  An explicit global and unique isometric embedding into hyperbolic 3-space,
H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is
given. In particular, this allows the embedding into H^3 of surfaces of
revolution having negative, but finite, Gaussian curvature at smooth fixed
points of the U(1) isometry. As an example, we exhibit the global embedding of
the Kerr-Newman event horizon into H^3, for arbitrary values of the angular
momentum. For this example, considering a quotient of H^3 by the Picard group,
we show that the hyperbolic embedding fits in a fundamental domain of the group
up to a slightly larger value of the angular momentum than the limit for which
a global embedding into Euclidean 3-space is possible. An embedding of the
double-Kerr event horizon is also presented, as an example of an embedding
which cannot be made global.
