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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!

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  • $\begingroup$ The field "geometric flows" is just too big to be covered/introduced by a book. $\endgroup$ – user99914 Sep 15 '16 at 9:09
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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.

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  • $\begingroup$ Thanks for your answer! :) in the comments you wrote that the field "geometric flows" is too big to be introduced by a book. Could you please explain better? I am very curious. What is the state of art of this field? What are the goals and applications (apart the famous example of the Poincaré Conjecture)? $\endgroup$ – Onil90 Sep 15 '16 at 9:28
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    $\begingroup$ Note that geometric flows correspond to a wide range of problems, besides MCF and Ricci flow. To name a few, there are Yamabe flow, Yang-Mill flow, harmonic map heat flow, Willmore flow....... . Basically the goal of these geometric flow is to find nice objects (for Ricci flow it is the space with constant curvature, and for MCF it is minimal submanifolds). In most of the case finding such objects directly is not easy, so one try to use geometric flows to "flow" bad objects to nicer one. @Onil90 (of course everything is oversimplified) $\endgroup$ – user99914 Sep 15 '16 at 9:43
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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

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  • $\begingroup$ Thanks for your answer. Could you please tell me the good features of those books in your opinion? :) $\endgroup$ – Onil90 Sep 15 '16 at 9:24
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In my opinion the easiest one for studying the Ricci flow is the book "The Ricci flow: An Introduction", written by Chow and Knoph.

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