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 h₀ be a hyperbolic metric on S. We show that there exists a unique quasifuchsian AdS spacetime with left metric isotopic to h₀, containing a past-convex Cauchy surface with induced metric isotopic to h.