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Convexity and cylindrical estimates for mean curvature flow in the sphere

Author: Huy The Nguyen
Journal: Trans. Amer. Math. Soc. 367 (2015), 4517-4536
MSC (2010): Primary 53C44
Published electronically: March 4, 2015
MathSciNet review: 3335392
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Abstract: We study mean curvature flow in the sphere with the quadratic curvature condition $ \vert A\vert^{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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Additional Information

Huy The Nguyen
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
Address at time of publication: The School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia

Received by editor(s): July 8, 2011
Received by editor(s) in revised form: July 19, 2012
Published electronically: March 4, 2015
Additional Notes: The author was supported by The Leverhulme Trust
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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