Asymptotically Locally Euclidean/Kaluza-Klein Stationary Vacuum Black Holes in 5 Dimensions

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in 5 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.