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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.17814 |
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| _version_ | 1866909799649640448 |
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| author | Tombinski, Marcin |
| author_facet | Tombinski, Marcin |
| contents | Let $Ω_1$, $Ω_2$ be two domains in $\mathbb{C}^n$ with Kobayashi metrics $k_{Ω_i}$ and consider $f \in \mathcal{O}(Ω_1,Ω_2)$ a holomorphic mapping. Let $\mathfrak{F}_1$ and $\mathfrak{F}_2$ be a family of geodesics defined on $Ω_1$ and $Ω_2$ respectively, where a geodesic between $z$ and $w$ in $Ω_i$ is the length minimizing curve between the two points for the metric $k_{Ω_i}$. We say that a holomorphic function \textit{preserves geodesics} if for any geodesic $γ_1$ in $\mathfrak{F}_1$ its image is a subset of a geodesic $γ_2$ in $\mathfrak{F}_2$ ($f(γ_1)\subset γ_2$). We aim to characterise the family of such functions between families of Kobayashi geodesics passing through a point in the unit disc $\mathbb{D}$ and in the unit ball $\mathbb{B}^n$. Some additional results in the complex plane $\mathbb{C}$ and $\mathbb{C}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterisation of geodesic preserving functions Tombinski, Marcin Complex Variables Let $Ω_1$, $Ω_2$ be two domains in $\mathbb{C}^n$ with Kobayashi metrics $k_{Ω_i}$ and consider $f \in \mathcal{O}(Ω_1,Ω_2)$ a holomorphic mapping. Let $\mathfrak{F}_1$ and $\mathfrak{F}_2$ be a family of geodesics defined on $Ω_1$ and $Ω_2$ respectively, where a geodesic between $z$ and $w$ in $Ω_i$ is the length minimizing curve between the two points for the metric $k_{Ω_i}$. We say that a holomorphic function \textit{preserves geodesics} if for any geodesic $γ_1$ in $\mathfrak{F}_1$ its image is a subset of a geodesic $γ_2$ in $\mathfrak{F}_2$ ($f(γ_1)\subset γ_2$). We aim to characterise the family of such functions between families of Kobayashi geodesics passing through a point in the unit disc $\mathbb{D}$ and in the unit ball $\mathbb{B}^n$. Some additional results in the complex plane $\mathbb{C}$ and $\mathbb{C}^n$. |
| title | Characterisation of geodesic preserving functions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2509.17814 |