Classification and Uniqueness of Einstein-Maxwell Black Holes

We consider the classification of the stationary and axisymmetric solutions to the five-dimensional Einstein-Maxwell equations with flat asymptotics. We discuss separately the static case and an integrable class of stationary and axisymmetric solutions possessing sigma-model structure of the field equations. Finely, we present the uniqueness theorem in the general case when the field equations are not completely integrable. We demonstrate that in general the conserved charges and the interval structure do not provide enough information to specify the solutions uniquely but we should consider in addition certain magnetic fluxes.