, Volume 299, Issue 1, pp 89-127
Date: 08 Jul 2010

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

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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).

Communicated by P.T. Chruściel
The first author was partially supported by a Clay research fellowship.
The second author was partially supported by a Packard Fellowship.
The third author was partially supported by NSF grant DMS-0070696.