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

Gregory J. Galloway, Richard Schoen

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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 languageEnglish (US)
Pages (from-to)571-576
Number of pages6
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Sep 1 2006


ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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