Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in five dimensions

Abstract

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space |$L(p,q)$| topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically |$S^1\times S^2$|⁠. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: |$S^3$|⁠, |$S^1\times S^2$|⁠, or |$L(p,q)$|⁠. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space |$SL(3,\mathbb{R})/SO(3)$|⁠. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.