Self-shrinkers to the mean curvature flow asymptotic to isoparametric cones

TL;DR

This paper constructs a self-shrinking solution to the mean curvature flow that asymptotically approaches an isoparametric cone, expanding understanding of singularity models in geometric flows.

Contribution

It introduces a new class of self-shrinkers asymptotic to isoparametric cones, linking the theory of isoparametric hypersurfaces with mean curvature flow singularities.

Findings

Constructed a self-similar shrinking solution asymptotic to an isoparametric cone.

Established the existence of such solutions outside the cone.

Connected isoparametric hypersurface theory with mean curvature flow analysis.

Abstract

In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface $\Gamma \subset S^n$. The theory of isoparametic hypersurfaces is extremely rich and there are infinitely many distinct classes of examples, each with infinitely many members.