This book introduces the field of analysis on Riemann manifolds and uses the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows with surgeries. It first discusses Sobolev inequalities in various settings, including the Euclidean case, the Riemannian case, and the Ricci flow case. The text 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 Poincare conjecture that clarifies and simplifies Perelman's original proof.

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

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 Solutions and Singularity Analysis. Sobolev Inequality with Surgeries. Applications to the Poincare Conjecture. Bibliography. Index.