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Gromov hyperbolicity and unbounded uniform domains

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Abstract

This paper focuses on Gromov hyperbolic characterizations of unbounded uniform domains. Let \(G\subsetneq \mathbb {R}^n\) be an unbounded domain. We prove that the following conditions are quantitatively equivalent: (1) G is uniform; (2) G is Gromov hyperbolic with respect to the quasihyperbolic metric and linearly locally connected; (3) G is Gromov hyperbolic with respect to the quasihyperbolic metric and there exists a naturally quasisymmetric correspondence between its Euclidean boundary and the punctured Gromov boundary equipped with a Hamenstädt metric (defined by using a Busemann function). As an application, we investigate the boundary quasisymmetric extensions of quasiconformal mappings, and of more generally rough quasi-isometries between unbounded domains with respect to the quasihyperbolic metrics.

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Qingshan Zhou, Yuehui He & Tiantian Guan

  2. Technion - Israel Institute of Technology, Guangdong Technion, Shantou, 515063, Guangdong, People’s Republic of China

    Antti Rasila

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  1. Qingshan Zhou
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  2. Yuehui He
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  3. Antti Rasila
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  4. Tiantian Guan
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Correspondence to Yuehui He.

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Qingshan Zhou was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012441). Yuehui He was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515111136). Antti Rasila was supported by NSF of China (No. 11971124), and NSF of Guangdong province (No. 2021A1515010326). Tiantian Guan was supported by NNSF of China (No. 12201115).

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Zhou, Q., He, Y., Rasila, A. et al. Gromov hyperbolicity and unbounded uniform domains. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01546-2

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