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The Gaussian image of mean curvature one surfaces in \(\mathbb{H}^3\) of finite total curvature. (English) Zbl 1030.53018
Fukaya, Kenji (ed.) et al., Minimal surfaces, geometric analysis and symplectic geometry. Based on the lectures of the workshop and conference, Johns Hopkins University, Baltimore, MD, USA, March 16-21, 1999. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 34, 9-14 (2002).
It was shown by Z. Yu [Proc. Am. Math. Soc. 125, 2997-3001 (1997; Zbl 0903.53004)] that if \(M\) is a complete surface of mean curvature one (CMC-1) in hyperbolic space \(\mathbb{H}^3\), and if the Gauss map \(G\) omits more than four points, then it omits all but one point, and \(M\) is a horosphere. In adjoining a hypothesis of finite total curvature, the authors succeed in replacing “four points” by “three points”. In view of the “canonical” isometric correspondence between CMC-1 surfaces in \(\mathbb{H}^3\) and minimal surfaces in \(\mathbb{R}^3\), this result complements a corresponding theorem of X. Mo and R. Osserman [J. Differ. Geom. 31, 343-355 (1990; Zbl 0666.53003)] (attributed by the authors to Osserman, without journal citation) for minimal surfaces in \(\mathbb{R}^3\).
The authors show further 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.
For the entire collection see [Zbl 0994.00028].
Reviewer: R.Finn (Stanford)

53A35 Non-Euclidean differential geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature