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The Ricci Flow : An Introduction : Mathematical Surveys and Monographs - Bennett Chow

The Ricci Flow : An Introduction

Mathematical Surveys and Monographs


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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.

Prefacep. vii
A guide for the readerp. viii
A guide for the hurried readerp. x
Acknowledgmentsp. xi
The Ricci flow of special geometriesp. 1
Geometrization of three-manifoldsp. 2
Model geometriesp. 4
Classifying three-dimensional maximal model geometriesp. 6
Analyzing the Ricci flow of homogeneous geometriesp. 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 isotropyp. 17
Notes and commentaryp. 19
Special and limit solutionsp. 21
Generalized fixed pointsp. 21
Eternal solutionsp. 24
Ancient solutionsp. 28
Immortal solutionsp. 34
The neckpinchp. 38
The degenerate neckpinchp. 62
Notes and commentaryp. 66
Short time existencep. 67
Variation formulasp. 67
The linearization of the Ricci tensor and its principal symbolp. 71
The Ricci-DeTurck flow and its parabolicityp. 78
The Ricci-DeTurck flow in relation to the harmonic map flowp. 84
The Ricci flow regarded as a heat equationp. 90
Notes and commentaryp. 92
Maximum principlesp. 93
Weak maximum principles for scalar equationsp. 93
Weak maximum principles for tensor equationsp. 97
Advanced weak maximum principles for systemsp. 100
Strong maximum principlesp. 102
Notes and commentaryp. 103
The Ricci flow on surfacesp. 105
The effect of a conformal change of metricp. 106
Evolution of the curvaturep. 109
How Ricci solitons help us estimate the curvature from abovep. 111
Uniqueness of Ricci solitonsp. 116
Convergence when X (M[superscript 2]) [less than sign] 0p. 120
Convergence when X (M[superscript 2]) = 0p. 123
Strategy for the case that X (M[superscript 2] [greater than sign] 0)p. 128
Surface entropyp. 133
Uniform upper bounds for R and [vertical bar down triangle, open]R[vertical bar]p. 137
Differential Harnack estimates of LYH typep. 143
Convergence when R(.,0) [greater than sign] 0p. 148
A lower bound for the injectivity radiusp. 149
The case that R(.,0) changes signp. 153
Monotonicity of the isoperimetric constantp. 156
An alternative strategy for the case X (M[superscript 2] [greater than sign] 0)p. 165
Notes and commentaryp. 171
Three-manifolds of positive Ricci curvaturep. 173
The evolution of curvature under the Ricci flowp. 174
Uhlenbeck's trickp. 180
The structure of the curvature evolution equationp. 183
Reduction to the associated ODE systemp. 187
Local pinching estimatesp. 189
The gradient estimate for the scalar curvaturep. 194
Higher derivative estimates and long-time existencep. 200
Finite-time blowupp. 209
Properties of the normalized Ricci flowp. 212
Exponential convergencep. 218
Notes and commentaryp. 221
Derivative estimatesp. 223
Global estimates and their consequencesp. 223
Proving the global estimatesp. 226
The Compactness Theoremp. 231
Notes and commentaryp. 232
Singularities and the limits of their dilationsp. 233
Classifying maximal solutionsp. 233
Singularity modelsp. 235
Parabolic dilationsp. 237
Dilations of finite-time singularitiesp. 240
Dilations of infinite-time singularitiesp. 246
Taking limits backwards in timep. 250
Notes and commentaryp. 251
Type I singularitiesp. 253
Intuitionp. 253
Positive curvature is preservedp. 255
Positive sectional curvature dominatesp. 256
Necklike points in finite-time singularitiesp. 262
Necklike points in ancient solutionsp. 271
Type I ancient solutions on surfacesp. 274
Notes and commentaryp. 277
The Ricci calculusp. 279
Component representations of tensor fieldsp. 279
First-order differential operators on tensorsp. 280
First-order differential operators on formsp. 283
Second-order differential operatorsp. 284
Notation for higher derivativesp. 285
Commuting covariant derivativesp. 286
Notes and commentaryp. 286
Some results in comparison geometryp. 287
Some results in local geometryp. 287
Distinguishing between local geometry and global geometryp. 295
Busemann functionsp. 303
Estimating injectivity radius in positive curvaturep. 312
Notes and commentaryp. 315
Bibliographyp. 317
Indexp. 323
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780821835159
ISBN-10: 0821835157
Series: Mathematical Surveys and Monographs
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 325
Publisher: American Mathematical Society
Country of Publication: US
Dimensions (cm): 25.4 x 17.78  x 2.54
Weight (kg): 0.82