Gromov hyperbolicity and unbounded uniform domains

Pub Date : 2024-03-16 DOI:10.1007/s00229-024-01546-2
Qingshan Zhou, Yuehui He, Antti Rasila, Tiantian Guan
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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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格罗莫夫双曲性与无界均匀域
本文主要研究无界均匀域的格罗莫夫双曲特征。让 \(G\subsetneq \mathbb {R}^n\) 是一个无界域。我们证明以下条件在量上是等价的:(1) G 是均匀的;(2) G 相对于准双曲度量是格罗莫夫双曲的,并且是线性局部相连的;(3) G 相对于准双曲度量是格罗莫夫双曲的,并且其欧几里得边界与配备哈门施塔特度量(通过使用布斯曼函数定义)的点状格罗莫夫边界之间存在自然的准对称对应关系。作为一种应用,我们研究了准共形映射的边界准对称扩展,以及更一般的无界域之间关于准超双曲度量的粗糙准等距。
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