The positive mass theorem for manifolds with distributional curvature

Understand

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 $W^{2,2}$.