The Differences of Ricci Flow between Perelman and Hamilton about the Poincaré Conjecture and the Geometrization Conjecture of Thurston for Mathematics Lectures

Karasawa Toshimitsu

(YPM Mara Education Foundation/Univertiti Kuala Lumpur, Japan)

Abstract: In this paper, it will be given the ideas of the proof the Poincaré conjecture and the geometrization conjecture of Thurston by Perelman who is a Russian mathematician posted the first of a series of three e-prints. Perelman’s proof uses a modified version of a Ricci flow program developed by Richard Streit Hamilton. Someone said that the major contributors of this work are Hamilton and Perelman.

Actually Ricci flow was pioneered by Hamilton. But the version of surgery needed for Perelman’s argument which is extremely delicate as one need to ensure that all the properties of Ricci flow used in the argument also hold for Ricci flow with surgery. Running the surgery in reverse, this establishes the full geometry conjecture of Thurston, and in particular the Poincaré conjecture as a special and simpler case. This is the excellent thing for the mathematics lecturer to find in his/her mathematics education.

Keywords: Ricci Flow, Poincaré conjecture, Geometrization conjecture, Perelman, Hamilton