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 $\Lambda \le 0$whosescalarisan$SU(2)$ valued field. The spacetime is set to be conformal to ${\mathcal{M}}^4\times {\mathcal{N}}^{N-4}$ where ${\mathcal{M}}^4$ and ${\mathcal{N}}^{N-4}$ are a four dimensional spacetime and a compact Einstein $(N-4)$-dimensional submanifold for $N\ge 5$, 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. At the end, we perform a linear stability analysis using perturbative method and give a remark about their stability.