Which is the best book for studying geometric flows?

Question

I have some knowledge about the basics in Riemannian Geometry (I used Do Carmo's and Petersen's books). Now I would like to focus my attention on geometric flows (mostly mean curvature flow and Ricci flow).

Where should I start? Which is the best introductory book?

Thank you!

Answer 1

For mean curvature flow, to me the easiest one is Zhu's lectures on mean curvature flow. It covers the simplest cases (hypersurfaces) and the "classical" techniques/results, for example,

• De-Turck trick for the existence of the flow,
• Calculations of evolution equations of geometric quantities, use of maximum principle (scalar and matrix),
• Huisken's monotonicity formula, basic classification of singularities,
• A version of a well-known iteration technique in elliptic PDE

This book was published in 2002 so it definitely does not cover the whole topics well. But as a first read you can give it a go.

Answer 2

I recommend these books. Because these books are not too old and actually useful for poincare conjecture. Ricci flow of Poincare conjecture The book of Hamilton

Answer 3

In my opinion the easiest one for studying the Ricci flow is the book "The Ricci flow: An Introduction", written by Chow and Knoph.