Sharp pinching estimates for mean curvature flow in the sphere

TL;DR

This paper establishes sharp curvature pinching and convexity estimates for mean curvature flow in the sphere, leading to classification results and new rigidity theorems for ancient solutions.

Contribution

It generalizes Simons' rigidity theorem to mean curvature flow in the sphere and introduces new estimates for singularity models and ancient solutions.

Findings

Sharp quadratic curvature pinching estimates established

Convexity and cylindrical estimates derived

Partial classification of singularity models achieved

Abstract

We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second fundamental form which we utilize, via a compactness argument, to obtain a convexity estimate. Together, the convexity and cylindrical estimates yield a partial classification of singularity models. We also obtain new rigidity results for ancient solutions.