Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian $σ_k$ Curvature Equations

Abstract

We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $σ_k(A[u])=u^p$, with $h_{ij}=-Wu_{ij}$ and $W=(1-|Du|^2)^{-1/2}$. For $2\leq k<n$ and $p\geq k$, we establish bounds for the height and the Lorentz factor that depend only on $n,k,p$, assuming pointwise strict spacelikeness and admissibility in the closed Gårding cone. The gradient estimate uses a block matrix inequality in the full Gårding cone. This inequality controls the transverse columns of the second fundamental form, including the mixed entries that arise when the height gradient is not a principal direction. A Lorentzian cutoff gives the gradient bound, and comparison with an explicit hyperbolic cap gives the uniform height bound. For $n>2k$ and $k\leq p<k(n+2)/(n-2k)$, we prove that every such solution vanishes identically, without symmetry, decay, integrability, or curvature pinching assumptions. The integral proof combines a quantitative Newton inequality with a weighted divergence identity and a finite sequence of integrations by parts, using the angle variable $2(W-1)$. We also obtain a corresponding rigidity result for complete spacelike immersions. At the upper endpoint, we identify the loss of quadratic gradient coercivity; our argument does not settle the critical Liouville problem.