On the extension of the mean curvature flow

Details On the extension of the mean curvature flow On the extension of the mean curvature flow

2009-05-07

arxiv.org/abs/0905.0936v2

Math. Z. 267 (2011), no. 3--4, 583--604

Mathematics

Differential Geometry

21 pages, presented self-contained proof of the main theorem

Scientific paper

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.

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