## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Minimal Surfaces, Geometric Analysis and Symplectic Geometry, K. Fukaya, S. Nishikawa and J. Spruck, eds. (Tokyo: Mathematical Society of Japan, 2002), 245 - 253

### Constant Mean Curvature 1 Surfaces with Low Total Curvature in Hyperbolic 3-Space

Wayne Rossman, Masaaki Umehara, and Kotaro Yamada

#### Abstract

Surfaces of constant mean curvature one in hyperbolic 3-space have quite similar properties to minimal surfaces in Euclidean 3-space. We shall list the possibilities of constant mean curvature one surfaces in hyperbolic 3-space with low total absolute curvature, or low dual total absolute curvature, and compare them with the known classification of minimal surfaces with low total curvature. Complete proofs of the new results will be published in two forthcoming papers (listed in the bibliography).

#### Article information

**Dates**

First available in Project Euclid:
31 December 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546230302

**Digital Object Identifier**

doi:10.2969/aspm/03410245

**Mathematical Reviews number (MathSciNet)**

MR1925744

**Zentralblatt MATH identifier**

1035.53018

**Subjects**

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Secondary: 53A35: Non-Euclidean differential geometry 53A42

#### Citation

Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro. Constant Mean Curvature 1 Surfaces with Low Total Curvature in Hyperbolic 3-Space. Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 245--253, Mathematical Society of Japan, Tokyo, Japan, 2002. doi:10.2969/aspm/03410245. https://projecteuclid.org/euclid.aspm/1546230302