The Gaussian Image of Mean Curvature One Surfaces in $\mathbb{H}^3$ of Finite Total Curvature

Abstract

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.