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

Thierry Barbot (1) , François Béguin (2) , Abdelghani Zeghib (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Gauss curvature; geometric time; Minkowski problem

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]; Géométrie différentielle [math.DG]; Physique mathématique [math-ph]

Thierry Barbot, François Béguin, Abdelghani Zeghib. Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space. Annales de l'Institut Fourier, 2011, 61 (2), pp.511-591. ⟨10.5802/aif.2622⟩. ⟨hal-00270851⟩