In this paper, we prove that if $D\subset R^n$ is a John domain which ishomeomorphic to a uniform domain via a quasiconformal mapping, then eachquasihyperbolic geodesic in $D$ is a cone arc, which shows that the answer toone of open problems raised by Heinonen in \cite{H} is affirmative. This resultalso shows that the answer to the open problem raised by Gehring, Hag andMartio in \cite{Gm} is positive for John domains which are homeomorphic touniform 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.
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机译：在本文中，我们证明如果$ D \ subset R ^ n $是一个John域，并且通过拟保形映射同形为一个均匀域，则$ D $中的每个拟双曲测地线都是圆锥弧，这表明对Heinonen在\ cite {H}中提出的公开问题是肯定的。该结果还表明，Gehring，Hag和Martio在\ cite {Gm}中提出的开放问题的答案对于John域是肯定的，而John域通过uasiconformal映射同胚到统一域。作为一个应用，我们证明如果$ D \ subset R ^ n $是一个John域，且同域对同一个域同胚，则$ D $必须是拟双曲$（b，\ lambda）$一致域。
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