Jerzy Lewandowski (University of Warsaw, Poland)
"Intrinsic uniqueness of extreme Horizons”

Killing horizon extremality condition, combined with Einstein's vacuum equations, induces equations that must be satisfied by the 2-dimensional geometry of the horizon section and the rotation vector field defined on it.   The study of these equations leads to the theory of the intrinsic uniqueness of extreme black holes. The zeroth law of thermodynamics, the spherical topology of the section, rigidity, no-hair, Kerr (Kerr-Newman, Kerr-(Anti) de Sitter)  horizon uniqueness - all those properties are proved in turn. These results are applicable in a proof of the uniqueness of the extreme Kerr solution, since the state-of-the-art methods of the mathematical theory of black holes work only in the non-extreme case.  The program of research on the equation of extreme horizons began in the early 2000s, culminating in this year's result of intrinsic rigidity.
The equation of extreme horizons is valid in any dimensional space-time, it applies not only to physical black holes but also to so-called near-horizontal geometries, topologically non-trivial horizons and spacetimes, and other solutions of Einstein's equations with special optical properties.