Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation
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 $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.