Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access July 1, 2020

Chordal Hausdorff Convergence and Quasihyperbolic Distance

  • David A. Herron , Abigail Richard and Marie A. Snipes EMAIL logo


We study Hausdorff convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdorff convergence of quasihyperbolic distances (and other conformal distances).


[1] P. Alestalo, D. A. Trotsenko, and J. Väisälä, Isometric approximation, Israel J. Math. 125 (2001), 61–82.10.1007/BF02773375Search in Google Scholar

[2] A.F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983.10.1007/978-1-4612-1146-4Search in Google Scholar

[3] A.F. Beardon and D. Minda, Carathéodory kernel theorem and the hyperbolic metric, in preparation (2022), 1–35.Search in Google Scholar

[4] M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001), 1–99.Search in Google Scholar

[5] M. Bonk and B. Kleiner, Rigidity for quasi-Möbius group actions, J. Differential Geom. 61 (2002), no. 1, 81–106.Search in Google Scholar

[6] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266–306.10.1007/s000390050009Search in Google Scholar

[7] S.M. Buckley, D.A. Herron, and X. Xie, Metric space inversions, quasihyperbolic distance and uniform spaces, Indiana Univ. Math. J. 57 (2008), no. 2, 837–890.Search in Google Scholar

[8] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, American Mathematical Society, Providence, RI, 2001.10.1090/gsm/033Search in Google Scholar

[9] F.W. Gehring and B.P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199.10.1007/BF02786713Search in Google Scholar

[10] J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, Duke Math. J. 113 (2002), no. 3, 465—529.Search in Google Scholar

[11] D.A. Herron, Gromov-Hausdorff distance for pointed metric spaces, J. Analysis 24 (2016), 1–38.10.1007/s41478-016-0001-xSearch in Google Scholar

[12] D.A. Herron, Z. Ibragimov, and D. Minda, Geodesics and curvature of Möbius invariant metrics, Rocky Mountain J. Math. 38 (2008), no. 3, 891–921.Search in Google Scholar

[13] D.A. Herron and P.K. Julian, Ferrand’s Möbius invariant metric, J. Anaylsis 21 (2013), 101–121.Search in Google Scholar

[14] D.A. Herron, W. Ma, and D. Minda, A Möbius invariant metric for regions on the Riemann sphere, Future Trends in Geometric Function Theory, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 92, Univ. Jyväskylä, Jyväskylä, 2003, RNC Workshop held in Jyväskylä, June 15-18, 2003, pp. 101–118.Search in Google Scholar

[15] D.A. Herron, W. Ma, and D. Minda, Estimates for conformal metric ratios, Comput. Methods Function Theory 5 (2005), no. 2, 323–345.Search in Google Scholar

[16] J.G. Hocking and G.S. Young, Topology, Dover, New York, 1988.Search in Google Scholar

[17] R. Kulkarni and U. Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), 89–129.10.1007/BF02572311Search in Google Scholar

[18] K. Kuratowski, Topology, vol. II, Academic Press, New York and London, 1968.Search in Google Scholar

[19] H. Luiro, On the uniqueness of quasihyperbolic geodesics, arXiv (2015), 1–39.Search in Google Scholar

[20] R. Luisto. A characterization of BLD-mappings between metric spaces., J. Geom. Anal. 27 (2017), no. 3, 2081-–2097.10.1007/s12220-016-9752-5Search in Google Scholar

[21] J.R. Munkres, Topology, Prentice Hall, Upper Saddle River, NJ, 2000.Search in Google Scholar

[22] A.H. Richard, Quasihyperbolic distance, pointed Gromov-Hausdorff distance, and bounded uniform convergence, Ph.D. thesis, University of Cincinnati, 2019.Search in Google Scholar

[23] J. Väisälä, Exhaustions of John domains, Ann. Acad. Sci. Fenn. Math. 19 (1994), 47–57.Search in Google Scholar

[24] J. Väisälä, The free quasiworld. Freely quasiconformal and related maps in Banach spaces., Quasiconformal Geometry and Dynamics (Lublin, 1996) (Warsaw), vol. 48, Inst. Math., Polish of Academy Sciences, Banach Center Publ., 1999, pp. 55– 118.10.4064/-48-1-55-118Search in Google Scholar

[25] J. Väisälä, Quasihyperbolic geometry of domains in Hilbert spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 2, 559–578.Search in Google Scholar

[26] J. Väisälä, Quasihyperbolic geometry of planar domains, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 447–473.Search in Google Scholar

[27] G.T. Whyburn, Analytical topology, Colloquium Publications, no. 28, American Mathematical Society, New York, 1942.10.1090/coll/028Search in Google Scholar

[28] G.T. Whyburn, Topological analysis, Princeton University Press, Princeton, N.J., 1964.Search in Google Scholar

Received: 2020-01-12
Accepted: 2020-04-23
Published Online: 2020-07-01

© 2020 David A. Herron et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 4.2.2023 from
Scroll Up Arrow