Type-II singularities of two-convex immersed mean curvature flow

TL;DR

This paper proves that certain two-convex mean curvature flow translators are rotationally symmetric, especially as blow-up limits, using cylindrical and gradient estimates, with implications for flows like the two-harmonic mean curvature flow.

Contribution

It establishes rotational symmetry of specific mean curvature flow translators under broad conditions, extending previous symmetry results to more general flows.

Findings

Any strictly mean convex translator with cylindrical and gradient estimates is rotationally symmetric.

Blow-up limits of two-convex mean curvature flows are rotationally symmetric.

Results apply to flows including the two-harmonic mean curvature flow.

Abstract

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension $n\geq 3$ is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of two-convex hypersurfaces by the two-harmonic mean curvature.