Quasihyperbolic geodesics in John domains in R^n

TL;DR

This paper proves that quasihyperbolic geodesics in certain John domains are cone arcs, resolving open problems and establishing new properties of these domains related to quasihyperbolic uniformity.

Contribution

It demonstrates that in John domains homeomorphic to uniform domains via quasiconformal maps, quasihyperbolic geodesics are cone arcs, confirming open conjectures.

Findings

Quasihyperbolic geodesics are cone arcs in specified John domains.

Open problems by Heinonen and by Gehring, Hag, and Martio are positively answered.

Such domains are quasihyperbolic (b, λ)-uniform domains.

Abstract

In this paper, we prove that if $D\subset R^n$ is a John domain which is homeomorphic to a uniform domain via a quasiconformal mapping, then each quasihyperbolic geodesic in $D$ is a cone arc, which shows that the answer to one of open problems raised by Heinonen in \cite{H} is affirmative. This result also shows that the answer to the open problem raised by Gehring, Hag and Martio in \cite{Gm} is positive for John domains which are homeomorphic to uniform domains via uasiconformal mappings. As an application, we prove that if $D\subset R^n$ is a John domain which is homeomorphic to a uniform domain, then $D$ must be a quasihyperbolic $(b, \lambda)$-uniform domain.