The Dirichlet Problem in Hyperbolic Space

Abstract

We consider the Dirichlet problem for the constant mean curvature equation in hyperbolic space \(\mathbb{H}^{3}\). Due to the type of umbilical surfaces in \(\mathbb{H}^{3}\) as well as the different notions of graphs, there is a variety of problems of Dirichlet type. In this chapter we study geodesic graphs defined in a domain Ω of a horosphere, a geodesic plane and an equidistant surface. In order to describe the techniques, we consider the Dirichlet problem when Ω is a bounded domain and the boundary curve is ∂Ω. As in Euclidean space, we shall prove existence of such graphs provided there is a certain relation between H and the value of the mean curvature H ∂Ω of ∂Ω as submanifold of Ω.