### Abstract

Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S
^{2} × S
^{1}. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

Original language | English (US) |
---|---|

Pages (from-to) | 571-576 |

Number of pages | 6 |

Journal | Communications in Mathematical Physics |

Volume | 266 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2006 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*266*(2), 571-576. https://doi.org/10.1007/s00220-006-0019-z

**A generalization of Hawking's black hole topology theorem to higher dimensions.** / Galloway, Gregory J; Schoen, Richard.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 266, no. 2, pp. 571-576. https://doi.org/10.1007/s00220-006-0019-z

}

TY - JOUR

T1 - A generalization of Hawking's black hole topology theorem to higher dimensions

AU - Galloway, Gregory J

AU - Schoen, Richard

PY - 2006/9

Y1 - 2006/9

N2 - Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S 2 × S 1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

AB - Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S 2 × S 1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

UR - http://www.scopus.com/inward/record.url?scp=33746214630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746214630&partnerID=8YFLogxK

U2 - 10.1007/s00220-006-0019-z

DO - 10.1007/s00220-006-0019-z

M3 - Article

AN - SCOPUS:33746214630

VL - 266

SP - 571

EP - 576

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -