The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

  • Luis J. Alías
  • Marcos Dajczer
  • Harold Rosenberg
Original Article

Abstract

We study constant mean curvature graphs in the Riemannian three- dimensional Heisenberg spaces \({\mathcal{H} = \mathcal{H}(\tau)}\) . Each such \({\mathcal{H}}\) is the total space of a Riemannian submersion onto the Euclidean plane \({\mathbb{R}^2}\) with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in \({\mathcal{H}}\) with respect to the Riemannian submersion over certain domains \({\Omega \subset \mathbb{R}^2}\) taking on prescribed boundary values.

Mathematics Subject Classification (2000)

35J60 53C42 

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References

  1. 1.
    Abresch U., Rosenberg H. (2004). A Hopf differential for constant mean curvature surfaces in \({S^2 \times \mathbb{R}}\) and \({H^2 \times \mathbb{R}}\). Acta Math. 193: 141– 174MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abresch U. and Rosenberg H. (2005). Generalized Hopf differentials. Mat. Contemp. 28: 1–28 MATHMathSciNetGoogle Scholar
  3. 3.
    Caffarelli L., Nirenberg L. and Spruck J. (1988). Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten hypersurfaces. Commun. Pure Appl. Math. 41: 47–70 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Daniel, B.: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. (to appear)Google Scholar
  5. 5.
    Figueroa C., Mercuri F. and Pedrosa R. (1999). Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl. 177: 173–194 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gilbarg D. and Trudinger N. (2001). Elliptic Partial Differential Equations of Second Order. Springer, Berlin MATHGoogle Scholar
  7. 7.
    Rosenberg H. (2002). Minimal surfaces in \({\mathbb{M}^2 \times \mathbb{R}}\). Illinois J Math. 46: 1177– 1195MATHMathSciNetGoogle Scholar
  8. 8.
    Rosenberg H. (2006). Constant mean curvature surfaces in homogeneously regular 3-manifolds. Bull. Aust. Math. Soc. 74: 227–238 MATHCrossRefGoogle Scholar
  9. 9.
    Serrin J. (1969). The problem of Dirichlet for quasilinear elliptic equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264: 413–496 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Luis J. Alías
    • 1
  • Marcos Dajczer
    • 2
  • Harold Rosenberg
    • 3
  1. 1.Departamento de MatematicasUniversidad de MurciaEspinardo, MurciaSpain
  2. 2.IMPARio de JaneiroBrazil
  3. 3.Département de MathématiquesUniversité de Paris VIIParisFrance

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