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 PoincarT 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 Poincar6 conjecture that clarifies and simplifies Perelman's original proof.

Features

Provides a user-friendly introduction to Sobolev inequality and differential geometry

Links Ricci flow with Sobolev inequality

Discusses the potential of the heat equation method to tackle other problems in Ricci flow

Solves an important conjecture in Ricci flow—the proof of little loop conjecture with surgery

Clarifies and simplifies Perelman's work on the PoincarT conjecture

Since Perelman solved the PoincarT 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.