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Extremal horizons in a spacetime can be studied through their near-horizon geometry. Building on work by Dunajski and Lucietti, we prove a near-horizon analogue of Hawking’s rigidity theorem: any rotating near-horizon geometry admits a Killing field tangent to the horizon cross-sections. This result holds in spacetimes of arbitrary dimension and with general matter content, assuming only that the cross-sections are compact. As a consequence, any near-horizon geometry admits an enhanced symmetry group containing either SO(2,1) or the two-dimensional Poincaré group.
The rigidity theorem enables a complete classification of four-dimensional near-horizon geometries in Einstein-Maxwell theory. We further construct new families of extremal horizons in five dimensions carrying charge and two independent angular momenta.