Existence of constant mean curvature graphs in hyperbolic space

Abstract.

We give an existence result for constant mean curvature graphs in hyperbolic space ${\Bbb H}^{n+1}$. Let $\Omega$ be a compact domain of a horosphere in ${\Bbb H}^{n+1}$ whose boundary $\partial\Omega$ is mean convex, that is, its mean curvature $H_{\partial\Omega}$ (as a submanifold of the horosphere) 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 $\partial\Omega$. Umbilical examples, when $\partial\Omega$ is a sphere, show that our hypothesis on H is the best possible.