  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.
