Static black holes in higher dimensional Einstein-Skyrme models

In this paper we construct a class of hairy static black holes of higher dimensional Einstein-Skyrme theories with the cosmological constant Λ ≤ 0 whose scalar is an SU(2) valued field. The spacetime is set to be conformal to M⁴ × N^N−4 where M⁴ and N^N−4 are a four dimensional spacetime and a compact Einstein (N − 4)-dimensional submanifold for N ≥ 5, respectively, whereas N = 4 is the trivial case. We discuss the behavior of solutions near the boundaries, namely, near the (event) horizon and in the asymptotic region. Then, we establish local-global existence of black hole solutions and show that black holes with finite energy exist if their geometries are asymptotically Ricci flat.