Focusing on Sobolev inequalities and their applications to analysis on manifolds and Ricci flow, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare 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

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.