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.
Categories: math.DG math.GT