Holomorphic maps acting as Kobayashi isometries on a family of geodesics
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913785114001408 |
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| author | Bracci, Filippo Kosiński, Łukasz Zwonek, Włodzimierz |
| author_facet | Bracci, Filippo Kosiński, Łukasz Zwonek, Włodzimierz |
| contents | Consider a holomorphic map $F: D \to G$ between two domains in ${\mathbb C}^N$. Let $\mathcal F$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal F$. This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that $F$ is a biholomorphism. Specifically, we establish this when $D$ is a complete hyperbolic domain, and $\mathcal F$ comprises all geodesic segments originating from a specific point. Another case is when $D$ and $G$ are $C^{2+α}$-smooth bounded pseudoconvex domains, and $\mathcal F$ consists of all geodesic rays converging at a designated boundary point of $D$. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10705 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Holomorphic maps acting as Kobayashi isometries on a family of geodesics Bracci, Filippo Kosiński, Łukasz Zwonek, Włodzimierz Complex Variables Consider a holomorphic map $F: D \to G$ between two domains in ${\mathbb C}^N$. Let $\mathcal F$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal F$. This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that $F$ is a biholomorphism. Specifically, we establish this when $D$ is a complete hyperbolic domain, and $\mathcal F$ comprises all geodesic segments originating from a specific point. Another case is when $D$ and $G$ are $C^{2+α}$-smooth bounded pseudoconvex domains, and $\mathcal F$ consists of all geodesic rays converging at a designated boundary point of $D$. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal. |
| title | Holomorphic maps acting as Kobayashi isometries on a family of geodesics |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2311.10705 |