Hamilton's Ricci Flow

374 Pages·2005·2.07 MB·english

by  Chow B.

Hamilton’s Ricci Flow
Volume 1
Preliminary Version
July 12, 2005
Bennett Chow
Peng Lu
Lei Ni
UC San Diego
E-mail address: [email protected]
University of Oregon
E-mail address: [email protected]
UC San Diego
E-mail address: [email protected]

Contents
A note to the reader of the preliminary version vii
Preface ix
Acknowledgments xiii
A guide to Volume 1 xv
Notation xxv
Chapter 1. Basic Riemannian geometry 1
1. Introduction 1
2. Basic conventions and formulas in Riemannian geometry 2
3. Laplacian and Hessian comparison theorems 29
4. Geodesic polar coordinates 37
5. First and second variation of arc length and energy formulas 50
6. Geometric applications of second variation 58
7. Green’s function 61
8. Comparison theory for the heat kernel 62
9. Parametrix for the heat equation 64
10. Eigenvalues and eigenfunctions of the Laplacian 69
11. The determinant of the Laplacian 78
12. Monotonicity for harmonic functions and maps 85
13. Lie groups and left invariant metrics 87
14. Bieberbach Theorem 89
15. Compendium of inequalities 94
16. Notes and commentary 95
Chapter 2. Elementary aspects of the Ricci flow equation 101
1. Some geometric flows predating Ricci flow 101
2. Ricci flow and geometrization: a short preview 101
3. Ricci flow and the evolution of scalar curvature 103
4. The maximum principle for heat-type equations 105
5. The maximum principle on noncompact manifolds 107
6. The Einstein-Hilbert functional 111
7. Evolution of geometric quantities - local coordinate calculations 112
8. DeTurck’s trick and short time existence 119
9. Notes and commentary 122
iii
iv CONTENTS
Chapter 3. Closed 3-manifolds with positive Ricci curvature 127
1. The maximum principle for tensors 127
2. Hamilton’s 1982 theorem 129
3. Evolution of curvature 130
4. The maximum principle for systems 134
5. Gradient of scalar curvature estimate 139
6. Curvature tends to constant 143
7. Exponential convergence to constant curvature of the normalized
flow 144
8. Notes and commentary 148
Chapter 4. Ricci solitons and other special solutions 151
1. Types of long existing solutions 151
2. Gradient Ricci solitons 151
3. Gaussian soliton 154
4. Cylinder shrinking soliton 154
5. Cigar steady soliton 155
6. Rosenau solution 158
7. An expanding soliton 160
8. Bryant soliton 162
9. Geometry at spatial infinity of ancient solutions 164
10. Homogeneous solutions 169
11. Isometry group 174
12. Notes and commentary 174
Chapter 5. Analytic results and techniques 177
1. Derivative estimates and long time existence 177
2. Cheeger-Gromov type compactness theorem for Ricci flow 181
3. The Hamilton-Ivey curvature estimate 187
4. Strong maximum principles and splitting theorems 192
5. 3-manifolds with nonnegative curvature 196
6. Manifolds with nonnegative curvature operator 197
7. Notes and commentary 204
Chapter 6. Some miscellaneous techniques for the Ricci, Yamabe and
cross curvature flows 207
1. Kazdan-Warner type identities and Yamabe and Ricci solitons 207
2. Andrews’ Poincar´e type inequality 210
3. The gradient of Hamilton’s entropy is the matrix Harnack 212
4. The Yamabe flow and Aleksandrov reflection 213
5. Isoperimetric estimate 220
6. Cross curvature flow 226
7. Notes and commentary 228
Chapter 7. Introduction to singularities 229
1. Dilating about a singularity and taking limits 229
CONTENTS v
2. Singularity types 231
3. Degenerate neck pinch 237
4. Classification of ancient solutions on surfaces 240
5. Extending noncompact ancient surface solutions to eternal
solutions 244
6. Dimension reduction 245
7. Hamilton’s partial classification of 3-dimensional singularities 252
8. Some conjectures about ancient solutions 255
9. Notes and commentary 258
Chapter 8. Harnack type estimates 259
1. Li-Yau estimate for the heat equation 259
2. Surfaces with positive curvature 264
3. Harnack estimate on complete surfaces with positive curvature 266
4. Linear trace and interpolated estimates for the Ricci flow on
surfaces 268
5. Hamilton’s matrix estimate 271
6. Sketch of the proof of the matrix Harnack estimate 275
7. Linear trace Harnack estimate in all dimensions 278
8. A pinching estimate for solutions of the linearized Ricci flow
equation 282
9. Notes and commentary 283
Chapter 9. Space-time geometry 287
1. A space-time solution to the Ricci flow for degenerate metrics 287
2. Space-time curvature is the matrix Harnack quadratic 294
3. Potentially infinite metrics and potentially infinite dimensions 295
4. Renormalizing the space-time metric yields the ℓ-length 308
5. Space-time DeTurck’s trick and fixing the measure 309
6. Notes and commentary 311
Index 319
Bibliography 325

