Convex ancient solutions of the mean curvature flow

Abstract

We study solutions of the mean curvature flow which are defined for all negative times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter, or a reverse isoperimetric inequality. We also study the behaviour of uniformly $k$-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.