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

TL;DR

This paper investigates the existence of surfaces with prescribed Gauss curvature in Lorentzian spacetimes, establishing foliations in certain cases and applying results to solve the Minkowski problem in Minkowski space.

Contribution

It proves the existence of Gauss curvature foliations in specific Lorentzian spacetimes and applies these to solve the Minkowski problem for invariant data.

Findings

Existence of Gauss curvature foliations in maximal globally hyperbolic spacetimes.

Foliations exist outside the convex core in negative curvature cases.

Solution to the Minkowski problem in Minkowski space for co-compact Fuchsian group-invariant data.

Abstract

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.