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.
Categories: math.DG math.AP