Gromov hyperbolicity and unbounded uniform domains

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.