempty. (ii) If C is compact, then f ≤ 0 on C. Consequently, unless f ≡ 0 on C, finf  inf {f (x) : x ∈ C} < 0. (iii) For any p ∈ C we can choose a generalized gradient ∇f (p) which is contained in T˜p C . (iv) (Uniqueness) If finf < 0, then whenever f (p) > finf , we have that (DV f ) (p) attains its negative minimum on T˜p C ∩ Spn−1 at a unique vector. Hence the generalized gradient ∇f (p) is unique and nonzero. Proof. (i) Suppose q ∈ C and r > 0 are such that (H.25)

B (q, r) ⊂ C.

Since C is connected and locally convex, Lemma H.15 implies that there is a smooth path γ : [0, 1] → C, joining p = γ (0) and q = γ (1), which is minimal among paths in C and which is a geodesic in M. Define ι0  inf inj (γ (s)) . s∈[0,1]

By Lebesgue’s number lemma, we can divide the geodesic γ into (short) segments γ|[si ,si+2 ] , i = 0, . . . , m − 2, where m ≥ 2, such that ι0 and (1) each segment is a smooth minimal geodesic with length ≤ 10 (2) γ|[si−2 ,si ] is contained in the convex set C ∩ U i , where Ui is an open set, for i = 2, . . . , m. Here s0 = 0 and sm = 1. We now prove inductively that each γ (si ), 1 ≤ i ≤ m, is in int (C). Since γ (sm ) = q, by (H.25) we know that γ (sm ) ∈ int (C). Suppose

γ (si ) ∈ B (γ (si ) , ri ) ⊂ int (C) ∩ U i for some i ≥ 2 and ri > 0. We now prove γ (si−1 ) ∈ int (C). Note that the open ‘geodesic cone’  ⎫ ⎧  ⎪ 0 < s < sW and expγ(si−2 ) (sW ) is ⎪ ⎬ ⎨ [0,s ] expγ(si−2 ) (sW ) : the minimal geodesic joining γ (s W i−2 ) ⎪ ⎪ ⎭ ⎩ and some point in B (γ (si ) , ri ) is contained in C ∩ U i . It is clear that the open geodesic cone contains γ (si−1 ). Hence γ (si−1 ) ∈ int (C).

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Let U1 be an open neighborhood of γ (s1 ) such that C ∩ U 1 is a convex set. Now B (γ (s1 ) , r1 ) ⊂ int (C) ∩ U 1 for some r1 > 0 and the open cone  ⎫ ⎧  ⎪ is ⎪ 0 < s < sW and expγ(0) (sW ) ⎬ ⎨ [0,s ] expγ(0) (sW ) : the minimal geodesic joining γ (0)W ⎪ ⎪ ⎭ ⎩ and some point in B (γ (s1 ) , r1 ) contains every point γ (s), 0 < s < s1 . Hence γ  (0) ∈ T˜p C and therefore T˜p C is nonempty. (ii) We prove f ≤ 0 on C by contradiction. Suppose that this statement is false. Since C is compact and f is continuous on C and vanishes on the boundary, there is a qsup ∈ int (C) such that f (qsup ) = sup f (q) > 0. q∈C

Consider any minimal geodesic segment γ : [−ε, ε] → int (C) with γ (0) = qsup . Then f ◦ γ is a convex function on [−ε, ε] which attains its maximum at s = 0. Hence f ◦ γ is a constant function and therefore f is a constant function in some neighborhood of qsup . Since we can prove that f is a locally constant function at any point q˜ with f (˜ q ) = supq∈C f (q) and since C is connected, we conclude that f is a constant function on C with f (qsup ) > 0. This is impossible because f = 0 on ∂C. The claim is proved. (iii) We only need to prove (iii) for p ∈ ∂C. By (ii) we have (DV f ) (p) ≤ 0 for all V ∈ T˜p C. So (iii) now follows from (i) and DV f (p) = 0 on T˜p C − T˜p C by Lemma H.33(i). (iv) For p ∈ ∂C, let γ : [0, 1] → C be a geodesic which is minimal in C among those paths connecting p to some point q ∈ int (C) with f (q) < 0. By the proof of (i) we have γ  (0) ∈ T˜p C. By the convexity of f ◦ γ (s), by f ◦ γ (0) = 0, and by f ◦ γ (1) < 0, we conclude that Dγ  (0) f (p) < 0. On the other hand, for p ∈ int (C), if (DV f ) (p) has a nonnegative minimum on T˜p C ∩Spn−1 = Spn−1 , then from Lemma H.31 we conclude that f has a local minimum at p. The convexity of f implies that p must be a global minimum point of f , which contradicts f (p) > finf . Hence   (H.26) min (DV f ) (p) : V ∈ T˜p C ∩ Spn−1 < 0. Now let p ∈ C and suppose that there are two distinct vectors V1 and V2 in T˜p C ∩ Spn−1 such that (DV1 f ) (p) = (DV2 f ) (p) =

min V ∈T˜p C∩Spn−1

(DV f ) (p) < 0.

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  Then  12 V1 + 12 V2  < 1, 12 V1 + 12 V2 ∈ T˜p C, and 12 V1 + 12 V2 = 0 by Lemma H.25(iii). From the convexity of (DV f ) (p) in V , we have   1 1 min (DV f ) (p) . D 1 V1 + 1 V2 f (p) ≤ (DV1 f ) (p) + (DV2 f ) (p) = 2 2 2 2 V ∈T˜p C∩Spn−1 It follows from ⎛ ⎝D |

1V +1V 2 1 2 2 1V +1V 2 1 2 2

1 V +1V 2 1 2 2 1 V +1V 2 1 2 2

| ⎞

f ⎠ (p) ≤ |

|

∈ T˜p C ∩ Spn−1 that

minV ∈T˜ C∩S n−1 (DV f ) (p)  p1 p 1  <  V1 + V2  2

2

min V ∈T˜p C∩Spn−1

(DV f ) (p) .

This is a contradiction and hence the minimum in (H.26) is attained by a
unique vector in T˜p C ∩ Spn−1 . This proves (iv) and the lemma. 3.2. Properties of generalized gradients of convex functions. Recall that CX0 (C) is given by Definition H.32 and finf  inf {f (q) : q ∈ C} . Lemma H.37 (Directional derivatives and the generalized gradient). Let f ∈ CX0 (C). Suppose finf < 0. (1) If p ∈ C is such that f (p) = finf , then (H.27)

(DV f ) (p) ≥ − ∇f (p) , V 

for any V ∈ Tp M,

where ∇f (p) is any generalized gradient vector. (2) If p ∈ C is such that f (p) > finf , then (H.28)

(DV f ) (p) ≥ − ∇f (p) , V 

for any V ∈ T˜p C,

where ∇f (p) is the unique generalized gradient vector. Proof. (1) If f (p) = finf , then p ∈ int (C) and (DV f ) (p) ≥ 0 for all V ∈ Tp M. Hence, by (H.18), we have    (DV f ) (p) : V ∈ Tp M − 0 . |∇f (p)| = min |V | Thus, for all V ∈ Tp M, (DV f ) (p) ≥ |V | · |∇f (p)| ≥ − ∇f (p) , V  . (2) If f (p) > finf , then there is a (unique) ∇f (p) ∈ T˜p C by Lemma H.36(iii). For any V ∈ T˜p C, by Lemma H.18(ii) we have ∇f (p) + λV ∈ T˜p C

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

for any λ ∈ (0, ∞). By the definition of the generalized gradient, we have  

D∇f (p)+λV f (p) = |∇f (p) + λV | D ∇f (p)+λV f (p) |∇f (p)+λV |

≥ − |∇f (p) + λV | · |∇f (p)| " = − |∇f (p)| |∇f (p)|2 + 2λ ∇f (p) , V  + O (λ2 )

= − |∇f (p)|2 − λ ∇f (p) , V  + O λ2 . Since Df (p) : T˜p C → R is convex (which follows from Lemma H.33(ii)), we have  

D∇f (p)+λV f (p) = (1 + λ) D 1 ∇f (p)+ λ V f (p) (1+λ) (1+λ)   λ 1 D (DV f ) (p) f (p) + ≤ (1 + λ) (1 + λ) ∇f (p) (1 + λ) = − |∇f (p)|2 + λ (DV f ) (p) . We have proved for any λ > 0, − ∇f (p) , V  + O (λ) ≤ (DV f ) (p) . Inequality (H.28) follows by taking λ → 0.




In general, if f : M → R is a C 1 function, then wherever ∇f = 0, the level sets {x ∈ M : f (x) = s} of f are C 1 hypersurfaces and the vector field ∇f is orthogonal to these level sets. Although convex functions are not necessarily C 1 , we have the following.10 Lemma H.38 (∇f makes acute angles with ‘f -nonincreasing’ geodesics). Let f ∈ CX0 (C). Suppose that finf < 0. If x, y ∈ C are such that f (x) > finf , f (x) ≥ f (y), and x = y, then for any constant speed geodesic γ : [0, 1] → C with γ (0) = x and γ (1) = y, we have π  (∇f (x) , γ˙ (0)) ≤ . 2 Proof. Clearly γ˙ (0) ∈ T˜p C, which, by Lemma H.37, implies

f (γ (s)) − f (x) . f (x) = lim − ∇f (x) , γ˙ (0) ≤ Dγ(0) ˙ + s s→0 On the other hand, since f is convex, f (γ (s)) ≤ (1 − s) f (γ (0)) + sf (γ (1)) = (1 − s) f (x) + sf (y) ≤ f (x) for s ∈ [0, 1]. We conclude that ∇f (x) , γ˙ (0) ≥ 0.




10 The geodesic γ joins a point at a higher ‘f -level’ to a point at a lower f -level; this is what we mean by γ being ‘f -nonincreasing’.

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We remark that, in the proof of the Sharafutdinov retraction Theorem H.59, we shall apply the above result a few times. 3.3. Differentiability of convex functions on Riemannian manifolds. Let V be an open subset of (Mn , g). We say that a continuous function f : V → R has nonnegative Hessian in the support sense if for any p ∈ V and any ε > 0 there is a neighborhood U ⊂ V of p and a C 2 ‘local lower barrier function’ ϕ : U → R such that ϕ (p) = f (p), f (x) ≥ ϕ (x) for all x ∈ U , and ∇∇ϕ (p) ≥ −ε. We denote this by Hess supp (f ) ≥ 0. Like convex functions on convex sets in Euclidean spaces, convex functions on convex sets in Riemannian manifolds have the following differentiability property. Note that Definition H.7 regarding second derivatives can easily be adapted to manifolds by using local coordinates. That is, a function f on (Mn , g) is twice differentiable in the sense of Stolz at x ∈ M if for any C ∞ local coordinates φ defined in a neighborhood of x we have f ◦ φ−1 is twice differentiable in the sense of Stolz at φ (x). Lemma H.39 (Convex functions have nonnegative Hessian). Let C ⊂ M be a connected locally convex set with nonempty boundary and interior. Suppose f : C → R is convex. Then for any p ∈ int (C) we have (H.29)

Hess supp (f ) ≥ 0

and hence f has second-order derivatives almost everywhere on int (C). Proof. For any p ∈ int (C) and V ∈ Tp M with |V | = 1, by (H.21) we have

f expp (sV ) ≥ f (p) + sDV f (p) for s ≥ 0. An adaptation of the proof of Lemma H.37 yields

f expp (sV ) ≥ f (p) − s ∇f (p) , V  , where the generalized gradient ∇f (p) is unique. Hence

ϕ expp (sV )  f (p) − ∇f (p) , sV  is a smooth function defined in a neighborhood of p and ϕ is a local lower barrier function for the continuous function f at p. Since Hess (ϕ) (p) = 0, we conclude that Hess supp (f ) (p) ≥ 0 and hence Hess supp (f ) ≥ 0

on int (C) .

Let {x} be normal coordinates centered at p. It follows from Lemma 7.122(iii) in Part I that there exists a C 2 function ψ defined in a neighborhood of p such that f + ψ is a convex function of x, in the Euclidean sense, for |x| sufficiently small. Hence, by the Aleksandrov theorem on convex functions (see Lemma 7.117(ii) in Part I), the second derivative D 2 f exists a.e. on int (C).


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4. Integral curves to gradients of concave functions In this section we shall discuss integral curves for the generalized gradients of concave functions and their properties. The main result is Sharafutdinov’s Theorem H.59. The proof of this theorem meshes better with concave functions as opposed to convex functions. Our prior discussion of convex functions applies to concave functions by changing the sign. Throughout this section (Mn , g) shall denote a connected complete Riemannian manifold. Let C ⊂ M be a closed connected locally convex set with nonempty interior and boundary. We denote by (H.30)

CV0 (C)

the collection of locally Lipschitz concave functions f : C → R satisfying f |∂C = 0; i.e., f ∈ CV0 (C) if and only if −f ∈ CX0 (C), where CX0 (C) is given by Definition H.32. For f ∈ CV0 (C) we define the generalized gradient by ∇f  ∇ (−f ) (by our convention, without the ‘extra’ minus sign on the rhs). By Lemma H.36, we have for p ∈ C, ∇f (p) = |(DU f ) (p)| U for some U ∈ T˜p C ∩ Spn−1 with   (H.31) (DU f ) (p) = max (DV f ) (p) : V ∈ T˜p C ∩ Spn−1  (Dmax f ) (p) . From Lemma H.30(i), if f ∈ CV0 (C) is differentiable at p, then ∇f (p) is equal to the (standard) gradient of f . For a concave function f ∈ CV0 (C), we say that a point p ∈ C is a critical point if (Dmax f ) (p) ≤ 0. 4.1. Right tangent vectors. Extending the definition of tangent vector for a C 1 path, we have the following. Definition H.40 (Right tangent vector). Let γ : [a, b] → (Mn , g) be a continuous path and let s ∈ [a, b). A vector γ˙ + (s) ∈ Tγ(s) M is said to be the right tangent vector to γ at s if for every C ∞ function h : M → R the right derivative h (γ (s + σ)) − h (γ (s)) d (h ◦ γ)  lim + + ds σ σ→0 exists and (H.32)

d (h ◦ γ) = γ˙ + (s) (h) . ds+

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Note that by (H.32), the right tangent vector, if it exists, is unique. We have the following characterization of the right tangent vector at a ‘Lipschitz point’ of a path. Lemma H.41 (Right tangent vector of a continuous path). Let γ : [a, b] → (Mn , g) be a continuous path and let γs,σ : [0, σ] → M be a constant speed minimal geodesic joining γ (s) to γ (s + σ). If γ is locally Lipschitz at s ∈ [a, b) in the sense that d (γ (s + σ) , γ (s)) = O (σ)

when σ → 0+ ,

then we have γ˙ + (s) = lim γ˙ s,σ (0) = lim

exp−1 (γ (s + σ)) γ(s)

σ provided either γ˙ + (s) exists or limσ→0+ γ˙ s,σ (0) exists. σ→0+

σ→0+

Proof. (1) If γ˙ + (s) exists. Suppose that, for some s ∈ [a, b), the right tangent vector γ˙ + (s) exists. Then, for every smooth function h : M → R,11 γ˙ + (s) (h) =

d h (γ (s + σ)) − h (γ (s)) . (h ◦ γ) (s) = lim + ds σ σ→0+

Since (H.33)

h (γ (s + σ)) − h (γ (s)) = γ˙ s,σ (0) (h) σ + o (d (γ (s + σ) , γ (s))) = γ˙ s,σ (0) (h) σ + o (σ) ,

we conclude γ˙ + (s) (h) = lim γ˙ s,σ (0) (h) , σ→0+

so that γ˙ + (s) = limσ→0+ γ˙ s,σ (0). (2) If limσ→0+ γ˙ s,σ (0) exists. Now suppose for s ∈ [a, b) the limit limσ→0+ γ˙ s,σ (0) exists. Then by (H.33) we have lim γ˙ s,σ (0) (h) = lim

σ→0+

σ→0+

h (γ (s + σ)) − h (γ (s)) = γ˙ + (s) (h) . σ


Remark H.42. Note that for σ sufficiently small there exists a unique small vector V ∈ Tγ(s) M such that expγ(s) (V ) = γ (s + σ) , i.e., exp−1 γ(s) (γ (s + σ)) is well defined. 11 In particular, h is defined on some convex neighborhood of γ (s) in M and h is differentiable at γ (s).

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

4.2. Integral curves for concave functions and their properties. Recall that CV0 (C) is defined in (H.30). We denote fsup  supy∈C f (y) for f ∈ CV0 (C). Definition H.43 (Integral curves for generalized gradient ∇f / |∇f |2 ). Suppose x ∈ C and f (x)  a. (1) A continuous path γx : [a, b] → C is an integral curve for (the generalized gradient) ∇f / |∇f |2 emanating from x if it satisfies the following properties: (i) γx (a) = x, (ii) γx |[a,b) is locally Lipschitz (and hence rectifiable), (iii) for all s ∈ [a, b) we have that f (γx (s)) < fsup , the right tangent vector (γ˙ x )+ (s) exists, and (H.34)

(γ˙ x )+ (s) =

∇f (γx (s)) |∇f (γx (s))|2

,

where ∇f is the unique and nonzero (see Lemma H.36(iv)) generalized gradient of f given by (H.17). (2) We say that such an integral curve is maximal if f (b) = fsup . Note that if f is smooth, then (γ˙ x )+ (f ) ≡ 1. Exercise H.44 (Example H.34 revisited). For C  [−1, 1] × [−1, 1] and the function f (x, y) = d ((x, y), ∂C) (which is the negative of the function in (H.24)), what is the maximal integral curve for ∇f / |∇f |2 emanating from (x, y) ∈ C? We prove some properties of integral curves assuming their existence, which is established in the next subsection. Lemma H.45 (f along integral curve for ∇f / |∇f |2 ). Let f ∈ CV0 (C). Let x ∈ C be a point such that f (x)  a < fsup . If b ∈ (a, fsup ] and γx : [a, b] → C is an integral curve for ∇f / |∇f |2 emanating from x, then (H.35)

f (γx (s)) = s

for all s ∈ [a, b]. Proof. Define φ (s)  f (γx (s)) . Since φ (s) is locally Lipschitz for s ∈ [a, b), its derivative φ (s) exists for a.e. s ∈ (a, b). We claim that (H.36)

d φ (s) = 1 for a.e. s ∈ (a, b). ds+

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This implies φ (s) = 1 a.e. Hence, by Lemma H.3 we have  s¯  s¯ φ (s) ds = ds = s¯ − a, φ (¯ s) − φ (a) = a

a

so that φ (¯ s) = s¯ as desired. We now prove the claimed equality (H.36). Since f is concave, for any p ∈ int (C) such that f (p) < fsup , by Lemma H.31 we have for U in (H.31)12 (DU f ) (p) = (Dmax f ) (p) > 0. Then, by applying (H.19) to −f , we have (H.37)

D

∇f (p) |∇f (p)|2

f = 1.

By the definition (H.8) of directional derivative, for s ∈ [a, b),   f expγx (s) (V ) − f (γx (s)) = DV f + o (|V |) as |V | → 0. As a special case of this, we have13 f (γx (s + σ)) − f (γx (s)) = Dexp−1

γx (s)

(γx (s+σ)) f

+ o (σ) .

Thus f (γx (s + σ)) − f (γx (s)) d φ (s) = lim + + ds σ σ→0 = lim D exp−1 (γx (s+σ)) f σ→0+

(H.38)

=D limσ→0+

γx (s)

σ exp−1 (γx (s+σ)) γx (s) σ

f,

where the last equality follows from the fact that DV f (γx (s)) is a continuous function of V . It follows from Lemma H.41 and (H.34) that for s ∈ (a, b), (H.39)

lim

σ→0+

exp−1 γx (s) (γx (s + σ)) σ

= (γ˙ x )+ (s) =

∇f |∇f |2

(γx (s)) .

