Experimental Mathematics

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


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