Geometry of horospheres in Kobayashi hyperbolic domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909956806017024 |
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| author | Chandel, Vikramjeet Singh Mandal, Nishith |
| author_facet | Chandel, Vikramjeet Singh Mandal, Nishith |
| contents | For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry of horospheres in Kobayashi hyperbolic domains Chandel, Vikramjeet Singh Mandal, Nishith Complex Variables 32F45, 53C23 (Primary), 32Q45, 32A40, 53C22 (Secondary) For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it. |
| title | Geometry of horospheres in Kobayashi hyperbolic domains |
| topic | Complex Variables 32F45, 53C23 (Primary), 32Q45, 32A40, 53C22 (Secondary) |
| url | https://arxiv.org/abs/2409.14114 |