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

Barbot, Thierry; Béguin, François; Zeghib, Abdelghani

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 511-591.

Nous étudions l’existence de surfaces à courbure de Gauss constante ou prescrite dans certains espaces-temps lorentziens. Nous montrons en particulier que tout espace-temps (non-élémentaire) globalement hyperbolique spatialement compact maximal à courbure constante positive ou nulle de dimension $3$ est feuilleté en surfaces de Cauchy à courbure de Gauss constante. Dans le cas des espaces-temps à courbure constante strictement négative, le complémentaire du cœur convexe est feuilleté par des surfaces de Cauchy à courbure de Gauss constante. On combinant ces résultats d’existence de feuilletages avec un théorème de C. Gerhardt, on obtient un certain nombre de corollaires. Par exemple, on résout le problème de Minkowski dans ${\mathrm{Min}}_3$ pour des données qui sont invariantes par l’action d’un groupe fuchsien cocompact.

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 ${\mathrm{Min}}_3$ for data that are invariant under the action of a co-compact Fuchsian group.

Classification : 53C50, 53C42, 53C80
Keywords: Gauss curvature, $K$-curvature, Minkowski problem
Mot clés : courbure de Gauss, $K$-courbure, problème de Minkowski

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