We study the existence of surfaces with constant or prescribed Gauss
curvature in certain Lorentzian spacetimes. We prove in particular that every
(non-elementary) 3-dimensional maximal globally hyperbolic spatially compact
spacetime with constant non-negative curvature is foliated by compact spacelike
surfaces with constant Gauss curvature. In the constant negative curvature
case, such a foliation exists outside the convex core. The existence of these
foliations, together with a theorem of C. Gerhardt, yield several corollaries.
For example, they allow to solve the Minkowski problem in the 3-dimensional
Minkowski space for datas that are invariant under the action of a co-compact
Fuchsian group