Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian σk Curvature Equations
Abstract
We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying σk(A[u])=up. For 2≤ k<n and p≥ k, pointwise strict spacelikeness and closed k-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full Gårding cone, a Lorentzian cutoff, and hyperbolic-cap comparison. For 2≤ k<n/2 and k≤ p≤ k(n+2)/(n−2k), every such solution vanishes, without symmetry, decay, finite-energy, curvature-pinching, or global Hessian assumptions. In the subcritical range we use a trace-free Newton tensor, a weighted divergence identity, and a finite descent. At the critical endpoint, the divergence identity retains an exact nonnegative defect and a positive quartic term. Direct estimates cover 2k<n≤4k+2. For n≥4k+3, we use compactness, concentration of the k-Hessian measure at the first crossing, a single-pole Pohozaev argument, core counting, a recursive defect estimate, and Souplet-type radius selection. A geometric corollary gives a rigidity result for complete spacelike immersions under the corresponding curvature equation.
Related
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BibTeX
@misc{shi2026universal,
title = {{Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian $σ_k$ Curvature Equations}},
author = {Shujun Shi and Yuzhou Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25170},
eprint = {2609.25170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25170},
}
RIS
TY - UNPB
TI - Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian σ_k Curvature Equations
AU - Shujun Shi
AU - Yuzhou Zhang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.25170
UR - https://arxiv.org/abs/2609.25170
AB - We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying σ_k(A[u])=u^p. For 2≤ k<n and p≥ k, pointwise strict spacelikeness and closed k-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full Gårding cone, a Lorentzian cutoff, and hyperbolic-cap comparison. For 2≤ k<n/2 and k≤ p≤ k(n+2)/(n−2k), every such solution vanishes, without symmetry, decay, finite-energy, curvature-pinching, or global Hessian assumptions. In the subcritical range we use a trace-free Newton tensor, a weighted divergence identity, and a finite descent. At the critical endpoint, the divergence identity retains an exact nonnegative defect and a positive quartic term. Direct estimates cover 2k<n≤4k+2. For n≥4k+3, we use compactness, concentration of the k-Hessian measure at the first crossing, a single-pole Pohozaev argument, core counting, a recursive defect estimate, and Souplet-type radius selection. A geometric corollary gives a rigidity result for complete spacelike immersions under the corresponding curvature equation.
ER -