The gaussian image of mean curvature one surfaces in hyperbolic space of finite total curvature
Résumé
The hyperbolic Gauss map G of a complete constant mean curvature one surface M in hyperbolic 3-space, is a holomorphic map from M to the Riemann sphere. When M has finite total curvature, we prove G can miss at most three points unless G is constant. We also prove that if M is a properly embedded mean curvature one surface of finite topology, then G is surjective unless M is a horosphere or catenoid cousin.