The Geometry and Topology of Stationary Multi-Axisymmetric Vacuum Black Holes in Higher Dimensions

Abstract

Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the reduced Einstein equations for vacuum black holes in $(n+3)$-dimensional spacetimes admitting the isometry group $\mathbb{R}\times U(1)^{n}$, with Kaluza-Klein asymptotics for $n\geq3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $\varphi: \mathbb{R}^3\setminus\Gamma\rightarrow SL(n+1,\mathbb{R})/SO(n+1)$ with prescribed blow-up along $\Gamma$, a subset of the $z$-axis in $\mathbb{R}^3$. We also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in the lower dimensional case. Furthermore, we provide a counterexample to a conjecture of Hollands-Ishibashi concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.