Mathematical Surveys and Monographs Volume 206 The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni American Mathematical Society The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics http://dx.doi.org/10.1090/surv/206 Mathematical Surveys and Monographs Volume 206 The Ricci Flow: Techniques and Applications Part IV: Long-Time Solutions and Related Topics Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 53C44, 53C21, 53C43, 58J35, 35K59, 35K05, 57Mxx, 57M50. For additional information and updates on this book, visit www.ams.org/bookpages/surv-206 Library of Congress Cataloging-in-Publication Data Chow, Bennett. The Ricci flow : techniques and applications / Bennett Chow... [et al.]. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 135) Includes bibliographical references and indexes. ISBN-13: 978-0-8218-3946-1 (pt. 1) ISBN-10: 0-8218-3946-2 (pt. 1) 1. Global differential geometry. 2. Ricci flow. 3. Riemannian manifolds. I. Title. QA670.R53 2007 516.362—dc22 2007275659 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. 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Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Contents Preface ix Acknowledgments xiii Contents of Volume One and Parts I, II, and III of Volume Two xv Notation and Symbols xvii Chapter 27. Noncompact Gradient Ricci Solitons 1 1. Basic properties of gradient Ricci solitons 1 2. Estimates for potential functions of gradient solitons 9 3. Lower bounds for the scalar curvature of nonflat nonexpanding gradient Ricci solitons 15 4. Volume growth of shrinking gradient Ricci solitons 17 5. Logarithmic Sobolev inequality 26 6. Gradient shrinkers with nonnegative Ricci curvature 29 7. Notes and commentary 33 Chapter 28. Special Ancient Solutions 35 1. Local estimate for the scalar curvature under Ricci flow 35 2. Properties of singularity models 40 3. Noncompact 2-dimensional ancient solutions with finite width 49 4. Ancient solutions with positive curvature 63 5. Notes and commentary 66 Chapter 29. Compact 2-Dimensional Ancient Solutions 69 1. Statement of the classification result and outline of its proof 69 2. The Ricci flow equation on S2 and some intuition 70 3. The King–Rosenau solution in the various coordinates 73 4. A priori estimates for the pressure function 76 5. The almost everywhere vanishing of R∞ 79 6. First properties of the backward limit v∞ 81 7. Isoperimetric constant of metrics on S2 83 8. Characterizing round solutions 87 9. Classifying the backward pointwise limit 100 10. An unrescaled cigar backward Cheeger–Gromov limit 106 11. Irreducible components of ∇3v 108 12. The heat-type equation satisfied by Q 111 13. That Q = 0 implies the solution is the King–Rosenau solution 117 14. The evolution equation for Q¯ 124 15. The quantity Q must be identically zero 125 v vi CONTENTS 16. The equivalence of Q and Q¯ 129 17. Notes and commentary 132 Chapter 30. Type I Singularities and Ancient Solutions 133 1. Reduced distance of Type A solutions 133 2. Reduced volume at the singular time for Type I solutions 145 3. Type I solutions have shrinker singularity models 154 4. Some results on Type I ancient solutions 159 5. Notes and commentary 169 Chapter 31. Hyperbolic Geometry and 3-Manifolds 171 1. Introduction to hyperbolic space 171 2. Topology and geometry of hyperbolic 3-manifolds 178 3. The Margulis lemma and hyperbolic cusps 185 4. Mostow rigidity 192 5. Seifert fibered manifolds and graph manifolds 193 6. Notes and commentary 194 Chapter 32. Nonsingular Solutions on Closed 3-Manifolds 197 1. Introduction 197 2. The main result on nonsingular solutions 200 3. The three cases of nonsingular solutions 203 4. The positive and zero cases of nonsingular solutions 207 5. The negative case—sequential limits must be hyperbolic 210 6. Notes and commentary 211 Chapter 33. Noncompact Hyperbolic Limits 213 1. Main results on hyperbolic pieces 214 2. Harmonic maps parametrizing almost hyperbolic pieces 219 3. Proof of the stability of hyperbolic limits 226 4. Incompressibility of boundary tori of hyperbolic pieces 237 5. Notes and commentary 254 Chapter 34. Constant Mean Curvature Surfaces and Harmonic Maps by IFT 257 1. Constant mean curvature surfaces 257 2. Harmonic maps near