Convexity and cylindrical estimates for mean curvature flow in the sphere
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- by Huy The Nguyen PDF
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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.References
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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
- Email: H.T.Nguyen@warwick.ac.uk, huy.nguyen@maths.uq.edu.au
- 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
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 4517-4536
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/S0002-9947-2015-05927-3
- MathSciNet review: 3335392