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

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. Sobolev Inequalities in the Euclidean Space. Basics of Riemann Geometry. Sobolev Inequalities on Manifolds. Basics of Ricci Flow. Perelman’s Entropies and Sobolev Inequality. Ancient κ Solutions and Singularity Analysis. Sobolev Inequality with Surgeries. Applications to the Poincaré Conjecture. Bibliography. Index.

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

Review :
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, MathSciNet, 2011 This is a very good book on Ricci flows. Anyone who is interested to know the most recent development in Ricci flows and the Poincare 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