the identity of Sn 260 3. Existence of harmonic maps near the identity of manifolds with negative Ricci curvature 266 4. Application of Mostow rigidity to the existence of isometries 273 5. Notes and commentary 278 Chapter 35. Stability of Ricci Flow 279 1. Linear stability of Ricci flow 280 2. Analytic semigroups and maximal regularity theory 287 3. Dynamic stability results obtained using linearization 296 4. Dynamic stability results obtained by other methods 304 CONTENTS vii Chapter 36. Type II Singularities and Degenerate Neckpinches 307 1. Numerical simulation of solutions with degenerate neckpinches 309 2. Matched asymptotic studies of degenerate neckpinches 318 3. Ricci flow solutions with degenerate neckpinch singularities 324 4. Concluding remarks 326 Appendix K. Implicit Function Theorem 327 1. The implicit function theorem 327 2. H¨older spaces and Sobolev spaces on manifolds 332 3. Harmonic maps and their linearization 336 n 4. Spectrum of Δd on p-forms on S 347 5. Notes and commentary 352 Bibliography 353 Index 371 Preface Keys to ignition, use at your discretion. – From “Starin’ Through My Rear View” by Tupac Shakur # This is Part IV (a.k.a. Rijk), the sequel to Volume One ([75]; a.k.a. gij ) ∂ and Parts I, II, III ([69], [70], [71]; a.k.a. Rijk, ∂tRijk,ΔRijk, respectively) of Volume Two on techniques and applications of the Ricci flow. For the reader’s convenience, we have included the titles of each chapter on the pages that follow. In this part we mainly discuss aspects of the long-time behavior of solutions to the Ricci flow, including the geometry of noncompact gradient Ricci solitons, ancient solutions, Hamilton’s classification of 3-dimensional nonsingular solutions, and the stability of the Ricci flow. Any theory about singularities of the Ricci flow requires an understanding of ancient solutions and, in particular, gradient Ricci solitons. Building on the success in dimensions at most 3, the study of higher-dimensional Ricci solitons is currently an active field; we discuss some of the progress in this direction. We also present recent progress on (1) the classifica- tion of ancient 2-dimensional solutions without the κ-noncollapsing hypothesis and (2) Type I ancient solutions and singularities. In a direction complementary to the study of singularities, we discuss 3-dimensional nonsingular solutions. These solu- tions underlie the Ricci flow approach to the geometrization conjecture; Hamilton’s work on this is a precursor to Perelman’s more general theory of immortal solutions to the Ricci flow with surgery. Finally, a largely unexplored direction in the Ricci flow concerns the sensitivity of solutions to their initial data; the study of stability of solutions represents an aspect of this. The choice of topics is based on our familiarity and taste. Due to the diversity of the field of Ricci flow, we have inevitably omitted many important works. We have also omitted some topics originally slated for this part, such as the linearized Ricci flow and the space-time formulation of the Ricci flow. We now give detailed descriptions of the chapter contents. Chapter 27. This chapter is a continuation of Chapter 1 of Part I. Here we discuss some recent progress on the geometry of noncompact gradient Ricci soli- tons (GRS), including some qualitatively sharp estimates for the volume growth, potential functions, and scalar curvatures of GRS. We also discuss the logarithmic Sobolev inequality for shrinking GRS as well as shrinking GRS with nonnegative Ricci curvature. Chapter 28. This chapter complements the discussion in Part III on Perel- man’s theory of 3-dimensional ancient κ-solutions. The topics discussed are a local lower bound for the scalar curvature under Ricci flow, some geometric properties of 3-dimensional singularity models, noncompact 2-dimensional ancient solutions ix xPREFACE without the κ-noncollapsed condition, and classifying certain ancient solutions with positive curvature. Chapter 29. In this chapter we present the results of Daskalopoulos, Hamil- ton, and Sesum that any simply-connected ancient solution to the Ricci flow on a closed surface must be either a round shrinking 2-sphere or the rotationally symmet- ric King–Rosenau solution.