On the classification of ancient solutions to curvature flows on the sphere

Abstract

We consider the evolution of hypersurfaces on the unit sphere SnC1 by smooth functions of the Weingarten map. We introduce the notion of “quasiancient” solutions for flows that do not admit nontrivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows, such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Aleksandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.