Now (H.37) with p = γx (s) and (H.38) imply d φ (s) = D ∇f (γx (s)) f = 1. ds+ |∇f |2
In view of (H.39), the following should be true. Problem H.46 (Right continuity of ∇f / |∇f |2 along its integral curves). Let f ∈ CV0 (C). Show that the vector field ∇f (and hence ∇f / |∇f |2 ) is right continuous along the integral curves to ∇f / |∇f |2 . 12

Indeed, by Lemma H.31, if p ∈ int (C) is such that (Dmax f ) (p) ≤ 0, then f (p) =

fsup . 13

Note that γx (s) ∈ int (C) for s ∈ (a, b).

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446

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Remark H.47. A nice example to keep in mind is the function given in Example H.34. The following result says that f is concave along the reparametrized integral curves for ∇f / |∇f |2 . Lemma H.48. Let γ be an integral curve for ∇f / |∇f |2 and let γ˜ be a reparametrization of γ by arc length. Then the function f ◦ γ˜ is concave. Proof. The existence of such a γ˜ follows from Lemma G.5 and the fact that γ is rectifiable. Now for s1 > s0 with s1 − s0 < injg (p), we have (the first equality follows from Lemma H.45) d (f ◦ γ˜ ) (s0 ) = |∇f | (˜ γ (s0 )) = Dmax f ds+ Dexp−1 (˜γ (s1 )) f γ ˜ (s0 ) ≥ d (˜ γ (s0 ) , γ˜ (s1 )) γ (s0 )) f (˜ γ (s1 )) − f (˜ ≥ d (˜ γ (s0 ) , γ˜ (s1 )) γ (s0 )) f (˜ γ (s1 )) − f (˜ ≥ s1 − s0     (note that exp−1 (˜ γ (s )) γ (s0 ) , γ˜ (s1 )) > 0 and, since γ˜ is paramet1  = d (˜ γ ˜ (s0 ) rized by arc length, that we have d (˜ γ (s0 ) , γ˜ (s1 )) ≤ s1 − s0 ). Thus f ◦ γ˜ is concave (see (H.7) and the sentence thereafter).
The next result implies the uniqueness of integral curves for ∇f / |∇f |2 . Roughly speaking, the result says that the distance between points ‘at the same level’ on two integral curves is nonincreasing. The main idea of its proof is to apply the first variation of arc length formula. Lemma H.49 (Monotonicity of distance between integral curves for the vector field ∇f/|∇f |2 ). Let C, x, a, and f be as in Lemma H.45. Let b ∈ (a, fsup ] and let y ∈ C with f (y) = a = f (x). If γx : [a, b] → C

and

γy : [a, b] → C

2

are integral curves for ∇f / |∇f | emanating from x and y, respectively, then the function s → dC (γx (s) , γy (s)) is nonincreasing for s ∈ [a, b]. Proof. Define ρ : [a, b] → [0, ∞) by ρ (s)  dC (γx (s) , γy (s)) . Since γx |[a,b) and γy |[a,b) are locally Lipschitz, we have that ρ|[a,b) is a locally Lipschitz function. To prove the lemma, it suffices to show that ρ (s + ∆s) − ρ (s) dρ ≤0 (s)  lim sup ds+ ∆s + ∆s→0

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4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS

447

for s ∈ (a, b). By Lemma H.15, for each s ∈ (a, b) there exists a constant speed geodesic (of (M, g)) αs : [0, 1] → C with αs (0) = γx (s), αs (1) = γy (s), and L (αs ) = dC (γx (s) , γy (s)) = ρ (s) . By hypothesis, f (x) = a = f (y). Hence we may apply Lemma H.38 to obtain   ∇f (γx (s)) π (H.40a) ≤ ,  α˙ s (0) , 2 2 |∇f (γx (s))|   ∇f (γy (s)) π (H.40b) ≤ .  −α˙ s (1) , 2 2 |∇f (γy (s))| We claim that by the first variation formula for arc length, this implies dρ (s) ≤ 0. ds+ To see the claim, consider any smooth vector field V along αs with V (αs (0)) =

∇f (γx (s)) |∇f (γx (s))|2

and

V (αs (1)) =

∇f (γy (s)) |∇f (γy (s))|2

.

Note that γx (s) , γy (s) ∈ int (C) since s ∈ (a, b). This implies αs (u) ∈ int (C) for all u ∈ [0, 1]. Thus there exists a smooth 1-parameter family of paths βs¯ : [0, 1] → C, defined for s¯ ∈ [s, s + ε) for some ε > 0, with

 ∂  βs¯ ∂¯ s 

βs (u) = αs (u)

for u ∈ [0, 1] ,

s) βs¯ (0) = γx (¯

for s¯ ∈ [s, s + ε),

s) βs¯ (1) = γy (¯

for s¯ ∈ [s, s + ε),

(u) = V (αs (u))

for u ∈ [0, 1] .

s¯=s

By the first variation formula, (H.41)

 0 1 0 1  d L (βs¯) = − β˙ s (0) , V (αs (0)) + β˙ s (1) , V (αs (1)) d¯ s s¯=s 6 ) 5 ( ∇f (γy (s)) ∇f (γx (s)) + α˙ s (1) , = − α˙ s (0) , |∇f (γx (s))|2 |∇f (γy (s))|2 ≤ 0,

s) for s¯ ∈ [s, s+ε) and L (βs ) = ρ (s), where we used (H.40). Since L (βs¯) ≥ ρ (¯ dρ (s) ≤ 0.
this implies ds +

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448

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Corollary H.50 (Uniqueness of integral curves for ∇f / |∇f |2 ). If x ∈ C with f (x) < fsup , then there is at most one integral curve for ∇f / |∇f |2 starting at x. Exercise H.51. Check directly for Example H.34 that Lemma H.49 holds. Roughly, the following result says that the distance from points on an integral curve to a fixed point at a ‘higher level’ is nonincreasing. Lemma H.52 (Distance from γx (s) to y ∈ C is nonincreasing). Let C, x, a, and f be as in Lemma H.45. Let y ∈ C with a < f (y)  b. If γx : [a, b] → C is an integral curve for ∇f / |∇f |2 emanating from x, then the function s → dC (γx (s) , y) is nonincreasing for s ∈ [a, b]. Proof. Let σ (s)  dC (γx (s) , y) . By Lemma H.15, for each s ∈ [a, b] there exists a constant speed geodesic αs : [0, 1] → C with αs (0) = γx (s), αs (1) = y, and L (αs ) = dC (γx (s) , y) = σ (s) . Since f (γx (s)) = s ≤ b = f (y), by Lemma H.38,   ∇f (γx (s)) π ≤  α˙ s (0) , 2 2 |∇f (γx (s))| for s ∈ [a, b]. We leave it as an exercise to show that, by the first variation formula (which is very similar to the proof of Lemma H.49), this implies dσ (s) ≤ 0 in the sense of the lim sup of forward difference quotients.
ds+ 4.3. The existence of maximal integral curves. In this subsection we prove that the maximal integral curves for the generalized gradients of concave functions always exist. 4.3.1. The main result. Proposition H.53 (Existence of maximal integral curve for ∇f / |∇f |2 ). Let C ⊂ (Mn , g) be a compact14 connected locally convex set with nonempty interior and boundary and let f ∈ CV0 (C). If x ∈ C is such that f (x)  a < fsup , then there exists a maximal integral curve γx : [a, fsup ] → C for ∇f / |∇f |2 emanating from x. 14 The use of the compactness assumption for C in the proof of Proposition H.53 may be traced back to Lemma H.55 below.

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4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS

449

Proof. Assuming Lemma H.58 below on the existence of integral curves (which we shall prove below), we prove that any integral curve can be extended to a maximal integral curve. Let {si }∞ i=0 be a sequence in [a, fsup ) with s0 = a, si < si+1 , and si → fsup . From Lemma H.58, we can define γ 0 : [s0 , s1 ] → C to be the integral curve of ∇f / |∇f |2 emanating from γ 0 (s0 ) = x and for i ≥ 1 we can inductively define γ i : [si , si+1 ] → C to be the integral curve of ∇f / |∇f |2 emanating from γ i (si ) = γ i−1 (si ). We define γ˜x : [a, fsup ) → C by γ˜x |[si ,si+1 ] (s) = γ i (s) . Clearly, for any b < fsup , the curve γ˜x |[a,b] is an integral curve of ∇f / |∇f |2 . We shall prove that lim γ˜x (s) s→fsup

exists, from which the proposition follows. Let q be any point in f −1 (fsup ). By Lemma H.52, we have γ˜x (s) ∈ B (q, dC (x, q)) . si ) → q˜. Hence there exist a sequence s˜i → fsup and q˜ ∈ C such that γ˜x (˜ si )) = s˜i that f (˜ q ) = fsup . Applying Lemma H.52 to It follows from f (˜ γx (˜ γx (s) , q˜), we conclude that γ˜x (s) → q˜ as s → fsup . This completes the dC (˜ proof of the proposition.
Next we turn to the proof of the existence of integral curves, i.e., Lemma H.58. 4.3.2. The proof of the existence of the integral curve. In this subsection let C, x, and f be as in Proposition H.53. Define the convex (superlevel) set Cσ  {y ∈ C : f (y) ≥ σ} . Fix y ∈ Cσ and given s ∈ (0, fsup − f (y)] (so that f (y) < f (y) + s ≤ fsup ), choose y s to be a point in Cf (y)+s closest to y (such a point always exists). Note that (H.42)

f (y s ) = f (y) + s.

The following gives an upper bound for the distance between level sets. Lemma H.54 (Distance to a superlevel set). Let y0 ∈ C be a point with f (y0 ) < fsup . For ε > 0 there exists a neighborhood U of y0 and number s& > 0 such that for all y ∈ C ∩ U and s ∈ (0, s& ) we have s ≥ |∇f (y0 )| − ε ; dC (y, y s )

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450

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

in particular, if |∇f (y0 )| > ε, then dC (y, y s ) ≤

s . |∇f (y0 )| − ε

Proof. By Lemma H.36(iv) and f (y0 ) < fsup it follows that |∇f (y0 )| = 0. Let W0  |∇f1(y0 )| ∇f (y0 ). Then W0 ∈ T˜y0 C ∩ Syn−1 0

DW0 f (y0 ) = |∇f (y0 )| .

and

Choose an ε1 > 0 such that 1 (|∇f (y0 )| − ε1 ) ≥ |∇f (y0 )| − ε. 1 + ε1 We identify a neighborhood of (y0 , W0 ) in T M with U0 × Ty0 M, where U0 is a neighborhood of y0 ,/U0 is compact, and U0 ∩ C is convex. By the lower semi-continuity of Df : p∈C T˜p C → R at (y0 , W0 ) (Lemma H.26(iii)), there is a neighborhood U1 ⊂ U1 ⊂ U0 of y0 and a neighborhood V1 in Ty0 M of W0 satisfying V1 ⊂ T˜y C for all y ∈ U1 ∩ C (by a natural identification, T˜y C may be considered as a subset of Ty0 M) such that DW f (y) ≥ DW0 f (y0 ) − ε1 = |∇f (y0 )| − ε1 and |W | < 1 + ε1 

. ˜ for all (y, W ) ∈ U1 × V1 ∩ C . T p∈C p ˆ (y0 ) = W0 . Define ˆ : U1 → V1 be a continuous map with W Let W h : U1 ∩ C → [0, ∞) by ⎫ ⎧   ˆ (y) ∈ U1 ∩ C for s ∈ (0, s1 ], ⎬ ⎨ expy sW   . h(y)  sup s1 : d ˆ (y) ∈ V1 for s ∈ (0, s1 ] ⎭ ⎩ exp s W y ds Clearly h(y) > 0 and h is lower semi-continuous on U1 ∩ C. Choose a neighborhood U2 ⊂ U2 ⊂ U1 of y0 ; we define s∗  inf h (y) , y ∈ U2 ∩ C . Note that s∗ > 0. Given a y ∈ U2 ∩ C, we define a concave function   g : 0, 12 s∗ → R    ˆ (y) . By the choice of U2 , U1 , and V1 , we have by g (s)  f expy sW  (s) = D d exp (sW g+ ˆ (y)) f (y) ≥ |∇f (y0 )| − ε1 ; y ds  1  hence for s ∈ 0, 2 s∗  s     ˆ g+ (s) ds ≥ s (|∇f (y0 )| − ε1 ) . f expy sW (y) − f (y) = 0

  ˆ (y) ∈ Cf (y)+s(|∇f (y )|−ε ) . On the other This implies that expy sW 0 1 ˆ hand, by the choice of V1 and W , we have      ˆ (y) = s W ˆ (y) ≤ s (1 + ε1 ) . d y, expy sW

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4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS

451

Hence, from the definition of y s , we have   d y, y s(|∇f (y0 )|−ε1 ) ≤ s (1 + ε1 ) .



By the choice of U0 we have d y, y s(|∇f (y0 )|−ε1 ) = dC y, y s(|∇f (y0 )|−ε1 ) ; we  1  have proved that for any s ∈ 0, 2 s∗ and y ∈ U2 ∩ C, |∇f (y0 )| − ε1 s (|∇f (y0 )| − ε1 )

≥ ≥ |∇f (y0 )| − ε. 1 + ε1 dC y, y s(|∇f (y0 )|−ε1 ) Hence the lemma holds for s&  s∗ (|∇f (y0 )| − ε1 ) and U = U2 .




We shall construct the integral curves of ∇f / |∇f |2 as the limits of broken geodesics, which we define below. Given a partition P = {f (x) = a  s0 < s1 < · · · < sm = T } of [f (x) , T ] with T < fsup , we define |P|  max (si − si−1 ) 1≤i≤m

and we define Γ (x, P) to be a broken geodesic joining x0 , x1 , . . . , xm (minimal between xi−1 and xi for i = 1, . . . , m), where x0 = x, xi = (xi−1 )si −si−1 , i.e., xi is a point in {y ∈ C : f (y) ≥ f (xi−1 ) + si − si−1 } closest to xi−1 . Furthermore, we assume that Γ (x, P) is parametrized as follows: Γ (x, P) : [f (x) , T ] → C, where Γ (x, P) (s) = γi (s) and γi : [si−1 , si ] → C is a constant speed minimal geodesic from xi−1 to xi . By (H.42) we have f (xi ) = si . As we shall see below, when |P| is small, the paths Γ (x, P) approximate the integral curves for ∇f / |∇f |2 . Note that the proof of the following is the only place where we use the assumption that C is compact. All other parts of the proof of Proposition H.53 work under the assumption that fsup > 0. Lemma H.55. Let C, x, and f be as in Proposition H.53. There exists a constant C < ∞ independent of P but depending on diam (C), fsup , and T < fsup such that for any broken geodesic Γ (x, P) and s, s ∈ [f (x) , T ] we have  

dC Γ (x, P) s , Γ (x, P) (s) ≤ C s − s .

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452

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Proof. Let C  diam(C) fsup −T . Since Γ (x, P) is a broken geodesic, where Γ (x, P)|[si−1 ,si ] is a constant speed minimal geodesic joining xi−1 to xi , the lemma follows from the claim that dC (xi−1 , xi ) ≤ C |si − si−1 |

for i = 1, . . . , m.

To see the claim, fix i and a point q ∈ f −1 (fsup ). Let γ : [0, 1] → C be the constant speed shortest path in C joining xi−1 and q. Define q1   si −si−1 γ fsup −si−1 . It follows from the concavity of f that fsup − si si − si−1 f (γ (0)) + f (γ (1)) fsup − si−1 fsup − si−1 fsup − si si − si−1 si−1 + fsup = fsup − si−1 fsup − si−1 = si .

f (q1 ) ≥

By the definition of xi = (xi−1 )si −si−1 , we have dC (xi−1 , xi ) ≤ dC (xi−1 , q1 ) . Hence the claim follows from dC (xi−1 , q1 ) = dC (xi−1 , q) ·

si − si−1 diam (C) (si − si−1 ) . = fsup − si−1 fsup − T


A simple consequence of Lemma H.55 and the Arzela–Ascoli theorem is Corollary H.56 (Limit Γ∞ of Γ (x, P) as |P| → 0). Let C, x, and f be k as in Proposition H.53. Given a sequence of partitions Pk = {sk,i }m i=0

with limk→∞ |Pk | = 0, there is a subsequence of broken geodesics Γ x, Pkj which converges as j → ∞ to a path Γ∞ : [a, T ] → C uniformly on [a, T ]. We have the following properties of Γ∞ . Lemma H.57. Let C, x, a, and f be as in Proposition H.53. Suppose T ∈ k (a, fsup ). If Pk = {sk,i }m i=0 is a sequence of partitions with limk→∞ |Pk | = 0 and such that as k → ∞ the broken geodesics Γ (x, Pk ) converge to a path Γ∞ : [a, T ] → C uniformly on [a, T ], then (1) Γ∞ (f (x)) = x, (2) Γ∞ is Lipschitz, (3) f (Γ∞ (s)) = s for all s ∈ [a, T ], (4) lim sup σ→0+

1 dC (Γ∞ (s0 + σ) , Γ∞ (s0 )) ≤ σ |∇f (Γ∞ (s0 ))|

for all s0 ∈ [a, T ).

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4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS

453

Proof. (1) This is trivial. (2) This follows from Lemma H.55 directly. (3) This follows from f (Γ (x, Pk ) (sk,i )) = f (γi (sk,i )) = sk,i for all k and i. (4) Given s0 ∈ [a, T ] and ε > 0, by Lemma H.54 there is a neighborhood U of Γ∞ (s0 ) and s& > 0 such that for all y ∈ C ∩ U and σ ∈ (0, s& ) we have s ≥ |∇f (Γ∞ (s0 ))| − ε. (H.43) dC (y, y σ ) Here we chose s& small enough such that Γ∞ ([s0 , s0 + s& ]) ⊂ U . By the uniform convergence of Γ (x, Pk ) there is a k0 such that for k ≥ k0 we have |Pk | < s& and Γ (x, Pk ) ([s0 , s0 + s& ]) ⊂ U . For any σ ∈ (0, s& ) and k ≥ k0 , we define imin (s0 , k)  min{i : s0 ≤ sk,i } and imax (s0 , σ, k)  max{i : s0 + σ ≥ sk,i }. Then it follows from (H.43) that for any i = imin (s0 , k)+1, . . . , imax (s0 , σ, k), we have



dC Γ (x, Pk ) sk,(i−1) , (Γ (x, Pk ) (sk,i ))  



s −s = dC Γ (x, Pk ) sk,(i−1) , Γ (x, Pk ) sk,(i−1) k,i k,(i−1) ≤

sk,i − sk,(i−1) . |∇f (Γ∞ (s0 ))| − ε

Hence







dC Γ (x, Pk ) sk,imin (s0 ,k) , Γ (x, Pk ) sk,imax (s0 ,σ,k) 

imax (s0 ,σ,k)

≤





dC Γ (x, Pk ) sk,(i−1) , (Γ (x, Pk ) (sk,i ))

i=imin (s0 ,k)+1

≤

sk,imax (s0 ,σ,k) − sk,imin (s0 ,k) . |∇f (Γ∞ (s0 ))| − ε

Now the lemma follows from taking the limit as k → ∞ of the above inequality.
The following says that Γ∞ is an integral curve for ∇f / |∇f |2 . Lemma H.58 (Existence of integral curve for ∇f / |∇f |2 ). Let C, x, a, k and f be as in Proposition H.53. Suppose T ∈ (a, fsup ). Let Pk = {sk,i }m i=0 be a sequence of partitions with limk→∞ |Pk | = 0 and such that the broken geodesics Γ (x, Pk ) converge as k → ∞ to a path Γ∞ : [a, T ] → C uniformly on [a, T ]. Then Γ∞ is the integral curve for ∇f / |∇f |2 emanating from x.

