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Inventiones mathematicae

, Volume 175, Issue 1, pp 137–221 | Cite as

Mean curvature flow with surgeries of two–convex hypersurfaces

  • Gerhard Huisken
  • Carlo Sinestrari
Open Access
Article

Keywords

Fundamental Form Extrinsic Curvature Convexity Estimate Normal Neck Singular Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Altschuler, S., Angenent, S.B., Giga, Y.: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5, 293–358 (1995)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Angenent, S.B., Velazquez, J.J.L.: Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. 482, 15–66 (1997)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  4. 4.
    Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation. J. Differ. Geom. 33, 749–786 (1991)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Colding, T., Kleiner, B.: Singularity structure in mean curvature flow of mean-convex sets. Electron. Res. Announc. Am. Math. Soc. 9, 121–124 (2003) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ecker, K.: Regularity Theory for Mean Curvature Flow. Birkhäuser, Boston (2004)zbMATHGoogle Scholar
  7. 7.
    Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature, I. J. Differ. Geom. 33, 635–681 (1991)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Fraser, A.M.: Minimal disks and two-convex hypersurfaces. Am. J. Math. 124, 483–493 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hamilton, R.S.: Four–manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hamilton, R.S.: Convex hypersurfaces with pinched second fundamental form. Commun. Anal. Geom. 2, 167–172 (1994)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hamilton, R.S.: The Harnack estimate for the mean curvature flow. J. Differ. Geom. 41, 215–226 (1995)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hamilton, R.S.: Formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995)MathSciNetGoogle Scholar
  14. 14.
    Hamilton, R.S.: Four–manifolds with positive isotropic curvature. Commun. Anal. Geom. 5, 1–92 (1997)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hirsch, M.W.: Differential Topology. Grad. Texts Math., vol. 33. Springer, Berlin (1976)zbMATHGoogle Scholar
  16. 16.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. Proc. Symp. Pure Math. 54, 175–191 (1993)MathSciNetGoogle Scholar
  18. 18.
    Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differ. Equ. 8, 1–14 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego (1980)zbMATHGoogle Scholar
  21. 21.
    Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187–197 (1969)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Matsumoto, Y.: An introduction to Morse theory. Trans. Math. Monogr., vol. 208. American Math. Soc., Providence (2002)Google Scholar
  23. 23.
    Mercuri, F., Noronha, M.H.: Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature. Trans. Am. Math. Soc. 348, 2711–2724 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Michael, J.H., Simon, L.M.: Sobolev and mean value inequalities on generalized submanifolds of ℝn. Commun. Pure Appl. Math. 26, 361–379 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint (2002). arXiv:math/0211159v1Google Scholar
  26. 26.
    Perelman, G.: Ricci flow with surgery on three-manifolds. Preprint (2003). arXiv:math/0303109v1Google Scholar
  27. 27.
    Sha, J.-P.: p-convex Riemannian manifolds. Invent. Math. 83, 437–447 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    White, B.: The topology of hypersurfaces moving by mean curvature. Commun. Anal. Geometry 3, 317–333 (1995)zbMATHGoogle Scholar
  29. 29.
    White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16, 123–138 (2002)CrossRefGoogle Scholar
  30. 30.
    Wu, H.: Manifolds of partially positive curvature. Indian. Univ. Math. J. 36, 525–548 (1987)zbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Albert-Einstein-InstitutMax-Planck-Institut für GravitationsphysikGolmGermany
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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