Author:
Stephen Lynch,Nguyen Huy The
Abstract
AbstractWe consider the flow by mean curvature of smooth n-dimensional submanifolds of $${\mathbb {R}}^{n+k}$$
R
n
+
k
, $$k \ge 2$$
k
≥
2
, which are compact and quadratically pinched. We establish that such flows are asymptotically convex, that is, the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. This generalises the convexity estimate of Huisken–Sinestrari to higher codimensions. By combining our estimate with work of Naff, we conclude that singularity models for the flow are convex ancient solutions of codimension one.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
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