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454

H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Proof. Note that the existence of partitions {Pk } is guaranteed by Corollary H.56. By Lemma H.57(1)   and (2), the lemma follows from verifyexists and that ing that f (Γ∞ (s0 )) < fsup , Γ˙ ∞  (H.44)

Γ˙ ∞



+

∇f (Γ∞ (s0 )) (s0 ) = + |∇f (Γ∞ (s0 ))|2

for all s0 ∈ [a, T ).

Let γs0 ,σ : [0, 1] → M be a minimal geodesic joining Γ∞ (s0 ) to Γ∞ (s0 + σ). By Lemma H.41, equation (H.44) follows from (H.45)

lim

σ→0+

γ˙ s0 ,σ (0) ∇f (Γ∞ (s0 )) = ; σ |∇f (Γ∞ (s0 ))|2

this is the equality we shall prove next. By Lemma H.57(3),    γ˙ s0 ,σ (0)   = lim sup dC (Γ∞ (s0 + σ) , Γ∞ (s0 ))  lim sup   σ σ + + σ→0

σ→0

≤

1 . |∇f (Γ∞ (s0 ))| γ˙

(0)

is uniformly bounded in TΓ∞ (s0 ) M. This implies that when σ is small, s0 ,σ σ Let σi → 0+ be any sequence such that (H.46)

γ˙ s0 ,σi (0) → V∞ ∈ TΓ∞ (s0 ) M and σi

|V

∞|

≤

1 |∇f (Γ∞ (s0 ))|

(such sequences clearly exist). Note that V∞ ∈ T˜Γ∞ (s0 ) C (we leave the proof of this as an exercise), which follows from the fact that ∇f (Γ∞ (s  0 )) is an in- γ˙ i (0) , terior point of T˜Γ∞ (s0 ) C. Hence, using Γ∞ (s0 + σi ) = expΓ∞ (s0 ) σi · s0 ,σ σi we have   dC Γ∞ (s0 + σi ) , expΓ∞ (s0 ) (σi V∞ ) → 0. σi It follows from the Lipschitz property of f that      f (Γ∞ (s0 + σi )) − f expΓ∞ (s0 ) (σi V∞ )  → 0. σi Since f (Γ∞ (s0 + σi )) = s0 + σi = f (Γ∞ (s0 )) + σi , we have       f exp   Γ∞ (s0 ) (σi V∞ ) − f (Γ∞ (s0 ))  − 1 → 0.  σi   This implies DV∞ f (Γ∞ (s0 )) = 1, which in turn implies 1 ≥ |∇f (Γ∞ (s0 ))| (H.47) D V∞ f (Γ∞ (s0 )) = |V∞ | |V∞ |

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4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS

455

by (H.46). Applying Lemma H.37 to the concave function −f and by applying the uniqueness of ∇f (Γ∞ (s0 )) (i.e., Lemma H.36(iv)), we have 1 = |∇f (Γ∞ (s0 ))| |V∞ |

and

V∞ ∇f (Γ∞ (s0 )) = . |∇f (Γ∞ (s0 ))| |V∞ |

Since we have proved that any sequential limit of the left-hand side of (H.45) is the same as the right-hand side, the left limit exists and (H.45) holds. This proves the lemma.
4.4. Sharafutdinov’s contraction map. Let C ⊂ Mn be a compact connected locally convex set with nonempty interior and boundary and let f ∈ CV0 (C), where CV0 (C) is defined as in (H.30). Let Cs  {y ∈ C : f (y) ≥ s}, where s ∈ [0, fsup ], as before. Let Γs : C → Cs be defined by

 Γs (x) 

(H.48)

x γx (s)

if f (x) ≥ s, if f (x) < s,

where γx : [f (x) , fsup ] → C is the maximal integral curve for ∇f / |∇f |2 emanating from x. Note that if x ∈ Cs , then Γs (x) = x, so by definition, the map Γs is a retraction. The following is an analogue of Corollary H.10. Theorem H.59 (Sharafutdinov’s distance nonincreasing retraction). Assume the notations above. (1) (Γs is length nonincreasing) For any s ∈ [0, fsup ] and any rectifiable path α : [0, 1] → C, we have L (α) ≥ L (Γs ◦ α) . (2) (Continuity in x and s) The map Γ : C × [0, fsup ] → C, defined by Γ (x, s)  Γs (x) , is continuous. Proof. (1) From the definition of rectifiable path, it suffices to show that for x, y ∈ C and s ∈ [a, b], (H.49)

dC (Γs (x) , Γs (y)) ≤ dC (x, y) .

Without loss of generality we may assume f (x) ≤ f (y). There are three cases to consider. (i) If s ≤ f (x), then Γs (x) = x and Γs (y) = y, so that dC (Γs (x) , Γs (y)) = dC (x, y) .

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H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(ii) If f (x) < s ≤ f (y), then Γs (y) = y, so that by Lemma H.52, dC (Γs (x) , Γs (y)) = dC (γx (s) , y) ≤ dC (x, y) . (iii) If s > f (y), then by Lemma H.49, dC (Γs (x) , Γs (y)) = dC (γx (s) , γy (s)) ≤ dC (x, y) . This completes the proof of (H.49). (2) This follows easily from the fact that Γs (x) is uniformly continuous in s and from the fact that for fixed s, Γs (x) is continuous in x.
5. Notes and commentary A classic reference for convex analysis is Rockafellar [160]. Another reference is Hiriart-Urruty and Lemar´echal [99]. §2. For Lemma H.15 see Lemma 1 in [171]. §3. Given a locally Lipschitz function f : M → R, we define the generalized Hessian to be the following difference quotient: for V ∈ Tx M, f (γV (s)) + f (γV (−s)) − 2f (x) ∈ [−∞, +∞] , (∇∇ f ) (V, V )  lim inf s→0 s2 where γV : (−ε, ε) → M is the constant speed geodesic with γ˙ V (0) = V . For example, consider the absolute value function f (x) = |x| on R. Then for 1 ∈ T0 R we have at x = 0, 2 |s| (∇∇ f ) (1, 1) = lim inf 2 = +∞. s→0 s For Lemma H.37 see Lemma 3 in [171]. For Lemma H.38 see the corollary to Lemma 3 of [171]. §4. We shall further discuss Sharafutdinov retraction in §1 of the next appendix. For Lemma H.41 see Lemma 4 in [171]. For Lemma H.45 see Lemma 5 in [171]. For Lemma H.48 see Lemma 2.1.3 in Petrunin [156] or Lemma 2.12 in Kapovitch, Petrunin, and Tuschmann [104]. For Lemma H.49 see Lemma 6 in [171]. For Lemma H.52 see Lemma 7 in [171]. For Proposition H.53 see Theorem 2 in [171]. For Lemma H.54 see Lemma 8 in [171]. For Lemma H.55 see Lemma 9 in [171]. For Lemma H.57 see Lemma 10 in [171]. For Lemma H.58 see Lemma 11 in [171]. For Theorem H.59 see Theorem 3 in [171].

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http://dx.doi.org/10.1090/surv/163/13

APPENDIX I

Asymptotic Cones and Sharafutdinov Retraction I keep on goin’ guess I’ll never know why. – From “Life’s Been Good” by Joe Walsh

This appendix mainly focuses on various results regarding the geometry of complete noncompact Riemannian manifolds with nonnegative sectional curvature. We shall find these results of interest for the Ricci flow since, by the Hamilton–Ivey curvature estimate, 3-dimensional finite time singularity models have nonnegative sectional curvature. Some of these results should also be useful in the study of 3-dimensional κ-solutions. In §1, as a continuation of our discussion of Sharafutdinov’s distance nonincreasing retraction in the previous appendix, we prove the Sharafutdinov retraction Theorem I.25. In §2 we prove the existence of asymptotic cones. In §3 we discuss a monotonicity property of distance spheres in nonnegatively curved manifolds when their radii are less than the injectivity radius. In §4 we discuss critical point theory for the distance function. As an application we prove that large radii distance spheres in a complete noncompact manifold with nonnegative sectional curvature are Lipschitz hypersurfaces. In §5 we discuss an almost distance-decreasing property of the nearest point projection map in a small tubular neighborhood of a hypersurface in any oriented Riemannian manifold. In §6 we discuss the mollified distance function and an approach, proposed by Gromov and considered by Kasue, toward constructing the space of points at infinity. 1. Sharafutdinov retraction theorem In this section we discuss the Sharafutdinov retraction theorem. We shall use some elementary comparison geometry, including the Busemann function and totally convex subsets, discussed in Appendix B of Volume One. In this section (Mn , g) denotes a (connected and oriented) complete noncompact Riemannian manifold. 457 Licensed to Chinese University of Hong Kong. Prepared on Mon Apr 4 02:01:05 EDT 2016for download from IP 137.189.171.235. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

458

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

1.1. Rays, Busemann functions and half-spaces. First we review a few notions, some of which have been used earlier in this book. In this subsection we do not assume the condition of nonnegative sectional curvature. The reader familiar with Busemann functions may skip to subsection 1.3, where we discuss the Sharafutdinov retraction theorem based on the theory of convex functions in the previous appendix. 1.1.1. Busemann functions. We begin with the following. Definition I.1 (Space of rays). Given a noncompact Riemannian manifold (Mn , g), we say that a unit speed geodesic γ : [0, ∞) → M is a ray if it is minimizing on each finite interval. Let RayM denote the set of all rays in M. Given p ∈ M, let RayM (p) denote the set of all rays γ in M emanating from p, i.e., γ (0) = p. An elementary fact is that for any point p ∈ M, there exists a ray γ emanating from p. Given a ray γ : [0, ∞) → M, the Busemann function bγ : M → R associated to γ is defined by bγ (x)  lim (s − d (γ (s) , x)) .

(I.1)

s→∞

We have the following bounds for the Busemann function in terms of the distance function (for a proof, see Lemma B.46 of Volume One for example). Lemma I.2. If γ : [0, ∞) → Mn is a ray emanating from p ∈ M, then (1) (bounded above by the distance function) for all x ∈ M, |bγ (x)| ≤ d (x, p)

(I.2)

and (2) (Lipschitz with Lipschitz constant 1) for all x, y ∈ M, (I.3)

|bγ (x) − bγ (y)| ≤ d (x, y) .

The Busemann function bp : Mn → R associated to a point p ∈ M is defined by bp  sup bγ , γ

where the supremum is taken over all rays γ emanating from p. Using Lemma I.2, we can prove the following (for a proof see Lemma B.48 of Volume One). Lemma I.3 (Busemann function bp — distance and Lipschitz). For any p ∈ Mn we have for all x ∈ M, |bp (x)| ≤ d (x, p) and for all x, y ∈ M, |bp (x) − bp (y)| ≤ d (x, y) .

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The following property concerning approximating long geodesics by rays is useful (for a proof see Lemma B.49 of Volume One). Lemma I.4 (Angle closeness of long geodesics to rays). Given p ∈ Mn , define θ : [0, ∞) → [0, π] by inf

θ (r) = sup

σ∈S(r) γ∈RayM (p)

p (σ˙ (0) , γ˙ (0)) ,

where S (r) is the set of all minimal geodesic segments σ of length L (σ) ≥ r emanating from p. Then lim θ (r) = 0. r→∞

1.1.2. Half-spaces in Riemannian manifolds. In this subsection we do not assume the condition of nonnegative sectional curvature. Now we use the distance function to define the Riemannian analogue of a half-space in Euclidean space. Definition I.5 (Half-spaces and geodesic balls). Let γ : [0, ∞) → Mn be a ray emanating from a point p ∈ M. (1) The open right geodesic half-space is . B (γ (s) , s) . Bγ  s∈(0,∞)

(2) The closed left geodesic half-space Hγ is Hγ  M − Bγ . The following lemma, which implies the equivalence of the Busemann function formulation and the half-space formulation, is easy to verify. Lemma I.6. The closed left geodesic half-space is a sublevel set of the Busemann function, i.e., bγ (x) ≤ 0 if and only if x ∈ Hγ . Given a ray γ : [0, ∞) → Mn and s ∈ [0, ∞), define γs : [0, ∞) → M by



γs s  γ s + s for all s ∈ [0, ∞). An elementary property is that if s ≤ t, then Bγt ⊂ Bγs . We also have Bγs is the open (t − s)-neighborhood of Bγt ; for a proof, see p. 138 of [30]. Lemma I.7 (ε-neighborhood of a half-space is a half-space). Whenever 0 < s < t, Bγs = {x ∈ Mn : d (x, Bγt ) < t − s} = Nt−s (Bγt ) . Given a point p ∈ Mn and s ∈ [0, ∞), the sublevel set of the Busemann function associated to p is defined by @ . Hγs = M − Bγs . Cs (p)  γ∈RayM (p)

γ∈RayM (p)

The terminology above is justified by the following (for a proof, see Lemma B.59 of Volume One).

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460

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

Lemma I.8 (Sublevel set of bp ). For every choice of basepoint p ∈ Mn and s ∈ [0, ∞), the sublevel set of the Busemann function associated to p is given by Cs (p) = {x ∈ Mn : bp (x) ≤ s} . Moreover, ∂Cs (p) = {x ∈ Mn : bp (x) = s} and if 0 ≤ s ≤ t, then Cs (p) ⊂ Ct (p) . Since bp (x) ≤ d (x, p), the sublevel sets of the Busemann function compare to geodesic balls in the following way (for a proof, see Corollary B.60 of Volume One). Corollary I.9 (Sublevel set of bp contains geodesic ball). For any p ∈ M and s > 0 we have B (p, s) ⊂ Cs (p) . 1.1.3. Convex and totally convex subsets. Recall that we defined convexity and local convexity for subsets in Riemannian manifolds in subsection 2.1.2 of Appendix H. We have a few variants on the notion of convex set. (1) A subset Σ ⊂ Mn is said to be totally convex if for all x, y ∈ Σ every geodesic γ (not necessarily minimizing) joining x and y is contained in Σ.1 ¯ (2) A subset Σ ⊂ Mn is said to be strongly convex if for any x, y ∈ Σ the minimal geodesic β from x to y is unique and its interior int (β) is contained in Σ. As we shall see, the above notions are particularly useful for complete noncompact manifolds with nonnegative sectional curvature. We first give some examples of such subsets. In the compact case we have the following. Example I.10 (Round sphere). Let Mn = S n be the unit n-sphere, which we may think of as the subset

S n = x ∈ Rn+1 : |x| = 1 . (1) A nonempty subset of S n is totally convex if and only if it is the whole S n . (2) For any a ∈ (0, 1] the subset Σa  {x ∈ S n : xn > a} is strongly convex. (3) The upper hemisphere Σ0 = {x ∈ S n : xn > 0} is not strongly convex. The following is a trivial noncompact example. Example I.11. A convex set in Euclidean space is totally convex. 1 See Definition 8.1 on p. 134 of [30]. Note that in Definition B.53 on p. 307 of Volume One the hypothesis that the geodesic α is minimizing should be removed.

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1. SHARAFUTDINOV RETRACTION THEOREM

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1.2. Manifolds with nonnegative sectional curvature. When complete noncompact manifolds have nonnegative sectional curvature, the notions introduced in the last subsection have the following properties. 1.2.1. Busemann functions. We begin with Busemann functions (see for example Exercises 1.167 and 1.161 in [45]). Proposition I.12. Let γ : [0, ∞) → Mn be a ray. (i) If (M, g) has nonnegative sectional curvature, then the Busemann function bγ is convex. Hence, for any p ∈ M, bp is convex. (ii) If Rcg ≥ 0, then the Busemann function bγ is subharmonic in the sense of distributions.2 As a consequence of Lemma I.4 we have that the Busemann function is ‘almost’ bounded below by the distance function (for a proof, see Corollary B.50 of Volume One). Corollary I.13 (bp is almost bounded below by distance). If (Mn , g) is a complete noncompact Riemannian manifold with nonnegative sectional curvature and p ∈ M, then for all x ∈ M, bp (x) ≥ d (x, p) (1 − θ (d (x, p))) . In particular, the Busemann function bp is bounded below (see also Corollary B.63 of Volume One). On the other hand, the Busemann function bγ associated to a ray is not, in general, bounded from below. Exercise I.14. Give an example of a complete noncompact Riemannian manifold (Mn , g) with positive sectional curvature and a ray γ in M such that bγ is not bounded from below. 1.2.2. Half-spaces. Next we discuss half-spaces in complete noncompact manifolds with nonnegative sectional curvature. By Lemma I.6, the next proposition follows from Proposition I.12; see also Theorem 8.2 of [30]. Proposition I.15 (sect ≥ 0 — left half-spaces are totally convex). Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. If γ is a ray, then the closed left half-space Hγ is totally convex. Sublevel sets of the Busemann function have the following nice property on manifolds with nonnegative sectional curvature. This follows from Proposition I.15 and Corollary I.13 (see also Proposition B.62 of Volume One). 2

See, for example, p. 373 in Part I for a definition.

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462

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

Proposition I.16 (Sublevel set of bp is compact and totally convex). Let be a complete noncompact Riemannian manifold with nonnegative sectional curvature and let p ∈ M. For any s ∈ [0, ∞), the set Cs (p) is compact and totally convex. (Mn , g)

We also recall the following (see Proposition 8.5 of [30] and compare with Lemma I.8). Proposition I.17 (Sublevel sets of bp are parallel). Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature and let p ∈ M. For any s < t, (I.4)

Cs (p) = {x ∈ Ct : d (x, ∂Ct (p)) ≥ t − s} .

In particular, p ∈ ∂C0 (p) and ∂Cs (p) = {x ∈ Ct : d (x, ∂Ct (p)) = t − s} . 1.3. Splitting and retraction theorems. There are a number of beautiful theorems concerning the geometry and topology of complete noncompact manifolds with nonnegative sectional curvature; we now summarize a few of them. We have already stated the Toponogov comparison theorem in §2 of Appendix G (see Theorem G.33). In addition, we have the following. 1.3.1. Cheeger–Gromoll splitting theorem. We have the following fundamental splitting theorem of Cheeger and Gromoll (see also Theorem 1.162 of [45]), which employs the fact that, when Rc ≥ 0, Busemann functions are subharmonic in the sense of distributions (see Proposition I.12(ii)). Theorem I.18 (Cheeger–Gromoll splitting). Suppose (Mn , g) is a complete Riemannian manifold with Rc ≥ 0 and suppose

there is a geodesic line n , g) is isometric to R× N n−1 , h with the product metric, in Mn . Then (M

where N n−1 , h is a Riemannian manifold with Rc ≥ 0. Remark I.19. For the case of Aleksandrov spaces, see Theorem G.45, which assumes the analogue of nonnegative sectional curvature. 1.3.2. Soul theorem and conjecture. We recall some basic definitions.

Definition I.20. Let N k , h be a Riemannian manifold and let Σ ⊂ N be a submanifold. (i) We say that Σ is totally geodesic if its second fundamental form is zero. (ii) The normal bundle of Σ is ν (Σ)  {V ∈ Tp N : p ∈ Σ and V, W  = 0 for all W ∈ Tp Σ} . For example, a slice M1 × {q} ⊂ M1 × M2 in a Riemannian product is totally geodesic.

