The optimal interior ball estimate for a $k$-convex mean curvature flow

Abstract:

In this note, we prove that at a singularity of an $(m+1)$-convex mean curvature flow, Andrews’ non-collapsing ratio improves as much as is allowed by the example of the shrinking cylinder $\mathbb {R}^m\times S^{n-m}$. More precisely, we show that for any $\varepsilon >0$ we have $\overline k\leq (1+\varepsilon )\frac {1}{n-m}H$ wherever the mean curvature $H$ is sufficiently large, where $\overline k$ is the interior ball curvature. When $(m+1)<n$, this estimate improves the inscribed radius estimate of Brendle, which was subsequently proved much more directly by Haslhofer-Kleiner by using the powerful new local blow-up method they developed in an earlier work. Our estimate is also based on their local blow-up method, but we do not require the structure theorem for ancient flows, instead making use of the gradient term which appears in the evolution equation of the two-point function which defines the interior and exterior ball curvatures. We also obtain an optimal exterior ball estimate for flows of convex hypersurfaces.