A conformal reduction for the X-ADM mass
Abstract
The X-ADM mass is a generalisation of the ADM mass for asymptotically flat manifolds, recently introduced by Mantegazza and Oronzio [arXiv:2602.11372], who proved a positive mass theorem for this quantity in dimension $3$ under a certain topological condition. In this article, we prove the X-positive mass theorem with rigidity in all dimensions n≥3 without any topological restrictions. To this end, we first prove that the scalar curvature condition appearing in the X-positive mass theorem implies the metric is of positive Yamabe type. We then show that the X-ADM mass of the given metric is equal to the ADM mass of a scalar-flat representative in its conformal class plus a term that is explicitly non-negative. From this, the X-positive mass theorem follows from the standard positive mass theorem. As a corollary, we obtain the Riemannian mass--charge inequality of general relativity in all dimensions, which was previously only known for spin manifolds. Finally, we give a version of the X-positive mass theorem for a manifold with boundary.
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BibTeX
@misc{mccormick2026conformal,
title = {{A conformal reduction for the X-ADM mass}},
author = {Stephen McCormick},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.03629},
eprint = {2607.03629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03629},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A conformal reduction for the X-ADM mass
AU - Stephen McCormick
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2607.03629
UR - https://arxiv.org/abs/2607.03629
AB - The X-ADM mass is a generalisation of the ADM mass for asymptotically flat manifolds, recently introduced by Mantegazza and Oronzio [arXiv:2602.11372], who proved a positive mass theorem for this quantity in dimension $3$ under a certain topological condition. In this article, we prove the X-positive mass theorem with rigidity in all dimensions n≥3 without any topological restrictions. To this end, we first prove that the scalar curvature condition appearing in the X-positive mass theorem implies the metric is of positive Yamabe type. We then show that the X-ADM mass of the given metric is equal to the ADM mass of a scalar-flat representative in its conformal class plus a term that is explicitly non-negative. From this, the X-positive mass theorem follows from the standard positive mass theorem. As a corollary, we obtain the Riemannian mass--charge inequality of general relativity in all dimensions, which was previously only known for spin manifolds. Finally, we give a version of the X-positive mass theorem for a manifold with boundary.
ER -