Convex surfaces with prescribed induced metrics in anti-de Sitter spacetimes

Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric of curvature $K<-1$ on $S$, and let $h_0$ be a hyperbolic metric on $S$. We show that there exists a unique quasifuchsian AdS spacetime with left metric isotopic to $h_0$, containing a past-convex Cauchy surface with induced metric isotopic to $h$.