A Gromov hyperbolic metric vs the hyperbolic and other related metrics. (English) Zbl 1402.30039
Summary: We mainly consider two metrics: a Gromov hyperbolic metric and a scale-invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls.
MSC:
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
| 51M05 | Euclidean geometries (general) and generalizations |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
References:
| [1] | Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, Oxford (1997) · Zbl 0885.30012 |
| [2] | Beardon, A.F.: Geometry of Discrete Groups. Springer, New York (1995) |
| [3] | Beardon, AF; Duren, P (ed.); Heinonen, J (ed.); Osgood, B (ed.); Palka, B (ed.), The Apollonian metric of a domain in \(\mathbb{R}^n\), 91-108, (1998), New York · Zbl 0890.30030 · doi:10.1007/978-1-4612-0605-7_8 |
| [4] | Borovikova, M; Ibragimov, Z, Convex bodies of constant width and the Apollonian metric, Bull. Malays. Math. Sci. Soc., 31, 117-128, (2008) · Zbl 1160.52003 |
| [5] | Bonk, M; Schramm, O, Embeddings in Gromov hyperbolic spaces, Geom. Funct. Anal., 10, 266-306, (2000) · Zbl 0972.53021 · doi:10.1007/s000390050009 |
| [6] | Chen, J; Hariri, P; Klén, R; Vuorinen, M, Lipschitz conditions, triangular ratio metric and quasiconformal maps, Ann. Acad. Sci. Fenn. Math., 40, 683-709, (2015) · Zbl 1374.30069 · doi:10.5186/aasfm.2015.4039 |
| [7] | Gehring, FW; Osgood, BG, Uniform domains and the quasihyperbolic metric, J. Anal. Math., 36, 50-74, (1979) · Zbl 0449.30012 · doi:10.1007/BF02798768 |
| [8] | Gehring, FW; Palka, BP, Quasiconformally homogeneous domains, J. Anal. Math., 30, 172-199, (1976) · Zbl 0349.30019 · doi:10.1007/BF02786713 |
| [9] | Gromov, M.: Hyperbolic Groups, Essays in Group Theory, 75-263, Mathematical Sciences Research Institute Publications, vol. 8. Springer, New York (1987) · Zbl 0634.20015 |
| [10] | Hariri, P; Klén, R; Vuorinen, M; Zhang, XH, Some remarks on the Cassinian metric, Publ. Math. Debrecen., 90, 269-285, (2017) · Zbl 1399.51009 · doi:10.5486/PMD.2017.7386 |
| [11] | Hästö, P, A new weighted metric: the relative metric. I, J. Math. Anal. Appl., 274, 38-58, (2002) · Zbl 1019.54011 · doi:10.1016/S0022-247X(02)00219-6 |
| [12] | Hästö, P, The Apollonian metric: uniformity and quasiconvexity, Ann. Acad. Sci. Fenn. Math., 28, 385-414, (2003) · Zbl 1029.30027 |
| [13] | Hästö, P, Gromov hyperbolicity of the \(j_G\) and \(\tilde{j}_G\) metrics, Proc. Am. Math. Soc., 134, 1137-1142, (2005) · Zbl 1089.30044 · doi:10.1090/S0002-9939-05-08053-6 |
| [14] | Hästö, P; Linden, H, Isometries of the half-Apollonian metric, Complex Var. Theory Appl., 49, 405-415, (2004) · Zbl 1063.30042 |
| [15] | Ibragimov, Z, On the Apollonian metric of domains in \(\overline{\mathbb{R}^n}\), Complex Var. Theory Appl., 48, 837-855, (2003) · Zbl 1043.30026 |
| [16] | Ibragimov, Z, The Cassinian metric of a domain in \(\bar{\mathbb{R}}^n\), Uzbek. Mat. Zh., 1, 53-67, (2009) |
| [17] | Ibragimov, Z, Hyperbolizing metric spaces, Proc. Am. Math. Soc., 139, 4401-4407, (2011) · Zbl 1244.30066 · doi:10.1090/S0002-9939-2011-10857-8 |
| [18] | Ibragimov, Z, Hyperbolizing hyperspaces, Michigan Math. J., 60, 215-239, (2011) · Zbl 1220.30075 · doi:10.1307/mmj/1301586312 |
| [19] | Ibragimov, Z, A scale-invariant Cassinian metric, J. Anal., 24, 111-129, (2016) · Zbl 1372.30045 · doi:10.1007/s41478-016-0018-1 |
| [20] | Ibragimov, Z., Sahoo, S.K.: Invariant Cassinian metrics, Manuscript |
| [21] | Ibragimov, Z; Mohapatra, MR; Sahoo, SK; Zhang, X-H, Geometry of the Cassinian metric and its inner metric, Bull. Malays. Math. Sci. Soc., 40, 361-372, (2017) · Zbl 1366.30007 · doi:10.1007/s40840-015-0246-6 |
| [22] | Klén, R; Mohapatra, MR; Sahoo, SK, Geometric properties of the Cassinian metric, Math. Nachr., 290, 1531-1543, (2017) · Zbl 1392.30005 · doi:10.1002/mana.201600117 |
| [23] | Seittenranta, P, Möbius-invariant metrics, Math. Proc. Camb. Philos. Soc., 125, 511-533, (1999) · Zbl 0917.30015 · doi:10.1017/S0305004198002904 |
| [24] | Väisälä, J, Gromov hyperbolic spaces, Expo. Math., 23, 187-231, (2005) · Zbl 1087.53039 · doi:10.1016/j.exmath.2005.01.010 |
| [25] | Vuorinen, M, Conformal invariants and quasiregular mappings, J. Anal. Math., 45, 69-115, (1985) · Zbl 0601.30025 · doi:10.1007/BF02792546 |
| [26] | Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Note in Mathematics. Springer, Berlin (1988) · Zbl 0646.30025 · doi:10.1007/BFb0077904 |
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