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1998 Constant mean curvature surfaces with two ends in hyperbolic space
Wayne Rossman, Katsunori Sato
Experiment. Math. 7(2): 101-119 (1998).

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.

Citation

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Wayne Rossman. Katsunori Sato. "Constant mean curvature surfaces with two ends in hyperbolic space." Experiment. Math. 7 (2) 101 - 119, 1998.

Information

Published: 1998
First available in Project Euclid: 24 March 2003

zbMATH: 0980.53081
MathSciNet: MR1677103

Subjects:
Primary: 53A10
Secondary: 65D17

Rights: Copyright © 1998 A K Peters, Ltd.

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Vol.7 • No. 2 • 1998
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