Preprint

Inserted: 12 feb 2026
Last Updated: 12 feb 2026

Year: 2026

Abstract:

For a given admissible vector field $X$, we define a geometric quantity for asymptotically flat $3$-manifolds, called $X$-ADM mass and we establish a relative positive mass theorem via a monotonicity formula along the level sets of a suitable Green's function. Under different assumptions on $X$, we obtain generalizations of the "classical'' positive mass theorem, like the one for weighted manifolds and the one "with charge'' under some topological restrictions. Finally, we also discuss the rigidity cases.

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• X_mass and PMT.pdf