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Definition I.21 (Soul). A submanifold S of a complete noncompact manifold (Mn , g) is called a soul if it is a compact, totally convex, totally geodesic submanifold. Since S is compact, we have dim (S) < dim (M). The following result, known as the soul theorem, is fundamental in the understanding of manifolds with nonnegative curvature. Theorem I.22 (Soul theorem). Let (Mn , g) be a complete noncompact Riemannian manifold. (1) (Cheeger and Gromoll [32], Theorem 1.11 and Theorem 2.2) If the sectional curvature of g is nonnegative, then there exists a soul S ⊂ M and M is diffeomorphic to its normal bundle ν (S). (2) (Sharafutdinov [172]) If sect (g) ≥ 0, then any two souls are isometric. (3) (Gromoll and Meyer [77]) If the sectional curvature of g is positive, then any soul is a point and hence Mn is diffeomorphic to Rn . Given p ∈ M, recall that (bp )min  minx∈M bp (x) > −∞ is attained since bp is a proper function bounded from below. Moreover, the subset (bp )min contains a soul. b−1 p Generalizing Theorem I.22(3), Perelman [148] proved the following, known as the soul conjecture. Theorem I.23 (Soul conjecture). If (Mn , g) is a complete noncompact Riemannian manifold with nonnegative sectional curvature everywhere and with positive sectional curvature at some point, then the soul is a point. 1.3.3. Sharafutdinov retraction map. An important tool in the study of manifolds with nonnegative curvature is the Sharafutdinov retraction map, which we define below. Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. Fix a point p ∈ M and let bp be the corresponding Busemann function. By Proposition I.12 and Corollary I.13, the function −bp is concave and bounded from above on M. Specializing Theorem H.59 to the Busemann function, we have the following (which is used in the proof of Proposition 18.33). Theorem I.24 (Sharafutdinov’s retraction for level sets of the Busemann function). Let (Mn , g) be a complete noncompact manifold with nonnegative sectional curvature and fix a point p ∈ M. Then for any a2 < a1 ≤ (−bp )max there exists a distance-nonincreasing map Φ : (−bp )−1 (a2 ) → (−bp )−1 (a1 ) between the level sets of the Busemann function bp . The same result holds for the Busemann function bγ associated to a ray γ.

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464

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Proof. Define f = −bp − a2 : M → R and C = (−bp )−1 ([a2 , ∞)) = f −1 ([0, ∞)) . Since −bp is concave and proper, we have f ∈ CV0 (C) and C is a compact connected locally convex set with nonempty interior and boundary. Then Cs = {y ∈ C : f (y) ≥ s} = (−bp )−1 ([s + a2 , ∞)) . Now apply Theorem H.59 with s = a1 − a2 (so that Cs = (−bp )−1 ([a1 , ∞))) to obtain a distance-nonincreasing map Φ = Γa1 −a2 |(−bp )−1 (a2 ) : (−bp )−1 (a2 ) → (−bp )−1 (a1 ) , where Γs : C → Cs is the distance-nonincreasing map defined by (H.48).
We now define a global distance-nonincreasing retraction map. Fix any integer m0 satisfying m0 ≤ supM (−bp )  (−bp )max . We define

(I.5) Cmax (p)  x ∈ M : −bp (x) = (−bp )max , Cm (p)  {x ∈ M : −bp (x) ≥ m} for any integer m ≤ m0 . Note that Cmax (p) ⊂ Cm (p) ⊂ Cm−1 (p) for any m ≤ m0 . We may apply Theorem H.59 to the concave function f = −bp − m0 , C = Cm0 (p) = f −1 ([0, ∞)), and s = (−bp )max − m0 to obtain a distancenonincreasing retraction map Γm0 ,max : Cm0 (p) → Cmax (p) . We may also apply Theorem H.59 to the concave function f = −bp − m, C = Cm (p) = f −1 ([0, ∞)), and s = 1 to obtain a distance-nonincreasing retraction map Γm−1,m : Cm−1 (p) → Cm (p)

for any m ≤ m0

(note that Γm0 ,max |Cmax (p) = id and Γm−1,m |Cm (p) = id). Recall that Cmax (p) contains a soul of (Mn , g). Now we can define the Sharafutdinov retraction map Shar : M → Cmax (p) . For any x ∈ M, either x ∈ Cm0 (p) or there is a unique integer m ≤ m0 such that x ∈ Cm−1 (p) − Cm (p). We define  if x ∈ Cm0 (p) , Γm0 ,max (x) Shar (x)  Γm,m+1 ◦ · · · ◦ Γm0 −1,m0 ◦ Γm0 ,max (x) if x ∈ Cm−1 (p) − Cm (p). The following result is an easy consequence of the properties of Γm0 ,max and Γm−1,m .

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Theorem I.25. Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature and let p ∈ M. Then there is a continuous distance-nonincreasing retraction map Shar : M → Cmax (p) .

(I.6)

That is, Shar (q) = q for all q ∈ Cmax (p) and d (Shar (x) , Shar (y)) ≤ d (x, y) for all x, y ∈ M. 2. The existence of asymptotic cones The main result of this section is the following. It is used in the proof of Theorem 20.1 on the ASCR and AVR of κ-solutions. Theorem I.26 (Asymptotic cone of noncompact manifold with sect ≥ 0). Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. Then the asymptotic cone exists and is a Euclidean metric cone as in Definition G.29. The proof depends on the notion of ideal boundary; to present this, we begin with the following. For any γ1 , γ2 ∈ RayM (p), we define (I.7)

˜ 1 (s) p γ2 (t) ∈ [0, π] , d˜∞ (γ1 , γ2 )  lim γ s,t→∞

˜ is the Euclidean comparison angle defined in (G.17). where  ˜ 1 (s) p γ2 (t) is a This limit exists since, by Lemma G.38, the angle γ nonincreasing function of both s and t. Lemma I.27 (Pseudo-metric on the space of rays). (i) For any γ1 , γ2 ∈ RayM (p) and for any positive numbers a and b, we have  1/2 d (γ1 (at) , γ2 (bt))  2 = a + b2 − 2ab cos d˜∞ (γ1 , γ2 ) , (I.8) lim t→∞ t where d is the distance function of (M, g). (ii) d˜∞ is a pseudo-metric on RayM (p). Proof. (i) By the Euclidean law of cosines, we have   ˜ 1 (at) p γ2 (bt) . d2 (γ1 (at) , γ2 (bt)) = (at)2 + (bt)2 − 2 (at) (bt) cos γ Dividing this by t2 and taking the limit as t → ∞, we obtain (I.8). (ii) It suffices to prove the triangle inequality for d˜∞ . Let γ1 , γ2 , γ3 ∈ RayM (p) and, given a, b, let c be a positive constant to be chosen below. By dividing the triangle inequality d (γ1 (at) , γ2 (bt)) ≤ d (γ1 (at) , γ3 (ct)) + d (γ3 (ct) , γ2 (bt))

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by t, taking the limit as t → ∞, and applying part (i), we obtain  1/2  a2 + b2 − 2ab cos d˜∞ (γ1 , γ2 )   1/2 2 2 ˜ ≤ a + c − 2ac cos d∞ (γ1 , γ3 )   1/2 + c2 + b2 − 2cb cos d˜∞ (γ3 , γ2 ) . We now show that (I.9)

d˜∞ (γ1 , γ2 ) ≤ d˜∞ (γ1 , γ3 ) + d˜∞ (γ3 , γ2 ) .

In the Euclidean plane, draw two adjacent triangles AOC and COB





with side lengths L OA = a, L OC = c, L OB = b and angles AOC = d˜∞ (γ1 , γ3 ) and COB = d˜∞ (γ3 , γ2 ). Then, by the law of cosines, 1/2

 L AC = a2 + c2 − 2ac cos d˜∞ (γ1 , γ3 )

and

1/2

 . L CB = c2 + b2 − 2cb cos d˜∞ (γ3 , γ2 )

Choosing c so that the points A, C, and B lie on one line, we have  1/2  a2 + b2 − 2ab cos d˜∞ (γ1 , γ2 )



≤ L AC + L CB

= L AB  1/2  . = a2 + b2 − 2ab cos d˜∞ (γ1 , γ3 ) + d˜∞ (γ3 , γ2 ) Thus

    cos d˜∞ (γ1 , γ2 ) ≥ cos d˜∞ (γ1 , γ3 ) + d˜∞ (γ3 , γ2 )

and the triangle inequality (I.9) follows since d˜∞ (γ1 , γ2 ) ≤ π.




The ideal boundary (M (∞), d∞ ) of (M, g) is defined to be the metric  space induced by RayM (p) , d˜∞ as given in (G.3). In particular, (I.10)

M (∞)  RayM (p) / ∼

and γ1 ∼ γ2 if and only if d˜∞ (γ1 , γ2 ) = 0. Note that d˜∞ (γ1 , γ2 ) = 0 if and only if d (γ1 (t) , γ2 (t)) = 0. lim t→∞ t

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467

Now we give a

Proof of Theorem I.26. We claim that M, λ2 g converges to the Euclidean metric cone Cone (M (∞) , d∞ ), in the pointed Gromov–Hausdorff topology as λ → 0. Fix an arbitrary constant (radius) r > 0 and any constant ε > 0. We need to show that for λ sufficiently close to 0 the Gromov– Hausdorff distance between the balls Bλ2 g (p, r) and BdCone(M(∞)) (O, r) is less than ε. ε -net (see Definition G.10) We take an 10 N = {([γi ] , ai )}m i=1 of BdCone(M(∞)) (O, r) ⊂ Cone (M (∞)), where γi ∈ RayM (p) and ai ∈ (0, r]. Then   ε ε ≤ . dGH BdCone(M(∞)) (O, r) , N ≤ 10 3 Let



m Nλ  γi λ−1 ai i=1 ⊂ Bλ2 g (O, r) = Bg O, λ−1 r ⊂ M. We claim that it follows from (I.8) that  .    ε ≤ (I.11) dGH Nλ , γ∈RayM (p) γ 0, λ−1 r 6  

/ −1 for λ sufficiently small, where γ∈RayM (p) γ 0, λ r ⊂ Bλ2 g (p, r). To prove (I.11), it suffices to show that for any

.  

0, λ−1 r γ λ−1 a ∈ γ∈RayM (p) γ

there exists γi λ−1 ai ∈ Nλ such that

ε

(I.12) d γi λ−1 ai , γ λ−1 a ≤ . 6 ε -net of BdCone(M(∞)) (O, r), Note ([γ] , a) ∈ Cone (M (∞)). Since N is an 10 there exists ([γi ] , ai ) ∈ N such that ε dCone(M(∞)) (([γi ] , ai ) , ([γ] , a)) ≤ . 10 This implies (I.12) for λ sufficiently small in view of (I.8). Now from (I.8) again, we have ε dGH (N, Nλ ) ≤ 3 when λ is sufficiently small. Recall that given any ε0 > 0, there exists T > 0 such that for t > T and x in the sphere S(p, t) there exists γ ∈ RayM (p) such that ε0 d (x, γ (t)) ≤ t 2 (see Lemma 7(i) in [43] for example). This implies that  ε .    −1 γ 0, λ r , B (O, r) ≤ dGH 2 γ∈RayM (p) λ g 6

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I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

when λ is sufficiently small. This and (I.11) imply dGH Nλ , Bλ2 g (O, r) ≤ ε 3 . Now we have for λ small enough   dGH BdCone(M(∞)) (O, r) , Bλ2 g (O, r)  

≤ dGH BdCone(M(∞)) (O, r) , N + dGH (N, Nλ ) + dGH Nλ , Bλ2 g (O, r) ≤ ε.
3. A monotonicity property of nonnegatively curved manifolds within the injectivity radius In this section we discuss a monotonicity property of distance spheres in nonnegatively curved manifolds when their radii are less than the injectivity radii of their centers. In particular, within the injectivity radii, there are relatively distance-nonincreasing maps between distance spheres of different radii centered at the same point which agree with the flow along rays emanating from that point. The proof of this uses the Rauch comparison theorem,3 which says that geodesics emanating from a point spread out slower than for the model Euclidean space. We may think of this result as motivation for the existence theorem for asymptotic cones in the previous section. Given p ∈ M, we have that any unit speed geodesic β : [0, inj (p)) → M with β (0) = p is minimal and the map   expp : B 0, inj (p) → B (p, inj (p)) is a diffeomorphism. Recall that S (p, s) denotes the distance sphere of radius s centered at p. For any 0 < s ≤ t < inj (p), define the map (I.13) by (I.14)

ϕs,t : S (p, s) → S (p, t) −1  t  , ϕs,t  expp ◦ ◦ expp B¯ (0,s) s

where st : Tp M → Tp M denotes multiplication by st .4 Note that ϕs,t is a diffeomorphism between smooth spheres for 0 < s ≤ t < inj (p). As a subset of the metric space (M, dg ), where dg denotes the metric induced by the Riemannian metric g, for any s > 0 the sphere S (p, s) is 3

Alternatively, use the Hessian comparison theorem. Note that ϕs,t (α (s)) = α (t) for any (minimal) unit speed geodesic α : [0, t] → M with α (0) = p. 4

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3. MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS

469

endowed with the subspace metric, which we denote by dS(p,s) . The metric dS(p,s) gives us the induced intrinsic metric defined by (see (G.6)) dS(p,s)  dLd

(I.15)

S(p,s)

.

We consider S (p, s) with the metric dS(p,s) . It is easy to see that dS(p,s) (x, y) is finite if and only if x and y lie in the same component of S (p, s). When S (p, s) is smooth, dS(p,s) is the same as the distance induced by the Riemannian metric g|S(p,s) on S (p, s). For intuition we note the (trivial) model case. Remark I.28 (Euclidean space). Given a point O (considered as the origin) in Euclidean space, for any s, t ∈ (0, ∞) there are natural maps ϕEs,t : S (O, s) → S (O, t) given by identifying points whichlie on   the same ray em 1 E anating from O. The maps ϕs,t : S (O, s) , s dS(O,s) → S (O, t) , 1t dS(O,t) are isometries. The following is the main result of this section. Proposition I.29 (Distance nonincreasing maps between small geodesic spheres). Let (Mn , g) be a complete Riemannian manifold with nonnegative sectional curvature. Given p ∈ M, for any s ≤ t ≤ u < inj (p), (1) the map     1 1 (I.16) ϕs,t : S (p, s) , dS(p,s) → S (p, t) , dS(p,t) , s t defined by (I.14), is distance nonincreasing and (2) (I.17)

ϕt,u ◦ ϕs,t = ϕs,u .

First proof. The composition property (I.17) of ϕs,t follows directly from (I.14). Below we shall argue using Jacobi fields and the Rauch comparison theorem that ϕs,t in (I.16) is distance nonincreasing for s ≤ t < inj (p). Let x, y ∈ S (p, s) be arbitrary points and let β : [0, ρ] → S (p, s) , where ρ  dS(p,s) (x, y), be a minimal geodesic, with respect to the (smooth) induced Riemannian metric on S (p, s), satisfying β (0) = x and β (ρ) = y. For every u ∈ [0, ρ], let αu : [0, t] → M be the unique unit speed minimal geodesic with αu (0) = p and αu (s) = β (u). Consider the path γ : [0, ρ] → S (p, t) defined by γ (u) = αu (t) .

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I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

Then γ (0) = α0 (t) = ϕs,t (x) and γ (ρ) = αρ (t) = ϕs,t (y). Clearly L (γ) ≥ dS(p,t) (ϕs,t (x) , ϕs,t (y)) . We shall estimate L (γ) from above. Let Ju be the Jacobi field along αu with Ju (0) = 0 and Ju (s) = β˙ (u). Then γ˙ (u) = Ju (t) . Since (Mn , g) has nonnegative sectional curvature, by the Rauch comparison theorem (see Chapter 1 of Cheeger and Ebin [30]), we have t |Ju (t)| ≤ . |Ju (s)| s Hence

 ρ |γ˙ (u)| du dS(p,t) (ϕs,t (x) , ϕs,t (y)) ≤ L (γ) = 0    t ρ  ˙ t ρ  |Ju (s)| du = ≤ β (u) du s 0 s 0 t t = L (β) = dS(p,s) (x, y) . s s This completes the proof of the proposition.




An equivalent way to see that ϕs,t in (I.16) is distance nonincreasing for s ≤ t < inj (p) is as follows. Second proof of Proposition I.29 — via Hessian comparison. Given x ∈ S (p, s) and U0 ∈ Tx S (p, s), let β : (−ε, ε) → S (p, s) be a path with β˙ (0) = U0 . For u ∈ (−ε, ε), define αu : [0, t] → M to be the unique unit speed minimal geodesic with αu (0) = p and αu (s) = β (u). Let ∂ ∂ αu (v) and V = αu (v) , ∂u ∂v so that ∇U V − ∇V U = [U, V ] = 0. By the Gauss lemma we have V = ∇r, where r  d (·, p). Furthermore, U0 = U (α0 (s)) and (ϕs,t )∗ (U0 ) = U (α0 (t)). We compute (I.18)

U=

d |U |2 (α0 (v)) = 2 ∇V U, U  = 2 ∇U V, U  = 2 ∇∇r, U ⊗ U  dv since V = ∇r. Because (M, g) has nonnegative sectional curvature, by the Hessian comparison theorem, 1 (I.20) ∇∇r ≤ g, r

(I.19)

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3. MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS

471

so that (I.21)

2 d |U |2 (α0 (v)) ≤ |U |2 (α0 (v)) . dv r (α0 (v))

Since r (α0 (v)) = v, we have d log dv

(I.22)



|U | (α0 (v)) v

 ≤ 0.

We conclude that (ϕs,t )∗ (U0 ) |U | (α0 (t)) |U | (α0 (s)) U0 = ≤ = . (I.23) t t s s This implies the distance-nonincreasing property of ϕs,t .




An example where one can see the proposition directly is when (Mn , g) is a rotationally symmetric complete Riemannian manifold with nonnegative sectional curvature, Mn is diffeomorphic to Rn , and p is the origin; in this case, inj (p) = ∞. In particular, writing the metric as g = dr2 + w2 (r) gS n−1 for 0 ≤ r < ∞ and where w (0) = 0 and w (0) = 1,5 we have that the sectional curvatures are given by (see (20.53) and (20.54)) ν1 =

1 − (w (r))2 w (r)2

and ν2 = −

w (r) . w (r)

Since the metric is complete and the sectional curvatures are nonnegative, we have 0 ≤ w (r) ≤ 1 and w (r) ≤ 0. This implies (rw (r) − w (r)) = rw (r) ≤ 0 and hence rw (r) − w (r) ≤ 0, so that   w (r)  rw (r) − w (r) = ≤0 r r2 for all r ∈ [0, ∞). Since     1 n−1 w (s) d n−1 , S (p, s) , dS(p,s) = S s s S (isometric), the distance-nonincreasing property of ϕs,t in (I.16) for s ≤ t follows. One expects that, on manifolds with nonnegative sectional curvatures, there are maps analogous to (I.16) when the radii of the spheres are allowed to be larger than the injectivity radii at the centers. Such maps would be useful for the study of the geometry at infinity. Working toward this end, we shall develop some techniques regarding the distance function and its mollifications. 5 This guarantees that the metric extends smoothly over the origin even though the coordinates are singular there.

