Stationary vacuum black holes in 5 dimensions

We study the problem of asymptotically flat bi-axially symmetric stationary solutions of the vacuum Einstein equations in 5-dimensional spacetime. In this setting, the cross section of any connected component of the event horizon is a prime 3-manifold of positive Yamabe type, namely the 3-sphere S ³ , the ring S ¹ × S ² , or the lens space L(p, q). The Einstein vacuum equations reduce to an axially symmetric harmonic map with prescribed singularities from R ³ into the symmetric space SL(3, R)=SO(3). In this paper, we solve the problem for all possible topologies, and in particular the first candidates for smooth vacuum non-degenerate black lenses are produced. In addition, a generalization of this result is given in which the spacetime is allowed to have orbifold singularities. We also formulate conditions for the absence of conical singularities which guarantee a physically relevant solution.