The Dirichlet Problem for constant mean curvature graphs in $\mathbb{M}\times\mathbb{R}$

Abstract

We study graphs of constant mean curvature $H>0$ in $\mathbb{M}\times ℝ$ for $\mathbb{M}$ a Hadamard surface, ie a complete simply connected surface with curvature bounded above by a negative constant $-a$. We find necessary and sufficient conditions for the existence of these graphs over bounded domains in $\mathbb{M}$, having prescribed boundary data, possibly infinite.