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.