
Mathematics 2008
Prescribing Gauss curvature of surfaces in 3dimensional spacetimes, Application to the Minkowski problem in the Minkowski spaceAbstract: We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (nonelementary) 3dimensional maximal globally hyperbolic spatially compact spacetime with constant nonnegative 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 3dimensional Minkowski space for datas that are invariant under the action of a cocompact Fuchsian group.
