Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature

  • Knut Smoczyk


This paper concerns the deformation by mean curvature of hypersurfaces M in Riemannian spaces Ñ that are invariant under a subgroup of the isometry-group on Ñ. We show that the hypersurfaces contract to this subgroup, if the cross-section satisfies a strong convexity assumption.

Mathematics subject classification (1991)

53C45 53C42 35B40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ang]
    Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equation And Their Equilibrium States 3 N.G. LLoyd, W.-M. Ni, L.A. Peletier, J. Serrin (eds.) Birkhäuser, Boston 1992 pp. 21–38Google Scholar
  2. [AAG]
    Altschuler, S.J., Angenent, S.B., Giga, Y.: Mean curvature flow through singularities for surfaces of rotation (Preprint)Google Scholar
  3. [CGG]
    Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Diff. Geom.33, 749–786 (1991)Google Scholar
  4. [DK]
    Dziuk, G., Kawohl, B.: On rotationally symmetric mean curvature flow. J. Differ. Eq.93, 142–149 (1991)Google Scholar
  5. [ES]
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom.33, 635–681 (1991)Google Scholar
  6. [GHL]
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Universitext, Springer (1990)Google Scholar
  7. [Ha]
    Hamilton, R.S. Three-manifolds with positive Ricci curvature. J. Differ. Geom.17, 256–306 (1982)Google Scholar
  8. [H1]
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom.20, 237–268 (1984)Google Scholar
  9. [H2]
    Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math.84, 463–480 (1986)Google Scholar
  10. [H3]
    Huisken, G. Asymptotic behaviour for singularities of the mean curvature flow. J. Differ. Geom.31, 285–299 (1991)Google Scholar
  11. [Ilm1]
    Ilmanen, T. The level-set flow on a manifold. Proceedings of Symposia in Pure Mathematics 54 (1), 193–204 (1993)Google Scholar
  12. [Ilm2]
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Memoirs AMSGoogle Scholar
  13. [I]
    Ishimura, N.: Limit shape of the cross-section of shrinking doughnuts. J. Math. Soc. Japan45-3, 569–582 (1993)Google Scholar
  14. [J]
    Jost, J.: A weak notion for mean curvature and a generalized mean curvature flow for singular sets (Preprint, Ruhr-University Bochum 1994)Google Scholar
  15. [LS]
    Luckhaus, S., Sturzenhecker, T.: Implizit time discretization for the mean curvature flow equation. Calc. Var.3, 253–271 (1995)Google Scholar
  16. [O'N]
    O'Neill, B.: The fundamental equations of a submersion. Michigan Math. J.13, 459–469 (1966)Google Scholar
  17. [S1]
    Smoczyk, K.: The evolution of special hypersurfaces by their mean curvature (Preprint, Ruhr-University Bochum 1993)Google Scholar
  18. [S2]
    Smoczyk, K.: The symmetric “doughnut” evolving by its mean curvature. Hokkaido Math. J.23, 523–547 (1994)Google Scholar
  19. [SS]
    Soner, G., Souganidis, P.E.: Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature (Preprint 1992)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Knut Smoczyk
    • 1
  1. 1.Department of MathematicsRuhr-UniversityBochumGermany

Personalised recommendations