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| Formato: | Preprint |
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2024
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| Acceso en línea: | https://arxiv.org/abs/2412.20560 |
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| author | Mocanu, Marcelina |
| author_facet | Mocanu, Marcelina |
| contents | We study in the setting of a metric space $\left( X,d\right) $ some generalizations of four hyperbolic-type metrics defined on open sets $G$ with nonempty boundary in the $n-$dimensional Euclidean space, namely Gehring-Osgood metric, Dovgoshey- Hariri-Vuorinen metric, Nikolov-Andreev metric and Ibragimov metric. In the definitions of these generalizations, the boundary $\partial G$ of $G$ and the distance from a point $x$ of $G$ to $\partial G$ are replaced by a nonempty proper closed subset $M$ of $X$ and by a $1-$Lipschitz function positive on $X\setminus M$, respectively. For each generalization $ρ$ of the hyperbolic-type metrics mentioned above we prove that $\left( X\setminus M,ρ\right) $ is a Gromov hyperbolic space and that the identity map between $\left( X\setminus M,d\right) $ and $% \left( X\setminus M,ρ\right) $ is quasiconformal. For the Gehring-Osgood metric and the Nikolov-Andreev metric we improve the Gromov constants known from the literature. For Ibragimov metric the Gromov hyperbolicity is obtained even if we replace the distance from a point $x$ to $\partial G$ by any positive function on $X\setminus M$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalizations of four hyperbolic-type metrics and Gromov hyperbolicity Mocanu, Marcelina Complex Variables 30C65, 30L10 We study in the setting of a metric space $\left( X,d\right) $ some generalizations of four hyperbolic-type metrics defined on open sets $G$ with nonempty boundary in the $n-$dimensional Euclidean space, namely Gehring-Osgood metric, Dovgoshey- Hariri-Vuorinen metric, Nikolov-Andreev metric and Ibragimov metric. In the definitions of these generalizations, the boundary $\partial G$ of $G$ and the distance from a point $x$ of $G$ to $\partial G$ are replaced by a nonempty proper closed subset $M$ of $X$ and by a $1-$Lipschitz function positive on $X\setminus M$, respectively. For each generalization $ρ$ of the hyperbolic-type metrics mentioned above we prove that $\left( X\setminus M,ρ\right) $ is a Gromov hyperbolic space and that the identity map between $\left( X\setminus M,d\right) $ and $% \left( X\setminus M,ρ\right) $ is quasiconformal. For the Gehring-Osgood metric and the Nikolov-Andreev metric we improve the Gromov constants known from the literature. For Ibragimov metric the Gromov hyperbolicity is obtained even if we replace the distance from a point $x$ to $\partial G$ by any positive function on $X\setminus M$ |
| title | Generalizations of four hyperbolic-type metrics and Gromov hyperbolicity |
| topic | Complex Variables 30C65, 30L10 |
| url | https://arxiv.org/abs/2412.20560 |