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Starshaped hypersurfaces and the mean curvature flow

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Abstract

Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the casen=2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs.

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References

  1. U. Abresch, J. Langer: The normalized curve shortening flow and homothetic solutions. J. Differential Geom.23, 175–196 (1986)

    MATH  MathSciNet  Google Scholar 

  2. S. Altschuler, S.B. Angenent, Y. Giga: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Analysis5, 293–358 (1995)

    MATH  MathSciNet  Google Scholar 

  3. S.B. Angenent, J.J.L. Velazquez: Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math.482, 15–66 (1997)

    MATH  MathSciNet  Google Scholar 

  4. R.S. Hamilton: Harnack estimate for the mean curvature flow. J. Differential Geom.41, 215–226 (1995)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  6. G. Huisken: Asymptotic behaviour for singularities of the mean curvature flow. J. Differential Geom.31, 285–299 (1990)

    MATH  MathSciNet  Google Scholar 

  7. G. Huisken: Local and global behaviour of hypersurfaces moving by mean curvature.Proceedings of Symposia in Pure Mathematics 54, 175–191 (1996)

    MathSciNet  Google Scholar 

  8. G. Huisken, C. Sinestrari:Mean curvature flow singularities for mean convex surfaces. Prepr. (1997)

  9. T. Ilmanen:Singularities of mean curvature flow of surfaces. Preprint, Northwestern University

  10. K. Smoczyk: Symmetric hypersurfaces in Riemannian manifolds contracting to Lie groups by their mean curvature. Calc. Var.4, 155–170 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. B. White:Partial regularity of mean convex hypersurfaces flowing by mean curvature. Prepr. Stanford University (1997)

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Authors and Affiliations

  1. Mathematics Department, ETH Zürich, HG E 18.2, CH-8092, Zürich, Switzerland

    Knut Smoczyk

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  1. Knut Smoczyk
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Smoczyk, K. Starshaped hypersurfaces and the mean curvature flow. Manuscripta Math 95, 225–236 (1998). https://doi.org/10.1007/BF02678027

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  • DOI: https://doi.org/10.1007/BF02678027

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