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

Author: Mat Langford
Journal: Proc. Amer. Math. Soc. 143 (2015), 5395-5398
MSC (2010): Primary 53C44, 35K55, 58J35
Published electronically: August 4, 2015
MathSciNet review: 3411154
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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.

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Additional Information

Mat Langford
Affiliation: Mathematical Sciences Institute, Australian National University, ACT 0200 Australia

Received by editor(s): October 27, 2013
Published electronically: August 4, 2015
Additional Notes: This research was partially supported by Discovery grant DP120102462 of the Australian Research Council, an Australian Postgraduate Award, and an Australian National University HDR Supplementary Scholarship.
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society