发表机构
Harbin Normal University（哈尔滨师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究Lorentz-Minkowski空间中满足$\sigma_k$方程的类空图，建立高度与Lorentz因子的先验界，并在一定参数范围内证明解恒为零的Liouville刚性定理。
AI 中文摘要
我们研究Lorentz-Minkowski空间中满足$\sigma_k(A[u])=u^p$的非负整体类空图，其中$h_{ij}=-Wu_{ij}$且$W=(1-|Du|^2)^{-1/2}$。对于$2\leq k<n$和$p\geq k$，在逐点严格类空性和闭Gårding锥中的可容许性假设下，我们建立了仅依赖于$n,k,p$的高度和Lorentz因子的界。梯度估计使用了全Gårding锥中的块矩阵不等式。该不等式控制第二基本形式的横向列，包括当高度梯度不是主方向时出现的混合项。Lorentzian截断函数给出梯度界，与显式双曲帽的比较给出均匀高度界。对于$n>2k$且$k\leq p<k(n+2)/(n-2k)$，我们证明每个这样的解恒为零，无需对称性、衰减性、可积性或曲率夹紧假设。积分证明结合了定量Newton不等式、加权散度恒等式和有限次分部积分，使用角度变量$2(W-1)$。我们还获得了完备类空浸入的相应刚性结果。在上端点，我们识别出二次梯度强制性的丧失；我们的论证未解决临界Liouville问题。
英文摘要
We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $σ_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, 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 $2\le k<n/2$ and $k\le p\le k(n+2)/(n-2k)$, 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 $2k<n\le4k+2$. For $n\ge4k+3$, 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.
CommentsThis version extends the Liouville theorem from the subcritical range to the critical endpoint $p=k(n+2)/(n-2k)$ for $2\le k<n/2$. The new proof uses an exact endpoint defect identity, direct estimates in lower dimensions, and concentration and core-counting arguments in higher dimensions. The exposition has been revised accordingly
