Partial regularity at the first singular time for hypersurfaces evolving by mean curvature

In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ implies that the $\mathcal{H}^{n+2-p}$ - measure of the first singular set vanishes inside $B_R(y)$. For $p=2$, this was established independently by Han and Sun. Our result furthermore generalizes previous work of Xu, Ye and Zhao and of Le and Sesum for $p\ge n+2$, in which case the singular set was shown to be empty. By a theorem of Ilmanen, our integrability condition is satisfied for $p=2$ and $n=2\,$ if the initial surface has finite genus. Thus, the first singular set has zero $\mathcal{H}^2$- measure in this case. This is the conclusion of Brakke's main regularity theorem for the special case of surfaces, but derived without having to impose the area continuity and unit density hypothesis. It follows from recent work of Head and of Huisken and Sinestrari that for the flow of closed, $k$ - convex hypersurfaces, that is hypersurfaces whose sum of the smallest $k$ principal curvatures is positive, our integrability criterion holds with exponent $p=n+3-k-\alpha$ for all small $\alpha >0$ as long as $1\le k\le n-1$. Therefore, the first singular set of such solutions is at most $(k-1)$ - dimensional, which is an optimal estimate in view of some explicit examples.