A volume-renormalized mass for asymptotically hyperbolic manifolds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914771963478016 |
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| author | Dahl, Mattias Kroencke, Klaus McCormick, Stephen |
| author_facet | Dahl, Mattias Kroencke, Klaus McCormick, Stephen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A volume-renormalized mass for asymptotically hyperbolic manifolds Dahl, Mattias Kroencke, Klaus McCormick, Stephen Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory 53C21, 53C25, 53E20 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. |
| title | A volume-renormalized mass for asymptotically hyperbolic manifolds |
| topic | Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory 53C21, 53C25, 53E20 |
| url | https://arxiv.org/abs/2307.06196 |