Harmonic maps and constant mean curvature surfaces in H2 x R

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H-2 x R with regular vertical projection, and prove that if the surface has constant mean curvature H = 1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere conformal harmonic map from an open simply connected Riemann surface I into the Poincare disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on Sigma can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H-2 x R. A similar result applies to minimal surfaces in the Heisenberg group Nib. Finally, we classify all complete minimal vertical graphs in H-2 x R.