Geometric and Functional Analysis

, Volume 20, Issue 4, pp 845–869

Hawking’s Local Rigidity Theorem Without Analyticity

  • Spyros Alexakis
  • Alexandru D. Ionescu
  • Sergiu Klainerman

DOI: 10.1007/s00039-010-0082-7

Cite this article as:
Alexakis, S., Ionescu, A.D. & Klainerman, S. Geom. Funct. Anal. (2010) 20: 845. doi:10.1007/s00039-010-0082-7


We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop.B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.

Keywords and phrases

Killing vector-field Einstein vacuum equations non-expanding bifurcatehorizon unique continuation 

2010 Mathematics Subject Classification

35A02 83C05 83C57 

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Spyros Alexakis
    • 1
    • 2
  • Alexandru D. Ionescu
    • 3
    • 4
  • Sergiu Klainerman
    • 3
  1. 1.Dept. of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.Dept. of MathematicsPrinceton UniversityPrincetonUSA
  4. 4.University of Wisconsin – MadisonMadisonUSA

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