Hyperbolizing metric spaces
HTML articles powered by AMS MathViewer
- by Zair Ibragimov
- Proc. Amer. Math. Soc. 139 (2011), 4401-4407
- DOI: https://doi.org/10.1090/S0002-9939-2011-10857-8
- Published electronically: April 25, 2011
- PDF | Request permission
Abstract:
It was proved by M. Bonk, J. Heinonen and P. Koskela that the quasihyperbolic metric hyperbolizes (in the sense of Gromov) uniform metric spaces. In this paper we introduce a new metric that hyperbolizes all locally compact noncomplete metric spaces. The metric is generic in the sense that (1) it can be defined on any metric space; (2) it preserves the quasiconformal geometry of the space; (3) it generalizes the $j$-metric, the hyperbolic cone metric and the hyperbolic metric of hyperspaces; and (4) it is quasi-isometric to the quasihyperbolic metric of uniform metric spaces. In particular, the Gromov hyperbolicity of these metrics also follows from that of our metric.References
- Mario Bonk, Juha Heinonen, and Pekka Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), viii+99. MR 1829896
- M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266–306. MR 1771428, DOI 10.1007/s000390050009
- F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
- F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
- Peter A. Hästö, Gromov hyperbolicity of the $j_G$ and $\~j_G$ metrics, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1137–1142. MR 2196049, DOI 10.1090/S0002-9939-05-08053-6
- Z. Ibragimov, A canonical $\delta$-hyperbolic metric for metric spaces, Abstracts, International Congress of Mathematicians, Hyderabad, India (2010), 209–210.
- Z. Ibragimov, Hyperbolizing hyperspaces, Michigan Math. J., vol. 60, no. 1 (2011) (to appear).
- Jussi Väisälä, Hyperbolic and uniform domains in Banach spaces, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 261–302. MR 2173365
Bibliographic Information
- Zair Ibragimov
- Affiliation: Department of Mathematics, California State University, Fullerton, California 92831
- Email: zibragimov@fullerton.edu
- Received by editor(s): September 9, 2010
- Received by editor(s) in revised form: October 20, 2010
- Published electronically: April 25, 2011
- Communicated by: Mario Bonk
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4401-4407
- MSC (2010): Primary 30F45; Secondary 53C23, 30C99
- DOI: https://doi.org/10.1090/S0002-9939-2011-10857-8
- MathSciNet review: 2823085
Dedicated: Dedicated to Fred Gehring on the occasion of his 85th birthday