Advertisement

manuscripta mathematica

, Volume 95, Issue 1, pp 225–236 | Cite as

Starshaped hypersurfaces and the mean curvature flow

  • Knut Smoczyk
Article
  • 338 Downloads

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.

Mathematics Subject Classification (1991)

53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    U. Abresch, J. Langer: The normalized curve shortening flow and homothetic solutions. J. Differential Geom.23, 175–196 (1986)zbMATHMathSciNetGoogle Scholar
  2. 2.
    S. Altschuler, S.B. Angenent, Y. Giga: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Analysis5, 293–358 (1995)zbMATHMathSciNetGoogle Scholar
  3. 3.
    S.B. Angenent, J.J.L. Velazquez: Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math.482, 15–66 (1997)zbMATHMathSciNetGoogle Scholar
  4. 4.
    R.S. Hamilton: Harnack estimate for the mean curvature flow. J. Differential Geom.41, 215–226 (1995)zbMATHMathSciNetGoogle Scholar
  5. 5.
    G. Huisken: Flow by mean curvature of convex surfaces into spheres. J. Differential Geom.20, 237–266 (1984)zbMATHMathSciNetGoogle Scholar
  6. 6.
    G. Huisken: Asymptotic behaviour for singularities of the mean curvature flow. J. Differential Geom.31, 285–299 (1990)zbMATHMathSciNetGoogle Scholar
  7. 7.
    G. Huisken: Local and global behaviour of hypersurfaces moving by mean curvature.Proceedings of Symposia in Pure Mathematics 54, 175–191 (1996)MathSciNetGoogle Scholar
  8. 8.
    G. Huisken, C. Sinestrari:Mean curvature flow singularities for mean convex surfaces. Prepr. (1997)Google Scholar
  9. 9.
    T. Ilmanen:Singularities of mean curvature flow of surfaces. Preprint, Northwestern UniversityGoogle Scholar
  10. 10.
    K. Smoczyk: Symmetric hypersurfaces in Riemannian manifolds contracting to Lie groups by their mean curvature. Calc. Var.4, 155–170 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. White:Partial regularity of mean convex hypersurfaces flowing by mean curvature. Prepr. Stanford University (1997)Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Knut Smoczyk
    • 1
  1. 1.Mathematics DepartmentETH Zürich, HG E 18.2ZürichSwitzerland

Personalised recommendations