Hyperbolic and Uniform Domains in Banach Spaces

A domain G in a Banach space is said to be δ-hyperbolic if it is a Gromov δ-hyperbolic space in the quasihyperbolic metric. Then G has the Gromov boundary ∂∗G and the norm boundary ∂G. We show that the following properties are quantitatively equivalent: (1) G is C-uniform. (2) G is δ-hyperbolic and there is a natural bijective map G ∪ ∂∗G → G ∪ ∂G, which is η-quasimöbius rel ∂∗G. (3) G is δ-hyperbolic and there is a natural η-quasimöbius homeomorphism ∂∗G→ ∂G. In a euclidean space, this improves a result of Bonk-Heinonen-Koskela, whose estimates depend on dimension and on a base point. MSC Subject Classification: 30C65, 53C23.