Sobolev inequalities, heat kernels under Ricci flow, and the Poincaré conjecture

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.

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.

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. --Book Jacket.