Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

TL;DR

This paper establishes curvature estimates for hypersurfaces evolving under mean curvature flow in complex and quaternionic projective spaces, showing that singularities are either convex or cylindrical under certain initial conditions.

Contribution

It generalizes previous Euclidean and spherical results to a broad class of symmetric spaces, providing new curvature bounds and singularity characterizations.

Findings

Asymptotic profiles are either strictly convex or cylindrical.

Curvature estimates hold under initial pinching conditions.

Results extend to complex and quaternionic projective spaces.

Abstract

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.