Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

An explicit global and unique isometric embedding into hyperbolic 3-space, H, of an axisymmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H 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, for arbitrary values of the angular momentum. For this example, considering a quotient of H 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.