TY - JOUR

T1 - The optimal interior ball estimate for a k-convex mean curvature flow

AU - Langford, Mat

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2015/12

Y1 - 2015/12

N2 - 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 Rm ×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.

AB - 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 Rm ×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.

UR - http://www.scopus.com/inward/record.url?scp=84944207339&partnerID=8YFLogxK

U2 - 10.1090/proc/12624

DO - 10.1090/proc/12624

M3 - Article

SN - 0002-9939

VL - 143

SP - 5395

EP - 5398

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -