Extreme black holes are those whose event horizons are also zero-temperature, or degenerate, Killing horizons. A certain limiting procedure defines a quasi-Einstein equation which these horizons must satisfy. Solutions are triples (M,g,X) where M is a closed manifold (the horizon), g is a Riemannian quasi-Einstein metric, and X is a 1-form. The case of a closed 1-form X is the so-called static case, which includes near horizon geometries of static extreme black holes. We give a partial rigidity theorem for quasi-Einsteain metrics with M a closed manifold and X a closed 1-form, specifying conditions under which the quasi-Einstein structure must be Einstein and correcting some of the earlier literature on near horizon geometries.

This talk is based on joint work with Eric Bahuaud, Sharmila Gunasekaran, and Hari K Kunduri.