Ancient solutions to mean curvature flow for isoparametric submanifolds

TL;DR

This paper classifies ancient solutions to mean curvature flow for isoparametric submanifolds and explores rigidity phenomena related to minimal hypersurfaces in spheres, connecting geometric flow behavior with classical conjectures.

Contribution

It proves all solutions for these submanifolds are ancient and investigates the rigidity of such flows, linking to Chern's conjecture.

Findings

All solutions are ancient for isoparametric submanifolds.

Rigidity results for ancient mean curvature flows in spheres.

Connection to Chern's conjecture on minimal hypersurfaces.

Abstract

Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern's conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.