Recent developments on the Hamilton’s Ricci Flow

Abstract

In 1982, Hamilton [41] introduced the Ricci flow to study compact three-manifolds with positive Ricci curvature. Through decades of works of many mathematicians, the Ricci flow has been widely used to study the topology, geometry and complex structure of manifolds. In particular, Hamilton’s fundamental works (cf. [12]) in the past two decades and the recent breakthroughs of Perelman [80, 81, 82] have made the Ricci flow one of the most intricate and powerful tools in geometric analysis, and led to the resolutions of the famous Poincare conjecture and Thurston’s geometrization conjecture in three-dimensional topology. In this survey, we will review the recent developments on the Ricci flow and give an outline of the Hamilton-Perelman proof of the Poincare conjecture, as well as that of a proof of Thurston’s geometrization conjecture.