Quasihyperbolic geodesics in John domains in R^n

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.