Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End

TL;DR

This paper investigates the long-term behavior of mean curvature flow starting from hypersurfaces asymptotic to cones with small entropy, showing convergence to self-expanding solutions and their stability.

Contribution

It establishes the asymptotic stability of self-expanders for mean curvature flow originating from conical hypersurfaces with small entropy.

Findings

Flow becomes asymptotically self-expanding

Limiting expanders are asymptotically stable

Results apply to hypersurfaces with small entropy

Abstract

If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow.