Advanced Studies in Pure Mathematics

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

Pascal Collin, Laurent Hauswirth, and Harold Rosenberg

Full-text: Open access

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.

Article information

Source
Minimal Surfaces, Geometric Analysis and Symplectic Geometry, K. Fukaya, S. Nishikawa and J. Spruck, eds. (Tokyo: Mathematical Society of Japan, 2002), 9-14

Dates
First available in Project Euclid: 31 December 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1546230290

Digital Object Identifier
doi:10.2969/aspm/03410009

Mathematical Reviews number (MathSciNet)
MR1925732

Zentralblatt MATH identifier
1030.53018

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53A35: Non-Euclidean differential geometry

Citation

Collin, Pascal; Hauswirth, Laurent; Rosenberg, Harold. The Gaussian Image of Mean Curvature One Surfaces in $\mathbb{H}^3$ of Finite Total Curvature. Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 9--14, Mathematical Society of Japan, Tokyo, Japan, 2002. doi:10.2969/aspm/03410009. https://projecteuclid.org/euclid.aspm/1546230290


Export citation