On the extension of the 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.