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


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)



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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

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