Constant mean curvature graphs on exterior domains of the hyperbolic plane

We prove an existence result for non-rotational constant mean curvature ends in ${mathbb{H}^2 times mathbb{R}}$ , where ${mathbb{H}^2}$ is the hyperbolic real plane. The value of the curvature is ${h in big(0, frac{1}{2} big)}$ . We use Schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of ${mathbb{H}^2}$ . We also prove a fine property of the asymptotic behavior of the rotational ends introduced by Sa Earp and Toubiana.