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Stability of translating solutions to mean curvature flow

  • Julie ClutterbuckEmail author
  • Oliver C. Schnürer
  • Felix Schulze
Article

Abstract

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

Keywords

Stability Mean curvature flow Translating solutions 

Mathematics Subject Classification (2000)

53C44 35B35 

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References

  1. 1.
    Altschuler S.J., Wu L.F. (1994) Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differential Equations 2(1): 101–111zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andrews, B., Clutterbuck, J.: Time-interior gradient estimates for graphical anisotropic mean curvature flows, forthcomingGoogle Scholar
  3. 3.
    Angenent S.B., Velázquez J.J.L. (1997) Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. 482, 15–66zbMATHMathSciNetGoogle Scholar
  4. 4.
    Barles G., Biton S., Bourgoing M., Ley O. (2003) Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differential Equations 18(2): 159–179zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cao H.-D., Shen Y., Zhu S. (1997) The structure of stable minimal hypersurfaces in R n+1. Math. Res. Lett. 4(5): 637–644zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chau A., Schnürer O.C. (2005) Stability of gradient Kähler-Ricci solitons. Commun. Anal. Geom. 13(4): 769–800zbMATHGoogle Scholar
  7. 7.
    Clutterbuck, J.: Parabolic equations with continuous initial data. Ph.D. Thesis, Australian National University, 2004Google Scholar
  8. 8.
    Colding T.H., Minicozzi W.P., II (2004) Sharp estimates for mean curvature flow of graphs. J. Reine Angew. Math. 574, 187–195zbMATHMathSciNetGoogle Scholar
  9. 9.
    Ecker K., Huisken G. (1991) Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105(3): 547–569zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics, Springer, Berlin Heidelberg New York, (2001), Reprint of the 1998 editionGoogle Scholar
  11. 11.
    Hamilton, R.S.: Eternal solutions to the mean curvature flow. (unpublished)Google Scholar
  12. 12.
    Huisken G., Sinestrari C. (1999) Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations 8(1): 1–14zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hungerbühler N., Smoczyk K. (2000) Soliton solutions for the mean curvature flow. Differential Integral Equations 13(10–12): 1321–1345zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kaabachi, S., Pacard, F.: Riemann minimal surfaces in higher dimensions. J. Inst. Math. Jussieu. (to appear) (2007)Google Scholar
  15. 15.
    Smoczyk K. (2001) A relation between mean curvature flow solitons and minimal submanifolds. Math. Nachr. 229, 175–186zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wang, X.-J.: Convex solutions to the mean curvature flow. arXiv:math.DG/0404326Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Julie Clutterbuck
    • 1
    Email author
  • Oliver C. Schnürer
    • 1
  • Felix Schulze
    • 1
  1. 1.Freie Universität BerlinBerlinGermany

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