AI 中文总结
该研究针对洛伦兹型给定平均曲率方程的整体严格类空解，在不同维数和参数范围证明了刘维尔定理，建立了全局高度与洛伦兹因子界，还结合张量恒等式得到几何半空间刚性定理。
AI 中文摘要
我们对\\(\mathbb R^n\\)中方程\\(\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0\\)的非负整体严格类空解证明了一个刘维尔定理。当\\(n\in\{1,2\}\\)且\\(p>1\\)，或当\\(n\ge3\\)且\\(1<p<\frac{n+2}{n-2}\\)时，每个满足\\(|\nabla u|<1\\)的非负\\(C^2\\)解均为零解，且不假设对称性、衰减性、可积性或均匀类空隙。独立地，对每个\\(n\ge2\\)和\\(p\ge1\\)，我们建立了全局高度和洛伦兹因子界，这些界在临界和超临界区域仍然有效。在二维情形下，对数容量论证完成了证明；对于\\(n\ge3\\)，我们结合两个加权无迹张量恒等式，其二次型恰好在次临界范围内是强制的，且在索伯列夫指数处退化。这将径向非存在性推广到任意整体解，并得到了完整类空超曲面的几何半空间刚性定理。
英文摘要
We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ 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 $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. 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 $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.
Commentsv3: Major revision. We establish a new Liouville theorem in the critical case, thereby settling the conjecture of Byeon, Ikoma, Malchiodi, and Mari in the classical-solution setting. The proof of the critical case in Section 6 is based on an invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument, and a Souplet-type feedback inequality
