Local version of the no-hair theorem

Abstract: Non-extremal isolated horizons embeddable in 4-dimensional spacetimes satisfying the vacuum Einstein equations with cosmological constant are studied. The horizons are assumed to be stationary to the second order. The Weyl tensor at the horizon is assumed to be of the Petrov type D. The corresponding equation on the intrinsic horizon geometry is solved in the axisymmetric case. The family of the solutions is $2$-dimensional, it is parametrized by the area and the angular momentum. The embeddability in the Kerr - de Sitter, the Kerr - anti de Sitter and the Near extremal Horizon spacetimes obtained by the Horowitz limit from the extremal Kerr - de Sitter and extremal Kerr - anti de Sitter is discussed. This uniqueness of the axisymmetric type D isolated horizons is a generalization of the similar earlier result valid in the cosmological constant free case.