Skip to main content

Convex ancient solutions to curve shortening flow

Calculus of Variations and Partial Differential Equations volume 59, Article number: 133 (2020) Cite this article

Abstract

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Šešum and X.-J. Wang.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2

References

  1. Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23(2), 175–196 (1986)

    MathSciNet  Article  Google Scholar 

  2. Andrews, B.: Harnack inequalities for evolving hypersurfaces. Math. Z. 217(2), 179–197 (1994)

    MathSciNet  Article  Google Scholar 

  3. Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equations and their Equilibrium States, 3. Proceedings from the 3rd Conference, held 20–29 August 1989 in Gregynog, Wales, UK, pp. 21–38. Birkhäuser, Boston (1992)

  4. Bourni, T., Langford, M., Tinaglia, G.: Collapsing ancient solutions of mean curvature flow. J. Differ. Geom. Preprint arXiv:1705.06981

  5. Bourni, T., Langford, M., Tinaglia, G.: On the existence of translating solutions of mean curvature flow in slab regions. Anal. PDE 13(4), 1051–1072 (2020)

    MathSciNet  Article  Google Scholar 

  6. Bryan, P., Louie, J.: Classification of convex ancient solutions to curve shortening flow on the sphere. J. Geom. Anal. 26(2), 858–872 (2016)

    MathSciNet  Article  Google Scholar 

  7. Chow, B.: On Harnack’s inequality and entropy for the Gaussian curvature flow. Commun. Pure Appl. Math. 44(4), 469–483 (1991)

    MathSciNet  Article  Google Scholar 

  8. Chow, B.: Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations. Commun. Anal. Geom. 5(2), 389–409 (1997)

    MathSciNet  Article  Google Scholar 

  9. Chow, B., Gulliver, R.: Aleksandrov reflection and geometric evolution of hypersurfaces. Commun. Anal. Geom. 9(2), 261–280 (2001)

    MathSciNet  Article  Google Scholar 

  10. Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of compact ancient solutions to the curve shortening flow. J. Differ. Geom. 84(3), 455–464 (2010)

    MathSciNet  Article  Google Scholar 

  11. Epstein, C.L., Weinstein, M.I.: A stable manifold theorem for the curve shortening equation. Commun. Pure Appl. Math. 40(1), 119–139 (1987)

    MathSciNet  Article  Google Scholar 

  12. Gage, M., Hamilton, R.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)

    MathSciNet  Article  Google Scholar 

  13. Gage, M.E.: Curve shortening makes convex curves circular. Invent. Math. 76(2), 357–364 (1984)

    MathSciNet  Article  Google Scholar 

  14. Hamilton, R.S.: Harnack estimate for the mean curvature flow. J. Differ. Geom. 41(1), 215–226 (1995)

    MathSciNet  Article  Google Scholar 

  15. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)

    MathSciNet  Article  Google Scholar 

  16. Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to motion by curvature. Commun. Pure Appl. Math. 59(3), 344–407 (2006)

    MathSciNet  Article  Google Scholar 

  17. Mullins, W.W.: Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27, 900–904 (1956)

    MathSciNet  Article  Google Scholar 

  18. Wang, X.-J.: Convex solutions to the mean curvature flow. Ann. Math. (2) 173(3), 1185–1239 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tennessee Knoxville, Knoxville, TN, 37996-1320, USA

    Theodora Bourni & Mat Langford

  2. Department of Mathematics, King’s College London, London, WC2R 2LS, UK

    Giuseppe Tinaglia

  3. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, NSW, 2308, Australia

    Mat Langford

Authors
  1. Theodora Bourni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mat Langford
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giuseppe Tinaglia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Theodora Bourni.

Additional information

Communicated by N. Trudinger.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bourni, T., Langford, M. & Tinaglia, G. Convex ancient solutions to curve shortening flow. Calc. Var. 59, 133 (2020). https://doi.org/10.1007/s00526-020-01784-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-020-01784-8

Mathematics Subject Classification

  • 53
  • 35