The Ricci Flow Equation and Poincaré Conjecture

Abstract

The Poincaré conjecture was formulated by the French mathematician Henri Poincaré more than hundred years ago. The conjecture states, when reformulated in modern language, that any simply connected closed 3-manifold is diffeomorphic to the standard 3-sphere $S^3$. This was the most famous open problem, and its solution turned out to be extraordinarily difficult. It had eluded all attempts at solution for more than hundred years. During 2002 and 2003, Grigoriy Perelman posted a proof of the conjecture on the Internet in three instalments, completing a program initiated in the 1980s by Richard Hamilton to solve a more general conjecture, called the geometrization conjecture of William Thurston. The key tool of Hamilton’s program is the Ricci flow, a differential equation on the space of Riemannian metrics of a 3-manifold. The equation is designed after the mathematical model for heat flow. As heat gradually flows from hotter to cooler parts of a metallic body until a uniform temperature is achieved throughout the body, it was expected that in Ricci flow, regions of higher curvature will tend to diffuse into regions of lower curvature to produce an equilibrium geometry for the 3-manifold for which Ricci curvature is uniform over the entire manifold. Thus in principle, a 3-manifold when subject to Ricci flow will produce a kind of normal form which will ultimately solve the geometrization conjecture. Although Hamilton established a number of beautiful geometric results using the Ricci flow equation, the progress in applying this program to the conjecture eventually came to a standstill mainly because of the formation of singularities, which defied solution of the problem. In his proof, Perelman constructed a program for getting around to these obstacles. He modified the Ricci flow used by Hamilton with “Ricci flow with surgery”. This expunges the singular regions as they develop in a controlled way and eventually solves the geometrization conjecture.