1 Introduction ................................................. 1
2 Sobolev inequalities in the Euclidean space .................. 7
2.1 Weak derivatives and Sobolev space Wk,p(D), D  Rn ...... 7
2.2 Main imbedding theorem for W0l,p(D) .................... 10
2.3 Poincare inequality and log Sobolev inequality ......... 23
2.4 Best constants and extremals of Sobolev inequalities ... 25
3 Basics of Riemann geometry .................................. 27
3.1 Riemann manifolds, connections, Riemann metric ......... 27
3.2 Second covariant derivatives, curvatures ............... 44
3.3 Common differential operators on manifolds ............. 52
3.4 Geodesies, exponential maps, injectivity radius etc. ... 56
3.5 Integration and volume comparison ...................... 80
3.6 Conjugate points, cut-locus and injectivity radius ..... 90
3.7 Bochner-Weitzenbock type formulas ...................... 98
4 Sobolev inequalities on manifolds .......................... 103
4.1 A basic Sobolev inequality ............................ 103
4.2 Sobolev, log Sobolev inequalities, heat kernel ........ 108
4.3 Sobolev inequalities and isoperimetric inequalities ... 127
4.4 Parabolic Harnack inequality .......................... 133
4.5 Maximum principle for parabolic equations ............. 151
4.6 Gradient estimates for the heat equation .............. 155
5 Basics of Ricci flow ....................................... 167
5.1 Local existence, uniqueness and basic identities ...... 167
5.2 Maximum principles under Ricci flow ................... 187
5.3 Qualitative properties of Ricci flow .................. 199
5.4 Solitons, ancient solutions, singularity models ....... 209
6 Perelman's entropies and Sobolev inequality ................ 225
6.1 Perelman's entropies and their monotonicity ........... 225
6.2 (Log) Sobolev inequality under Ricci flow ............. 238
6.3 Critical and local Sobolev inequality ................. 248
6.4 Harnack inequality for the conjugate heat equation .... 272
6.5 Fundamental solutions of heat type equations .......... 281
7 Ancient к solutions and singularity analysis ............... 291
7.1 Preliminaries ......................................... 291
7.2 Heat kernel and k solutions ........................... 297
7.3 Backward limits of k solutions ........................ 308
7.4 Qualitative properties of k solutions ................. 316
7.5 Singularity analysis of 3-dimensional Ricci flow ...... 331
8 Sobolev inequality with surgeries .......................... 341
8.1 A brief description of the surgery process ............ 341
8.2 Sobolev inequality, little loop conjecture with
surgeries ............................................. 354
9 Applications to the Poincare conjecture .................... 381
9.1 Evolution of regions near surgery caps ................ 382
9.2 Canonical neighborhood property with surgeries ........ 394
9.3 Summary and conclusion ................................ 405
Bibliography .................................................. 409
Index ......................................................... 421
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