Stationary Vacuum Black Holes in 5 Dimensions

The problem of bi-axially symmetric stationary solutions of the vacuum Einstein equations in 5-dimensional spacetimes reduces to an axially symmetric harmonic map with prescribed singularities from ℝ^3 into the symmetric space SL(3,ℝ)/SO(3). 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). In this talk, I will describe recent results we obtained solving the reduced problem for all possible topologies, and in particular the first candidates for smooth vacuum non-degenerate black lenses are produced. Our results apply to asymptotically flat as well as to asymptotically locally Euclidean and asymptotically Kaluza-Klein geometries. We also formulate conditions for the absence of conical singularities which guarantee a physically relevant solution.

This is joint work with Marcus Khuri and Sumio Yamada