Gromov hyperbolicity of the 𝑗̃_{𝐺} metric and boundary correspondence

Abstract

Let G ⊊ R n G\subsetneq \mathbb {R}^n be an open set. It is shown by Hästö that G G equipped with the j ~ G \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 , j ~ G ) (G, \tilde {j}_G) and its Euclidean boundary ∂ G \partial G . Both bounded and unbounded cases are in our considerations.