Liouville Rigidity for Entire Strictly Spacelike Solutions of a Lorentzian Prescribed Mean Curvature Equation

Abstract

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n\in \{1,2\}$ and $p>1$, or if $n\ge 3$ and \[ 1<p<\frac{n+2}{n-2}, \] every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ is zero. No symmetry, decay, integrability, or uniform spacelike gap is assumed. Independently, for every $n\ge 2$ and $p\ge 1$ we establish global height and Lorentz-factor bounds that remain valid in the critical and supercritical regimes. In dimension two, a logarithmic-capacity argument completes the proof. For $n\ge 3$, we combine two weighted trace-free tensor identities whose quadratic form is coercive exactly in the subcritical range and degenerates at the Sobolev exponent. This extends radial nonexistence to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.