$X$-ADM Mass and $X$-Positive Mass Theorem

TL;DR

This paper introduces the $X$-ADM mass for asymptotically flat 3-manifolds, proving a positive mass theorem and its generalizations using a monotonicity formula, with discussions on rigidity cases.

Contribution

It defines the $X$-ADM mass for a broad class of vector fields and establishes a generalized positive mass theorem with new monotonicity techniques.

Findings

$X$-ADM mass is well-defined for asymptotically flat 3-manifolds.

A monotonicity formula along Green's function level sets proves the positive mass theorem.

Generalizations include weighted manifolds and charged cases with topological restrictions.

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.