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Communications in Mathematical Physics

, Volume 299, Issue 1, pp 89–127 | Cite as

Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces

  • S. Alexakis
  • A. D. Ionescu
  • S. Klainerman
Article

Abstract

The goal of the paper is to prove a perturbative result, concerning the uniqueness of Kerr solutions, a result which we believe will be useful in the proof of their nonlinear stability. Following the program started in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009), we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike (Ionescu and Klainerman in Invent. Math. 175:35–102, 2009), which was based on a tensorial characterization of the Kerr solutions, due to Mars (Class. Quant. Grav. 16:2507–2523, 1999), we rely here on Hawking’s original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009), in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in Alexakis et al. (Hawking’s local rigidity theorem without analyticity. http://arxiv.org/abs/0902.1173v1[gr-qc], 2009).

Keywords

Black Hole Stationary Black Hole Carleman Estimate Null Hypersurface Kerr Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of Wisconsin – MadisonMadisonUSA
  3. 3.Princeton UniversityPrincetonUSA

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