陶哲轩评述佩雷尔曼的庞加莱猜想证明——从非线性偏微分方程观点.pdf

´ PERELMAN’S PROOF OF THE POINCARE CONJECTURE: A 6 NONLINEAR PDE PERSPECTIVE 0 0 2 TERENCE TAO t c Abstract. We discuss some of the key ideas of Perelman’s proof of Poincar´e’s O conjecture via the Hamilton program of using the Ricci flow, from the perspec- 9 tive of the modern theory of nonlinear partial differential equations. 2 ] G 1. Introduction D . 1.1. Perelman’s theorem. In three remarkable papers [28], [29], [30] in 2003, h Grisha Perelman significantly advanced the theory of the Ricci flow, which was a t a quasilinear PDE introduced by Richard Hamilton [16] and then extensively studied m by Hamilton and others in a program to understand the topology of manifolds. In [ particular, in [29] Perelman showed that in three spatial dimensions there was a well- defined Ricci flow with surgery procedure, which is the Ricci flow PDE punctuated 1 v at a discrete set of times by carefully chosen surgery operations which may possibly 3 change the topology of the manifold, with the following remarkable property: 0 9 Theorem 1.2 (Perelman’s theorem, finite time extinction case). [28], [29], [30] Let M = (M , g ) be a closed smooth three-dimensional compact Riemmanian mani- 0 0 0 1 fold whose fundamental group