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4. Critical point theory and properties of distance spheres Continuing with the topic of the previous section, to study distance spheres whose radii are not necessarily within the injectivity radius, we discuss some issues related to the nonsmoothness of the distance function r. The main result of this section is that distance spheres with large enough radii in complete noncompact manifolds with nonnegative sectional curvatures are Lipschitz hypersurfaces. 4.1. Critical point theory for the distance function. In this subsection we briefly discuss critical point theory for the distance function on a complete Riemannian manifold (Mn , g). Fix p ∈ M and let r (x)  d (x, p) . The gradient of r, as a set (or set gradient), is defined by   |V | = 1 and for all u ∈ [0, r (x)] we . (∇
r) (x)  V ∈ Tx M : have u + r (expx (−uV )) = r (x) By definition, ∇
r (p) = Spn−1 ⊂ Tp M is the unit tangent sphere. Observe the following properties of ∇
r (p). (1) If x is a smooth point of r, then (∇
r) (x) = (∇r) (x) is the usual gradient. (2) A vector V ∈ Sxn−1 is in (∇
r) (x) if and only if there is a unit speed geodesic γ : [0, ∞) → M such that γ (r (x)) = x and V = γ˙ (r (x)) . (3) The set (∇
r) (x) ⊂ Tx M is compact. To see the only if part of (2), we note the following. Given V ∈ (∇
r) (x), we have r (expx (−r (x) V )) = 0, so that expx (−r (x) V ) = p. Hence d (expx (−r (x) V ) , x) = r (x) , so that u → expx (−uV ) ,

u ∈ [0, r (x)] ,

is a minimal geodesic joining x to p. Property (3) follows from (2) easily. Let . (∇
r) (x) ⊂ SM, ∇
r  where SM 

/

x∈M n−1 x∈M Sx

⊂ T M denotes the unit tangent bundle.

Lemma I.30 (Local compactness of ∇
r). If {Vi } is a sequence in ∇
r and V∞ ∈ T M are such that Vi → V∞ , then V∞ ∈ ∇
r.

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The set ∇
r ⊂ SM is locally compact. In particular, if M is compact, then so is ∇
r. Proof. Let xi ∈ M denote the basepoints of Vi , which converge to the basepoint x∞ of V∞ . Let ri = d (xi , p). Then γi (s)  expxi (− (ri − s) Vi ) , where s ∈ [0, ri ], is a unit speed minimal geodesic joining p to xi such that γ˙ i (ri ) = Vi . Let γ∞ (s)  expx∞ (− (r∞ − s) V∞ ) , where s ∈ [0, r∞ ] and r∞ = d (x∞ , p). Then γi → γ∞ and so γ∞ is a unit speed minimal geodesic joining p to x∞ with γ˙ ∞ (r∞ ) = V∞ . Hence
V∞ ∈ ∇
r. Having defined the gradient of r, the following definition is natural. Definition I.31 (Regular point of a distance function). (i) We say that a point x ∈ M is a regular point of r  d (x, p) if there exists V ∈ Sxn−1 such that for every W ∈ (∇
r) (x) we have π (I.24)  (V, W ) < , 2 i.e., V, W  > 0. (ii) We say that x ∈ M is a critical point of r if it is not a regular point of r; i.e., x is a critical point of r if and only if for every V ∈ Sxn−1 there exists W ∈ (∇
r) (x) such that π  (V, W ) ≥ . 2 Note that if x ∈ M is a regular point of r, then (∇
r) (x) ⊂ Sxn−1 is contained in an open hemisphere. Actually, since (∇
r) (x) is compact, for the vector V in (I.24), there exists ε > 0 such that π (I.25)  (V, W ) ≤ − ε 2 for all W ∈ (∇
r) (x). Since ∇
r is a locally compact set, the set of critical points of r is a closed set. In other words, the set of regular points of r is an open set. The following are trivial examples of critical points: (1) the point p itself is a critical point of r since, by definition, (∇
r) (p) = Spn−1 , (2) any farthest point from p is a critical point of r, (3) more generally, any local maximum of r is a critical point of r, (4) if there exists W ∈ Sxn−1 such that W, −W ∈ ∇
r, then x is a critical point of r. Indeed, for every V ∈ Sxn−1 , either  (V, W ) ≥ π2 or  (V, −W ) ≥ π2 .

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Example I.32 (Spheres and tori). (i) The sphere S n (ρ) of radius ρ. Given p ∈ S n (ρ), the set of regular points of r is S n (ρ) − {p, −p} and the only critical points are p and −p. (ii) For 2-dimensional rectangular tori we have that for each point p the distance function r has exactly 3 critical points (Exercise).6 In view of (I.25), we give the following definitions. Given θ ∈ (0, π2 ], define

Cθ (x)  V ∈ Sxn−1 :  (V, W ) < θ for all W ∈ (∇
r) (x) . Since Cθ (x) is the intersection of convex subsets of Sxn−1 (balls of radii ≤ π2 ), it is convex inside Sxn−1 . Equivalently, the cone on Cθ (x) with vertex at the origin is convex inside Tx M. We say that x is a θ-regular point of r if Cθ (x) is nonempty. In particular, x is a regular point of r if and only if x is a π/2-regular point of r. Note that, since (∇
r) (x) is compact, if x is a θ-regular point of r, then for ε > 0 sufficiently small, x is also a (θ − ε)-regular point of r. Lemma I.30 also yields the following. Lemma I.33 (Regular points form an open set). For any θ ∈ (0, π2 ], the set of θ-regular points of r is an open set. Moreover, . Cθ (x) ⊂ SM Cθ  x∈M

is an open set. Proof. We prove that SM − Cθ is a closed set. Let Vi ∈ SM − Cθ be a convergent sequence with Vi → V∞ ; then the corresponding basepoints converge xi → x∞ . By the definition of Cθ , there exist Wi ∈ (∇
r) (xi ) such that  (Vi , Wi ) ≥ θ. Passing to a subsequence, by Lemma I.30 we may assume Wi → W∞ ∈ (∇
r) (x) . Since  (V∞ , W∞ ) ≥ θ, we have V∞ ∈ Sxn−1 − Cθ (x∞ ).




4.2. Geometric characterization of regular points. The following is the main result of this subsection. Lemma I.34 (Characterization of regular points, I). A point x ∈ M−{p} is a regular point of r if and only if there exist V ∈ Sxn−1 , c > 0, and ε > 0 such that r (expx (sV )) − r (x) ≥ cs for all s ∈ [0, ε), i.e., there exists a direction in which the distance is growing at least at a positive rate.7 6 7

See also Examples 1.6–1.8 on pp. 3–4 of [29]. We may think of such a V as a gradient-like vector.

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Proof. (1) If case. Suppose that V ∈ Sxn−1 , c > 0, and ε > 0 are such that r (expx (sV )) − r (x) ≥ cs for all s ∈ [0, ε). Let {γs }s∈[0,ε) be any smooth family of paths joining p to expx (sV ) with γ0 being a minimal geodesic. Then L (γs ) ≥ r (expx (sV )) , while L (γ0 ) = r (x). On the other hand, by the first variation of arc length formula, we have  r (expx (sV )) − r (x) d  ≥ c > 0. L (γs ) ≥ lim inf V, γ ˙ =  + ds s s→0 s=0

Since γ0 is an arbitrary minimal geodesic joining p to x, we conclude that x is a regular point of r. (2) Only if case. Suppose that x is a regular point of r. By definition, there exists V ∈ Sxn−1 such that for every W ∈ (∇
r) (x) we have π  (V, W ) < . 2 Let α (s)  expx (sV ) for s ∈ [0, ε). By Lemma I.33 and α˙ (0) = V , provided ε > 0 is sufficiently small, we have π (I.26)  (α˙ (s) , W ) ≤ − ε 2 for all W ∈ (∇
r) (α (s)) and s ∈ [0, ε). Since r is Lipschitz, we have r ◦ α : [0, ε) → [0, ∞) is Lipschitz and hence (r ◦ α) (s) exists for a.e. s and for any s¯ ∈ (0, ε) we have8  s¯ (r ◦ α) (¯ s) − r (x) = (r ◦ α) (s) ds. 0 

Given s0 ∈ (0, ε) such that (r ◦ α) (s0 ) exists, we have (r ◦ α) (s0 ) = lim

∆s→0+

(r ◦ α) (s0 ) − (r ◦ α) (s0 − ∆s) . ∆s

For δ > 0 sufficiently small, let {β∆s }∆s∈[0,δ) be any smooth family of paths joining p to α (s0 − ∆s) with β0 being a minimal geodesic. Clearly L (β∆s ) ≥ (r ◦ α) (s0 − ∆s). Hence 1 L (β0 ) − L (β∆s ) 0 ˙ = β0 (α (s0 )) , α˙ (s0 ) , (r ◦ α) (s0 ) ≥ lim ∆s ∆s→0+ where the last equality follows from the first variation of arc length formula. Since β˙ 0 (α (s0 )) ∈ (∇
r) (α (s0 )) and by (I.26), we conclude that  π −ε >0 (r ◦ α) (s0 ) ≥ cos 2 8

Note that, a priori, r may not be differentiable at α (s) for a.e. s.

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for all s0 ∈ (0, ε) such that (r ◦ α) (s0 ) exists. Therefore  π −ε (r ◦ α) (¯ s) − r (x) ≥ s¯ cos 2 for all s¯ ∈ [0, ε).




We leave it to the reader to trace through the proof of the ‘only if’ part of the lemma to obtain the following slight improvement (the ‘if’ part is obvious from Lemma I.34). Exercise I.35 (Characterization of regular points, II). Show that x ∈ M − {p} is a regular point of r if and only if there exist V ∈ Sxn−1 such that for any smooth unit speed path γ : [0, ε) → M, where ε > 0, with γ˙ (0) = V , there exists δ ∈ (0, ε] such that r (γ (s)) − r (x) ≥ δs for all s ∈ [0, δ). 4.3. Regular points on complete noncompact manifolds with nonnegative curvature. Next we discuss regular points on complete noncompact manifolds with nonnegative sectional curvature. Define the maximum angle function of r ang p : M → [0, π] by (I.27)

ang p (x)  max { (V, W ) : V, W ∈ (∇
r) (x)} .

/ Cut (p). Clearly, if ang p (x) < π2 , then x is a Note that ang p (x) = 0 for x ∈ regular point of r. Remark I.36. There is a similar definition for the set gradient of the Busemann function associated to a ray or a point and the corresponding maximum angle function (see [107]). The following is the main result of this subsection. Lemma I.37 (Maximum angle function tends to zero). Let (Mn , g) be a complete noncompact manifold with nonnegative sectional curvature. Given p ∈ M, we have (I.28)

lim d(x,p)→∞

ang p (x) = 0.

In particular, for any ε ∈ (0, π2 ], there exists s0 (p, ε) < ∞ such that x is an ε-regular point of r for all x ∈ M − B (p, s0 (p, ε)). Proof.

We prove the lemma by contradiction. Suppose that there exist c ∈ 0, π2 and a sequence of points {xi }∞ i=1 in M such that ri  d (xi , p) → ∞ and ang p (xi ) > 2c

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477

for all i ∈ N. Then for all i there exist Vi , Wi ∈ (∇
r) (xi ) such that  (Vi , Wi ) > 2c.

(I.29)

Note that the unit speed geodesics γVi : [0, ri ] → M, γWi : [0, ri ] → M, defined by γVi (s)  expxi (− (ri − s) Vi ) , γWi (s)  expxi (− (ri − s) Wi ) for s ∈ [0, ri ], both join p to xi . Fix an ε > 0 and an i ∈ N. Suppose that x ∈ M is such that ri . (I.30) r (x) = d (x, p) ≥ 1−ε Let di  d (x, xi ) and let αi : [0, di ] → M be a unit speed minimal geodesic joining xi to x. Note that from the triangle inequality and (I.30), we have ε ri . (I.31) di ≥ r (x) − d (xi , p) ≥ 1−ε By (I.29), we may assume, without loss of generality (i.e., by switching Vi and Wi if necessary), that (I.32)





 (α˙ i (0) , Vi ) > c,

is as above. where c ∈ 0, Now define θi  π −  (α˙ i (0) , Vi ) . π 2

By (I.32), (I.33) let

0 ≤ θi < π − c.

Let β : [0, r (x)] → M be a unit speed minimal geodesic from p to x and   φi   β˙ (0) , γ˙ Vi (0) .

We have a geodesic triangle ∆pxi x in M with sides αi , β, and γVi and corresponding opposite vertices p, xi , and x, respectively. Consider the ˜ in Euclidean space with side lengths r (x), ri , and di comparison triangle ∆ which is associated to the geodesic triangle ∆pxi x in M. We denote the ˜ corresponding to φi and θi by φ˜i and θ˜i . angles of ∆ By the triangle version of the Toponogov comparison theorem (Theorem G.33(1)), we have (I.34)

φ˜i ≤ φi

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and θ˜i ≤ θi .

(I.35)

We shall prove a uniform positive lower bound for φ˜i . In this regard, without loss of generality, we may assume φ˜i ≤ π2 (so that cos φ˜i ≥ 0). Case (1). Suppose θ˜i ≥ π2 . By (I.33) and (I.35), we then have sin θ˜i > sin c. In the Euclidean comparison triangle, by the law of sines, di di sin θ˜i > sin c sin φ˜i = r (x) di + ri since r (x) ≤ di + ri . Hence, by (I.31), we have sin φ˜i > ε sin c, so that φ˜i ≥ sin−1 (ε sin c) since φ˜i ≤ π2 . Case (2). Now suppose θ˜i < π2 . By the Euclidean law of cosines applied ˜ we have to ∆, d2i = ri2 + r2 (x) − 2ri r (x) cos φ˜i , so that

1 2 ri + r2 (x) − d2i ri r (x) cos φ˜i = 2 ≤ ri2 since θ˜i
0. Now, by passing to a subsequence, we may assume ri+1 ≥ 2ri for all i. By (I.36) with ε = 12 , we then have for any j = i,9  

−1 sin c .  γ˙ Vj (0) , γ˙ Vi (0) ≥ sin 2 This is a contradiction to the compactness of the unit sphere Spn−1 and hence the lemma is proved.
The following consequence of the Toponogov comparison theorem is due to Abresch [1]. 9

Note that cos−1

1 2

=

π 3

≥ sin−1

 sin c  . 2

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4. CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES

479

Exercise I.38. Under the hypotheses in the proof of Lemma I.37 above, we have  (I.37) r (x) < di + ri 1 − ε2 sin2 c. (For a proof see for example Fact 1.3(ii) of [107].) 4.4. Distance spheres as Lipschitz hypersurfaces. Since the distance function is only Lipschitz and the sphere S (p, s) is not necessarily a smooth manifold, we recall the following. Definition I.39 (Lipschitz hypersurface). A Lipschitz hypersurface in a smooth manifold is a subset which is locally the graph of a Lipschitz function. That is, a subset N ⊂ Mn is a Lipschitz hypersurface if for every q ∈ N there exists a smooth local coordinate chart U , x = xi with q ∈ U and x (U ) = V ×(a, b), where V ⊂ Rn−1 is an open set and −∞ < a < b < ∞ and where x (N ∩ U ) is the graph of a Lipschitz function on V. If N is a connected Lipschitz hypersurface of a complete Riemannian manifold (Mn , g), then any two points x, y ∈ N may be connected by a Lipschitz path in N ,10 which in turn has finite length. Hence the induced distance dN  dLdN , as defined in (G.6), is finite. We now show the following. Lemma I.40. If N is a connected Lipschitz hypersurface of a complete Riemannian manifold (Mn , g) such that N is a closed subset of M, then the length space N is complete. Proof. Given x, y ∈ N , let γi : [0, 1] → N , i ∈ N, be a length minimizing sequence of constant speed paths joining x to y, i.e., L (γi ) → dN (x, y) and for any s1 , s2 ∈ [0, 1] we have   L γi |[s1 ,s2 ] = L (γi ) |s1 − s2 | . Hence dM (γi (s1 ) , γi (s2 )) ≤ L (γi ) ≤ C |s1 − s2 | for s1 = s2 and where C < ∞ is independent of i, s1 , s2 . By the Arzela– Ascoli theorem, there exists a subsequence such that γi converges to a path γ∞ : [0, 1] → N joining x to y. By definition, dN (x, y) ≤ L (γ∞ ). On the other hand, by the lower semi-continuity of L, we have L (γ∞ ) ≤ lim L (γi ) = dN (x, y) , i→∞

so that L (γ∞ ) = dN (x, y).




10 Locally, such a path may be obtained by intersecting N with a smooth transversal surface.

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480

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

Our interest in Lipschitz hypersurfaces stems from the following. For the definition of topological end, see p. 143 of Benedetti and Petronio [11]. Proposition I.41 (Large radii distance spheres are Lipschitz hypersurfaces). Let (Mn , g) be a complete noncompact manifold with nonnegative sectional curvature. For s sufficiently large, we have the following. (i) The distance sphere hypersurface.   S (p, s) is a Lipschitz (ii) The length space S (p, s) , LdS(p,s) is complete. (iii) The number of components of S (p, s) is equal to the number of topological ends of (M, g). In particular, if (M, g) has exactly one topological end, then S (p, s) is connected for s sufficiently large. Proof. First we shall prove that if x is a regular point of r, then there is an open neighborhood W of x such that S (p, r (x)) ∩ W is a Lipschitz hypersurface. By Lemma I.33, if x is a ( π2 -)regular point of r, then there exists an open neighborhood U of x, a smooth unit vector field V defined on U , and an ε > 0 such that for every y ∈ U we have π (I.38)  (V (y) , W ) ≤ − ε 2 for all W ∈ (∇
r) (y). Let γ be an integral curve of V in U with γ (0) = y for some point y ∈ U . By the proof of Exercise I.35, we may deduce r (γ (s)) − r (y) ≥ sin ε · s for all s ∈ (0, δ (y)] and some δ (y) > 0. In fact, (I.39)

r (γ (t2 )) − r (γ (t1 )) ≥ sin ε · (t2 − t1 )

for any t1 < t2 in the domain of γ.11 Now let Hn−1 ⊂ U be a smooth hypersurface which passes through x and which is transversal to V ,12 i.e., for every y ∈ H we have Ty H + RV (y) = Ty M. Near x, by using V , we shall write S (p, r (x)) as a ‘graph’ over H. It follows from (I.39) that there exists an open neighborhood V ⊂ U of x such that for every point y ∈ H ∩ V,13 the maximal integral curve γy : (ay , by ) → V to V in V, with γy (0) = y, also intersects r−1 (r (x)) = S (p, r (x)) at exactly one point; call this point Φ (y). That is, we have defined a map Φ : H ∩ V → S (p, r (x)) ∩ V, where the image Φ (H ∩ V) is an open subset of S (p, r (x)). 11 12

expx .

13

We leave it as an exercise to verify (I.39). For example, we can take Hn−1 to be the image of a small (n − 1)-ball in V ⊥ of Note that H ∩ V is nonempty since x ∈ H ∩ V.

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4. CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES

481

We define the function f : H ∩ V → R by Φ (y)  γy (f (y)) .

