Positive mass theorem for initial data sets with arbitrary ends

TL;DR

This paper proves a positive mass theorem for initial data sets with arbitrary ends, extending results to asymptotically hyperbolic manifolds and their symmetry cases.

Contribution

It introduces a positive energy theorem for asymptotically flat and hyperbolic initial data sets using spectral PSC and Jang equation techniques, including new quantitative shielding results.

Findings

Established positive mass theorem for asymptotically hyperbolic manifolds.

Extended results to manifolds with asymptotically locally hyperbolic ends.

Provided corollaries for symmetric end cases.

Abstract

We showed a positive energy theorem for asymptotically flat initial data sets with the concept of spectral PSC by He-Shi-Yu, Bi-Hao-He-Shi-Zhu and Brendle-Wang; and the Jang equation in Schoen-Yau, Eichmair and Jang. Then, we proved a quantitative shielding theorem concerning the causal property of the energy-momentum vector of an asymptotically hyperbolic manifold. As a result, we established the positive mass theorem for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition. As corollaries, we also obtained corresponding results for manifolds with asymptotically locally hyperbolic ends with a certain symmetry.