The Positive Mass Theorem with Arbitrary Ends

TL;DR

This paper proves a positive mass theorem for manifolds with one asymptotically flat end and arbitrary other ends, allowing negative scalar curvature and incompleteness, by compensating with positive scalar curvature on an annulus.

Contribution

It extends the positive mass theorem to manifolds with multiple ends, including incomplete and negatively curved ones, confirming a conjecture of Schoen and Yau in the complete case.

Findings

Proved positive mass theorem for manifolds with arbitrary ends.

Established conditions under which negative scalar curvature is compensated.

Confirmed a long-standing conjecture of Schoen and Yau.

Abstract

We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. 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.