Resumen de Harmonic maps and constant mean curvature surfaces

Isabel Fernández Delgado , Pablo Mira Carrillo

• We introduce a {\it hyperbolic Gauss map\/} into the Poincar\'e disk for any surface in ${\Bbb H}^2\times {\Bbb R}$ 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 $\Sigma$ into the Poincar\'e 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 ${\Bbb H}^2 \times {\Bbb R}$. A similar result applies to minimal surfaces in the Heisenberg group ${\rm Nil_3}$. Finally, we classify all complete minimal vertical graphs in ${\Bbb H}^2\times {\Bbb R}$.