Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture

by Zhang; Qi S.

• ISBN13: 9781439834596
• ISBN10: 1439834598
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-07-02
• Publisher: CRC Press

Summary

Focusing on Sobolev inequalities and their applications to analysis on manifolds and Ricci flow, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture introduces the field of analysis on Riemann manifolds and uses the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows, especially with surgeries. The author explains key ideas, difficult proofs, and important applications in a succinct, accessible, and unified manner.The book first discusses Sobolev inequalities in various settings, including the Euclidean case, the Riemannian case, and the Ricci flow case. It then explores several applications and ramifications, such as heat kernel estimates, Perelman's W entropies and Sobolev inequality with surgeries, and the proof of Hamilton's little loop conjecture with surgeries. Using these tools, the author presents a unified approach to the Poincaré conjecture that clarifies and simplifies Perelman's original proof.Since Perelman solved the Poincaré conjecture, the area of Ricci flow with surgery has attracted a great deal of attention in the mathematical research community. Along with coverage of Riemann manifolds, this book shows how to employ Sobolev imbedding and heat kernel estimates to examine Ricci flow with surgery.

Table of Contents

Introduction	p. 1
Sobolev inequalities in the Euclidean space	p. 7
Weak derivatives and Sobolev space Wk k,p (D), D ⊂ Rn	p. 7
Main imbedding theorem for W01,p(D)	p. 10
Poincaré inequality and log Sobolev inequality	p. 23
Best constants and extremals of Sobolev inequalities	p. 25
Basics of Riemann geometry	p. 27
Riemann manifolds, connections, Riemann metric	p. 27
Second covariant derivatives, curvatures	p. 44
Common differential operators on manifolds	p. 52
Geodesies, exponential maps, injectivity radius etc	p. 56
Integration and volume comparison	p. 80
Conjugate points, cut-locus and injectivity radius	p. 90
Bochner-Weitzenbock type formulas	p. 98
Sobolev inequalities on manifolds	p. 103
A basic Sobolev inequality	p. 103
Sobolev, log Sobolev inequalities, heat kernel	p. 108
Sobolev inequalities and isoperimetric inequalities	p. 127
Parabolic Harnack inequality	p. 133
Maximum principle for parabolic equations	p. 151
Gradient estimates for the heat equation	p. 155
Basics of Ricci flow	p. 167
Local existence, uniqueness and basic identities	p. 167
Maximum principles under Ricci flow	p. 187
Qualitative properties of Ricci flow	p. 199
Solitons, ancient solutions, singularity models	p. 209
Perelman's entropies and Sobolev inequality	p. 225
Perelman's entropies and their monotonicity	p. 225
(Log) Sobolev inequality under Ricci flow	p. 238
Critical and local Sobolev inequality	p. 248
Harnack inequality for the conjugate heat equation	p. 272
Fundamental solutions of heat type equations	p. 281
Ancient k solutions and singularity analysis	p. 291
Preliminaries	p. 291
Heat kernel and k solutions	p. 297
Backward limits of k solutions	p. 308
Qualitative properties of k solutions	p. 316
Singularity analysis of 3-dimensional Ricci flow	p. 331
Sobolev inequality with surgeries	p. 341
A brief description of the surgery process	p. 341
Sobolev inequality, little loop conjecture with surgeries	p. 354
Applications to the Poincaré conjecture	p. 381
Evolution of regions near surgery caps	p. 382
Canonical neighborhood property with surgeries	p. 394
Summary and conclusion	p. 405
Bibliography	p. 409
Index	p. 421
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