The positive mass theorem for manifolds with distributional curvature

TL;DR

This paper proves a positive mass theorem for spin manifolds with metrics of low regularity, where curvature is understood distributionally, extending previous results requiring higher regularity.

Contribution

It introduces a generalized notion of ADM mass for metrics with Sobolev regularity and proves its non-negativity under distributional scalar curvature, broadening the theorem's applicability.

Findings

The ADM mass is well-defined and non-negative for low-regularity metrics.

Vanishing ADM mass implies the manifold is Euclidean space.

The method extends Witten’s spinor approach to distributional curvature.

Abstract

We formulate and prove a positive mass theorem for n-dimensional spin manifolds whose metrics have only the Sobolev regularity $C^0 \cap W^{1,n}$. At this level of regularity, the curvature of the metric is defined in the distributional sense only, and we propose here a (generalized) notion of ADM mass for such a metric. Our main theorem establishes that if the manifold is asymptotically flat and has non-negative scalar curvature distribution, then its (generalized) ADM mass is well-defined and non-negative, and vanishes only if the manifold is isometric to Euclidian space. Prior applications of Witten's spinor method by Lee and Parker and by Bartnik required the much stronger regularity $W^{2,2}$. Our proof is a generalization of Witten's arguments, in which we must treat the Dirac operator and its associated Lichnerowicz-Weitzenbock identity in the distributional sense and cope with certain averages of first-order derivatives of the metric over annuli that approach infinity. Finally, we observe that our arguments are not specific to scalar curvature and also allow us to establish a universal positive mass theorem.