A proof of the uniqueness theorem for axionic black holes

Points out a discrepancy in the proof of the uniqueness theorem for axionic black holes that was provided by Bowick and co-workers in 1988, and gives a more appropriate proof. The author solves directly the field equations of the theory in which gravity is coupled to a Kalb-Ramond field and proceed to prove that, under spherical symmetry, the unique static asymptotically flat solution is the Schwarzschild black hole, which has vanishing axion field but non-vanishing potential.