Entire hypersurfaces of constant scalar curvature in Minkowski space

Abstract

We show that every regular domain $\mathcal{D}$ in Minkowski space ${\mathbb{R}}^{n,1}$ which is not a wedge admits an entire hypersurface whose domain of dependence is $\mathcal{D}$ and whose scalar curvature is a prescribed constant (or function, under suitable hypotheses) in $(-\infty ,0)$. Under rather general assumptions, these hypersurfaces are unique and provide foliations of $\mathcal{D}$. As an application, we show that every maximal globally hyperbolic Cauchy compact flat spacetime admits a foliation by hypersurfaces of constant scalar curvature, generalizing to any dimension previous results of Barbot-Béguin-Zeghib (for $n=2$) and Smith (for $n=3$).

30 pages