Constant mean curvature foliation of domains of dependence in AdS₃

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