The Positive Mass Theorem for Manifolds with Distributional Curvature

Abstract

We formulate and prove a positive mass theorem for n-dimensional spin manifolds whose metrics have only the Sobolev regularity ${C^0 \cap W_{\rm loc}^{1,n}}$. 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 ${W_{\rm loc}^{2,p}}$ with p > n. Our proof is a generalization of Witten’s arguments, in which we must treat the Dirac operator and its associated Lichnerowicz-Weitzenböck 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.