GENERALIZED JOHN GROMOV HYPERBOLIC DOMAINS AND EXTENSIONS OF MAPS

Let Omega subset of R-n be a Gromov hyperbolic, phi-length John domain. We show that there is a uniformly continuous identification between the inner boundary of Omega and the Gromov boundary endowed with a visual metric. By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.

Automated summary

A uniformly continuous identification between the inner boundary of Omega and the Gromov boundary with a visual metric is shown, leading to boundary continuity for both quasiconformal homeomorphisms and rough quasi-isometries between domains equipped with quasihyperbolic metrics.