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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
DOI: https://doi.org/10.1090/S0002-9947-2015-05927-3
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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  • [1] Fernando Codá Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math. (2) 176 (2012), no. 2, 815-863. MR 2950765, https://doi.org/10.4007/annals.2012.176.2.3
  • [2] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497 (84a:53050)
  • [3] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. MR 772132 (86j:53097)
  • [4] Gerhard Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463-480. MR 837523 (87f:53066), https://doi.org/10.1007/BF01388742
  • [5] Gerhard Huisken, Deforming hypersurfaces of the sphere by their mean curvature, Math. Z. 195 (1987), no. 2, 205-219. MR 892052 (88d:53058), https://doi.org/10.1007/BF01166458
  • [6] Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175-191. MR 1216584 (94c:58037)
  • [7] Gerhard Huisken and Carlo Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45-70. MR 1719551 (2001c:53094), https://doi.org/10.1007/BF02392946
  • [8] Gerhard Huisken and Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009), no. 1, 137-221. MR 2461428 (2010a:53138), https://doi.org/10.1007/s00222-008-0148-4
  • [9] H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. MR 0238229 (38 #6505)
  • [10] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $ R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361-379. MR 0344978 (49 #9717)
  • [11] Guido Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965), Les Presses de l'Université de Montréal, Montreal, Que., 1966 (French). MR 0251373 (40 #4603)
  • [12] Ben Weinkove, Singularity formation in the Yang-Mills flow, Calc. Var. Partial Differential Equations 19 (2004), no. 2, 211-220. MR 2034580 (2005b:53111), https://doi.org/10.1007/s00526-003-0217-x

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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

DOI: https://doi.org/10.1090/S0002-9947-2015-05927-3
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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