The volume-renormalized mass of any orientable 3-manifold asymptotic to hyperbolic space with scalar curvature at least -6 and no spherical second homology classes is nonnegative, vanishing only for hyperbolic space.
Bray, Demetre P
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Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds
The volume-renormalized mass of any orientable 3-manifold asymptotic to hyperbolic space with scalar curvature at least -6 and no spherical second homology classes is nonnegative, vanishing only for hyperbolic space.