Uniqueness of smooth stationary black holes in vacuum: Small perturbations of the Kerr spaces

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).