Geometric characterizations of inner uniformity through Gromov hyperbolicity
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866916735991414784 |
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| author | Huang, Manzi Rasila, Antti Wang, Xiantao Zhou, Qingshan |
| author_facet | Huang, Manzi Rasila, Antti Wang, Xiantao Zhou, Qingshan |
| contents | In this paper, we study the characterization of inner uniformity of bounded domains $G$ in $\IR^n$, and prove that the following three conditions are equivalent: $(1)$ $G$ is inner uniform; $(2)$ $G$ is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; $(3)$ $G$ is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions $(1)$ and $(2)$, and the implication from $(2)$ to $(3)$ affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1706_05494 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Geometric characterizations of inner uniformity through Gromov hyperbolicity Huang, Manzi Rasila, Antti Wang, Xiantao Zhou, Qingshan Complex Variables Primary: 30C65, 30F45, 30L10, Secondary: 30C20 In this paper, we study the characterization of inner uniformity of bounded domains $G$ in $\IR^n$, and prove that the following three conditions are equivalent: $(1)$ $G$ is inner uniform; $(2)$ $G$ is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; $(3)$ $G$ is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions $(1)$ and $(2)$, and the implication from $(2)$ to $(3)$ affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001. |
| title | Geometric characterizations of inner uniformity through Gromov hyperbolicity |
| topic | Complex Variables Primary: 30C65, 30F45, 30L10, Secondary: 30C20 |
| url | https://arxiv.org/abs/1706.05494 |