(I.40)

If we prove that f is a Lipschitz function, then / (I.40) implies that S (p, r (x)) near x is a Lipschitz hypersurface. (Let W = y∈H∩V γy ((ay , by )), which is an open subset of V; then S (p, r (x)) ∩ W is a Lipschitz graph over H ∩ V (using the vector field V ).) To see that f is a Lipschitz function, we argue as follows. For any y ∈ H ∩ V, we have |r (γy (0)) − r (γy (f (y)))| = |r (y) − r (x)| ≤ d (x, y) since γy (f (y)) ∈ r−1 (r (x)) and by the triangle inequality. From (I.39), with γ = γy , we obtain |r (γy (0)) − r (γy (f (y)))| ≥ sin ε |f (y)| , where ε > 0 is as in (I.38). Hence using f (x) = 0, we have 1 d (x, y) , (I.41) |f (x) − f (y)| = |f (y)| ≤ sin ε which implies f is Lipschitz continuous at x. Now we show the Lipschitz continuity of f at other points in H ∩ V. Let V be the same smooth unit vector field in V as before. For any point y1 ∈ H ∩ V, let x1 = Φ (y1 ) ∈ S (p, r (x)) ∩ V. Consider the smooth hypersurface H1n−1 ⊂ V defined by H1 = {γy (f (y1 )) : y ∈ H ∩ V such that f (y1 ) ∈ (ay , by )} . (The parts of the integral curves γ to V between H∩V and H1 have constant length f (y1 ).)14 We have that H1 passes through x1 and is transversal to V . Repeating the above construction at x1 (instead of x), we have a map Φ1 : H1 → S (p, r (x)) ∩ V (note that r (x1 ) = r (x)) and a Lipschitz function f1 : H1 → R with Φ1 (y)  γy (f1 (y)) . Note that f1 (x1 ) = 0. For the same reason that (I.41) is true, we have 1 d (x1 , y) (I.42) |f1 (x1 ) − f1 (y)| ≤ sin ε for any y ∈ H1 . Given y1 as above, let H0  {y ∈ H ∩ V : f (y1 ) ∈ (ay , by )} ⊂ H ∩ V. Note that the map Φ2 : H0 → H1 , defined by Φ2 (y) = γy (f (y1 )) , Let H
{y ∈ H ∩ V : f (y1 ) ∈ (ay , by )}. Then H1 is a ‘constant height’ f (y1 ) graph over H with respect to V . 14

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482

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

is a smooth invertible map. By originally choosing U sufficiently small (independent of y1 ), we may assume that d (Φ2 (y) , Φ2 (y1 )) ≤ 2d (y, y1 )

(I.43) for y ∈ H0 . Moreover

f (y) = f (y1 ) + f1 (Φ2 (y)) . On the other hand, it follows from (I.42) and (I.43) that 1 d (Φ2 (y) , Φ2 (y1 )) |f1 (Φ2 (y)) − f1 (Φ2 (y1 ))| ≤ sin ε 2 d (y, y1 ) ≤ sin ε for y ∈ H0 . Hence 2 d (y, y1 ) |f (y) − f (y1 )| ≤ sin ε and therefore f is Lipschitz continuous on H ∩ V. (i) By Lemma I.37, there exists s0 (p) < ∞ such that x is a π4 -regular point of r for all x ∈ M − B (p, s0 (p)). Hence S (p, s) is a Lipschitz hypersurface for all s > s0 (p). (ii) By Lemma I.40, the Lipschitz hypersurface S (p, s) is complete as a length space. (iii) The last part of the proposition follows from the definition of topo
logical end and the assumption that s0 (p) is large. 5. Approximate Busemann–Feller theorem In this section we prove an almost distance-decreasing property of the nearest point projection map in a small tubular neighborhood of a hypersurface in a Riemannian manifold. Let (Mn , g) be a complete orientable Riemannian manifold with sectional curvature bounded above by a constant κ2 > 0. Let Hn−1 ⊂ M be a complete smooth orientable hypersurface and let ν be a choice of a smooth unit normal vector field to H. Then H is two-sided in M, i.e., there exists a smooth embedding Φ : H × [−1, 1] → M such that Φ (x, 0) = x for x ∈ H. Let Ω+  Φ (H × [0, 1)) , Ω−  Φ (H × (−1, 0]) , so that Φ (H × (−1, 1)) = Ω+ ∪ Ω− and Ω+ ∩ Ω− = H. Without loss of generality, we may assume that the sets Ω+ and Ω− are on the sides of ν and −ν, respectively. Let M+ be the set of x ∈ M such that either: (1) x ∈ H or

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5. APPROXIMATE BUSEMANN–FELLER THEOREM

483

(2) for any y ∈ H with d (x, y) = d (x, H) and for any unit speed minimal geodesic γ : [0, d (x, y)] → M joining y to x there exists ε > 0 such that γ ([0, ε)) ⊂ Ω+ . We may define the set M− in the same way using Ω− (so that M+ ∩ M− = H). We leave it as an exercise to show that for any compact subset K of H, there exists δ > 0 such that Φ (K × [0, δ)) ⊂ M+ . The second fundamental form IIH of H with respect to ν is defined by IIH (X, Y ) = DX ν, Y  = − ν, DX Y 

for X, Y ∈ T H.

Given ε > 0, let Nε+ (H) denote the following subset of the open ε-neighborhood of H on the side of ν: Nε+ (H) = {x ∈ M+ : d (x, H) < ε} . Assuming the exercise in the previous paragraph, it is easy to see that, for any ε > 0 and for any compact subset K of H, there exists η > 0 such that Φ (K × [0, η)) ⊂ Nε+ (H). Suppose that (I.44)

IIH ≥ −Λ IH

for some constant 0 ≤ Λ < ∞, where IH is the first fundamental form of H. Lemma I.42 (Nearest point projection map in small tubular neighborhood). There exists a constant ι (Λ, κ) > 0 with the following property. If + (H), then there exists a unique point π (x) ∈ H closest to x, so x ∈ Nι(Λ,κ) that the nearest point projection map + (H) → H π : Nι(Λ,κ)

is well defined. Moreover, π is smooth. Proof. Let N+ H  {aν (x) : x ∈ H and a ≥ 0} denote the ‘nonnegative normal bundle’ of H, where ν (x) is the above choice of unit normal at x. Consider the exponential map restricted to this bundle, i.e., exp|N+ H : N+ H → M. Then consider the parallel hypersurfaces Hr  {x ∈ M+ : d (x, H) = r} , where r ≥ 0. Let IIr denote the second fundamental form of Hr , which is well defined as long as Hr is smooth. Given x ∈ H, let αx : [0, ∞) → M be the unit speed geodesic with α˙ x (0) = ν (x). Recall the Riccati equation ∇α˙ x IIr = − Rm (α˙ x , ·) ·, α˙ x  − II2r

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484

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

(see (1.139) of [45] for example). Since the sectional curvature of (M, g) is bounded above by κ2 > 0, we have − Rm (α˙ x , ·) ·, α˙ x  ≥ −κ2 I, so that along αx we have the differential inequality ∇α˙ x IIr ≥ −κ2 Ir − II2r ,

(I.45)

where Ir denotes the first fundamental form of Hr . Given r¯ ≥ 0, suppose that Vr¯ ∈ Tαx (¯r) Hr¯ is a unit vector and let V (r) ∈ Tαx (r) Hr denote the parallel translation of Vr¯ along αx .15 We have |V (r)| ≡ 1 and V (¯ r) = Vr¯. If Vr¯ is an eigenvector of IIr¯, then (I.45) implies  d  (IIr (V (r) , V (r))) ≥ −κ2 − (IIr¯ (V (¯ r) , V (¯ r)))2 . dr  r=¯ r

In particular, if we define λ (r)  min IIr (W, W ) , |W |=1

then, by applying Lemma 10.29 in Part II to −λ, we have along αx ,     d  d+  λ (r) = min (IIr (V (r) , V (r))) : IIr¯ (Vr¯, Vr¯) = λ (¯ r) dr r=¯r dr r=¯r   r) ≥ min −κ2 − (IIr¯ (Vr¯, Vr¯))2 : IIr¯ (Vr¯, Vr¯) = λ (¯ r)2 . = −κ2 − λ (¯ r), then Vr¯ is Here we have used the elementary fact that if IIr¯ (Vr¯, Vr¯) = λ (¯ an eigenvector of IIr¯. Furthermore, by (I.44) we have λ (0) ≥ −Λ, where Λ ≥ 0. Hence, by the Sturmian comparison theorem for differential inequalities (see Angenent [6] for example), we conclude that16

IIr ≥ λ (r) Ir ≥ κ cot κr + cot−1 −κ−1 Λ Ir

as long as the rhs is finite (i.e., > −∞). (Note that cot−1 −κ−1 Λ ∈ [ π2 , π).) Thus there are no focal points of distance less than

(I.46) ι (Λ, κ)  κ−1 π − cot−1 −κ−1 Λ > 0 to H. This implies that ι(Λ,κ)

exp|N ι(Λ,κ) H : N+ +

15 16

+ H → Nι(Λ,κ) (H)

Note that parallel translation is an isometry between Tαx(¯r) H and Tαx (r) Hr .   Note that the function φ (r) = κ cot κr + cot−1 −κ−1 C is a solution to the ode dφ = −κ2 − φ2 , dr φ (0) = −C.

We leave it as an exercise for the reader to verify that the Sturmian comparison theorem holds for differential inequalities using the lim inf of forward difference quotients. Actually this last fact is used for the conventional ‘pde to ode formulation’ of the scalar maximum principle.

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5. APPROXIMATE BUSEMANN–FELLER THEOREM

485

is a diffeomorphism, where N+ε H  {aν (x) : x ∈ H and expx (sν (x)) ⊂ M+ for 0 ≤ s ≤ a < ε} . + Moreover, for every x ∈ Nι(Λ,κ) (H), there exists a unique point π (x) ∈ H closest to x. This point π (x) also has the property that the unit speed geodesic απ(x) [0,d(x,H)] is minimal and joins π (x) to x. It is easy to see that π is smooth.


Remark I.43. Note that we have shown that Hr is a smooth hypersurface for 0 ≤ r < ι (Λ, κ). The following gives an almost distance-decreasing property of the nearest point projection map in a small tubular neighborhood. It may be thought of as an approximate version of the Busemann–Feller theorem (see Corollary H.10) for closed convex sets. We would like to thank Deane Yang for showing us both the statement and the proof of the theorem. Theorem I.44 (Approximate Busemann–Feller theorem). Let (Mn , g) and Hn−1 ⊂ M be as above, so that in particular we have sect (g) ≤ κ2 and IIH ≥ −Λ IH . For any ε ∈ (0, ι (Λ, κ)], where ι (Λ, κ) > 0 is given by (I.46), we have for any smooth path γ ⊂ Nε+ (H) that −1  Λ L (γ) . L (π ◦ γ) ≤ cos (εκ) − sin (εκ) κ + (H) be a smooth path and let π ◦ γ be Proof. Let γ : [0, 1] → Nι(Λ,κ) its projection into H. Without loss of generality, by reparametrizing γ, we may assume that π ◦ γ : [0, 1] → H has constant speed (= L (π ◦ γ)). Define a smooth map (homotopy) + (H) Γ : [0, 1] × [0, 1] → Nι(Λ,κ)

by Γ (s, t)  expπ◦γ(t) sd (π ◦ γ (t) , γ (t)) ν (π ◦ γ (t)) . Then we have Γ (0, ·) = π ◦ γ, Γ (1, ·) = γ, and, for each t, the path s → Γ (s, t) is a constant speed geodesic. Let ∂Γ ∂Γ and Γt  . Γs  ∂s ∂t Note that Γs , Γt  (0, t) = 0

(I.47) for all t since Γt (0, t) = (I.48)

d dt

(π ◦ γ) (t) is tangent to H whereas

Γs (0, t) = d (π ◦ γ (t) , γ (t)) ν (π ◦ γ (t))

is normal to H.

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486

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

We have |Γt (0, t)| ≡ L (π ◦ γ) .

(I.49)

Claim. If γ ∈ Nε+ (H), where 0 < ε ≤ ι (Λ, κ), then (I.50)

(i)

(I.51)

(ii)

∂ |Γt | (0, t) ≥ −Λε L (π ◦ γ) , ∂s ∂2 |Γt | (s, t) ≥ −ε2 κ2 |Γt | (s, t) ∂s2

for all s and t. Proof of claim. (i) Using ∇Γs Γt −∇Γt Γs = Γ∗ |Γt |

∂

∂ ∂s , ∂t



= 0, we compute

∂ 1 ∂ |Γt | = |Γt |2 ∂s 2 ∂s = ∇Γs Γt , Γt  = ∇Γt Γs , Γt  ∂ Γs , Γt  − Γs , ∇Γt Γt  . = ∂t

At s = 0 we have |Γt | ≡ L (π ◦ γ) and (I.47), which implies at s = 0. Hence, using (I.48), we have L (π ◦ γ)

∂ ∂t

Γs , Γt  ≡ 0

∂ |Γt | (0, t) = −d (π ◦ γ (t) , γ (t)) ν (π ◦ γ (t)) , ∇Γt Γt  (0, t) ∂s = d (π ◦ γ (t) , γ (t)) II 0 (Γt (0, t) , Γt (0, t)) .

Since II 0 = IIH ≥ −Λ IH and |Γt (0, t)| ≡ L (π ◦ γ), we have ∂ |Γt | (0, t) ≥ −Λd (π ◦ γ (t) , γ (t)) L (π ◦ γ) ∂s ≥ −Λε L (π ◦ γ) because d (π ◦ γ (t) , γ (t)) ≤ ε by assumption. (ii) We compute ∂2 1 ∂2 |Γt |2 − |Γt | 2 |Γt | = ∂s 2 ∂s2



2 ∂ |Γt | ∂s

 2

= ∇Γs ∇Γs Γt , Γt  + |∇Γs Γt | −

(I.52) Since

 2

|∇Γs Γt | −

∂ |Γt | ∂s

2 =

∂ |Γt | ∂s

2 .

 1  2 2 2 Γ | |Γ | − ∇ Γ , Γ  |∇ ≥0 t t Γs t Γs t |Γt |2

and since ∇Γs (∇Γs Γt ) = ∇Γs (∇Γt Γs ) = R (Γs , Γt ) Γs − ∇Γt (∇Γs Γs ) = R (Γs , Γt ) Γs

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6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY

487

(note that [Γs , Γt ] ≡ 0 and ∇Γs Γs ≡ 0), equation (I.52) implies |Γt |

∂2 |Γt | ≥ R (Γs , Γt ) Γs , Γt  ∂s2 ≥ −κ2 |Γt |2 |Γs |2 ≥ −ε2 κ2 |Γt |2

since, given any t, we have |Γs | (s, t) ≡ d (π ◦ γ (t) , γ (t)) ≤ ε. Inequality (ii) follows immediately. Thus the claim has been proven. Now we may apply the Sturmian comparison theorem to (I.50)–(I.51) to obtain   Λ |Γt | (s, t) ≥ cos (εκs) − sin (εκs) L (π ◦ γ)  ψ (s) ; κ note that ψ satisfies d2 ψ (s) = −ε2 κ2 ψ (s) , ds2 d ψ (s) = −Λε L (π ◦ γ) , ds ψ (0) = L (π ◦ γ) . In particular,

 |Γt | (1, t) ≥

Thus

cos (εκ) −



1

L (γ) = 

 Λ sin (εκ) L (π ◦ γ) . κ

|Γt | (1, t) dt

 Λ cos (εκ) − sin (εκ) L (π ◦ γ) . κ

0

≥


Finally, we leave it as an exercise for the reader to formulate local versions of Lemma I.42 and Theorem I.44. 6. Equivalence classes of rays and points at infinity The monotonicity property of distance spheres in nonnegatively curved manifolds given by Proposition I.29 suggests another approach to the existence of asymptotic cones in Theorem I.26, originally suggested by Gromov [9] and later considered by Kasue [107]. On pp. 593–594 of [107] it is written:

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488

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

“... Gromov has defined, in his lectures [3], the Tits’ metric on the points at infinity, the equivalence classes of rays, of a Hadamard manifold. Moreover, he has suggested that there is a counterpart to Tits’ metric for nonnegative curvature and proposed several interesting exercises on such manifolds (cf. [3], pp. 58–59).”17 Throughout the discussion in this section, (Mn , g) shall denote a complete noncompact Riemannian manifold with nonnegative sectional curvature. Note that such a manifold either has one topological end or has two topological ends, and in the latter case it contains a line. To see that M has at most two topological ends, we argue as follows. If M has at least two topological ends, then (Mn , g) contains a line, so by the Toponogov

splitting

n n−1 , h , where N n−1 , h theorem, (M , g) is isometric to a product R × N is a complete Riemannian manifold with sect ≥ 0. If N is compact, then M has exactly two topological ends. If N is noncompact, then N has exactly the same number of topological ends as M. By induction on the dimension, we see that M has exactly two topological ends. 6.1. The mollified distance function. Fix p ∈ M and s1 ∈ (0, ∞). Suppose that the sectional curvatures of g are bounded from above by k0 ≥ 1 in the ball B p, s1 + π2 . Recall from (12.31) of Part II that for r (x)  d (x, p) the mollified distance functions rε : B (p, s1 ) → [0, ∞) are defined by

 v  1 η r (expx (v)) dµg(x) (v) εn B (0,ε) ε   for x ∈ B (p, s1 ) and ε ∈ 0, 2√πk and where η : Tx M → R is the standard (I.53)

rε (x) 

0

mollifier.18 ¯ (p, s1 ) → R is a C ∞ function. MoreBy Lemma 12.28 of Part II, rε : B over, by the proof of Lemma 12.29 in Part II, we have the following. (1) The functions Lemma I.45 (Convergence of both rε and ∇rε ). rε converge uniformly to r as ε → 0 in any compact subset of M. (2) In any compact subset of the complement of Cut (p), the gradient ∇rε converges uniformly to ∇r as ε → 0. By the proof of Lemma 12.30 in Part II, we have for ε > 0 sufficiently small, (I.54) r − 1 ≤ rε ≤ r + 1 and |∇rε | ≤ C ¯ (p, s1 ), where C is independent of ε (see also Lemma 1.5 in [107]). in B 17

[3] in [107] is [9] in the current book, for the reader’s convenience. Here we isometrically identified Euclidean space En = (Rn , gE ) with (Tx M, g (x)) (note that η is rotationally symmetric). 18

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6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY

489

We begin with an example on a compact manifold which contrasts to the following discussion regarding a positive lower bound of |∇rε |. Example I.46 (Mollifying distance on the sphere). Consider the unit sphere S n . Let p ∈ S n and let r (x)  d (x, p). The mollified distance functions rε , where ε ∈ (0, π/2), are rotationally symmetric with respect to the axis spanned by p. Moreover, rε attains its maximum at the antipodal point −p, so that |∇rε | (−p) = 0 for all ε ∈ (0, π/2). From the rotational symmetry of rε , it is not difficult to show that ∇rε (x) = ∇r (x) |∇rε (x)| for all x ∈ S n − {p, −p} and ε ∈ (0, π/2). Now we return to the general case of complete noncompact manifolds with nonnegative sectional curvature. Define θ1 : (0, ∞) → [0, ∞) by θ1 (s)  max {ang p (x) : r (x) ≥ s} , where ang p (x) is defined in (I.27). This is the maximum angle of r in the complement of the ball of radius s. By (I.28), we have θ1 is well defined and finite and lim θ1 (s) = 0.

(I.55)

s→∞

On manifolds with nonnegative sectional curvature, rε has the following properties, which may be used to approach Presumed Theorem I.50. Lemma I.47 (Properties of the derivatives of rε ). Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. For any p ∈ M and s1 > 0 there exist an ε0 > 0 and functions κi (ε), defined for ε ∈ (0, ε0 ) and i = 1, 2, 3, with lim κi (ε) = 0

(I.56)

ε→0

such that for any ε ∈ (0, ε0 ) we have the following: ¯ (p, s1 ), (a) (gradient estimate) for x ∈ B (I.57)

1 − κ1 (ε) − θ1 (r (x)) ≤ |∇rε | (x) ≤ 1 + κ1 (ε) ,

(b) (improved gradient estimate outside cut locus) for d (x, Cut (p)) ≥ ¯ (p, s1 ), κ2 (ε) and x ∈ B (I.58)

1 − κ2 (ε) ≤ |∇rε | (x) ,

and ¯ (p, s1 ),19 (c) (Hessian comparison) for x ∈ B (I.59) 19

∇∇rε (x) ≤

1 + κ3 (ε) . rε (x)

Compare with (I.20).

