AI 中文总结
本文针对任意维数$n\geqslant3$的渐近平坦黎曼流形，通过向量场$X$的加权分解建立$X$-ADM质量的分解式，结合$X$-正质量定理得到高维多电荷质量不等式，解决了相关质量刚性与磁荷刻画问题。
AI 中文摘要
我们研究任意维数$n\geqslant 3$下无边界、完备且一端渐近平坦的黎曼流形上的$X$-ADM质量。从向量场$X$的加权梯度-无散分解出发，我们构造了一个共形度量，其标量曲率由与$X$相关的临界修正标量曲率$\u200b\mathrm{R}_{X}^{(1-n)}$支配。这将$X$-ADM质量精确分解为共形度量的ADM质量加上一个非负缺陷项，当且仅当$X$是梯度场时该缺陷项消失，从而基于分解给出了带完整刚性陈述的正性的另一种证明。明确的负质量例子表明，修正标量曲率$\u200b\mathrm{R}_{X}^{(k)}$条件中参数$k$的取值范围是最优的。随后我们将$X$-正质量定理应用于向量场系统，得到了带刚性和等式情形下全局对齐性质的高维多电荷质量不等式，该结果包含经典三维电磁不等式及其高维类似物。在高维情形下，磁数据$\u200b\beta$是一个2-形式，在渐近平坦端没有规范标量电荷；我们证明当$\u200bd\beta \in L^1$时，选定的张量通量消失，在给定的临界渐近假设下，该通量定义了非平凡的张量磁荷，我们还将此表述为对$X$-正质量定理的条件约化。
英文摘要
We study the $X$-ADM mass on complete, one-ended asymptotically flat Riemannian manifolds without boundary in arbitrary dimension $n\geqslant 3$. Starting from a weighted gradient--divergence-free decomposition of the vector field $X$, we construct a conformal metric whose scalar curvature is governed by the critical modified scalar curvature $\mathrm{R}_{X}^{(1-n)}$ associated with $X$. This yields an exact decomposition of the $X$-ADM mass into the ADM mass of the conformal metric plus a nonnegative defect term, which vanishes precisely when $X$ is a gradient, thereby giving an alternative decomposition-based proof of positivity with a full rigidity statement. Explicit negative-mass examples show that the range of parameters $k$ in the modified scalar curvature $\mathrm{R}_{X}^{(k)}$ condition is sharp. We then apply the $X$-positive mass theorem to systems of vector fields, obtaining higher-dimensional multi-charge mass inequalities with rigidity and global alignment in the equality case. This yields the classical three-dimensional electric-magnetic inequality and its higher-dimensional analogue. In higher dimensions, the magnetic datum $β$ is a $2$-form and does not have a canonical scalar charge on the asymptotically flat end. We prove that a selected tensorial flux vanishes when $dβ\in L^1$. Under a prescribed critical asymptotic ansatz, this flux defines a non-trivial tensorial magnetic charge and we formulate a conditional reduction to the $X$-positive mass theorem.
