Skip to main content
SpringerLink
Log in

On an inverse curvature flow in two-dimensional space forms

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the \(n=2\) case of a conjecture of Girão–Pinheiro.

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

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Agostiniani, V., Fogagnolo, M., Mazzieri, L.: Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math. 222(3), 1033–1101 (2020)

    Article  MathSciNet  Google Scholar 

  2. Andrews, B.: Evolving convex curves. Calc. Var. Partial. Differ. Equ. 7(4), 315–371 (1998)

    Article  MathSciNet  Google Scholar 

  3. Andrews, B., Bryan, P.: Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson’s theorem. J. Reine Angew. Math. 2011(653), 179–187 (2011)

    Article  MathSciNet  Google Scholar 

  4. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1987)

    MATH  Google Scholar 

  5. Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publications Mathématiques de l’IHÉS 117(1), 247–269 (2013)

    Article  MathSciNet  Google Scholar 

  6. Brendle, S., Guan, P., Li, J.: An inverse curvature type hypersurface flow in space forms. Preprint (2020)

  7. Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the anti-de Sitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)

    Article  MathSciNet  Google Scholar 

  8. Chen, C., Guan, P., Li, J., Scheuer, J.: A fully-nonlinear flow and quermassintegral inequalities in the sphere. arXiv preprint arXiv:2102.07393 (2021). To appear in Pure and Applied Mathematics Quarterly

  9. Chen, M., Sun, J.: Alexandrov–Fenchel type inequalities in the sphere. arXiv preprint arXiv:2101.09419 (2021)

  10. de Lima, L.L., Girão, F.: An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality. Ann. Henri Poincaré 17(4), 979–1002 (2016)

    Article  MathSciNet  Google Scholar 

  11. Ge, Y., Wang, G., Wu, J.: Hyperbolic Alexandrov–Fenchel quermassintegral inequalities II. J. Differ. Geometry 98(2), 237–260 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gerhardt, C.: Curvature Problems, vol. 39. International Press, Somerville (2006)

    MATH  Google Scholar 

  13. Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geometry 32(1), 299–314 (1990)

    Article  MathSciNet  Google Scholar 

  14. Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geometry 89(3), 487–527 (2011)

    Article  MathSciNet  Google Scholar 

  15. Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geometry 100(2), 301–347 (2015)

    Article  MathSciNet  Google Scholar 

  16. Girão, F., Pinheiro, N.M.: An Alexandrov–Fenchel-type inequality for hypersurfaces in the sphere. Ann. Glob. Anal. Geometry 52(4), 413–424 (2017)

    Article  MathSciNet  Google Scholar 

  17. Guan, P., Li, J.: The quermassintegral inequalities for \(k\)-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)

    Article  MathSciNet  Google Scholar 

  18. Guan, P., Li, J.: A mean curvature type flow in space forms. Int. Math. Res. Not. IMRN 13, 4716–4740 (2015)

    Article  MathSciNet  Google Scholar 

  19. Guan, P., Li, J.: Isoperimetric type inequalities and hypersurface flows. J. Math. Study 54(1), 56–80 (2021)

    Article  MathSciNet  Google Scholar 

  20. Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geometry 24(2), 153–179 (1986)

    Article  MathSciNet  Google Scholar 

  21. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press (1952)

    MATH  Google Scholar 

  22. Hu, Y., Li, H.: Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space. Calc. Var. Partial Differ. Equ. 58(2), 1–20 (2019)

    Article  MathSciNet  Google Scholar 

  23. Hu, Y., Li, H., Wei, Y.: Locally constrained curvature flows and geometric inequalities in hyperbolic space. Mathematische Annalen (2020). https://doi.org/10.1007/s00208-020-02076-4

    Article  MATH  Google Scholar 

  24. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geometry 59(3), 353–437 (2001)

    Article  MathSciNet  Google Scholar 

  25. Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geometry 80(3), 433–451 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Kröner, H.: A note on expansion of convex plane curves via inverse curvature flow. Nonlinear Differ. Equ. Appl. 26(2), 1–11 (2019)

    Article  MathSciNet  Google Scholar 

  27. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order, vol. 7. Springer (1987)

    Book  Google Scholar 

  28. Kwong, K.-K., Miao, P.: A new monotone quantity along the inverse mean curvature flow in \(\mathbb{R}^n\). Pac. J. Math. 267(2), 417–422 (2014)

    Article  Google Scholar 

  29. Kwong, K.-K., Miao, P.: Monotone quantities involving a weighted \(\sigma _k\) integral along inverse curvature flows. Commun. Contemp. Math. 17(05), 1550014 (2015)

    Article  MathSciNet  Google Scholar 

  30. Li, H., Wei, Y., Xiong, C.: A geometric inequality on hypersurface in hyperbolic space. Adv. Math. 253, 152–162 (2014)

    Article  MathSciNet  Google Scholar 

  31. Makowski, M., Scheuer, J.: Rigidity results, inverse curvature flows and Alexandrov–Fenchel type inequalities in the sphere. Asian J. Math. 20(5), 869 (2016)

    Article  MathSciNet  Google Scholar 

  32. Nirenberg, L.: An extended interpolation inequality. Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze 20(4), 733–737 (1966)

    MathSciNet  MATH  Google Scholar 

  33. Qiu, G., Xia, C.: A generalization of Reilly’s formula and its applications to a new Heintze–Karcher type inequality. Int. Math. Res. Not. 2015(17), 7608–7619 (2015)

    Article  MathSciNet  Google Scholar 

  34. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Revista Matemática Iberoamericana 3(3), 447–453 (1987)

    Article  MathSciNet  Google Scholar 

  35. Scheuer, J.: Minkowski inequalities and constrained inverse curvature flows in warped spaces. Adv. Calculus Var. (2020). https://doi.org/10.1515/acv-2020-0050. (ahead-of-print)

    Article  Google Scholar 

  36. Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(1), 355–372 (1990)

    Article  MathSciNet  Google Scholar 

  37. Wei, Y., Xiong, C.: Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere. Pac. J. Math. 277(1), 219–239 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Kwok-Kun Kwong would like to thank Man-Chun Lee for useful discussions. Yong Wei was supported by the National Key R and D Program of China 2020YFA0713100 and Research Grant KY0010000052 from University of Science and Technology of China. Valentina-Mira Wheeler was supported in part by Discovery Project DP180100431 and DECRA DE190100379 of the Australian Research Council. We would also like to thank the referees for their comments and useful advice on our paper.

Author information

Authors and Affiliations

  1. University of Wollongong, Northfields Ave, Wollongong, NSW, 2522, Australia

    Kwok-Kun Kwong, Glen Wheeler & Valentina-Mira Wheeler

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Yong Wei

Authors
  1. Kwok-Kun Kwong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Glen Wheeler
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Valentina-Mira Wheeler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kwok-Kun Kwong.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by Giga.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kwong, KK., Wei, Y., Wheeler, G. et al. On an inverse curvature flow in two-dimensional space forms. Math. Ann. 384, 1–24 (2022). https://doi.org/10.1007/s00208-021-02285-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02285-5

Mathematics Subject Classification

Search

Navigation