Prescribed mean curvature graphs in hyperbolic space

In this work we examine mean curvature in hyperbolic space. We seek the existence and uniqueness of smooth hypersurfaces with prescribed mean curvature H and given ideal boundary Gamma. We prove that in the half-space model Hn+1=Rn x (0, infinity) of hyperbolic space, unique radial graphs over the semi-sphere Sn∩Hn+1 exist provided H ∈ C 1 Sn+ , |H| ≤ 1, |H| < 1 on 6Sn+ , and Gamma is the radial graph in Rn x {lcub}0{rcub} of a continuous function on 6Sn+ . Classical methods in the theory of elliptic partial differential equations are employed. If one varies the mean curvature or the ideal boundary in an appropriate way, the resulting surfaces are shown to foliate Hn+1 .