Prescribing metrics on the boundary of AdS 3-manifolds

We prove that given two metrics $g_{+}$ and $g_{-}$ with curvature $\kappa <-1$ on a closed, oriented surface $S$ of genus $\tau\geq 2$, there exists an $AdS$ manifold $N$ with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of $\partial N$ are equal to $g_{+}$ and $g_{-}$. Using the duality between convex space-like surfaces in $AdS_{3}$, we obtain an equivalent result about the prescription of the third fundamental form.