We have recently reported (arXiv:1403.2757) the existence of Kerr black holeswith scalar hair in General Relativity minimally coupled to a massive, complexscalar field. These solutions interpolate between boson stars and Kerr blackholes. 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) ina region of parameter space. We show that hairy black holes always have anergo-region, and that this region is delimited by either an ergo-sphere or anergo-Saturn -- i.e. a S^2\oplus (S^1*S^1) ergo-surface. In the phase space ofsolutions, the ergo-torus can either appear disconnected from the ergo-sphereor pinch off from it. We provide a heuristic argument, based on a measure ofthe size of the ergo-region, that superradiant instabilities - which are likelyto be present - are weaker for hairy black holes than for Kerr black holes withthe same global charges. We observe that Saturn-like, and even more remarkableergo-surfaces, should also arise for other rotating `hairy' black holes.
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