ABSTRACT
The book is centered around Sobolev inequalities and their applications to analysis on manifolds, and in particular to Ricci flow. There are two objectives. One is to serve as an introduction to the field of analysis on Riemann manifolds. The other is to use the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows, especially in the case with surgeries, a research field that has attracted much attention. Rather than making a comprehensive presentation, the aim is to present key ideas, to explain the hard proofs and most important applications in a succinct, accessible and unified manner.
