Positive mass theorem for initial data sets with arbitrary ends

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.