Ricci flow and the Poincare conjecture

Overview of Perelman's argument......Page 10
Volume and injectivity radius......Page 13
Canonical Neighborhoods......Page 14
First results......Page 16
Generalized Ricci flows......Page 17
The maximum principle......Page 19
Geometric Limits......Page 20
The reduced length function......Page 21
Application to non-collapsing results......Page 22
Application to ancient -non-collapsed solutions......Page 23
Bounded Curvature at Bounded Distance......Page 25
The standard solution......Page 26
Ricci Flows with surgery......Page 27
The Inductive Conditions Necessary for doing Surgery......Page 28
Topological effect of surgery......Page 29
Surgery and canonical neighborhoods for generalized Ricci flows......Page 30
Finite Extinction......Page 32
Acknowledgements......Page 35
List of related papers......Page 36
Riemannian metrics and the Levi-Civitá connection......Page 37
Curvature of a riemannian manifold......Page 39
Consequences of the Bianchi identities......Page 41
First Examples......Page 42
Cones......Page 43
Geodesics and the energy functional......Page 44
Families of geodesics and Jacobi fields......Page 45
Minimal geodesics......Page 46
The exponential mapping......Page 48
Computations in Gaussian normal coordinates......Page 49
Basic Curvature Comparison Results......Page 51
Manifolds of non-negative curvature......Page 53
Busemann Functions......Page 54
Comparison results in non-negative curvature......Page 55
The Soul Theorem......Page 57
The Splitting Theorem......Page 59
-necks......Page 60
Forward Difference Quotients......Page 63
The Definition of the Ricci Flow......Page 65
Einstein Manifolds......Page 66
Solitons......Page 67
Local Existence and Uniqueness......Page 68
Evolution of Curvatures......Page 70
Curvature Evolution in an Evolving Orthonormal Frame......Page 71
Variation of distance under Ricci Flow......Page 74
Shi's derivative estimates......Page 79
Space-Time......Page 88
More definitions for generalized Ricci flows......Page 89
Maximum Principle for Scalar Curvature......Page 91
The global version......Page 93
Flows with normalized initial conditions......Page 94
Extending Flows......Page 95
Non-negative curvature is preserved......Page 96
The Strong Maximum Principle for Curvature......Page 97
Applications of the Strong Maximum Principle......Page 98
Pinching toward positive curvature......Page 101
Application of the Pinching Result......Page 106
Harnack Inequality......Page 107
Geometric convergence of riemannian manifolds......Page 109
Geometric convergence of manifolds in the case of Ricci flow......Page 114
Geometric convergence of Ricci Flows......Page 115
Gromov-Hausdorff Limits......Page 117
Precompactness......Page 120
The Tits Cone......Page 121
Blow-up Limits......Page 123
Splitting limits at infinity......Page 125
L-length and L-geodesics......Page 127
L-geodesics......Page 128
The L-Jacobi Equation......Page 130
Relation to the second order variation of L......Page 131
The L-exponential map and its first-order properties......Page 134
The differential of Lexp......Page 135
Gradient of L"0365L......Page 136
Local Diffeomorphism near the initial......Page 137
Minimizing L-geodesics and the Injectivity Domain......Page 138
Monotonicity of the Ux() with respect to......Page 139
The second variation formula for L......Page 141
Inequalities for the Hessian of Lx......Page 144
Inequalities for Lx......Page 149
The reduced length function lx on space-time......Page 151
Local Lipschitz estimates for lx......Page 154
Statement and Corollaries......Page 155
The Proof of Proposition 6.57......Page 156
Rescaling......Page 161
The integrand in the reduced volume integral......Page 162
Monotonicity of reduced volume......Page 165
Existence of L-geodesics......Page 168
Results about lx and Ux()......Page 170
A bound for min lx......Page 171
Extension of the 2nd and 3rd inequalities in Corollary 6.52......Page 174
Reduced Volume......Page 182
Converse to Lemma 7.25......Page 183
Non-collapsing for compact Ricci flows......Page 185
Upper bound for V"0365Vx(Wsm(1))......Page 187
Upper Bound for V"0365Vx( Wlg(1))......Page 190
Application to Ricci Flows......Page 191
Definition......Page 193
Examples......Page 194
Asymptotic Curvature......Page 195
A consequence of Hamilton's Harnack Inequality......Page 197
The Asymptotic Gradient Shrinking Soliton for -solutions......Page 199
Bounding the reduced length and the curvature......Page 201
Analyzing the limit......Page 202
Preliminary results toward the proof of Proposition 9.20......Page 203
Extension to non-compactly supported functions......Page 206
Completion of the proof of Proposition 9.20......Page 211
The Gradient Shrinking Soliton Equation......Page 212
Completion of the proof of Theorem 9.14......Page 216
Point-picking......Page 217
Splitting Results......Page 218
Two-dimensional Ricci Flows......Page 220
Bounded curvature in dimension three......Page 221
Classification of Gradient Shrinking Solitons in dimension  3......Page 222
The Case of Compact Asymptotic Soliton......Page 223
