Convexity and cylindrical estimates for mean curvature flow in the sphere

Nguyen, Huy

doi:10.1090/S0002-9947-2015-05927-3

Convexity and cylindrical estimates for mean curvature flow in the sphere
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Trans. Amer. Math. Soc. 367 (2015), 4517-4536 Request permission

Abstract:

We study mean curvature flow in the sphere with the quadratic curvature condition $|A|^{2} \leq \frac { 1}{n-2} H^{2} + 4 K$ which is related to but different from two-convexity for submanifolds of the sphere. We classify type $I$ singularities with no further hypotheses. If $H> 0$, then we apply the Huisken-Sinestrari convexity estimates to this situation and show that we can classify type $II$ singularities. This shows that at a singularity the surface is asymptotically convex. We then prove cylindrical estimates for the mean curvature flow and a pointwise gradient estimate which shows that near a singularity the surface is quantitatively convex.

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