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

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space