Phases of five-dimensional black holes

We argue that the configurations that approach maximal entropy in five-dimensional asymptotically flat vacuum gravity, for fixed mass and angular momentum, are 'black Saturns' with a central, close to static, black hole and a very thin black ring around it. For any value of the angular momentum, the upper bound on the entropy is equal to the entropy of a static black hole of the same total mass. FOr fixed mass, spin and area there are families of multi-ring solutions with an arbitrarily large number of continuous parameters so the total phase space is infinite-dimensional. Somewhat surprisingly, the phases of highest entropy are not in thermal equilibrium. Imposing thermodynalical equilibrium drastically reduces the phase space to a finite, small number of different phases.