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

TL;DR

This paper proves the existence and uniqueness of entire spacelike hypersurfaces in Minkowski space that are asymptotic to the light-cone and have a prescribed scalar curvature depending on direction, using a graph approach over hyperbolic space.

Contribution

It establishes existence and uniqueness results for hypersurfaces with prescribed scalar curvature in Minkowski space, extending previous Cartesian results to a Lorentzian setting.

Findings

Existence of solutions for the prescribed scalar curvature problem.

Uniqueness of these hypersurfaces under given conditions.

Construction of radial upper and lower solutions.

Abstract

Existence and uniqueness in ${\Bbb R}^{n,1}$ 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 $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (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.