Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian $\sigma_k$ Curvature Equations

Abstract

We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $\sigma_k(A[u])=u^p$, 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 G\r{a}rding cone. The gradient estimate uses a block matrix inequality in the full G\r{a}rding 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.