The Ricci Flow : An Introduction

By: Bennett Chow, Dan Knopf

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The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to ''flow'' a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics. Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program. The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture. The authors also provide a ''Guide for the hurried reader'', to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called ''fast track''. The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Calculus & Mathematical Analysis

Preface	p. vii
A guide for the reader	p. viii
A guide for the hurried reader	p. x
Acknowledgments	p. xi
The Ricci flow of special geometries	p. 1
Geometrization of three-manifolds	p. 2
Model geometries	p. 4
Classifying three-dimensional maximal model geometries	p. 6
Analyzing the Ricci flow of homogeneous geometries	p. 8
The Ricci flow of a geometry with maximal isotropy SO (3)	p. 11
The Ricci flow of a geometry with isotropy SO (2)	p. 15
The Ricci flow of a geometry with trivial isotropy	p. 17
Notes and commentary	p. 19
Special and limit solutions	p. 21
Generalized fixed points	p. 21
Eternal solutions	p. 24
Ancient solutions	p. 28
Immortal solutions	p. 34
The neckpinch	p. 38
The degenerate neckpinch	p. 62
Notes and commentary	p. 66
Short time existence	p. 67
Variation formulas	p. 67
The linearization of the Ricci tensor and its principal symbol	p. 71
The Ricci-DeTurck flow and its parabolicity	p. 78
The Ricci-DeTurck flow in relation to the harmonic map flow	p. 84
The Ricci flow regarded as a heat equation	p. 90
Notes and commentary	p. 92
Maximum principles	p. 93
Weak maximum principles for scalar equations	p. 93
Weak maximum principles for tensor equations	p. 97
Advanced weak maximum principles for systems	p. 100
Strong maximum principles	p. 102
Notes and commentary	p. 103
The Ricci flow on surfaces	p. 105
The effect of a conformal change of metric	p. 106
Evolution of the curvature	p. 109
How Ricci solitons help us estimate the curvature from above	p. 111
Uniqueness of Ricci solitons	p. 116
Convergence when X (M[superscript 2]) [less than sign] 0	p. 120
Convergence when X (M[superscript 2]) = 0	p. 123
Strategy for the case that X (M[superscript 2] [greater than sign] 0)	p. 128
Surface entropy	p. 133
Uniform upper bounds for R and [vertical bar down triangle, open]R[vertical bar]	p. 137
Differential Harnack estimates of LYH type	p. 143
Convergence when R(.,0) [greater than sign] 0	p. 148
A lower bound for the injectivity radius	p. 149
The case that R(.,0) changes sign	p. 153
Monotonicity of the isoperimetric constant	p. 156
An alternative strategy for the case X (M[superscript 2] [greater than sign] 0)	p. 165
Notes and commentary	p. 171
Three-manifolds of positive Ricci curvature	p. 173
The evolution of curvature under the Ricci flow	p. 174
Uhlenbeck's trick	p. 180
The structure of the curvature evolution equation	p. 183
Reduction to the associated ODE system	p. 187
Local pinching estimates	p. 189
The gradient estimate for the scalar curvature	p. 194
Higher derivative estimates and long-time existence	p. 200
Finite-time blowup	p. 209
Properties of the normalized Ricci flow	p. 212
Exponential convergence	p. 218
Notes and commentary	p. 221
Derivative estimates	p. 223
Global estimates and their consequences	p. 223
Proving the global estimates	p. 226
The Compactness Theorem	p. 231
Notes and commentary	p. 232
Singularities and the limits of their dilations	p. 233
Classifying maximal solutions	p. 233
Singularity models	p. 235
Parabolic dilations	p. 237
Dilations of finite-time singularities	p. 240
Dilations of infinite-time singularities	p. 246
Taking limits backwards in time	p. 250
Notes and commentary	p. 251
Type I singularities	p. 253
Intuition	p. 253
Positive curvature is preserved	p. 255
Positive sectional curvature dominates	p. 256
Necklike points in finite-time singularities	p. 262
Necklike points in ancient solutions	p. 271
Type I ancient solutions on surfaces	p. 274
Notes and commentary	p. 277
The Ricci calculus	p. 279
Component representations of tensor fields	p. 279
First-order differential operators on tensors	p. 280
First-order differential operators on forms	p. 283
Second-order differential operators	p. 284
Notation for higher derivatives	p. 285
Commuting covariant derivatives	p. 286
Notes and commentary	p. 286
Some results in comparison geometry	p. 287
Some results in local geometry	p. 287
Distinguishing between local geometry and global geometry	p. 295
Busemann functions	p. 303
Estimating injectivity radius in positive curvature	p. 312
Notes and commentary	p. 315
Bibliography	p. 317
Index	p. 323
Table of Contents provided by Rittenhouse. All Rights Reserved.

The Ricci Flow : An Introduction

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