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490

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

In Lemma I.47, parts (a), (b), and (c) correspond to parts (ii), (iii), and (iv) of Lemma 1.5 of [107], respectively. In fact, a check of the proof of Lemma 12.29 and the proof of Lemma 12.30, both in Part II, yields the upper bounds in (a), (b), and (c). As for the proof of the lower bound in (a) and (b), it uses Lemma I.37 and Lemma I.34 regarding ∇
r. We leave the interested reader to either work out the proof as an exercise or see pp. 602–603 of [107]. One may ask how the gradients of the (smooth) mollified distance functions are related to the (set-valued) generalized gradient of the (Lipschitz) distance function. In this regard, we venture to guess the following first approximation to a conjecture. Problem I.48 (Does ∇rε approach ∇
r in some sense?). As ε → 0, does the gradient ∇rε approach the set gradient ∇
r in some sense? For example, one may ask whether at any point x ∈ M, ∇rε (x) limits to a vector in the convex hull of (∇
r) (x) in Tx M. If so, is this limit uniform (on compact subsets)? Note that away from the cut locus we have ∇rε → ∇r. Perhaps another possibility is that ∇rε approaches a vector in the cone (with vertex at 0) over the convex hull of (∇
r) (x) in the sphere Sxn−1 . Toward generalizing Proposition I.29 to large radii spheres (see Presumed Theorem I.50 below), one approach is to prove an approximate distance-nonincreasing result for the (smooth) level surfaces of rε and then take the limit as ε → 0 to obtain the distance-nonincreasing result for geodesic spheres (i.e., level surfaces of r). One concern is how well the intrinsic distance function on a level surface of rε approximates the intrinsic distance function dS(p,s) on a nearby level surface S (p, s) of r. The following example illustrates a potential issue. Example I.49. (1) (Bumpy hypersurfaces limiting to hyperplane) Consider the family of smooth curves



Cε  x, ε sin ε−2 x : x ∈ R in the plane, defined for ε > 0. We have lim Cε = R × {0} .

ε→0



The length along Cε from (0, 0) to 1, ε sin ε−2 is given by  1 1 + ε−2 cos2 (ε−2 x)dx Lε  0  1 

 −1 cos ε−2 x  dx. ≥ε 0

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6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY

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We have lim L ε = ∞,

ε→0

whereas the length along R × {0} from (0, 0) to (1, 0) is equal to 1. Note that we may take the product of this example with Euclidean space to get higher-dimensional examples. In the above example, although the limit is smooth (flat), the ‘Lipschitz constants’ of the curves Cε tend to ∞ as ε → 0. (2) (Less bumpy hypersurfaces limiting to a hyperplane) Let !x" denote the greatest less than or equal to x. Consider the function   integer f : R → − 12 , 12 defined by    1   f (x)  x − !x" −  . 2 Then f is Lipschitz with Lipschitz constant 1 (f is not smooth exactly at the half-integers). Consider the Lipschitz plane curves



Sε  x, εf ε−1 x : x ∈ R defined for ε > 0. We have lim Sε = R × {0} .

ε→0



On the other hand, the length along Sε from (0, 0) to 1, εf ε−1 √ is equal to 2 independent of ε. 6.2. An approach toward the space of points at infinity. In this subsection we discuss an approach toward constructing the space of points at infinity. 6.2.1. A presumed result on distance-nonincreasing maps. This discussion hinges on the validity of the following presumed result (see Proposition 2.2 in Kasue [107]). Presumed Theorem I.50 (Distance nonincreasing maps between large spheres). Suppose (Mn , g) is a complete Riemannian manifold with nonnegative sectional curvature. For any p ∈ M there exists s0 (p) < ∞ such that if t ≥ s ≥ s0 (p), then there exists a map φs,t : S (p, s) → S (p, t) such that (1) ((relatively) distance nonincreasing) 1 1 d (φs,t (x) , φs,t (y)) ≤ dS(p,s) (x, y) , (I.60) t S(p,t) s where dS(p,s) is defined by (I.15) on the sphere S (p, s); that is, the map of metric spaces     1 1 φs,t : S (p, s) , dS(p,s) → S (p, t) , dS(p,t) s t

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492

I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

is distance nonincreasing (in particular, the map φs,t is continuous); (2) (composition rule) for any u ≥ t ≥ s ≥ s0 (p), φt,u ◦ φs,t = φs,u ; (3) (preserves rays emanating from p) for any α ∈ RayM (p) and t ≥ s ≥ s0 (p), φs,t (α (s)) = α (t) . 6.2.2. Equivalence of rays. Two rays α, β ∈ RayM are said to be equivalent if d (α (s) , β (s)) = 0; s in this case we write α ∼ β. Clearly ∼ is an equivalence relation.20 Let [α] = {β ∈ RayM : β ∼ α} denote the equivalence class of a ray α. Fix a point p ∈ M and define the distance dR on RayM / ∼ by (I.61)

lim

s→∞

dS(p,s) (α ∩ S (p, s) , β ∩ S (p, s)) . s→∞ s Regarding the issue of the existence of the limit on the rhs of (I.62), see Lemma I.53(iii) below.

(I.62)

dR ([α] , [β]) = lim

Example I.51 (Euclidean space). If (Mn , g) = En is Euclidean space, then ( RayM / ∼, dR ) = S n−1 , the unit (n − 1)-sphere. Note that here α ∼ β if and only if α and β are parallel and pointing in the same direction. That is, RayM = En × S n−1 (a basepoint and a direction) and (x, v) ∼ (y, w) if and only if v = w; so the quotient identifies the factor En to a point. Let RayM (p) be the set of rays emanating from a fixed point p.21 The equivalence relation ∼ on RayM induces an equivalence relation ≈ on RayM (p). Lemma I.52. For any α ∈ RayM and p ∈ M there exists αp ∈ RayM (p) such that αp ∼ α. Proof. Given t ∈ (0, ∞), let αt : [0, d (p, α (t))] → M be a unit speed minimal geodesic joining p to α (t). For the sequence of unit vectors {α˙ i (0)}i∈N , there exists a subsequence such that limi→∞ α˙ i (0)  V ∈ Tp M exists. Let αp : [0, ∞) → M be the unit speed geodesic with α˙ p (0) = V . Then αp ∈ RayM (p) and αp ∼ α. (Exercise: Prove that αp is a ray and
αp ∼ α.) The following lemma justifies the definition (I.62). 20

Reflexivity and symmetry are obvious, whereas transitivity follows from the triangle inequality. 21 Note that there is an injection j : RayM (p) → Spn−1 ⊂ Tp M, where Spn−1 denotes the unit sphere.

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6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY

493

Lemma I.53. Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. For any α1 , α2 ∈ RayM , we have the following. (i) dR ([α1 ] , [α2 ]) = 0 is equivalent to lims→∞ d(α1 (s),s α2 (s)) = 0. (ii) dR is well defined independent of the choice of representatives of [α1 ] and [α2 ]. (iii) Assuming Presumed Theorem I.50 is true, we have dR ([α1 ] , [α2 ]) exists. Proof. (i) (=⇒). By the triangle inequality, we have (I.63)

u − d (αi (0) , p) ≤ d (αi (u) , p) ≤ u + d (αi (0) , p) .

Thus, if αi (si ) ∈ αi ∩ S (p, s), then si ∈ [s − d (αi (0) , p) , s + d (αi (0) , p)]. Hence d (α1 (s) , α2 (s)) lim s→∞ s d (α1 (s) , α1 (s1 )) + d (α1 (s1 ) , α2 (s2 )) + d (α2 (s2 ) , α2 (s)) ≤ lim s→∞ s dS(p,s) (α1 ∩ S (p, s) , α2 ∩ S (p, s)) d (α1 (0) , p) + d (α2 (0) , p) + lim ≤ lim s→∞ s→∞ s s = 0. (⇐=). Using the modified distance function rε and following the idea of the proof of Proposition I.41, one can show that there is a constant C independent of s such that (exercise) (I.64)

dS(p,s) (x, y) ≤ C · d (x, y) for x, y ∈ S (p, s) .

Hence, for any s let si be defined by αi (si ) ∈ αi ∩ S (p, s), dS(p,s) (α1 ∩ S (p, s) , α2 ∩ S (p, s)) s→∞ s d (α1 (s1 ) , α2 (s2 )) ≤ C lim s→∞ s d (α1 (s1 ) , α1 (s)) + d (α1 (s) , α2 (s)) + d (α2 (s) , α2 (s2 )) ≤ C lim s→∞ s d (α1 (0) , p) + d (α2 (0) , p) d (α1 (s) , α2 (s)) + C lim ≤ C lim s→∞ s→∞ s s = 0. lim

(ii) If αi ∼ βi , then by the triangle inequality     d  d (α ∩ S (p, s) , α ∩ S (p, s)) (β ∩ S (p, s) , β ∩ S (p, s)) 1 2 1 2 S(p,s)   S(p,s) −     s s ≤

dS(p,s) (α1 ∩ S (p, s) , β1 ∩ S (p, s)) dS(p,s) (α2 ∩ S (p, s) , β2 ∩ S (p, s)) + . s s

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I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

By (i) we have dS(p,s) (αi ∩ S (p, s) , βi ∩ S (p, s)) = 0; s→∞ s lim

hence (ii) follows. (iii) From (ii) and Lemma I.52, it suffices to prove the existence of dS(p,s) (α (s) , β (s)) s→∞ s lim

for any α, β ∈ RayM (p). It follows from parts (1) and (3) of Presumed Theorem I.50 that dS(p,s) (α (s) , β (s)) s −→ s is monotonically nonincreasing for s ≥ s0 (p); hence the limit exists and the lemma is proved.
6.2.3. Points at infinity. Based on the above discussion, one expects that the following is true. Presumed Theorem I.54 (Points at infinity). Let (Mn , g) be a complete noncompact Riemannian manifold with nonnegative sectional curvature. If Presumed Theorem I.50 is true, then (I.65)

M (∞)  ( RayM / ∼, dR )

is a compact metric space and diam (M (∞)) < ∞. We call the metric space M (∞) in (I.65) the space of points at infinity. Proof. To see that M (∞) is a metric space, note that by Lemma I.53(i) we have dR ([α] , [β]) = 0 if and only if [α] = [β]. On the other hand, that dR ([α] , [β]) = dR ([β] , [α]) is trivial, whereas the triangle inequality, i.e., dR ([α] , [γ]) ≤ dR ([α] , [β]) + dR ([β] , [γ]) for any [α] , [β] , [γ] ∈ RayM / ∼, follows from the triangle inequality for dS(p,s) . From the proof of Lemma I.53(iii), the fact that diam (M (∞)) < ∞ d

(α1p ∩S(p,s), α2p ∩S(p,s)) s

S(p,s) follows from  the monotonicity of  that diam S (p, s0 (p)) , dS(p,s (p)) < ∞.

and the property

0

Next we prove that M (∞) is compact. By Lemma I.52 each element α in M (∞) can be represented by αp ∈ RayM (p). But elements in RayM (p) are in one-to-one correspondence to a closed subset R of the unit tangent sphere Sp (1) ⊂ Tp M. Since M (∞) is a quotient space of RayM (p) and R is compact, M (∞) is compact.


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7. NOTES AND COMMENTARY

495

Assuming Presumed Theorem I.50 is true, the definition of M (∞) in (I.10) is equivalent to the definition of M (∞) in (I.65); this follows from (I.8) and (I.61). Moreover, by (I.64) we have that dR ([α] , [β]) ≤ C d˜∞ ([α] , [β]) for any [α] , [β] ∈ M (∞). Actually, we expect that dR = d˜∞ . Remark I.55. Analogous to the notion of points at infinity for complete Riemannian manifolds with nonnegative sectional curvature, for simplyconnected complete Riemannian manifolds of nonpositive curvature there is a metric, called the Tits metric, on the space of ‘points at infinity’, which, as a point-set, is also defined by putting an equivalence relation on the space of rays and taking the quotient. The metric cone over the space of points at infinity is called the Tits cone. 7. Notes and commentary A standard reference for some of the comparison geometry in this appendix is the book by Cheeger and Ebin [30], especially Chapter 8 on ‘complete manifolds of nonnegative curvature’. We also refer the reader to the seminal papers of Cheeger, Gromoll, Meyer, Perelman, and Sharafutdinov, i.e., [32], [31], [77], [148], and [171]. §1. For the Definition I.21 of soul, see p. 422 of [32] or the beginning of Chapter 8 of [30]. For a proof of Theorem I.22(1) and (3), see also Theorem 8.11 and Corollary 8.12 of [30]. §2. For Theorem I.26 see Theorem 5.3 in Shiohama [175] or Lemma 3.4 in Guijarro and Kapovitch [87]. §4. Critical point theory for the distance function was first developed by Grove and Shiohama [83]. Excellent references are Meyer [127], Cheeger [29], Grove [81], and Karcher [105]. For Lemma I.34, see pp. 360–361 of [81] (see also Lemma 3.6 of [127] and Proposition 47 on pp. 335–337 of Petersen [155]).22 For Lemma I.37 see Lemma 1.4(ii) of [107].

22 Note that there was a typographical error in the proof of Proposition 1.2 on pp. 320–321 in the first edition of [155] which is corrected in the second edition.

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http://dx.doi.org/10.1090/surv/163/14

APPENDIX J

Solutions to Selected Exercises It’s not what you know; it’s what you can prove. – From the movie “Training Day”.

Solution to Exercise 22.2. We prove this by contradiction. If the result is not true, then for all j ∈ N there exists (xj , tj ) ∈ P (xj−1 , tj−1 , (εQj−1 )−1/2 , −(εQj−1 )−1 ) with Qj  | Rm |(xj , tj ) > 2Qj−1 > 2j Q and

  T

−1 1 + 2−j 1 + 2−1 + · · · + 2−(j−1) ≥ tj ≥ tj−1 − HQ−1 j−1 ≥ t0 − HQ 4 T T −1 since t0 ≥ 2 and HQ = 8 . This contradicts supM×[0,T ] | Rm |(x, t) < ∞. α (we verified this Now assume by induction that | Rm |(xj−1 , tj−1 ) > tj−1 for j = 2, 3). We then have 2α α ≥ | Rm |(xj , tj ) > 2| Rm |(xj−1 , tj−1 ) > tj−1 tj (since tj ≥ tj−1 − HQ−1 2−(j−1) = tj−1 − T8 2−(j−1) ≥ 12 tj−1 ). Solution to Exercise 22.5. Suppose that, inductively, for some m ∈ N the points (xk , tk ) ∈ S with xk ∈ Bg(tk ) (x0 , (2A + 1)ε) and tk > 0 have been defined for 1 ≤ k ≤ m. Moreover, suppose that the point (¯ x, t¯) in the claim cannot be taken to be (xm , tm ) (otherwise we are done). Then we may define (xm+1 , tm+1 ) ∈ S to be a point such that (J.1)

| Rm |(xm+1 , tm+1 ) > 4| Rm |(xm , tm ),

(J.2) 1 0 < tm+1 ≤ tm and dg(tm+1 ) (xm+1 , x0 ) ≤ dg(tm ) (xm , x0 )+A| Rm |− 2 (xm , tm ). By induction we also have that (J.1) and (J.2) hold with m replaced by k ∈ [1, m − 1]. Thus, for all 1 ≤ k ≤ m + 1 we have 0 < tk ≤ t1 ≤ ε2 and the curvatures at (xk , tk ) are increasing at least geometrically: (J.3)

| Rm |(xk , tk ) > 4k−1 | Rm |(x1 , t1 ),

so that (J.4)

| Rm |− 2 (xk , tk ) < 1

1 2k−1

| Rm |− 2 (x1 , t1 ). 1

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498

J. SOLUTIONS TO SELECTED EXERCISES

Hence, by (J.2) and (J.4), for all 1 ≤ k ≤ m + 1 the distance of xk to x0 at time tk has the following upper bound: dg(tk ) (xk , x0 ) ≤ dg(tk−1 ) (xk−1 , x0 ) + A| Rm |− 2 (xk−1 , tk−1 ) 1

≤ dg(t1 ) (x1 , x0 ) + A| Rm |− 2 (x1 , t1 ) 1

+ · · · + A| Rm |− 2 (xk−1 , tk−1 )   1 1 1 ≤ ε + A 1 + + · · · + k−2 | Rm |− 2 (x1 , t1 ). 2 2 1

(J.5)

Since | Rm |− 2 (x1 , t1 ) < ε, we have 1

dg(tk ) (xk , x0 ) < (2A + 1)ε for 1 ≤ k ≤ m + 1. In particular, xm+1 ∈ Bg(tm+1 ) (x0 , (2A + 1)ε). By (J.3), if the point (¯ x, t¯) in the claim cannot be taken to be (xk , tk ) for any k ∈ N, then we have lim | Rm |(xk , tk ) = ∞,

k→∞

which contradicts the fact metrics g (t) are continuous in time so / that the ¯g(t) (x0 , (2A + 1)ε) × {t} is compact and that the space-time set t∈[0,ε2 ] B |Rm| has a uniform upper bound on this set. Hence there exists  ∈ N such that (¯ x, t¯) can be taken to be (x
, t
). Solution to Exercise 22.18. By (22.93) and the co-area formula, we have   ∞  −1 n |∇h| dσ ds, F (t) = Vol {y ∈ R : h (y) ≥ t} = t

{h=s}

where dσ is the area form of {y ∈ Rn : h (y) = s}. Hence  dF (s) = |∇h|−1 dσ for a.e. s > 0. − ds {h=s} Applying (22.95) to this and again applying the co-area formula yields  ∞   ∞ dλ (s) F (s) ds = λ (s) |∇h|−1 dσds ds 0 0 {h=s}  λ (h) dµRn . = {y∈Rn :h(y)>0}

 Solution to Exercise 25.7. With ri  r0 1 +

γ−1 2i−1

 we have

(γ − 1) r0 , 2i (γ − 1) 2 ≥ r , 2i−1 0

ri − ri+1 = 2 ri2 − ri+1

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J. SOLUTIONS TO SELECTED EXERCISES

ηi in (25.30) now satisfies 0 ≤ ηi ≤ 0≤

∂ψi 2i ≤ ∂τ (γ − 1) r02

2i , (γ−1)r02

and ψi in (25.31) now satisfies

|∇ψi |g˜ ≤

and

499

2i+1 . (γ − 1) r0

Instead of (25.33), we now have ψi

4i+2 r0−2 ∂ψi + |∇ψi |2g˜ ≤ Li  . ∂τ γ−1

Corresponding to (25.27) now define (instead of (25.34)) √     2 γ − 1 2 ˜ n+1 4i+2 eC (1+ Kri ) ˜0n 4i+2 r−2 n . Mi  1 + + 1 2 C C 0 0 2/n 2i−1 γ−1 Volg˜ Bg˜ (x0 , ri ) Analogous to (25.35) and (25.36), we have

v L∞ (Qr ) ≤ C˜ v L2 (Qγr ) , 0 0 where

√

n C˜  eC (1+γ Kr0 ) 4 (Volg˜ Bg˜ (x0 , r0 ))− 2 −i  ∞   2  12 ( n+2 i+3 ; n ) 4 n 2C˜0n 4i+2 r0−2 × . 1 + C˜0n+1 γ−1 1

i=1

Again assuming C˜0n+1 44 ≥ 1, we have C˜ ≤ Cˆ (γ), where Cˆ (γ) is given by (25.40). Solution to Exercise G.3. It is clear that f ∗ dY is nonnegative. If x1 , x2 , x3 ∈ X, then (f ∗ dY ) (x1 , x3 ) = dY (f (x1 ) , f (x3 )) ≤ dY (f (x1 ) , f (x2 )) + dY (f (x2 ) , f (x3 )) = (f ∗ dY ) (x1 , x2 ) + (f ∗ dY ) (x2 , x3 ) , so the triangle inequality holds. If f is injective, then (f ∗ dY ) (x1 , x2 ) = dY (f (x1 ) , f (x2 )) > 0 provided x1 = x2 . Example of a solution to Exercise G.24. First consider the subset of R2 defined by the zigzagging set ∞    . 1 1 , × {0} Y  {(0, 0)} ∪ (2k+1)! (2k)!  ∪

k=0 ∞ .



