  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 $\phi$.
