Advertisement

Mathematische Annalen

, 342:309 | Cite as

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in \({\mathbb H^2 \times \mathbb R}\)

  • R. Sa EarpEmail author
  • E. Toubiana
Article

Abstract

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in \({\mathbb H^2 \times \mathbb R}\). As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ is a Jordan curve homologous to zero in \({\partial_\infty\mathbb H^2\times \mathbb R}\) such that Γ is contained in a slab between two horizontal circles of \({\partial_\infty\mathbb H^2\times \mathbb R}\) with width equal to π. We construct vertical minimal graphs in \({\mathbb H^2\times \mathbb R}\) over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in \({\mathbb H^2\times \{0\}}\) are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.

Mathematics Subject Classification (2000)

53C42 

References

  1. 1.
    Anderson M.: Complete minimal varieties in hyperbolic space. Invent. Math. 69, 477–494 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson M.: Complete minimal hypersurfaces in hyperbolic n-manifolds. Comment. Math. Helv. 58, 264–290 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Collin, P., Rosenberg, H.: Construction of harmonic diffeomorfisms and minimal graphs (preprint)Google Scholar
  4. 4.
    Courant R., Hilbert D.: Methods of Mathematical Physics, vol. 2. Inter-Science, New York (1962)Google Scholar
  5. 5.
    Daniel, B.: Isometric immersions into \({\mathbb{S} ^n \times \mathbb R}\) and \({\mathbb H^n \times \mathbb R}\) and applications to minimal surfaces. Trans. Am. Math. Soc. (to appear)Google Scholar
  6. 6.
    Fernández I., Mira P.: Harmonic maps and constant mean curvature surfaces in \({\mathbb H^2\times\mathbb R}\). Am. J. Math. 129(4), 1145–1181 (2007)zbMATHCrossRefGoogle Scholar
  7. 7.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg (1983)zbMATHGoogle Scholar
  8. 8.
    Han Z.-C., Tam L.-T., Treibergs A., Wan T.: Harmonic maps from the complex plane into surfaces with nonpositive curvature. Commun. Anal. Geom. 3(1), 85–114 (1995)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Hauswirth L.: Minimal surfaces of Riemann type in three-dimensional product manifolds. Pac. J. Math. 224, 91–117 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hauswirth, L., Sa Earp, R., Toubiana, E.: Associate and conjugate minimal immersions in M 2 ×  R. Tohoku Math. J. 60, 267–286 (2008)Google Scholar
  11. 11.
    Hauswirth, L., Rosenberg, H., Spruck, J.: Infinite boundary value problems for constant mean curvature graphs in \({\mathbb H^2 \times \mathbb R}\) and \({ \mathbb{S}^2 \times \mathbb R}\). Am. J. Math. (to appear). http://www.math.jhu.edu/~js/jsfinal.pdf
  12. 12.
    Morrey C.B.: The problem of plateau on a Riemannian manifold. Ann. Math. No. 2(49), 807–851 (1948)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Nelli B., Rosenberg H.: Minimal surfaces in H 2 × R. Bull. Braz. Math. Soc. 33, 263–292 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nelli B., Sa Earp R., Santos W., Toubiana E.: Uniqueness of H-surfaces in \({\mathbb H^2 \times \mathbb R}\), \({\vert H \vert \leq 1/2}\), with boundary one or two parallel horizontal circles. Ann. Glob. Anal. Geom. 33, 307–321 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Radó: Some remarks on the problem of Plateau. Proc. Natl. Acad. Sci. USA 16, 242–248 (1930)zbMATHCrossRefGoogle Scholar
  16. 16.
    Sa Earp, R.: Parabolic and hyperbolic screw motion in \({\mathbb H^2\times\mathbb R}\). J. Aust. Math. Soc. (to appear). http://www.mat.puc-rio.br/~earp/pscrew.pdf
  17. 17.
    Sa Earp R., Toubiana E.: Existence and uniqueness of minimal graphs in hyperbolic space. Asian J. Math. 4, 669–694 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Sa Earp R., Toubiana E.: Screw Motion Surfaces in \({\mathbb H^2\times\mathbb R}\) and \({\mathbb S^2\times\mathbb R}\). Ill. J. Math. 49(4), 1323–1362 (2005)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Simon L.: Equations of mean curvature type in 2 independent variables. Pac. J. Math. 69(1), 245–268 (1977)Google Scholar
  20. 20.
    Spruck, J.: Interior gradient estimates and existence theorems for constant mean curvature graphs in \({M^n\times\mathbb R}\). http://www.math.jhu.edu/~js/grad2.pdf
  21. 21.
    Tam L-F., Wan T.Y.: Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials. Commun. Anal. Geom. 2(4), 593–625 (1994)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticaPontifí cia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Institut de Mathématiques de JussieuUniversité Paris VII, Denis DiderotParis Cedex 05France

Personalised recommendations