On the singular set of mean curvature flows with Neumann free boundary conditions

TL;DR

This paper proves that the singular set of mean curvature flow with Neumann free boundary conditions on a smooth support surface has zero Hausdorff measure, with results depending on the interaction strength and mean convexity of the support.

Contribution

It establishes the measure-zero property of the singular set for mean curvature flows with free boundary conditions, considering different interaction types and the mean convexity assumption.

Findings

Singular set has Hausdorff measure zero.

Result holds under weaker interaction without extra assumptions.

Mean convexity of support surface is necessary for stronger interaction cases.

Abstract

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of weaker interaction, we need make no further assumptions than in the case without boundary to achieve our result. In the case of stronger interaction, we need only make the additional assumption that $H_{\Sigma}>0$, that is, that the support surface be mean convex. We go on, in this case, to show that the result is not, in general, true without the mean convexity assumption.