{0} ×

1 1 (2k+2)! , (2k+1)!

Now define a subset of

 ∪

∞ .

t 1−t n! , n!



: t ∈ [0, 1] .

n=0

k=0

R3



by

X = {(r, θ, z) : (r, z) ∈ Y and θ ∈ [0, 2π)} ,

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500

J. SOLUTIONS TO SELECTED EXERCISES

where r, θ, z are cylindrical coordinates. That is, X is obtained by revolving Y about the z-axis. Endow X with the intrinsic metric induced from the ambient space R3 . Note that X is compact. (Why does X have finite diameter?)1 Let p = (0, 0) be the origin and take for example  αi  (2i + 1)! · (2i)!,  βi  (2i)! · (2i − 1)!, for i ∈ N. Then lim (X, αi d, p) = R2 × {0} ,

i→∞

lim (X, βi d, p) = {(0, 0)} × [0, ∞).

i→∞

We may also consider the sequences defined by γi  (2i)! and δi  (2i − 1)!. What are the corresponding limits for these sequences? Solution to Exercise H.44. Go parallel to the x- or y-axis to the closest of the two diagonals {y = x} and {y = −x} and then go along this diagonal to the origin (0, 0). Solution to Exercise I.14. We first consider a 2-dimensional flat cone and then approximate it by smooth complete Riemannian surfaces 2π) and with positive curvature. Let Sα1 denote the circle

of length α ∈ (0, α 1 consider the Euclidean metric cone Cone Sα . If d (θ1 , θ2 ) = 2 , then by (G.12) we have " dCone(Sα1 ) ([(θ1 , r1 )] , [(θ2 , r2 )]) = r12 + r22 − 2βr1 r2 ,

where β  cos α2 ∈ (−1, 1). Now consider the 2-dimensional positively curved rotationally symmetric expanding gradient solitons discussed in §4 of Chapter 2 of Volume One; we do not need that they satisfy the Ricci flow, just that they approximate the

cone well. For any α ∈ (0, 2π) 1there

2 , g (t) , which is asymptotic to Cone Sα . In is such a solution, call it R α

1

2 distance fact, R , gα (t) converges to Cone Sα in the Gromov–Hausdorff

+ , while the incomplete solutions R2 − {0} , g (t) converge to as t → 0 α

Cheeger–Gromov sense as t → 0+ . For the Cone Sα1 − {origin} in the C ∞

2 Riemannian surface R , gα (t) , where α ∈ (π, 2π) and t > 0 is sufficiently small depending on α, consider the ray γ : [0, ∞) → R2



emanating from the origin defined by γ (s)  (θ2 , s). On the cone Cone Sα1 consider the Busemann function for the corresponding ray γ (s) = [(θ2 , s)], 1

Answer: Essentially because

∞

1 n=0 n!

= e < ∞.

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J. SOLUTIONS TO SELECTED EXERCISES

501

which we also denote by bγ . We have

bγ ([(θ1 , r)]) = lim s − dCone(Sα1 ) ([(θ1 , r)] , γ (s)) s→∞    = lim s − r2 + s2 − 2βrs s→∞

= βr, which tends to −∞ as r → ∞ since β < 0. We leave it to the reader to show that, for any α ∈ (π, and t > 0 sufficiently small depending on α, 2π) 2 we have for the ray γ in R , gα (t) that lim bγ (θ1 , r) = −∞.

r→∞

Solution to Exercise I.38. Consider the hinge with geodesic sides γVi and αi and angle θi at the common vertex xi . By (I.33), we have cos θi > cos (π − c) = − cos c > −1. Applying the hinge version of the Toponogov comparison theorem (comparing to Euclidean space and using the law of cosines), i.e., Theorem G.33(2), we then have r2 (x) < d2i + ri2 + 2di ri cos c. √ Let a = sin c, so that cos c = 1 − a2 and  (J.7) r2 (x) < d2i + ri2 + 2di ri 1 − a2 . (J.6)

Now by (J.7), inequality (I.37) follows from establishing    1 − a2 ε 2 − 1 − a2 . a2 ε2 ri ≤ 2di Since ri ≤

1−ε ε di ,

this is implied by  √ √ 1 − a2 ε 2 − 1 − a2 2 a2 ε (1 − ε)

≥ 1,

which in turn follows from   2 (1 + ε) ≥ 2 ≥ 1 − a2 ε 2 + 1 − a2 . ε Inequality (I.37) has been proved.

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Bibliography [1] Abresch, Uwe. Lower curvature bounds, Toponogov’s theorem, and bounded topology. ´ Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), no. 4, 651–670. [2] Alexakis, Spyros. Unique continuation for the vacuum Einstein equations. arXiv:0902.1131. [3] Alexakis, S.; Ionescu, A. D.; Klainerman, S. Uniqueness of smooth stationary black holes in vacuum: Small perturbations of the Kerr spaces. arXiv:0904.0982. [4] Alexander, J. W. On the subdivision of 3-space by a polyhedron. Proc. Nat. Acad. Sci., USA, 10, 6-8, 1924. [5] Andersson, Lars; Galloway, Gregory J.; Howard, Ralph. A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry. Comm. on Pure and Applied Math. 51 (1998), 581–624. [6] Angenent, Sigurd B. The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390 (1988), 79–96. [7] Angenent, Sigurd B.; Knopf, Dan. An example of neckpinching for Ricci flow on S n+1 . Math. Res. Lett. 11 (2004), no. 4, 493–518. [8] Aubin, Thierry. Probl`emes isop´erim´etriques et espaces de Sobolev. (French) J. Differential Geom. 11 (1976), no. 4, 573–598. [9] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor. Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkh¨ auser Boston, Inc., Boston, MA, 1985. [10] Bando, Shigetoshi. Real analyticity of solutions of Hamilton’s equation, Math. Zeit. 195 (1987), 93–97. [11] Benedetti, Riccardo; Petronio, Carlo. Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992. [12] Berestovskii, V.; Nikolaev, I. Multidimensional generalized Riemannian spaces. In Geometry IV. Non-regular Riemannian geometry. Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1993, 165–244. [13] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond. Le spectre d’une vari´et´e riemannienne. (French) Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, BerlinNew York, 1971. [14] Berline, Nicole; Getzler, Ezra; Vergne, Mich`ele. Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften 298. Springer-Verlag, Berlin, 1992. [15] Besse, Arthur, Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, 10. Springer-Verlag, Berlin, 1987. [16] B¨ ohm, Christoph; Wilking, Burkhard. Manifolds with positive curvature operators are space forms. Annals of Math. 167 (2008), 1079–1097. [17] Branson, Thomas P.; Gilkey, Peter B.; Vassilevich, Dmitri V. Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary. J. Math. Phys. 39 (1998), 1040–1049. Erratum. J. Math. Phys. 41 (2000), 3301. [18] Burago, D.; Burago, Y.; Ivanov, S. A course in metric geometry, Grad Studies Math. 33, Amer. Math. Soc., Providence, RI, 2001. Corrections of typos and

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Index

adjoint heat kernel, 171 operator, 3, 90 Aleksandrov space, 403 cut locus, 409 exponential map, 408 splitting theorem, 407 α-large curvature point, 168, 184 α-small curvature point, 184 angle, 404 comparison, 404 arc length evolution, 270 asymptotic scalar curvature ratio, 85, 124 asymptotic shrinker existence of, 113 asymptotic volume ratio, 85, 124 is time-independent, 87 of Type I ancient solutions, 124

injectivity radius estimate, 81 cigar soliton, 153 Claim 1 on point picking, 170, 186 Claim 2 on point picking, 170, 188 compact modulo scaling, 137 compactness theorem Perelman’s, 137 complete geodesically, 409 concatenation of two paths, 390 concave function, 425 λ-, 410 semi-, 410 cone Euclidean metric, 399 Riemannian, 86 topological, 399 convex function, 415, 425 convex set, 419 locally, 419 convolution space-time, 229 curvature bump, 57 cut locus, 409 cylinder parabolic, 184

backward limit, 118 solution to Ricci flow, 117 Bonnet–Myers theorem, 407 boundary of Aleksandrov space, 409 boundedly compact, 397 Busemann function, 458 associated to a point, 458 Busemann–Feller theorem, 417

Davies’ upper bound for heat kernel, 361 defining equation for HN , 218 derivative directional, 428 derivative estimates Perelman’s scaled, 150 Shi’s local, 144

changing distances estimate, 45 application of, 101 Cheeger–Gromov convergence, 174, 191 limits and heat kernels, 193 Cheeger–Gromov–Taylor 513

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514

difference quotient, 428 differential Harnack estimate, 318 dilatation, 414 dimension classic, xi reduces, 53 reduction, 54 direction of a path at a point, 405 directional derivative, 428 directions space of, 408 Dirichlet heat kernel, 290 distance function mollified, 488 time derivative of, 41 under Ricci flow, 42 distance-like function, 378 distance-preserving, 390 distortion of a map, 394 domain regular, 158 Duhamel’s principle, 335 energy functional minimizer, 2 Perelman’s, 2 entropy existence of a minimizer, 23 maximum value of a minimizer, 31 minimizer may not be unique, 22 monotonicity formula, 3 monotonicity formula, localized, 201 entropy functional Perelman’s, 2, 90 ε-isometry, 394 ε-neck, 68 embedded, 62 in Bryant soliton, 83 ε-neighborhood, 349, 393 ε-net, 394 ε-pointed Hausdorff approximation, 395 Euclidean isoperimetric inequality, 159 exponential map, 408 extremal set, 411 forward difference quotient lim inf of, 41 function semi-concave, 410 fundamental solution heat equation, 215

INDEX

minimal positive, 216 fundamental theorem of calculus, 414 generalized gradient, 433 geodesic, 391 geodesically complete, 409 gradient generalized, 433 Gromoll–Meyer theorem, 63 Gromov–Hausdorff convergence, 395 distance, 393 pointed convergence, 396 pointed distance, 395 half-spaces, 459 Harnack estimate, 318 Perelman’s, 201 Harnack quadratic linear trace, 31 matrix, 32 trace, 32 Hausdorff dimension, 405 distance, 393 measure, 405 heat ball, 365, 368 heat equation fundamental solution, 215 heat kernel, 216, 334 asymptotics, 251 characterization of Ricci flow, 286 Davies’ upper bound, 361 Dirichlet, 290 existence, 216 expansion, 232 first approximation to, 217 for time-dependent metric, 271 good approximation to, 217 L1 -norm is preserved, 342 lower bound, 361 on noncompact manifold, 301 transplanted, 217 under Cheeger–Gromov limits, 193 upper bound, 356 heat operator adjoint, 3, 266 heat sphere, 367, 373 Hessian generalized, 456 hole fixing, 387 homogenous

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INDEX

of degree 1, 428 injectivity radius estimate Cheeger–Gromov–Taylor, 81 integral curve for ∇f / |∇f |2 , 444 interior tangent cone, 418 intrinsic metric, 391 induced, 392 strictly, 391 isometric, 390 embedding, 390 isometry, 390 isometry group is preserved, 212 isoperimetric inequality Euclidean, 159 Jensen’s inequality, 29 κ-collapsed strongly, 112 κ-gap theorem, 117 κ-noncollapsed at all scales, 80 below the scale ρ, 80 κ-solution, 81 2-dimensional , 104 Perelman’s, 94 satisfying trace Harnack, 89 with Harnack, 104 King–Rosenau solution, 82 L-distance, 107 L-length, 107 λ-concave, 410 λ-invariant, 2 large curvature points, 168 length space, 391 complete, 391 quasi-, 390 length structure induced by a metric, 392 Levi parametrix method, 263 Li–Schoen argument, 315 Li–Yau inequality, 318 limit backward, 114 line, 407 linear trace Harnack quadratic, 31 Lipschitz constant, 414 map, 413

515

localization of Li–Yau inequality, 324 locally Lipschitz map, 414 logarithmic Sobolev inequality, 4, 206 Euclidean, 16, 206 Matrix The, ix matrix Harnack estimate, 89 quadratic, 32 maximum angle function, 476 mean value inequality parabolic, 305 mean value property space-time Euclidean , 366 space-time Riemannian, 369 metric intrinsic, 391 strictly intrinsic, 391 subspace, 388 metric cone Euclidean, 399 metric space, 388 complete, 388 induced by a pseudo-metric space, 389 pseudo-, 389 quasi-, 389 Minakshisundaram–Pleijel method, 263 minimizer existence of, 23 nonuniqueness of, 22 mollifier, 95 monotonicity formula for entropy, 3 localized, 201 Moser iteration, 307 µ-invariant, 2 as τ → 0, 16 continuous dependence on g, 12 is negative for τ small, 15 lower bound, 7 upper bound, 5 nearest point projection map, 416 necks rough monotonicity in positively curved manifolds, 68 no local collapsing, 80

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516

localized weakened, 69 weakened, 113 normal bundle, 462 notation, xvii ν-invariant, 3 parabolic cylinder, 184, 306 parabolic mean value inequality, 305 parametrix convolution series, 230 convergence of, 233 derivative of convolution with, 239 for heat operator, 224 for time-dependent heat operator, 272 parametrix method Levi, 263 Perelman §9.6 of [152], 255, 285 §11.1 of [152], 90 §11.4 of [152], 124 §11.7 of [152], 137 §11.9 of [152], 117, 153 Perelman’s entropy functional, 90 Harnack estimate, 201 κ-solution on S n , 94 stability theorem, 411 Poincar´e-type inequality reverse, 308 point picking Claim 1, 170, 186 Claim 2, 170, 188 general formulation of , 76 space-time, 57 when ASCR = ∞, 50 when change in R is unbounded, 52 when sup R = ∞, 50 pointed Gromov–Hausdorff distance, 395 precompact modulo scaling, 136 product metric, 390 projection map nearest point, 416 pseudo-metric space, 389 pseudolocality theorem, 159 quasi-geodesic, 411 quasi-length space, 390 quasi-metric space, 389 ray, 458 equivalent, 492 rectifiable, 390 reduced distance, 107

INDEX

estimate for, 107 reduced volume for Ricci flow, 112 mock, 115 monotonicity, 112 regular domain, 158 point, 408 reverse Poincar´e-type inequality, 308 RG flow, 161, 162, 212 Ricci flow backward uniqueness, 211 forward uniqueness, 212 unique continuation, 212 right tangent vector, 442 sausage model, 82 scaled derivative estimates, 150 Schoenflies conjecture, 63 semi-concave, 410 function, 410 semigroup property, 338 set gradient, 472 Sharafutdinov map, 464 retraction, 68, 463 Shi’s estimate, 144 shifted ray, 459 shortest path, 391 shrinking Ricci soliton compact, 116 singular point, 408 singularity model, 9 compact implies shrinker, 10 existence of, 81 on closed 3-manifold is round, 13 Sobolev inequality, 309 logarithmic, 206 soul, 463 conjecture, 463 theorem, 463 space of directions, 408 spherical symmetrization, 207 stable geodesic estimate for Rc along, 48 strainer, 406 strata, 409 stratification into topological manifolds, 409 strong maximum principle for weak solutions, 26 strongly

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INDEX

convex subset, 460 κ-collapsed, 112 sublevel set of a Busemann function, 459 sublinear, 432 submanifold totally geodesic, 462 subset strongly convex, 460 totally convex, 460 tangent cone, 397, 398 interior, 418 tangent vector right, 442 Tits cone, 495 Toponogov comparison theorem, 402 monotonicity principle, 56 totally convex subset, 460 geodesic submanifold, 462 trace Harnack quadratic, 32 triangle, 403 unique continuation for Ricci flow, 212 uniqueness backward, 211 forward, 212 variation formula for F , 32 volume comparison, 407 lower bound for solutions, 8 weakened no local collapsing, 113 localized, 69

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517

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Titles in This Series 163 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 III: Geometric-analytic aspects, 2010 urgen Rossmann, Elliptic equations in polyhedral domains, 162 Vladimir Mazya and J¨ 2010 161 Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan, Morse theoretic aspects of p-Laplacian type operators, 2010 160 Alexander S. Kechris, Global aspects of ergodic group actions, 2010 159 Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line, 2010 158 D. R. Yafaev, Mathematical scattering theory: Analytic theory, 2010 157 Xia Chen, Random walk intersections: Large deviations and related topics, 2010 156 Jaime Angulo Pava, Nonlinear dispersive equations: Existence and stability of solitary and periodic travelling wave solutions, 2009 155 Yiannis N. Moschovakis, Descriptive set theory, 2009 ˇ 154 Andreas Cap and Jan Slov´ ak, Parabolic geometries I: Background and general theory, 2009 153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, 2009 152 J´ anos Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alc´ ala lectures, 2009 151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications in signal theory, optics, quantization, and field quantization, 2008 150 Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, 2008 149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008 148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, 2008 147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008 146 Murray Marshall, Positive polynomials and sums of squares, 2008 145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, 2008 144 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 II: Analytic aspects, 2008 143 Alexander Molev, Yangians and classical Lie algebras, 2007 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Mazya and Gunther Schmidt, Approximate approximations, 2007 140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007 139 Michael Tsfasman, Serge Vlˇ adut ¸, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007 138 Kehe Zhu, Operator theory in function spaces, 2007 137 Mikhail G. Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 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 I: Geometric aspects, 2007 134 Dana P. Williams, Crossed products of C ∗ -algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006

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TITLES IN THIS SERIES

130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006 129 William M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlev´ e transcendents, 2006 127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 123 Barbara Fantechi, Lothar G¨ ottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups I. Structure of strongly quasithin K-groups, 2004 110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004 108 107 106 105

Michael Farber, Topology of closed one-forms, 2004 Jens Carsten Jantzen, Representations of algebraic groups, 2003 Hiroyuki Yoshida, Absolute CM-periods, 2003 Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003

104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 e, 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanr´ Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 100 99 98

Eli Glasner, Ergodic theory via joinings, 2003 Peter Duren and Alexander Schuster, Bergman spaces, 2004 Philip S. Hirschhorn, Model categories and their localizations, 2003 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002

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The Ricci flow uses methods from analysis to study the geometry and topology of manifolds. With the third part of their volume on techniques and applications of the theory, the authors give a presentation of Hamilton’s Ricci flow for graduate students and mathematicians interested in working in the subject, with an emphasis on the geometric and analytic aspects. The topics include Perelman’s entropy functional, point picking methods, aspects of Perelman’s theory of κ -solutions including the κ -gap theorem, compactness theorem and derivative estimates, Perelman’s pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices, we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in the previous volumes, the authors have endeavored, as much as possible, to make the chapters independent of each other. The book makes advanced material accessible to graduate students and nonexperts. It includes a rigorous introduction to some of Perelman’s work and explains some technical aspects of Ricci flow useful for singularity analysis. The authors give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

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