The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

Abstract

We study constant mean curvature graphs in the Riemannian 3-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.