Annals of Global Analysis and Geometry

, Volume 34, Issue 1, pp 39–53 | Cite as

Minimal graphs in \({M \times \mathbb{R}}\)

  • Maria Fernanda ElbertEmail author
  • Harold Rosenberg
Original Paper


We study minimal graphs in \({M \times \mathbb{R}}\) . First, we establish some relations between the geometry of the domain and the existence of certain minimal graphs. We then discuss the problem of finding the maximal number of disjoint domains Ω ⊂ M that admit a minimal graph that vanishes on ∂Ω. When M is two-dimensional and has non-negative sectional curvature, we prove that this number is 3. This was proved by Tkachev in \({\mathbb{R}^3}\) .


Minimal surface Minimal graph Unbounded domain 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Collin, P., Krust, R.: Le Probleme de Dirichlet pour L’Equation des Surfaces Minimales sur des Domaines non Bornes. Bull. Soc. Math. France 119, 443–462 (1991)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Li, P., Wang, J.: Finiteness of disjoint minimal graphs. Math. Res. Lett. 8, 771–777 (2001)zbMATHGoogle Scholar
  3. 3.
    Meeks, W.H., Rosenberg, H.: The uniqueness of the helicoid. Ann. Math. 161(2), 723–754 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Rosenberg, H.: Minimal surfaces in \({M^{2} \times \mathbb{R}}\) . Illinois J. Math. 46(4), 1177–1195 (2002)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Schoen, R.: Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds, vol. 103. Ann. of Math. Studies, Princeton Univ. Press (1983)Google Scholar
  6. 6.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry, vol. I. International Press Incorporated (1994)Google Scholar
  7. 7.
    Spruck, J.: Two dimensional minimal graphs over unbounded domains. J. Inst. Math. Jussieu 1(4), 631–640 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Tkachev, V.: Disjoint minimal graphs. Ann. Glob. Anal. Geom. (To appear)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Instituto de MatematicaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Institut de MathématiquesUniversité Paris VIIParisFrance

Personalised recommendations