On the Weyl problem in Minkowski space

Let $S$ be a closed surface of hyperbolic type. We show that, for every pair $(g_{+},g_{-})$ of negatively curved metrics over $S$ there exists a unique GHMC Minkowski spacetime $X$ into which $(S,g_{+})$ and $(S,g_{-})$ isometrically embed as Cauchy surfaces in the future and past components respectively.