Ancient solutions of geometric flows with curvature pinching

TL;DR

This paper establishes rigidity theorems for ancient solutions of geometric flows, identifying conditions under which these solutions are characterized as shrinking spheres, especially in higher codimension and certain nonlinear flows.

Contribution

It introduces new curvature pinching conditions that uniquely characterize shrinking spheres among ancient solutions in various geometric flows.

Findings

Pinching conditions characterize shrinking spheres in mean curvature flow.

Rigidity theorems apply to certain nonlinear curvature flows.

Results extend to higher codimension cases.

Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.