Classical instability of Kerr-AdS black holes and the issue of final state

It is now established that small Kerr\char21{}anti-de Sitter (Kerr-AdS) black holes are unstable against scalar perturbations, via the superradiant amplification mechanism. We show that small Kerr-AdS black holes are also unstable against gravitational perturbations and we compute the features of this instability. We also describe with great detail the evolution of this instability. In particular, we identify its end-point state. It corresponds to a Kerr-AdS black hole whose boundary is an Einstein universe rotating with the light velocity. This black hole is expected to be slightly oblate and to coexist in equilibrium with a certain amount of outside radiation.