Ergo-spheres, ergo-tori and ergo-Saturns for Kerr black holes with scalar hair

We have recently reported (arXiv:1403.2757) the existence of Kerr black holes with scalar hair in General Relativity minimally coupled to a massive, complex scalar field. These solutions interpolate between boson stars and Kerr black holes. The latter have a well known topologically S^2 ergo-surface (ergo-sphere) whereas the former develop a S^1*S^1 ergo-surface (ergo-torus) in a region of parameter space. We show that hairy black holes always have an ergo-region, and that this region is delimited by either an ergo-sphere or an ergo-Saturn -- i.e. a S^2\oplus (S^1*S^1) ergo-surface. In the phase space of solutions, the ergo-torus can either appear disconnected from the ergo-sphere or pinch off from it. We provide a heuristic argument, based on a measure of the size of the ergo-region, 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 ergo-surfaces, should also arise for other rotating `hairy' black holes.