Blow up of subcritical quantities at the first singular time of the mean curvature flow

Details Blow up of subcritical quantities at the first singular time of the mean curvature flow Blow up of subcritical quantities at the first singular time of the mean curvature flow

2010-02-25

arxiv.org/abs/1002.4669v2

Geom. Dedicata, 151 (2011), no. 1, 361--371

Mathematics

Differential Geometry

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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 $t\in [0,T)$. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example $\int_{0}^{t} \int_{M_{s}} \frac{\abs{A}^{n + 2}}{log (2 + \abs{A})} d\mu ds,$ blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.

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