The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.
Sakovich,The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting, Communications in Mathematical Physics386(2021), no
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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.