Skip to main content
Log in

The Björling problem for prescribed mean curvature surfaces in \(\mathbb {R}^3\)

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In this paper we prove existence and uniqueness of the Björling problem for the class of immersed surfaces in \(\mathbb {R}^3\) whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in \(\mathbb {R}^3\).

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Aledo, J.A., Chaves, R.M.B., Gálvez, J.A.: The Cauchy problem for improper affine spheres and the Hessian one equation. Trans. Amer. Math. Soc. 359, 4183–4208 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large, I. Vestnik Leningrad Univ. 11 (1956), 5–17. (English translation): Amer. Math. Soc. Transl. 21 (1962), 341–354

  3. Alías, L.J., Mira, P.: A Schwarz-type formula for minimal surfaces in Euclidean space \(\mathbb{R}\). C. R. Acad. Sci. Paris Ser. I 334, 389–394 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Björling, E.G.: In integrazionem aequationis derivatarum partialum superfici cujus inpuncto uniquoque principales ambos radii curvedinis aequales sunt sngoque contrario. Arch. Math. Phys. 4(1), 290–315 (1844)

    Google Scholar 

  5. Brander, D., Dorfmeister, J.F.: The Björling problem for non-minimal constant mean curvature surfaces. Comm. Anal. Geom. 18, 171–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bueno, A., Gálvez, J.A., Mira, P.: The global geometry of surfaces with prescribed mean curvature in \(\mathbb{R}^3\). Preprint arXiv:1802.08146

  7. Christoffel, E.B.: Über die Bestimmung der Gestalt einer krummen Oberfläche durch lokale Messungen auf derselben. J. Reine Angew. Math. 64, 193–209 (1865)

    Article  MathSciNet  Google Scholar 

  8. Cintra, A.A., Mercuri, F., Onnis, I.: The Björling problem for minimal surfaces in a Lorentzian three-dimensional Lie group. Ann. Mat. 195, 95–110 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clutterbuck, J., Schnurer, O., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differential Equations 29, 281–293 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dierkes, U., Hildebrant, S., Küster, A., Wohlrab, O.: Minimal Surfaces I. A Series of Comprehensive Studies in Mathematics, vol. 295. Springer, Berlin (1992)

    Google Scholar 

  11. Gálvez, J.A., Mira, P.: Dense solutions to the Cauchy problem for minimal surfaces. Bull. Braz. Math. Soc. 35, 387–394 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gálvez, J.A., Mira, P.: The Cauchy problem for the Liouville equation and Bryant surfaces. Adv. Math. 195, 456–490 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gálvez, J.A., Mira, P.: Embedded isolated singularities of flat surfaces in hyperbolic 3-space. Calc. Var. Partial Differential Equations 24, 239–260 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gálvez, J.A., Mira, P.: A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov. Math. Ann. 366, 909–928 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)

    MathSciNet  MATH  Google Scholar 

  16. López, R., Webber, M.: Explicit Björling surfaces with prescribed geometry. Michigan Math. J. 67, 561–584 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. Partial Differential Equations 54, 2853–2882 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meeks III, W.H.: The classification of complete minimal surfaces in \({\mathbb{R}}^3\) with total curvature greater than \(-8\pi \). Duke Math. J. 48, 523–535 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Meeks III, W.H., Weber, M.: Bending the helicoid. Math. Ann. 339, 783–798 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mercuri, F., Montaldo, S., Piu, P.: A Weierstrass representation formula of minimal surfaces in \(\mathbb{H}^3\) and \(\mathbb{H}^2\times \mathbb{R}\). Acta Math. Sinica 22, 1603–1612 (2006)

    Article  MATH  Google Scholar 

  21. Mercuri, F., Onnis, I.: On the Björling problem in a three-dimensional Lie group. Illinois J. Math. 53, 431–440 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mira, P.: Complete minimal Möbius strips in \({\mathbb{R}}^n\) and the Björling problem. J. Geom. Phys. 56, 1506–1515 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Petrovsky, I.G.: Lectures on Partial Differential Equations. Interscience Publishers, New York (1954)

    MATH  Google Scholar 

  24. Pogorelov, A.V.: Extension of a general uniqueness theorem of A.D. Aleksandrov to the case of nonanalytic surfaces. Dokl. Akad. Nauk SSSR 62, 297–299 (1948). (in Russian)

    MathSciNet  Google Scholar 

  25. Schwarz, H.A.: Gesammelte mathematische abhandlungen. Band I. Springer, Berlin (1890)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P, Junta de Andalucía Grant No. FQM325 and FPI-MINECO Grant No. BES-2014-067663. The author wants to express his gratitude to his Ph.D. advisor Pablo Mira, for fruitful conversations about this topic.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Bueno.

Additional information

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

Bueno, A. The Björling problem for prescribed mean curvature surfaces in \(\mathbb {R}^3\). Ann Glob Anal Geom 56, 87–96 (2019). https://doi.org/10.1007/s10455-019-09657-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-019-09657-w

Keywords

Mathematics Subject Classification

Navigation