The axial symmetry of Kerr without the rigidity theorem

Abstract: Local condition that imply the no-hair property of black holes are completed. The conditions take the form of constraints on the geometry of the 2-dimensional crossover surface of black hole horizon. They imply also the axial symmetry without the rigidity theorem. This is the new result contained in this letter. The family of the solutions to our constraints is 2-dimensional and can be parametrized by the area and angular momentum. The constraints are induced by our assumption that the horizon is of the Petrov type D. Our result applies to all the bifurcated Killing horizons: inner/outer black hole horizons as well as cosmological horizons. Vacuum spacetimes with a given cosmological constant can be reconstructed from our solutions via Racz's black hole holograph.