The geometry and topology of stationary multiaxisymmetric vacuum black holes in higher dimensions

Vishnu Kakkat, Marcus Khuri, Jordan Rainone and Gilbert Weinstein

Vol. 322 (2023), No. 1, 59–97

DOI: 10.2140/pjm.2023.322.59

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 $ℝ\times U<!--FUNCTION\; APPLICATION-->{(1)}^n$, with Kaluza–Klein asymptotics for $n\ge 3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $\phi :{ℝ}^3\setminus \Gamma \to \mathrm{SL}<!--FUNCTION\; APPLICATION-->(n+1,ℝ)∕\mathrm{SO}<!--FUNCTION\; APPLICATION-->(n+1)$ with prescribed blow-up along $\Gamma$, a subset of the $z$-axis in ${ℝ}^3$. 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.

Keywords

stationary solutions, black holes, domain of outer communication

Mathematical Subject Classification

Primary: 53C43, 53C50, 53C80, 83C05, 83C57

Milestones

Received: 19 March 2022

Revised: 6 July 2022

Accepted: 23 November 2022

Published: 3 May 2023