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Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature. (English) Zbl 1169.53040
Summary: We prove the existence and uniqueness in \(\mathbb R^{n,1}\) of entire space-like 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 \(\vec{OM}\), divided by the square of the norm of \(\vec{OM}\) (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.

53C40 Global submanifolds
35J65 Nonlinear boundary value problems for linear elliptic equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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