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 div((∇ u)/(√(1−|∇ u|2)))+up=0 in ℝn. If n=2 and pgeqslant1, or if ngeqslant3 and 1leqslant pleqslant(n+2)/(n−2), every nonnegative C2 solution satisfying |∇ 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 ngeqslant2 and pgeqslant1, for both the height u and the Lorentz factor (1−|∇ 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 ngeqslant3, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.
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BibTeX
@misc{ma2026liouville,
title = {{Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation}},
author = {Xi-Nan Ma and Tian Wu and Wangzhe Wu and Bao Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07231},
eprint = {2608.07231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07231},
}
RIS
TY - UNPB
TI - Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation
AU - Xi-Nan Ma
AU - Tian Wu
AU - Wangzhe Wu
AU - Bao Yu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.07231
UR - https://arxiv.org/abs/2608.07231
AB - We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of div((∇ u)/(√(1−|∇ u|^2)))+u^p=0 in ℝ^n. If n=2 and pgeqslant1, or if ngeqslant3 and 1leqslant pleqslant(n+2)/(n−2), every nonnegative C^2 solution satisfying |∇ 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 ngeqslant2 and pgeqslant1, for both the height u and the Lorentz factor (1−|∇ 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 ngeqslant3, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.
ER -