Closed mean curvature flows with asymptotically conical singularities

TL;DR

This paper constructs mean curvature flows that develop singularities modeled on asymptotically conical self-shrinkers, using topological methods, and demonstrates the existence of fattening level set flows from smooth hypersurfaces.

Contribution

It introduces a new technique to produce closed hypersurfaces with prescribed conical singularities in mean curvature flow, extending previous work in Ricci flow.

Findings

Existence of singularity models for mean curvature flow from asymptotically conical self-shrinkers.

Construction of fattening level set flows from smooth closed hypersurfaces.

Application of Ważewski's topological method to geometric flow problems.

Abstract

In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Wa\.zewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent--Ilmanen--Vel\'azquez and Chodosh--Daniels-Holgate--Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans--Spruck.