Weighted decomposition of vector fields, $X$-ADM mass, and higher-dimensional mass-charge inequalities

Abstract

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 $\beta$ 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 $d\beta \in L^1$. 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.