The Dirichlet problem for constant mean curvature surfaces in Heisenberg space. (English) Zbl 1210.53010
Summary: 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.
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 35J60 | Nonlinear elliptic equations |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
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