Uniqueness of near-horizon geometries of rotating extremal AdS ₄ black holes

We consider stationary extremal black hole solutions of the Einstein–Maxwell equations with a negative cosmological constant in four dimensions. We determine all non-static axisymmetric near-horizon geometries and all static near-horizon geometries for black holes of this kind. This allows us to deduce that the most general near-horizon geometry of an asymptotically globally AdS 4 rotating extremal black hole is the near-horizon limit of extremal Kerr–Newman–AdS 4 . We also identify the subset of near-horizon geometries which are supersymmetric. Finally, we show which physical quantities of extremal black holes may be computed from the near-horizon limit alone, and point out a simple formula for the entropy of the known supersymmetric AdS 4 black hole. Analogous results are presented in the case of a vanishing cosmological constant.