The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.
The Hyperboloidal and Spacetime Positive Mass Theorem in All Dimensions
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abstract
Using the recent work of Brendle--Wang on the Riemannian positive mass theorem, we prove the spacetime positive mass theorem for asymptotically flat and asymptotically hyperboloidal initial data sets in arbitrary dimension $n$.
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Positive mass theorem for initial data sets with arbitrary ends
The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.