Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

Abstract

We prove the existence and uniqueness in Rn,1 of entire spacelike hypersurfaces contained in the future of the origin O and asymptotic to the light-cone, with scalar curvature prescribed at their generic point M as a negative function of the unit vector pointing in the direction of OM→, divided by the square of the norm of OM→ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from O (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence.