Quasi-isometric and weakly quasisymmetric maps between locally compact non-complete metric spaces
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The aim of this paper is to show that there exists a weakly quasisymmetric homeomorphism $f:(X, d)\to (Y, d')$ between two locally compact non-complete metric spaces such that $f:(X, d_h)\to (Y, d'_h)$ is not quasi-isometric, where $d_h$ denotes the Gromov hyperbolic metric with respect to the metric $d$ introduced by Ibragimov in $2011$. This result shows that the answer to the related question asked by Ibragimov in 2013 is negative.
Abstract
The aim of this paper is to show that there exists a weakly quasisymmetric homeomorphism $f:(X, d)\to (Y, d')$ between two locally compact non-complete metric spaces such that $f:(X, d_h)\to (Y, d'_h)$ is not quasi-isometric, where $d_h$ denotes the Gromov hyperbolic metric with respect to the metric $d$ introduced by Ibragimov in $2011$. This result shows that the answer to the related question asked by Ibragimov in 2013 is negative.
참고문헌 (2)
- Z. Ibragimov, "Hyperbolizing metric spaces", Proc. Amer. Math. Soc., 139(12), pp.4401~4407 (2011)[DOI]
- Z. Ibragimov, "Hyperbolization of locally compact non-complete metric spaces, in In the tradition of Ahlfors-Bers. VI, Contemp. Math., 590", pp.51~59 (2013)