On higher dimensional black holes with Abelian isometry group

We consider $(n+1)$-dimensional, stationary, asymptotically flat, or Kaluza–Klein asymptotically flat black holes with an Abelian $s$-dimensional subgroup of the isometry group satisfying an orthogonal integrability condition. Under suitable regularity conditions, we prove that the area of the group orbits is positive on the domain of outer communications $⟨⟨Mext⟩⟩$⁠, vanishing only on the boundary $∂⟨⟨Mext⟩⟩$ and on the “symmetry axis” $A$⁠. We further show that the orbits of the connected component of the isometry group are timelike throughout the domain of outer communications. Those results provide a starting point for the classification of such black holes. Finally, we show nonexistence of zeros of static Killing vectors on degenerate Killing horizons, as needed for the generalization of the static no-hair theorem to higher dimensions.