Boundary Properties of Gromov Hyperbolic Hölder Domains

Let Ω ⊂ Rn be a Gromov hyperbolic domain which satisfies a quasihyperbolic boundary condition. In this paper we prove that there is a bi-Hölder identification between the internal boundary of Ω and the Gromov boundary endowed with a visual metric by using a diameter type Gehring–Hayman inequality and also the uniformization of Bonk–Heinonen–Koskela. As an application, we establish the internal boundary continuity not only for quasiconformal homeomorphisms, but also for rough quasi-isometries between the domains with respect to the quasihyperbolic metrics.