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A volume-renormalized mass for asymptotically hyperbolic manifolds
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We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein--Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.
Forward citations
Cited by 2 Pith papers
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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.
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The Volume-Renormalized Mass from a Hamiltonian Perspective
The volume-renormalized mass of an asymptotically hyperbolic initial data set equals a Fischer-Moncrief-style reduced Hamiltonian for asymptotically Milne-like spacetimes and is monotone under Einstein evolution.
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