The Dirichlet problem for CMC surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces H = H(τ). Each such H is the total space of a Rieman-nian submersion onto the Euclidean plane R2 with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in H with respect to the Riemannian submersion over certain domains Ω ⊂ R2 taking on prescribed boundary values.