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

TL;DR

The paper introduces the X--ADM mass m_X for asymptotically flat 3--manifolds and proves a generalized positive mass theorem relative to m_X, unifying classical ADM mass with weighted and charged PMT variants under a topological no-spherical-class condition on H_2(M;Z). The authors develop a monotonicity formula along level sets of the minimal positive Green's function for the drift operator L_X = Δ - (1/2) ∇_X, obtaining existence and precise near-pole and infinity asymptotics for the Green's function. The main result, the X--PMT, follows from this monotonicity by comparing limits as t → 0^+ and t → ∞, yielding m_X ≥ 0 and rigidity statements: either X vanishes and (M,g) is Euclidean or X is a gradient and (M,g) is conformally Euclidean. The approach extends to weighted manifolds and manifolds with boundary, and clarifies how charged or weighted analogues of the PMT arise within the same framework, enhancing the geometric understanding of mass and rigidity in low dimensions.

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.

Theorems & Definitions (16)

• Definition 1.1
• Definition 1.2: $X$--ADM mass
• Theorem 1.3: $X$--Positive mass theorem
• Theorem 1.4
• proof : Proof of Theorem \ref{['GenPMTmon']}
• Remark 3.1
• Proposition 3.2
• proof
• Lemma 4.1
• proof