Ergosurfaces for Kerr black holes with scalar hair

We have recently reported the existence of Kerr black holes with scalar hair in General Relativity minimally coupled to a massive, complex scalar field [C. Herdeiro and E. Radu, Phys. Rev. Lett. 112, 221101 (2014)]. These solutions interpolate between boson stars and Kerr black holes. The latter have a well-known topologically S-2 ergosurface (ergosphere) whereas the former develop a S-1 x S-1 ergosurface (ergotorus) in a region of parameter space. We show that hairy black holes always have an ergoregion, and that this region is delimited by either an ergosphere or an ergo-Saturn-i.e. a S-2 circle times(S-1 x S-1) ergosurface. In the phase space of solutions, the ergotorus can either appear disconnected from the ergosphere or pinch off from it. We provide a heuristic argument, based on a measure of the size of the ergoregion, that superradiant instabilities-which are likely to be present-are weaker for hairy black holes than for Kerr black holes with the same global charges. We observe that Saturn-like, and even more remarkable, ergosurfaces should also arise for other rotating hairy black holes.