Entire hypersurfaces of constant scalar curvature in Minkowski space

TL;DR

This paper constructs and characterizes entire hypersurfaces with constant scalar curvature in Minkowski space, providing unique foliations of certain domains and applications to flat spacetimes, extending previous low-dimensional results.

Contribution

It establishes existence, uniqueness, and foliation properties of hypersurfaces with prescribed constant scalar curvature in Minkowski space for general dimensions.

Findings

Existence of hypersurfaces with prescribed constant scalar curvature in Minkowski space.

Uniqueness of these hypersurfaces under general conditions.

Foliations of maximal globally hyperbolic flat spacetimes by such hypersurfaces.

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-B\'eguin-Zeghib (for $n=2$) and Smith (for $n=3$).