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Gromov hyperbolicity of the \(\tilde{j}_G\) metric and boundary correspondence. (English) Zbl 1493.30088

Summary: Let \(G\subsetneq \mathbb{R}^n\) be an open set. It is shown by P. A. Hästö [Proc. Am. Math. Soc. 134, No. 4, 1137–1142 (2006; Zbl 1089.30044)] that \(G\) equipped with the \(\tilde{j}_G\) metric is Gromov hyperbolic. The purpose of this paper is to show that there is a natural quasisymmetric correspondence between the Gromov boundary of \((G,\tilde{j}_G)\) and its Euclidean boundary \(\partial G\). Both bounded and unbounded cases are in our considerations.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
30C99 Geometric function theory

Citations:

Zbl 1089.30044
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References:

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