Superradiance and Black Holes with a Single Killing Field

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It is well known that superradiance can extract energy from a black hole and, in an asymptotically global AdS background, it drives the black hole unstable. The onset of superradiance also signals a bifurcation to a new family of AdS black holes in a phase diagram of stationary solutions. We construct non-linearly the hairy black holes, solitons and boson stars associated to scalar superradiance. We present both charged and rotating solutions with scalar hair. In the charged case, the structure of phase diagram varies considerably, depending on the charge of the condensate. In the rotating case, the hairy solutions give the first examples of black holes with only a Killing field: the black holes are neither stationary nor axisymmetric, but are invariant under a single Killing field which is tangent to the null generators of the horizon.
We discuss the role of these solutions in a full time evolution of the superradiant instability. We emphasize how scarce is our knowledge of the rotating superradiant instability endpoint, and that this instability will compete with the turbulent instability of AdS.