Geometric flows and 3-manifolds

Abstract

The current article arose from a lecture1 given by the author in October 2005 on the work of R. Hamilton and G. Perelman on Ricci-flow and explains central analytical ingredients in geometric parabolic evolution equations that allow the application of these flows to geometric problems including the Uniformisation Theorem and the proof of the Poincare conjecture. Parabolic geometric evolution equations of second order are nonlinear extensions of the ordinary heat equation to a geometric setting, so we begin by reminding the reader of the linear heat equation and its properties. We will then introduce key ideas in the simpler equations of curve shortening and 2-d Ricci-flow before discussing aspects of three-dimensional Ricci-flow.