A note on the selfsimilarity of limit flows

TL;DR

This paper proves that for mean convex surfaces under mean curvature flow, all limit flows are selfsimilar if and only if there are finitely many spherical singularities, with a local version extending to neck singularities in higher dimensions.

Contribution

It establishes a precise condition linking the selfsimilarity of limit flows to the finiteness of spherical singularities, advancing understanding of singularity formation in mean curvature flow.

Findings

All limit flows are selfsimilar if and only if finitely many spherical singularities exist.

A local equivalence for neck singularities in arbitrary dimensions is established.

Ancient ovals occur as limit flows if and only if spherical singularities accumulate at neck singularities.

Abstract

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity.