Stability and singular set of the two-convex level set flow

TL;DR

This paper investigates the stability and structure of singularities in the level set flow of two-convex hypersurfaces, showing that under certain conditions, the perturbed flow preserves the nature of singularities.

Contribution

It establishes stability of the level set flow off singularities and characterizes the singular set structure for two-convex flows with finitely many singularities.

Findings

Singular set has finitely many components, each a point or a $C^{1}$ curve.

Perturbed flows preserve the type of singularities near original singular components.

Flow stability holds away from singularities under perturbations.

Abstract

The level set flow of a mean-convex closed hypersurface is stable off singularities, in the sense that the level set flow of the perturbed hypersurface would be close in the smooth topology to the original flow wherever the latter is regular. To study the behavior near singularities, we further assume that the initial hypersurface is two-convex and that the flow has finitely many singular times. In this case, the singular set of the flow would have finitely many connected components, each of which is either a point or a compact $C^{1}$ embedded curve. Then under additional conditions, we show that near each connected component of the singular set of the original flow, the perturbed flow would have "the same type" of singular set as that of the singular component.