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

  • Antonio BuenoEmail author


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.


Prescribed mean curvature Half-space theorem Asymptotic behavior 

Mathematics Subject Classification

53A10 53C42 34A26 



  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.
  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

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© 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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