We introduce a hyperbolic Gauss map into the Poincar´e disk for any surface in H2×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 Σ into the Poincaré 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 Σ can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H2 x R. A similar result applies to minimal surfaces in the Heisenberg group Nil3. Finally, we classify all complete minimal vertical graphs in H2 × R.


Additional Information

Print ISSN
pp. 1145-1181
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.