Skip to main content
SpringerLink
Log in

Uniqueness of convex ancient solutions to mean curvature flow in \({\mathbb {R}}^3\)

  • Published:
Inventiones mathematicae Aims and scope

Abstract

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are \(\kappa \)-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in \({\mathbb {R}}^3\), and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in \({\mathbb {R}}^3\) which is strictly convex and noncollapsed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Altschuler, S., Wu, L.F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. 2, 101–111 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angenent, S., Daskalopoulos, P., Šešum, N.: Unique asymptotics of ancient convex mean curvature flow solutions. arXiv:1503.01178v3

  3. Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brendle, S.: A sharp bound for the inscribed radius under mean curvature flow. Invent. Math. 202, 217–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brendle, S., Choi, K.: Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. arXiv:1804.00018

  6. Brendle, S., Huisken, G.: Mean curvature flow with surgery of mean convex surfaces in \({\mathbb{R}}^3\). Invent. Math. 203, 615–654 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colding, T.H., Minicozzi, W.P.: Uniqueness of blowups and Lojasiewicz inequalities. Ann. Math. 182, 221–285 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Daskalopoulos, P., Hamilton, R., Šešum, N.: Classification of ancient compact solutions to the Ricci flow on surfaces. J. Differ. Geom. 91, 171–214 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Haslhofer, R.: Uniqueness of the bowl soliton. Geom. Topol. 19, 2393–2406 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. 70, 511–546 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haslhofer, R., Kleiner, B.: Mean curvature flow with surgery. Duke Math. J. 166, 1591–1626 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. 8, 1–14 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huisken, G., Sinestrari, C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175, 137–221 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huisken, G., Sinestrari, C.: Convex ancient solutions of the mean curvature flow. J. Differ. Geom. 101, 267–287 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kleene, S., Møller, N.M.: Self-shrinkers with a rotational symmetry. Trans. Am. Math. Soc. 366, 3943–3963 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Merle, F., Zaag, H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Commun. Pure Appl. Math. 51, 139–196 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159

  23. Sheng, W., Wang, X.J.: Singularity profile in the mean curvature flow. Methods Appl. Anal. 16, 139–155 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. White, B.: The size of the singular set in mean curvature flow of mean convex sets. J. Am. Math. Soc. 13, 665–695 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. White, B.: The nature of singularities in mean curvature flow of mean convex sets. J. Am. Math. Soc. 16, 123–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Simon Brendle

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02138, USA

    Kyeongsu Choi

Authors
  1. Simon Brendle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kyeongsu Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simon Brendle.

Additional information

Publisher's Note

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

The first author was supported by the National Science Foundation under Grant DMS-1649174 and by the Simons Foundation. The second author was supported by the National Science Foundation under Grant DMS-1811267. The authors gratefully acknowledge the hospitality of Tübingen University, where part of this work was carried out.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brendle, S., Choi, K. Uniqueness of convex ancient solutions to mean curvature flow in \({\mathbb {R}}^3\). Invent. math. 217, 35–76 (2019). https://doi.org/10.1007/s00222-019-00859-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-019-00859-4

Search

Navigation