## Abstract

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.

Original language | English |
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Pages (from-to) | 1205-1241 |

Number of pages | 37 |

Journal | Communications in Partial Differential Equations |

Volume | 43 |

Issue number | 8 |

DOIs | |

State | Published - 3 Aug 2018 |

## Keywords

- Lens space
- non-spherical event horizons
- singular harmonic maps
- stationary black hole spacetime