Stability of mean convex cones under mean curvature flow

TL;DR

This paper proves that homothetically expanding solutions to mean curvature flow originating from positive mean curvature cones are stable under small perturbations at infinity, using barrier constructions.

Contribution

It establishes the stability of specific expanding solutions under perturbations, extending understanding of mean curvature flow behavior from cones of positive mean curvature.

Findings

Homothetically expanding solutions are stable under perturbations.

Solutions close at infinity remain close for large times.

Barrier methods are effective in proving stability.

Abstract

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.