Entire spacelike radial graphs with prescribed mean curvature in the Lorentz--Minkowski space

Abstract

In this paper we address the existence and uniqueness of entire spacelike hypersurfaces in the Lorentz--Minkowski space ${\mathbb{L}}^{m+1}$ with prescribed mean curvature that are star-shaped with respect to a point and asymptotic to a light cone. We also establish a Willmore-type inequality and prove a non-existence result for spacelike radial graphs asymptotic to the light cone whose mean curvature belongs to $L^p$ for $1\le p\le m$, in particular in the case of compactly supported mean curvature.