The positive mass theorem with arbitrary ends

Abstract

We prove a Riemannian positive mass theorem which allows for incompleteness and negative scalar curvature. The manifolds are assumed to have one asymptotically Schwarzschild end, but the complement of this end is otherwise arbitrary. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau.