Fuchsian polyhedra in Lorentzian space-forms

Let S be a compact surface of genus > 1, and g be a metric on S of constant curvature \({K\in\{-1,0,1\}}\) with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are > 2π. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S, g). Moreover, the pair (P, G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of Alexandrov and Rivin–Hodgson (Rec Math Mat Sbornik] NS 11(53):15–65, 1942; Invent Math 111(1):77–111, 1993) concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie and Schlenker (Math Ann 316(3):465–483, 2000).