Liouville Rigidity and Universal Spacelikeness Estimates for 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 $n=2$ and $p⩾1$, or if $n⩾3$ and $1⩽p⩽\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n⩾2$ and $p⩾1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u{|}^2{)}^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n⩾3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

v3: Major revision. We establish a new Liouville theorem in the critical case, thereby settling the conjecture of Byeon, Ikoma, Malchiodi, and Mari in the classical-solution setting. The proof of the critical case in Section 6 is based on an invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument, and a Souplet-type feedback inequality