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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.21271 |
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| _version_ | 1866929454237876224 |
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| author | Arango, Juan Arbeláez, Hugo Mejía, Diego |
| author_facet | Arango, Juan Arbeláez, Hugo Mejía, Diego |
| contents | A proper subdomain $G$ of the unit disk $\mathbb{D}$ is horocyclically convex (horo-convex) if, for every $ω\in \mathbb{D}\cap \partial G$, there exists a horodisk $H$ such that $ω\in \partial H$ and $G\cap H=\emptyset$. In this paper we give an internal characterization of these domains, namely, that $G$ is horo-convex if and only if any two points can be joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic metric of horo-convex regions and some consequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk Arango, Juan Arbeláez, Hugo Mejía, Diego Complex Variables A proper subdomain $G$ of the unit disk $\mathbb{D}$ is horocyclically convex (horo-convex) if, for every $ω\in \mathbb{D}\cap \partial G$, there exists a horodisk $H$ such that $ω\in \partial H$ and $G\cap H=\emptyset$. In this paper we give an internal characterization of these domains, namely, that $G$ is horo-convex if and only if any two points can be joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic metric of horo-convex regions and some consequences. |
| title | On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2407.21271 |