THE OPTIMAL INTERIOR BALL ESTIMATE FOR A <i>k</i>-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 x Sn-m. More precisely, we show that for any epsilon > 0 we have (k) over bar <= (1 + epsilon)1/n-m H 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.