Averaging one-point hyperbolic-type metrics
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- by Asuman Güven Aksoy, Zair Ibragimov and Wesley Whiting PDF
- Proc. Amer. Math. Soc. 146 (2018), 5205-5218 Request permission
Abstract:
It is known that the $\tilde \jmath$-metric, the half-Apollonian metric, and the scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point), and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that is Gromov hyperbolic. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of boundary points used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic.References
- A. F. Beardon, The Apollonian metric of a domain in $\textbf {R}^n$, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) Springer, New York, 1998, pp. 91–108. MR 1488447
- A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9–56. MR 2492498
- Mario Bonk, Quasiconformal geometry of fractals, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1349–1373. MR 2275649
- Mario Bonk, Juha Heinonen, and Pekka Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), viii+99. MR 1829896
- Oleksiy Dovgoshey, Parisa Hariri, and Matti Vuorinen, Comparison theorems for hyperbolic type metrics, Complex Var. Elliptic Equ. 61 (2016), no. 11, 1464–1480. MR 3513361, DOI 10.1080/17476933.2016.1182517
- 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
- Jacqueline Ferrand, A characterization of quasiconformal mappings by the behaviour of a function of three points, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 110–123. MR 982077, DOI 10.1007/BFb0081247
- 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
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- 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
- Peter Hästö, Zair Ibragimov, and Henri Lindén, Isometries of relative metrics, Comput. Methods Funct. Theory 6 (2006), no. 1, 15–28. MR 2241030, DOI 10.1007/BF03321114
- Peter Hästö and Henri Lindén, Isometries of the half-Apollonian metric, Complex Var. Theory Appl. 49 (2004), no. 6, 405–415. MR 2073171, DOI 10.1080/02781070410001712702
- David A. Herron, Universal convexity for quasihyperbolic type metrics, Conform. Geom. Dyn. 20 (2016), 1–24. MR 3463280, DOI 10.1090/ecgd/288
- David A. Herron and Poranee K. Julian, Ferrand’s Möbius invariant metric, J. Anal. 21 (2013), 101–121. MR 3408021
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Zair Ibragimov, On the Apollonian metric of domains in $\overline {\Bbb R}{}^n$, Complex Var. Theory Appl. 48 (2003), no. 10, 837–855. MR 2014392, DOI 10.1080/02781070310001015107
- Zair Ibragimov, Hyperbolizing hyperspaces, Michigan Math. J. 60 (2011), no. 1, 215–239. MR 2785872, DOI 10.1307/mmj/1301586312
- Zair Ibragimov, A scale-invariant Cassinian metric, J. Anal. 24 (2016), no. 1, 111–129. MR 3755814, DOI 10.1007/s41478-016-0018-1
- Z. Ibragimov, Möbius invariant Cassinian metric, Bulletin, Malaysian Math. Sci. Soc. (to appear). DOI 10.1007/s40840-017-0550-4.
- Ravi S. Kulkarni and Ulrich Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), no. 1, 89–129. MR 1273468, DOI 10.1007/BF02572311
- Henri Lindén, Hyperbolic-type metrics, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 151–164. MR 2492502
- Pasi Seittenranta, Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 511–533. MR 1656825, DOI 10.1017/S0305004198002904
- Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
- Jussi Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231. MR 2164775, DOI 10.1016/j.exmath.2005.01.010
Additional Information
- Asuman Güven Aksoy
- Affiliation: Department of Mathematics, Claremont McKenna College, Claremont, California 91711
- MR Author ID: 24095
- Email: aaksoy@cmc.edu
- Zair Ibragimov
- Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92831
- Email: zibragimov@fullerton.edu
- Wesley Whiting
- Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92831
- Email: weswhiting@gmail.com
- Received by editor(s): September 12, 2017
- Received by editor(s) in revised form: February 5, 2018, and March 22, 2018
- Published electronically: September 4, 2018
- Communicated by: Jeremy T. Tyson
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5205-5218
- MSC (2010): Primary 30F45; Secondary 51F99, 30C99
- DOI: https://doi.org/10.1090/proc/14173
- MathSciNet review: 3866859