Constant mean curvature foliation of domains of dependence in 𝐴𝑑𝑆₃

We prove that, given an acausal curve Γ \Gamma in the boundary at infinity of A d S 3 AdS_{3} which is the graph of a quasi-symmetric homeomorphism ϕ \phi , there exists a unique foliation of its domain of dependence D ( Γ ) D(\Gamma ) by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of ϕ \phi .