A conformal reduction for the X-ADM mass

Abstract

The X-ADM mass is a geometric invariant for asymptotically flat manifolds, recently introduced by Mantegazza and Oronzio [arXiv:2602.11372], generalising the weighted mass of Baldauf and Ozuch [arXiv:2201.04475] which itself is a generalisation of the well-known ADM mass. We show that the positivity of the X-ADM mass is in fact equivalent to the standard positive mass theorem for the ADM mass by way of a conformal reduction argument, which in particular proves the X-positive mass theorem in all dimensions, which previously was only established in dimension 3 under a topological condition. As a corollary, we obtain the Riemannian mass--charge inequality of general relativity in all dimensions. Finally, we prove a version of the X-positive mass theorem for a manifold with boundary.