A note on minimal graphs over certain unbounded domains of Hadamard manifolds

Given an unbounded domain $\mathrm{\Omega }$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $\mathrm{\Omega }$ satisfies certain convexity condition at infinity and if $\mathrm{\partial }\mathrm{\Omega }$ is mean convex. We also prove that mean convexity of $\mathrm{\partial }\mathrm{\Omega }$ is a necessary condition, extending to unbounded domains some results that are valid on bounded ones.