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Half-space theorems for properly immersed surfaces in \({\mathbb {R}}^{3}\) with prescribed mean curvature

  • Antonio BuenoEmail author
Article
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Abstract

Motivated by the large amount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in \({\mathbb {R}}^3\) whose mean curvature is given as a prescribed function of its Gauss map. In order to achieve this purpose, we will study the behavior at infinity of a one-parameter family of properly embedded annuli that are analogous to the usual minimal catenoids.

Keywords

Prescribed mean curvature Half-space theorem Asymptotic behavior 

Mathematics Subject Classification

53A10 53C42 34A26 

Notes

References

  1. 1.
    Alexandrov, A.D.: Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad Univ. 11, 5–17 (1956). (English translation): Am. Math. Soc. Transl. 21 (1962), 341–354Google Scholar
  2. 2.
    Bueno, A., Gálvez, J.A., Mira, P.: Rotational hypersurfaces of prescribed mean curvature, preprint. arxiv:1902.09405
  3. 3.
    Bueno, A., Gálvez, J.A., Mira, P.: The global geometry of surfaces with prescribed mean curvature in \({{\mathbb{R}}}^3\), preprint. arxiv:1802.08146
  4. 4.
    Bueno, A.: The Björling problem for prescribed mean curvature surfaces in \({\mathbb{R}}^3\), Ann. Glob. Ann. Geom.  https://doi.org/10.1007/s10455-019-09657-w
  5. 5.
    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
  6. 6.
    Daniel, B., Hauswirth, L.: Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. (3) 98(2), 445–470 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daniel, B., Meeks III, W.H., Rosenberg, H.: Half-space theorems for minimal surfaces in \({{\rm Nil}}_3\) and \({{\rm Sol}}_3\). J. Differ. Geom. 88(1), 41–59 (2011)CrossRefGoogle Scholar
  8. 8.
    Hauswirth, L., Rosenberg, H., Spruck, J.: On complete mean curvature \(1/2\) surfaces in \({\mathbb{H}}^2\times {{\mathbb{R}}}\). Commun. Anal. Geom. 16(5), 989–1005 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hoffman, D., Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101(2), 373–377 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pogorelov, A.V.: Extension of a general uniqueness theorem of A.D. Aleksandrov to the case of nonanalytic surfaces. Doklady Akad. Nauk SSSR 62, 297–299 (1948). (in Russian)MathSciNetGoogle Scholar
  11. 11.
    Rodriguez, L., Rosenberg, H.: Half-space theorems for mean curvature one surfaces in hyperbolic space. Proc. Am. Math. Soc. 126(9), 2755–2762 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

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