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

Introduction

Sobolev Inequalities in the Euclidean Space
Weak derivatives and Sobolev space W^k,p(D), D subset Rⁿ
Main imbedding theorem for W₀^1,p(D)
Poincaré inequality and log Sobolev inequality
Best constants and extremals of Sobolev inequalities

Basics of Riemann Geometry
Riemann manifolds, connections, Riemann metric
Second covariant derivatives, curvatures
Common differential operators on manifolds
Geodesics, exponential maps, injectivity radius etc.
Integration and volume comparison
Conjugate points, cut-locus, and injectivity radius
Bochner–Weitzenbock type formulas

Sobolev Inequalities on Manifolds
A basic Sobolev inequality
Sobolev, log Sobolev inequalities, heat kernel
Sobolev inequalities and isoperimetric inequalities
Parabolic Harnack inequality
Maximum principle for parabolic equations
Gradient estimates for the heat equation

Basics of Ricci Flow
Local existence, uniqueness and basic identities
Maximum principles under Ricci flow
Qualitative properties of Ricci flow
Solitons, ancient solutions, singularity models

Perelman’s Entropies and Sobolev Inequality
Perelman’s entropies and their monotonicity
(Log) Sobolev inequality under Ricci flow
Critical and local Sobolev inequality
Harnack inequality for the conjugate heat equation
Fundamental solutions of heat type equations

Ancient κ Solutions and Singularity Analysis
Preliminaries
Heat kernel and κ solutions
Backward limits of κ solutions
Qualitative properties of κ solutions
Singularity analysis of 3-dimensional Ricci flow

Sobolev Inequality with Surgeries
A brief description of the surgery process
Sobolev inequality, little loop conjecture, and surgeries

Applications to the Poincaré Conjecture
Evolution of regions near surgery caps
Canonical neighborhood property with surgeries
Summary and conclusion

Bibliography

Index

Biography

Qi S. Zhang is a professor of mathematics at the University of California, Riverside.

The approach here is somewhat different from that of Perelman. The author shows that the W-entropy and its monotonicity imply certain uniform Sobolev inequalities along Ricci flows. These are used in the proofs of the two steps mentioned above, bypassing the use of the reduced volume and reduced distance, which simplifies Perelman’s proof considerably.
—John Urbas, Mathematical Reviews, Issue 2011m

This is a very good book on Ricci flows. Anyone who is interested to know the most recent development in Ricci flows and the Poincaré conjecture should take a look at the book.
—Zentralblatt MATH

It is clear as vodka that, as Zhang advertises in the Preface, ‘the first half of the book is aimed at graduate students and the second half is intended for researchers.’ With some good timing, the same reader can start as one and continue as the other. … a very important contribution to the genre.
—MAA Reviews, September 2010