Mathematics – Differential Geometry
Scientific paper
2008-04-26
J. Exp. Math. 7(2) (1998), 101-119
Mathematics
Differential Geometry
Scientific paper
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 (when the surfaces are considered globally). 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.
Rossman Wayne
Sato Katsunori
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