We study the thermodynamics of four-dimensional Kerr-Newman-AdS black holes both in the canonical and the grand-canonical ensemble. The stability conditions are investigated, and the complete phase diagrams are obtained, which include the Hawking-Page phase transition in the grand-canonical ensemble. In the canonical case, one has a first order transition between small and large black holes, which disappears for sufficiently large electric charge or angular momentum. This disappearance corresponds to a critical point in the phase diagram. Via the AdS/CFT conjecture, the obtained phase structure is also relevant for the corresponding conformal field theory living in a rotating Einstein universe, in the presence of a global background U (1) current. An interesting limit arises when the black holes preserve some supersymmetry. These BPS black holes correspond to highly degenerate zero temperature states in the dual CFT, which lives in an Einstein universe rotating with the *

We examine supersymmetry of four-dimensional asymptotically Anti-de Sitter (AdS) dyonic black holes in the context of gauged N = 2 supergravity. Our calculations concentrate on black holes with unusual topology and their rotating generalizations, but we also reconsider the spherical rotating dyonic Kerr-Newman-AdS black hole, whose supersymmetry properties have previously been investigated by Kostelecký and Perry within another approach. We find that in the case of spherical, toroidal or cylindrical event horizon topology, the black holes must rotate in order to preserve some supersymmetry; the non-rotating supersymmetric configurations representing naked singularities. However, we show that this is no more true for black holes whose event horizons are Riemann surfaces of genus g > 1, where we find a nonrotating extremal solitonic black hole carrying magnetic charge and permitting one Killing spinor. For the nonrotating supersymmetric configurations of various topologies, all Killing spinors are explicitely constructed.

We construct solutions for thin black rings in Anti-deSitter and deSitter spacetimes using approximate methods. Black rings in AdS exist with arbitrarily large radius and satisfy a bound |J| ≤ LM , which they saturate as their radius becomes infinitely large. For angular momentum near the maximum, they have larger area than rotating AdS black holes. Thin black rings also exist in deSitter space, with rotation velocities varying between zero and a maximum, and with a radius that is always strictly below the Hubble radius. Our general analysis allows us to include black Saturns as well, which we discuss briefly. We present a simple physical argument why supersymmetric AdS black rings must not be expected: they do not possess the necessary pressure to balance the AdS potential. We discuss the possible existence or absence of 'large AdS black rings' and their implications for a dual hydrodynamic description. An analysis of the physical properties of rotating AdS black holes is also included.

We review the classification of supersymmetric solutions to minimal gauged supergravity in four dimensions. After a short introduction to the main features of the theory, we explain how to obtain all its solutions admitting a Killing spinor. Then, we analyze the rich mathematical structure behind them and present the supersymmetric field configurations. Among them, we find supersymmetric black holes, quarter and half BPS traveling waves, kink solutions, and supersymmetric Kundt and Robinson-Trautman solutions. Finally, we generalize the classification to include external sources, and show a particular solution describing a supersymmetric Gödel-type universe.