I will present a recent proof which shows that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon, possibly with a cosmological constant, must admit a Killing vector field.
In particular, this implies that the extremal Kerr horizon is the most general such horizon in four-dimensional General Relativity and completes the classification of the associated near-horizon geometries. I will also discuss a recent uniqueness proof which shows that any analytic Einstein spacetime, that contains a static extremal horizon with a maximally symmetric compact cross-section, is the extremal Schwarzschild de Sitter spacetime or its near-horizon geometry.
James Lucietti (Edinburgh): Rigidity of extremal horizons and black hole uniqueness
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