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Constant mean curvature surfaces with two ends in hyperbolic space

By Wayne Rossman and Katsunori Sato

Abstract

We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces 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 3-space, there do exist complete connected immersed constant mean curvature 1 surfaces with two ends in hyperbolic space that are not surfaces of revolution – the genus 1 catenoid cousins. The genus 1 catenoid cousins are of interest because they show that, although minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space are intimately related, there are essential differences between these two sets of surfaces. The proof we give of existence of the genus 1 catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.

Year: 1998
OAI identifier: oai:CiteSeerX.psu:10.1.1.312.8806
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