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

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 R^m ×^Sn−m. More precisely, we show that for any ε > 0 we have (formula presented) wherever the mean curvature H is sufficiently large, where 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.