Uniqueness of the asymptotic AdS 3 geometry

We explicitly show that in (2+1) dimensions the general solution of the Einstein equations with negative cosmological constant on a neigbourhood of timelike spatial infinity can be obtained from BTZ metrics by coordinate transformations corresponding geometrically to deformations of their spatial infinity surface. Thus, whatever the topology and geometry of the bulk, the metric on the timelike extremities is BTZ. Graham and Lee [1] proved that, under suitable topological assumptions, Eu-clidean Einstein spaces with negative cosmological constant Λ are completely defined by the geometry on their boundary. Furthermore, Fefferman and Gra-ham (FG) [2] showed that, whatever the signature, there exists an asymptotic expansion of the metric, which formally solves the Einstein equations with Λ = −1/ℓ 2 < 0; we choose hereafter the length units such that ℓ = 1. The first terms of this expansion may be given by even powers of a radial coordinate r : ds 2 ≃