Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Universal convexity for quasihyperbolic type metrics

Author: David A. Herron
Journal: Conform. Geom. Dyn. 20 (2016), 1-24
MSC (2010): Primary 53A30; Secondary 53C22, 51F99, 30C65, 30F45
Published electronically: February 24, 2016
MathSciNet review: 3463280
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the open sets in the sphere that are geodesically convex in any containing domain with respect to various conformal metrics.

References [Enhancements On Off] (What's this?)

  • [Bea83] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • [BM08] A. F. Beardon and D. Minda, Conformal automorphisms of finitely connected regions, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 37-73. MR 2458798 (2010c:30061),
  • [Ber87] M. Berger, Geometry I, Springer, Berlin, 1987.
  • [BH99] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [Fer88] 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 (89m:30040),
  • [Fli83] Barbara Brown Flinn, Hyperbolic convexity and level sets of analytic functions, Indiana Univ. Math. J. 32 (1983), no. 6, 831-841. MR 721566 (85b:30010),
  • [HIM08] David A. Herron, Zair Ibragimov, and David Minda, Geodesics and curvature of Möbius invariant metrics, Rocky Mountain J. Math. 38 (2008), no. 3, 891-921. MR 2426525 (2009h:30079),
  • [HJ13] David A. Herron and Poranee K. Julian, Ferrand's Möbius invariant metric, J. Anal. 21 (2013), 101-121. MR 3408021
  • [HMM03] D. A. Herron, W. Ma, and D. Minda, A Möbius invariant metric for regions on the Riemann sphere, Future trends in geometric function theory, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 92, University of Jyväskylä, Jyväskylä, 2003. MR 2060255 (2004k:30001)
  • [HMM05] David A. Herron, William Ma, and David Minda, Estimates for conformal metric ratios, Comput. Methods Funct. Theory 5 (2005), no. 2, 323-345. MR 2205417 (2006j:30082),
  • [HMM08] David A. Herron, William Ma, and David Minda, Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric, Conform. Geom. Dyn. 12 (2008), 67-96. MR 2410919 (2009b:30090),
  • [Jør56] Vilhelm Jørgensen, On an inequality for the hyperbolic measure and its applications in the theory of functions, Math. Scand. 4 (1956), 113-124. MR 0084584 (18,885a)
  • [Jul12] P. K. Julian, Geometric Properties of the Ferrand Metric, Ph.D. thesis, University of Cincinnati, 2012.
  • [KP94] 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 (95b:53017),
  • [Mar85] Gaven J. Martin, Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric, Trans. Amer. Math. Soc. 292 (1985), no. 1, 169-191. MR 805959 (87a:30037),

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 53A30, 53C22, 51F99, 30C65, 30F45

Retrieve articles in all journals with MSC (2010): 53A30, 53C22, 51F99, 30C65, 30F45

Additional Information

David A. Herron
Affiliation: Department of Mathematical Sciences, French Hall West, PO Box 210025, Cincinnati Ohio 45221-0025

Keywords: Geodesic convexity, quasihyperbolic metric, Ferrand metric, Kulkarni-Pinkall metric, conformal metrics
Received by editor(s): November 10, 2015
Received by editor(s) in revised form: June 23, 2016
Published electronically: February 24, 2016
Dedicated: Dedicated to Taft Professor David Minda on the occasion of his retirement.
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society