Case 2: M is non-compact and strictly positively curved......Page 224
Case 3: Non-strictly positively curved.......Page 230
Universal......Page 231
Asymptotic Volume......Page 232
Volume Comparison......Page 233
Compactness of the space of three-dimensional -solutions......Page 236
Strong canonical neighborhoods......Page 240
Canonical neighborhoods for -solutions......Page 245
The statement of the theorem......Page 252
The sequence of tubes......Page 254
Extracting a limit of a subsequence of the tubes......Page 257
Properties of the limiting tube......Page 259
Cone Limits near the End E for Rescalings of U......Page 261
Directions at E......Page 262
The Metric on the space of directions at E......Page 264
Comparison Results for distances......Page 265
Completion of the proof of a cone limit at E......Page 266
Comparison of the Gromov-Hausdorff limit and the smooth limit......Page 268
Completion of the comparison of the blow-up limits......Page 269
The final contradiction......Page 270
A smooth blow-up limit defined for a small time......Page 271
Long-time blow-up limits......Page 275
Assumptions......Page 281
Canonical neighborhoods for (M"0362M,G"0362G)......Page 283
The ends of (,g(T))......Page 285
Existence of strong -necks sufficiently deep in a 2-horn......Page 288
The standard solution......Page 293
Uniqueness and properties: The Statement......Page 295
Completeness, Positive Curvature, and Asymptotic Behavior......Page 296
Standard solutions are rotationally symmetric......Page 299
Non-collapsing......Page 303
From Ricci flow to Deturck-Ricci flow......Page 304
Solutions of harmonic map flow......Page 306
The properties of r as a function of  and t......Page 308
The harmonic map flow equation......Page 309
An equation equivalent to the harmonic map flow equation......Page 311
The short time existence......Page 312
The asymptotic behavior of the solutions......Page 316
Completion of the Argument......Page 317
The uniqueness for the solutions of Deturck-Ricci flow......Page 318
Existence of Canonical Neighborhoods......Page 320
Completion of the Proof of Theorem 12.5......Page 321
Some corollaries......Page 322
Notation and the Statement of the Result......Page 325
Preliminary Computations......Page 327
Proof of the first two items for s< 4......Page 333
Surgery Space-time......Page 334
An exotic chart......Page 335
Coordinate charts for a surgery space-time......Page 336
Definition and basic properties of surgery space-time......Page 337
Horizontal Metrics......Page 339
Examples of Ricci flows with surgery......Page 340
Scaling and translating generalized Ricci Flows......Page 341
More Basic Definitions......Page 342
Normalized Initial Conditions......Page 343
First Assumptions......Page 344
Topological Consequences of Assumptions 1 -- 7......Page 346
The surgery parameters......Page 348
The process of surgery......Page 350
Statements about the existence of Ricci flow with surgery......Page 351
The Statement of the Non-collapsing Result......Page 354
The proof of non-collapsing when R(x)=r-2 with rri+1......Page 355
Minimizing L-geodesics exist when R(x)r-2i+1: The Statement......Page 356
Evolution of neighborhoods of surgery caps......Page 357
A length estimate......Page 362
Paths with short L+-length avoid the surgery caps......Page 364
Paths with small energy avoid the disappearing regions......Page 368
Limits of a sequence of paths with short L-length......Page 369
Completion of the proof of Proposition 16.4......Page 374
Completion of the proof of Proposition 16.1......Page 377
Completion of the Proof of Theorem 15.9......Page 380
Proof of the Canonical Neighborhood Assumption......Page 381
Completion of the Proof of Theorem 15.9......Page 391
History of this approach......Page 394
Existence of the Ricci flow with surgery......Page 396
Disappearance of components with non-trivial 2......Page 398
Two-sphere surgeries are trivial after finite time......Page 399
For all T sufficiently large 2(MT)=0......Page 401
Forward difference quotient for 3......Page 405
Proof of Theorem 18.1 assuming Proposition 18.17......Page 407
A further reduction of Proposition 18.17......Page 408
Curve Shrinking Flow......Page 411
The proof of Proposition 18.22 in a simple case......Page 412
Basic estimates for curve shortening......Page 414
Ramp solutions in M=MS1......Page 418
Proof of Proposition 18.22......Page 421
The case of a single cS2......Page 422
The completion of the proof of Proposition 18.22......Page 428
Proof of Lemma 18.49: Annuli of small area......Page 431
First reductions......Page 432
Focusing triangles......Page 433
No Dx is an embedded arc with both ends in c0......Page 438
For every xX, the geodesic Dx is embedded......Page 441
Far apart Dx's don't meet......Page 442
Completion of the proof......Page 443
Proof of the first inequality in Lemma 18.42......Page 444
A bound for kds......Page 445
Writing the curve flow as a graph......Page 448
t3=t2......Page 450
t2=t'+r2......Page 452
The geometry of an -neck......Page 454
Overlapping -necks......Page 459
Chains of -necks......Page 461
Subsets of the union of cores of (C,)-caps and -necks.......Page 464