Universal convexity for quasihyperbolic type metrics
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- by David A. Herron
- Conform. Geom. Dyn. 20 (2016), 1-24
- DOI: https://doi.org/10.1090/ecgd/288
- Published electronically: February 24, 2016
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Abstract:
We characterize the open sets in the sphere that are geodesically convex in any containing domain with respect to various conformal metrics.References
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- 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, DOI 10.1017/CBO9780511735233.004
- M. Berger, Geometry I, Springer, Berlin, 1987.
- 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, DOI 10.1007/978-3-662-12494-9
- 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
- Barbara Brown Flinn, Hyperbolic convexity and level sets of analytic functions, Indiana Univ. Math. J. 32 (1983), no. 6, 831–841. MR 721566, DOI 10.1512/iumj.1983.32.32056
- 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, DOI 10.1216/RMJ-2008-38-3-891
- David A. Herron and Poranee K. Julian, Ferrand’s Möbius invariant metric, J. Anal. 21 (2013), 101–121. MR 3408021
- David Herron (ed.), Future trends in geometric function theory, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 92, University of Jyväskylä, Jyväskylä, 2003. MR 2060255
- 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, DOI 10.1007/BF03321101
- 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, DOI 10.1090/S1088-4173-08-00178-1
- Vilhelm Jørgensen, On an inequality for the hyperbolic measure and its applications in the theory of functions, Math. Scand. 4 (1956), 113–124. MR 84584, DOI 10.7146/math.scand.a-10460
- P. K. Julian, Geometric Properties of the Ferrand Metric, Ph.D. thesis, University of Cincinnati, 2012.
- 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
- 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, DOI 10.1090/S0002-9947-1985-0805959-2
Bibliographic Information
- David A. Herron
- Affiliation: Department of Mathematical Sciences, French Hall West, PO Box 210025, Cincinnati Ohio 45221-0025
- MR Author ID: 85095
- Email: David.Herron@UC.edu
- Received by editor(s): November 10, 2015
- Received by editor(s) in revised form: June 23, 2016
- Published electronically: February 24, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Conform. Geom. Dyn. 20 (2016), 1-24
- MSC (2010): Primary 53A30; Secondary 53C22, 51F99, 30C65, 30F45
- DOI: https://doi.org/10.1090/ecgd/288
- MathSciNet review: 3463280
Dedicated: Dedicated to Taft Professor David Minda on the occasion of his retirement.