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Blow-up of the mean curvature at the first singular time of the mean curvature flow

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Abstract

It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal than 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small \(L^2\)-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the \(L^2\)-norm.

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References

  1. Aubin, T.: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer, Berlin (1998)

    Book  Google Scholar 

  2. Chen, B.: On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof. Math. Ann. 194, 19–26 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chow, B., Peng, L., Ni, L.: Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence (2006)

    Google Scholar 

  4. Colding, T.H., Minicozzi, W.P., II.: Generic mean curvature flow I: generic singularities. Ann. Math. 175(2), 755–833 (2012)

  5. Andrew, A.: Cooper, A characterization of the singular time of the mean curvature flow. Proc. Amer. Math. Soc. 139(8), 2933–2942 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Lellis, C., Müller, S.: Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom. 69(1), 75–110 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Gerhardt, C.: Closed immersed umbilic hypersurfaces in \(\mathbb{R} ^{n+1}\) are spheres. http://www.math.uni--heidelberg.de/studinfo/gerhardt/spheres.pdf

  8. Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986)

    MathSciNet  MATH  Google Scholar 

  9. Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26(2), 285–314 (1987)

    MathSciNet  MATH  Google Scholar 

  10. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Huang, Z., Lin, L.: Stability of the surface area preserving mean curvature flow in euclidean space. J. Geom. 106(3), 483–501 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differ. Equations 8(1), 1–14 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)

    MathSciNet  MATH  Google Scholar 

  14. Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Differ. Geom. 57(3), 409–441 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Kong, W., Sigal, I.M.: Stability of spherical collapse under mean curvature flow (2012) (preprint)

  17. Lin, L.: Mean curvature flow of star-shaped hypersurfaces. arXiv:1508.01225 (2015) (preprint)

  18. Le, N.Q., Sesum, N.: The mean curvature at the first singular time of the mean curvature flow. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(6), 1441–1459 (2010)

  19. Liu, K., Xu, H., Ye, F., Zhao, E.: The extension and convergence of mean curvature flow in higher codimension (2011) (preprint)

  20. Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \(R^{n}\). Comm. Pure Appl. Math. 26, 361–379 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  21. Perez, D.R.: On nearly umbilical hypersurfaces, Ph.D. thesis, Universität Zürich (2011)

  22. Smoczyk, K.: Starshaped hypersurfaces and the mean curvature flow. Manuscr. Math. 95(2), 225–236 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Topping, P.: Relating diameter and mean curvature for submanifolds of Euclidean space. Comment. Math. Helv. 83(3), 539–546 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The research of the Natasa Sesum is partially supported by NSF Grant 1056387. The authors would like to thank Zheng Huang and Nam Le for many helpful discussions.

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Authors and Affiliations

  1. Mathematics Department, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064, USA

    Longzhi Lin

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Natasa Sesum

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  1. Longzhi Lin
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  2. Natasa Sesum
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Correspondence to Natasa Sesum.

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Communicated by A. Neves.

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Lin, L., Sesum, N. Blow-up of the mean curvature at the first singular time of the mean curvature flow. Calc. Var. 55, 65 (2016). https://doi.org/10.1007/s00526-016-1003-x

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  • DOI: https://doi.org/10.1007/s00526-016-1003-x

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