Volume functions and boundary data of 3-dimensional hyperbolic manifolds
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916993483931648 |
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| author | Schlenker, Jean-Marc |
| author_facet | Schlenker, Jean-Marc |
| contents | We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the hyperbolic metric on such a manifold is uniquely determined by either of two possible geometric data on their boundary. The second aspect is the ``volume'' associated to such a manifold, such as the renormalized volume of a convex co-compact hyperbolic manifold. The relation between the two is provided by the first variation of the volume functions, which involves the two kinds of boundary data as ``conjugate'' variables.
While progress has recently been made on some questions, others remain open. New connections have recently emerged, with physics (and in particular the AdS/CFT correspondence) as well as with probability theory (the Loewner energy). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05627 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Volume functions and boundary data of 3-dimensional hyperbolic manifolds Schlenker, Jean-Marc Geometric Topology Mathematical Physics Complex Variables Differential Geometry We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the hyperbolic metric on such a manifold is uniquely determined by either of two possible geometric data on their boundary. The second aspect is the ``volume'' associated to such a manifold, such as the renormalized volume of a convex co-compact hyperbolic manifold. The relation between the two is provided by the first variation of the volume functions, which involves the two kinds of boundary data as ``conjugate'' variables. While progress has recently been made on some questions, others remain open. New connections have recently emerged, with physics (and in particular the AdS/CFT correspondence) as well as with probability theory (the Loewner energy). |
| title | Volume functions and boundary data of 3-dimensional hyperbolic manifolds |
| topic | Geometric Topology Mathematical Physics Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2510.05627 |