Entire hypersurfaces of constant scalar curvature in Minkowski space

Abstract We show that every regular domain 𝒟 in Minkowski space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathbb{R}^{n,1} which is not a wedge admits an entire hypersurface whose domain of dependence is 𝒟 and whose scalar curvature is a prescribed constant (or function, under suitable hypotheses) in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (-\infty,0) . Under rather general assumptions, these hypersurfaces are unique and provide foliations of 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n=2 ) and Smith (for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n=3 ).