Liouville theorems for nonlinear parabolic equations in geometry

Speaker: Mat Langford
Affiliation: Australian National University

Abstract

The parabolic analogue of Liouville’s theorem, due to Appel and Hirschman, states that every positive solution to the heat equation on $\mathbb{R}^n \times (-\infty,\infty)$ with subexponential spatial growth is constant. Motivated in part by Appel’s theorem, many recent attempts have been made to classify ancient solutions to nonlinear parabolic equations arising in geometry, such as the heat equation on manifolds, the mean curvature flow of hypersurfaces, and the Ricci flow of Riemannian metrics. We describe some of this progress, some of our own contributions (joint with many others), and some of the outstanding problems.