Constant mean curvature surfaces with two ends in hyperbolic space

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dc.creatorRossman, Wayne
dc.creatorSato, Katsunori
dc.date2008-04-26
dc.date.accessioned2026-07-07T09:35:29Z
dc.date.available2026-07-07T09:35:29Z
dc.descriptionWe 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.
dc.identifierhttps://arxiv.org/abs/0804.4211
dc.identifierhttp://arxiv.org/abs/0804.4211
dc.identifierJ. Exp. Math. 7(2) (1998), 101-119
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/159848
dc.subjectDifferential Geometry
dc.subject53A10; 53A35; 53C42
dc.titleConstant mean curvature surfaces with two ends in hyperbolic space
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