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Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 87–96 | Cite as

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

  • Antonio BuenoEmail author
Article
  • 56 Downloads

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 \(\mathbb {R}^3\).

Keywords

Björling problem Prescribed mean curvature surfaces Möbius strip 

Mathematics Subject Classification

53A10 53C42 

Notes

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.

References

  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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), 341–354Google Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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. 5.
    Brander, D., Dorfmeister, J.F.: The Björling problem for non-minimal constant mean curvature surfaces. Comm. Anal. Geom. 18, 171–194 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 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)MathSciNetCrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clutterbuck, J., Schnurer, O., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differential Equations 29, 281–293 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 11.
    Gálvez, J.A., Mira, P.: Dense solutions to the Cauchy problem for minimal surfaces. Bull. Braz. Math. Soc. 35, 387–394 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gálvez, J.A., Mira, P.: The Cauchy problem for the Liouville equation and Bryant surfaces. Adv. Math. 195, 456–490 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)MathSciNetzbMATHGoogle Scholar
  16. 16.
    López, R., Webber, M.: Explicit Björling surfaces with prescribed geometry. Michigan Math. J. 67, 561–584 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meeks III, W.H., Weber, M.: Bending the helicoid. Math. Ann. 339, 783–798 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)CrossRefzbMATHGoogle Scholar
  21. 21.
    Mercuri, F., Onnis, I.: On the Björling problem in a three-dimensional Lie group. Illinois J. Math. 53, 431–440 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mira, P.: Complete minimal Möbius strips in \({\mathbb{R}}^n\) and the Björling problem. J. Geom. Phys. 56, 1506–1515 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Petrovsky, I.G.: Lectures on Partial Differential Equations. Interscience Publishers, New York (1954)zbMATHGoogle Scholar
  24. 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) MathSciNetGoogle Scholar
  25. 25.
    Schwarz, H.A.: Gesammelte mathematische abhandlungen. Band I. Springer, Berlin (1890)CrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

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