The phase structure of higher-dimensional black rings and black holes

We construct an approximate solution for an asymptotically flat, neutral, thin rotating black ring in any dimension D >= 5 by matching the near-horizon solution for a bent boosted black string, to a linearized gravity solution away from the horizon. The rotating black ring solution has a regular horizon of topology S-1 X SD-3 and incorporates the balancing condition of the ring as a zero-tension condition. For D = 5 our method reproduces the thin ring limit of the exact black ring solution. For D >= 6 we show that the black ring has a higher entropy than the Myers-Perry black hole in the ultra-spinning regime. By exploiting the correspondence between ultra-spinning black holes and black membranes on a two-torus, we take steps towards qualitatively completing the phase diagram of rotating blackfolds with a single angular momentum. We are led to propose a connection between MP black holes and black rings, and between MP black holes and black Saturns, through merger transitions involving two kinds of 'pinched' black holes. More generally, the analogy suggests an infinite number of pinched black holes of spherical topology leading to a complicated pattern of connections and mergers between phases.