CONSTANT MEAN-CURVATURE SURFACES WITH 2 ENDS IN HYPERBOLIC SPACE

We investigate the close relationship between minimal surfaces in Eucl idean three-space and surfaces of constant mean curvature 1 in hyperbo lic 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-understoo d surfaces of revolution: the catenoid cousins. In contrast to this, w e show that, unlike the case of minimal surfaces in Euclidean three-sp ace, there do exist complete connected immersed surfaces of constant m ean curvature 1 with two ends in hyperbolic space that are not surface s of revolution: the genus-one catenoid cousins. These surfaces are of interest because they show that, although minimal surfaces in Euclide an three-space and surfaces of constant mean curvature 1 in hyperbolic three-space are intimately related, there are essential differences b etween these two sets of surfaces. The proof we give of existence of t he genus-one catenoid cousins is a mathematically rigorous verificatio n that the results of a computer experiment are sufficiently accurate to imply existence.