The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.
Eichmair,The Jang equation reduction of the spacetime positive energy theorem in dimen- sions less than eight, Communications in Mathematical Physics319(2013), 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.