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

Understand

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.