Lectures on geometric evolution equations

Abstract

We present a number of evolution equations which arise in differential geometry starting with the linear heat equation on a Riemannian manifold and proceeding to the curve shortening flow, mean curvature flow and Hamilton's Ricci flow for metrics. We shall first show that a solution of the heat equation on a compact Riemannian manifold converges smoothly to its average value as $t \rightarrow \infty$, using only techniques which carry over to the nonlinear evolution equations presented in the lectures. We will then concentrate mainly on curve shortening and mean curvature flow which exhibit many of the features particular to a variety of nonlinear parabolic equations.