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Mathematics > Differential Geometry
Title: A volume-renormalized mass for asymptotically hyperbolic manifolds
(Submitted on 12 Jul 2023 (v1), last revised 26 Apr 2024 (this version, v2))
Abstract: 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.
Submission history
From: Klaus Kroencke [view email][v1] Wed, 12 Jul 2023 14:36:24 GMT (40kb)
[v2] Fri, 26 Apr 2024 15:11:47 GMT (41kb)
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