A note to the reader of the preliminary version
This is a preliminary version of a book to be published by Science Press,
China. Comments or corrections are very welcome. Please email any of
these to [email protected] before September 1, 2005. Thanks!
vii

Preface
The subject of Hamilton’s Ricci flow lies in the more general field of
geometricflows,whichinturnliesintheevenmoregeneralfieldofgeometric
analysis. RicciflowdeformsRiemannianmetricsonmanifoldsbytheirRicci
tensor, an equation which turns out to exhibit many similarities with the
heat equation. Other geometric flows, such as the mean curvature flow of
submanifolds demonstrate similar smoothing properties. The aim for many
geometric flows is to produce canonical geometric structures by deforming
rathergeneralinitialdatatothesestructures. Dependingontheinitialdata,
the solutions to geometric flows may encounter singularities where at some
time the solution can no longer be defined smoothly. For various reasons,
in Ricci flow the study of the qualitative aspects of solutions, especially
ones which form singularities, is at present more amenable in dimension
3. This is precisely the dimension in which the Poincar´e Conjecture was
originallystated; thehigherdimensionalgeneralizationshavebeensolvedby
Smale in dimensions at least 5 and by Freedman in dimension 4. Remaining
in dimension 3, a vast generalization of this conjecture was proposed by
Thurston, called the Geometrization Conjecture, which roughly speaking,
says that each closed 3-manifold admits a geometric decomposition, i.e.,
can be decomposed into pieces which admit complete locally homogeneous
metrics.
Hamilton’s program is to use Ricci flow to approach this conjecture.
Perelman’s work aims at completing this program. Through their works
one hopes/expects that the Ricci flow may be used to infer the existence of
a geometric decomposition by taking any initial Riemannian metric on any
closed 3-manifold and proving enough analytic, geometric and topological
resultsaboutthecorrespondingsolutionoftheRicciflowwithsurgery. Note
on the other hand that, in this regard, one does not expect to need to prove
the convergence of the solution to the Ricci flow with surgery to a (possibly
disconnected) homogeneous Riemannian manifold. The reason for this is
Cheeger and Gromov’s theory of collapsing manifolds and its extension to
case where the curvature is only bounded from below. Furthermore, if one
is only interested in approaching the Poincar´e Conjecture, then one does
not expect to need the theory of collapsing manifolds. For these geometric
and topological reasons, the study of the Ricci flow as an approach to the
Poincar´e and Geometrization Conjectures are reduced to proving certain
analytic and geometric results. In many respects the Ricci flow appears to
ix
x PREFACE
be a very natural equation and we feel that the study of its analytic and
geometric properties is of interest in its own right. Independent of the res-
olution of the above conjectures, there remain a number of interesting open
problems concerning the Ricci flow in dimension 3. In higher dimensions,
the situation is perhaps even more interesting in that, in general, much less
is known.
So in this book we emphasize the more analytic and geometric aspects
of Ricci flow rather than the topological aspects. We also attempt to convey
some of the relations and formal similarities between Ricci flow and other
geometric flows such as mean curvature flow. The interaction of techniques
and ideas between Ricci flow and other geometric flows is a two-way street.
So we hope the reader with a more general interest in geometric flows will
benefit from the usefulness of ideas originating in Ricci flow to the study
of other geometric flows. We have not aimed at completeness, even in the
realm of the limited material that we cover. A more extensive coverage of
the subject of Ricci flow is planned in the book by Dan Knopf and one of
the authors [153] and its multi-authored sequel [143].
At places we follow the informal style of lecture notes and have at-
tempted to cover some of the basic material in a relatively direct and effi-
cient way. At the same time we take the opportunity to expose the reader
to techniques, some of which lie outside of the subject of Ricci flow per se,
which they may find useful in pursuing research in Ricci flow. So metaphor-
ically speaking this book is hybrid between rushing to work in the morning
on a cold and blistery winter day and a casual stroll through the park on
a warm and sunny midsummer afternoon. As much as possible, we have
attempted to construct the book so that the chapters, and in some cases,
individual sections, are relatively independent. In this way we hope that
the book may be used as both an introduction and as a reference. We have
endeavored to include some open problems which are aimed at conveying
to the reader what are the limits of our current knowledge and to point to
some interesting directions. We have also attempted to give the appropriate
references so that the reader may further pursue the statements and proofs
of the various results. To use real estate jargon, we hope that the refer-
ences are reliable but they are not guaranteed; in particular, sometimes the
references given may not be the first place a particular result is proven.
Theyear1982markedthebeginningofRicciflowwiththeappearanceof
Hamilton’s paper on 3-manifolds with positive Ricci curvature. Since then,
the development of Hamilton’s program is primarily scattered throughout a
series of several of his papers (see [79] for a collection of some Ricci flow pa-
perseditedbyCao, Chu, Yauandoneoftheauthors). InHamilton’spapers
(sometimesimplicitlyandsometimesbyanalogy)awell-developedtheoryof
Ricci flow is created as an approach toward the Geometrization Conjecture.
We encourage the reader to go back to these original papers which contain a
wealth of information and ideas. Hamilton’s program especially takes shape
in the three papers [267], [270] and [271]. The first two papers discuss