A Hilbert metric for bounded symmetric domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911816751251456 |
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| author | Falbel, Elisha Guilloux, Antonin Will, Pierre |
| author_facet | Falbel, Elisha Guilloux, Antonin Will, Pierre |
| contents | Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish-Chandra realization in projective spaces. We describe this construction on the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hilbert metric for bounded symmetric domains Falbel, Elisha Guilloux, Antonin Will, Pierre Differential Geometry Complex Variables Metric Geometry 32M15, 53C35 Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish-Chandra realization in projective spaces. We describe this construction on the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ. |
| title | A Hilbert metric for bounded symmetric domains |
| topic | Differential Geometry Complex Variables Metric Geometry 32M15, 53C35 |
| url | https://arxiv.org/abs/2403.18634 |