Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture: 1st Edition (Hardback) book cover

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

1st Edition

By Qi S. Zhang

CRC Press

432 pages

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pub: 2010-07-02
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Description

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.

Reviews

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

Table of Contents

Introduction

Sobolev Inequalities in the Euclidean Space

Weak derivatives and Sobolev space Wk,p(D), D subset Rn

Main imbedding theorem for W01,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

About the Author

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

Subject Categories

BISAC Subject Codes/Headings:
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis