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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.

MSC:
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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