## Experimental Mathematics

- Experiment. Math.
- Volume 7, Issue 2 (1998), 101-119.

### Constant mean curvature surfaces with two ends in hyperbolic space

Wayne Rossman and Katsunori Sato

#### Abstract

We investigate the close relationship between minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space. Just as in the case of minimal surfaces in Euclidean three-space, the only complete connected embedded surfaces of constant mean curvature 1 with two ends in hyperbolic space are well-understood surfaces of revolution: the catenoid cousins.

In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean three-space, there do exist complete connected immersed surfaces of constant mean curvature 1 with two ends in hyperbolic space that are not surfaces of revolution: the genus-one catenoid cousins. These surfaces are of interest because they show that, although minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space are intimately related, there are essential differences between these two sets of surfaces. The proof we give of existence of the genus-one catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.

#### Article information

**Source**

Experiment. Math., Volume 7, Issue 2 (1998), 101-119.

**Dates**

First available in Project Euclid: 24 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1048515658

**Mathematical Reviews number (MathSciNet)**

MR1677103

**Zentralblatt MATH identifier**

0980.53081

**Subjects**

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

Secondary: 65D17: Computer aided design (modeling of curves and surfaces) [See also 68U07]

#### Citation

Rossman, Wayne; Sato, Katsunori. Constant mean curvature surfaces with two ends in hyperbolic space. Experiment. Math. 7 (1998), no. 2, 101--119. https://projecteuclid.org/euclid.em/1048515658