Existence of constant mean curvature graphs in hyperbolic space (Q1290386)

In this paper, the authors study graphs of constant mean curvature over a horosphere in hyperbolic space \(H^{n+1}\). The main result is: let \(\Omega \) be a compact domain of a horosphere in \(H^{n+1},\) whose boundary \(\partial \Omega \) is ``mean convex'', that is, its mean curvature \(H_{\partial \Omega }\) is positive with respect to the inner orientation. If \(H\) is a number such that \(-H_{\partial \Omega }<H<1,\) then there exists a graph over \(\Omega \) with constant mean curvature \(H\) and boundary \(\partial \Omega .\)