Kerr black holes with parity-odd scalar hair

We study Kerr black holes with synchronized nontrivial parity-odd massive scalar hair in four dimensional asymptotically flat space-time. These axially symmetric stationary spinning solutions of the minimally coupled Einstein-Klein-Gordon theory provide yet another example of bound states in synchronous rotation with the event horizon. We discuss the properties of these parity-odd hairy black holes and boson stars and exhibit their domain of existence. Considering the ergoregions of these hairy black holes, we show that apart from the previously discussed ergosphere and ergo-Saturn, they support a new type of composite ergosurfaces with the topology of a double-torus-Saturn (S-1 x S-1) circle plus (S-1 x S-1) circle plus S-2.