Ricci flow–a nonlinear PDE perspective

The important thing in life is to have a great aim, and dermination to attain it. ——Johan wolfgang von Goethe.

There is an elegant and profound paper by Terence Tao. In that paper the author discusses some of the key ideas of Perelman’s proof of Poincare’s conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

As the author of the article suggests a large part of Perelman’s work is actually conducted in the arena of nonlinear PDE arguments, would already be the most technically impressive and significant result in the field of nonlinear PDE in recent years; the fact that this PDE result also gives the Poincar´e conjecture and the more general geometrization conjecture makes it the best piece of mathematics we have seen in the last ten years. It is truly a landmark achievement for the entire discipline.

There are three excellently detailed expositions of Perelman’s work Kleiner-Lott , Morgan-Tian and Cao-Zhu , in addition to Perelman’s original papers 1 2 3. The author suggests that reading all these papers in parallel (in particular, switching from one paper to another whenever we were stuck on a particular point) we were able to obtain a far richer view of the argument than we could have obtained just from reading one of them. At last, the author’s lecture notes were very heuristic and valuable.

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