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A classification of near-horizon geometries of extremal vacuum black holes

We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in four dimensions (4D), two commuting rotational symmetries in five dimensions (5D), and in both cases nontoroidal horizon topology. In 4D we determine the most general near-horizon geometry of such a black hole and prove it is the same as the near-horizon limit of the extremal Kerr-AdS4 black hole. In 5D, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S1×S2 horizon and two distinct families with topologically S3 horizons. The S1×S2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black...