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 \({t\in [0,T)}\). Cooper (Mean curvature blow up in mean curvature flow, arxiv.org/abs/0902.4282) 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.