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.

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

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 GeometryRiemann 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 InequalityPerelman’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