Original Article

Calculus of Variations and Partial Differential Equations

, Volume 30, Issue 4, pp 513-522

First online:

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

  • Luis J. AlíasAffiliated withDepartamento de Matematicas, Universidad de Murcia
  • , Marcos DajczerAffiliated withIMPA
  • , Harold RosenbergAffiliated withDépartement de Mathématiques, Université de Paris VII Email author 

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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