On the extension of the mean curvature flow

TL;DR

This paper proves that under certain boundedness conditions on the second fundamental form, the mean curvature flow of hypersurfaces can be extended beyond singularities, establishing an optimal integral bound criterion.

Contribution

It introduces a new extension criterion for mean curvature flow based on the boundedness of the second fundamental form and an integral mean curvature bound, improving understanding of flow continuation.

Findings

Flow can be extended if second fundamental form is bounded from below and mean curvature integral is bounded.

The mean curvature integral bound is shown to be optimal for flow extension.

Provides conditions under which singularities can be bypassed in mean curvature flow.

Abstract

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. In \cite{Cooper} Cooper has recently proved that the mean curvature blows up at the singular time $T$. We show that if the second fundamental form stays bounded from below all the way to $T$, then the scaling invariant mean curvature integral bound is enough to extend the flow past time $T$, and this integral bound is optimal in some sense explained below.