Phases of five-dimensional black holes

We argue that the configurations that approach maximal entropy infive-dimensional asymptotically flat vacuum gravity, for fixed mass and angularmomentum, are `black Saturns' with a central, close to static, black hole and avery thin black ring around it. For any value of the angular momentum, theupper bound on the entropy is equal to the entropy of a static black hole ofthe same total mass. For fixed mass, spin and area there are families ofmulti-ring solutions with an arbitrarily large number of continuous parameters,so the total phase space is infinite-dimensional. Somewhat surprisingly, thephases of highest entropy are not in thermal equilibrium. Imposingthermodynamical equilibrium drastically reduces the phase space to a finite,small number of different phases.Comment: 16 pages, 2 figures; v2: comments and ref added; v3: ref adde