Surveys in Differential Geometry (Volume 12): Geometric Flows

✍ Scribed by Huai-Dong Cao, Shing-Tung Yau

• Publisher: International Press
• Year: 2008
• Tongue: English
• Leaves: 354
• Series: Surveys in Differential Geometry volume 12
• Category: Library

✦ Synopsis

Geometric flows are non-linear parabolic differential equations which describe the evolution of geometric structures. Inspired by Hamilton's Ricci flow, the field of geometric flows has seen tremendous progress in the past 25 years and yields important applications to geometry, topology, physics, nonlinear analysis, and so on. Of course, the most spectacular development is Hamilton's theory of Ricci flow and its application to three-manifold topology, including the Hamilton-Perelman proof of the Poincaré conjecture. This twelfth volume of the annual Surveys in Differential Geometry examines recent developments on a number of geometric flows and related subjects, such as Hamilton's Ricci flow, formation of singularities in the mean curvature flow, the Kähler-Ricci flow, and Yau's uniformization conjecture. Contents On the conformal scalar curvature equation and related problems A survey of the Kähler-Ricci Flow and Yau's Uniformization Conjecture Recent developments on the Hamilton's Ricci Flow Curvature flows in semi-Riemannian manifolds Global regularity and singularity development for wave maps Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1 dimensional Einstein gravity Monotonicity and Li-Yau-Hamilton inequalities Singularities of mean curvature flow and flow with surgeries Some recent developments in Lagrangian mean curvature flows

✦ Table of Contents

Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface......Page 4
Contents......Page 6
1. The Yamabe problem......Page 8
2. Compactness of the set of constant scalar curvature metrics in a given conformal class......Page 10
3. Non-compactness and non-uniqueness results......Page 13
4. Deformation of Riemannian metrics by their scalar curvature......Page 17
5. Convergence of the Yamabe flow for n > 6......Page 21
References......Page 24
1. Introduction......Page 28
2. Kahler-Ricci flow......Page 30
3. Curvature decay rate......Page 33
4. Steinness of nonnegatively curved manifolds......Page 36
5. Gradient Kahler-Ricci solitons......Page 40
6. Eternal solutions to the normalized Kahler-Ricci flow......Page 42
7. A Theorem of Mok-Siu-Yau and its generalizations......Page 48
References......Page 50
1.1. Short-time Existence and Uniqueness.......Page 54
1.2. Shi's Local Derivative Estimates......Page 55
1.3. Advanced Maximum Principles.......Page 56
1.4. Hamilton-Ivey Pinching Estimate......Page 58
1.5. Li-Yau-Hamilton inequalities......Page 59
2.1. Differential Sphere Theorems......Page 62
2.2. Kahler Manifolds with Nonnegative Holomorphic Bisectional Curvature......Page 66
3.1. Perelman's Conjugate Heat Equation Approach......Page 73
3.2. Perelman's Reduced Volume Approach......Page 76
4.1. Hamilton's Compactness Theorem......Page 80
4.2. Hamilton's Classification of Singularities......Page 81
4.3. Ancient K-solutions......Page 83
4.4. Singularity Structure Theorem......Page 90
5.1. The Solution at the First Singular Time......Page 93
5.2. Definition of Surgical Solutions......Page 96
5.3. Long-Time Existence of Surgical Solutions......Page 98
5.4. Topological Implications......Page 108
Thurston's Geometrization Conjecture......Page 109
THEOREM 6.1 (Thick-thin decomposition theorem).......Page 111
THEOREM 6.3 (Collapsing Theorem)......Page 113
References......Page 115
CONTENTS......Page 120
2. Notations and preliminary results......Page 121
3. Evolution equations for some geometric quantities......Page 124
LEMMA 3.2 (Evolution of the metric).......Page 125
LEMMA 3.4 (Evolution of the second fundamental form).......Page 126
LEMMA 3.5 (Evolution of (phi - f).......Page 127
4. Essential parabolic flow equations......Page 129
5. Stability of the limit hypersurfaces......Page 136
6. Existence results......Page 147
7. The inverse mean curvature flow......Page 162
8. The IMCF in ARW spaces......Page 164
References......Page 171
1. The wave maps problem......Page 174
Motivation for studying wave maps......Page 176
2. Well-posedness typology for nonlinear wave equations......Page 177
Energy constraint......Page 178
3. The local existence theory for wave maps......Page 180
Abstract function space requirements:......Page 181
4. Small data global existence theory......Page 183
4.1. Strong global wellposedness in the critical Besov space......Page 184
4.2. Weak global well-posedness in the critical Sobolev space......Page 186
5. Approaching the large data problem in the critical dimension n = 2 and hyperbolic target......Page 194
Goal:......Page 195
6.1. Radial wave maps......Page 196
6.2. Equivariant wave maps......Page 197
Conjectured general bubbling off scenario......Page 199
7.1. The co-rotational case......Page 200
References......Page 205
1. Introduction......Page 210
2. Preliminary Computations......Page 213
3. The Dirichlet energy of the Gauss map......Page 222
4. The reduced Hamiltonian......Page 224
5. Hamilton-Jacobi theory and the Dirichlet energy......Page 231
6. Analysis of the Gauss Map equation......Page 235
7. Some basic properties of the At equation......Page 240
8. Existence of solutions to the At equation......Page 244
9. Einstein solution curves and ray structures on Teichmiiller space......Page 248
10. Lagrangian foliations of T*r(~)......Page 250
Concluding remarks......Page 253
References......Page 255
1. Introduction......Page 258
2. Li-Yau-Hamilton type inequalities......Page 259
A. Li-Yau's gradient estimate for the linear heat equation......Page 260
B. The differential estimates on the fundamental solution......Page 261
C. The matrix LYH inequalities......Page 262
D. LYH inequality on (1,1) forms......Page 263
A. Matrix LYH inequalities on curvature......Page 264
B. The linear trace LYH inequalities......Page 265
D. Interpolations......Page 267
E. LYH for the conjugate heat equation......Page 269
2.3. Hypersurface flows in R^n+l.......Page 270
A. The support function......Page 271
B. The LYH inequality......Page 274
C. PDE satisfied by the speed function......Page 277
A. From the matrix LYH inequalities......Page 280
B. Thermodynamical consideration......Page 284
A. Hamilton's entropy for surfaces......Page 285
B. From matrix LYH inequalities......Page 286
C. Perelman's entropy......Page 290
3.3. Interpolation and localization......Page 291
A. The general formulation......Page 294
B. The entropy formulae......Page 295
4.1. Linear heat equation......Page 298
4.2. Ricci flow......Page 301
4.3. Dual version and the matrix inequality......Page 303
5. Comments......Page 305
References......Page 306
1. Introduction......Page 310
EXAMPLE 2.1. Homothetically shrinking solutions......Page 314
EXAMPLE 2.4. The degenerate neckpinch......Page 315
3. Invariance properties......Page 316
4. Convergence to a point of convex surfaces......Page 320
5. Convexity estimates for mean convex surfaces......Page 322
6. Cylindrical and gradient estimates for two-convex surfaces......Page 327
7. The surgery procedure......Page 331
8. Neck detection......Page 333
9. The surgery algorithm......Page 335
References......Page 338
1.1. The mean curvature flow.......Page 340
1.3. What is special about being Lagrangian......Page 342
1.4. Calabi-Yau case......Page 343
1.5. Overview of the article.......Page 344
2.1. Global existence and convergence......Page 346
2.2. Characterization of first-time singularities......Page 350
2.3. Constructions of self-similar solutions......Page 351
3. Prospects......Page 352
References......Page 353