Black holes with unusual topology

Einstein’s equations with a negative cosmological constant admit solutions which are asymptotically anti–de Sitter space. Matter fields in anti–de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition. Hence there is no fundamental reason that black hole horizons should have a spherical topology. In anti–de Sitter space Einstein’s equations admit black hole solutions where the horizon can be a Riemann surface with genus $g.\mathrm{}$ The case $g=0\mathrm{}$ is the asymptotically anti–de Sitter black hole first studied by Hawking and Page, which has a spherical topology. The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs. The genus $g>\mathrm{}1\mathrm{}$ black hole has no other free parameters apart from the mass and the charge. All such black holes exhibit a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite.