Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric


Authors: David A. Herron, William Ma and David Minda
Journal: Conform. Geom. Dyn. 12 (2008), 67-96
MSC (2000): Primary 30F45; Secondary :, 30C55, 30F30
DOI: https://doi.org/10.1090/S1088-4173-08-00178-1
Published electronically: June 10, 2008
MathSciNet review: 2410919
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.


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

  • [Ahl73] L.V. Ahlfors, Conformal invariants: Topics in geometric function theory, McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
  • [Ahl79] -, Complex analysis: An introduction to the theory of analytic functions of one complex variable, third ed., McGraw-Hill, New York, 1979. MR 0054016 (14:857a)
  • [Bet08] D. Betsakos, Estimation of the hyperbolic metric by using the punctured plane, Math. Z. (2008), to appear. MR 2377748
  • [BP78] A.F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. 18 (1978), no. 2, 475-483. MR 518232 (80a:30020)
  • [Car60] C. Carathéodory, Theory of functions of a complex variable, $ 2^{\rm nd}$ English ed., vol. 2, Chelsea Publ. Co., New York, 1960.
  • [Fer88] J. Ferrand, A characterization of quasiconformal mappings by the behavior of a function of three points, Complex Analysis, Joensuu 1987 (Berlin), Lecture Notes in Math., no. 1351, Springer-Verlag, 1988, pp. 110-123. MR 982077 (89m:30040)
  • [GL01] F.P. Gardiner and N. Lakic, Comparing Poincaré densities, Ann. of Math. (2) 154 (2001), no. 2, 245-267. MR 1865971 (2003c:30046)
  • [Hei62] M. Heins, On a class of conformal metrics, Nagoya Math. J. 30 (1962), 1-60. MR 0143901 (26:1451)
  • [Hej74] D.A. Hejhal, Universal covering maps for variable regions, Math. Z 137 (1974), 7-20. MR 0349989 (50:2482)
  • [Hem79] J.A. Hempel, The Poincaré metric on the twice punctured plane and the theorems of Landau and Schottky, J. London Math. Soc. 20 (1979), 435-445. MR 561135 (81c:30025)
  • [HIM08] D.A. Herron, Z. Ibragimov, and D. Minda, Geodesics and curvature of Möbius invariant metrics, Rocky Mountain J. Math. (2008), vol. 38, no. 3, pp. 891-921.
  • [Jen81] J.A. Jenkins, On explicit bounds in Landau's theorem II, Canad. J. Math. 33 (1981), 559-562. MR 627642 (83a:30026)
  • [Min82] D. Minda, Lower bounds for the hyperbolic metric in convex regions, Rocky Mountain J. Math. 12 (1982), 471-479. MR 692577 (84j:30039)
  • [Min87] -, Inequalities for the hyperbolic metric and applications to geometric function theory, Complex Analysis, I (College Park, MD, 1985-1986) (Berlin), Lecture Notes in Math., no. 1275, Springer-Verlag, 1987, pp. 235-252. MR 922304 (89d:30029)
  • [Neh75] Z. Nehari, Conformal mapping, Dover Publ., Inc., New York, 1975. MR 0377031 (51:13206)
  • [Pom79] Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192-199. MR 534933 (80j:30073)
  • [Pom84] -, On uniformly perfect sets and fuchsian groups, Analysis 4 (1984), no. 3-4, 299-321. MR 780609 (86e:30044)
  • [Sol97] A.Yu. Solynin, Functional inequalities via polarization, St. Petersburg Math. J. 8 (1997), 1015-1038. MR 1458141 (98e:30001a)
  • [Sol99a] -, Ordering of set, hyperbolic metrics, and harmonic measures, J. Math. Sci. 95 (1999), no. 3, 2256-2266. MR 1691288 (2000d:30068)
  • [Sol99b] -, Radial projection and the Poincaré metric, J. Math. Sci. 95 (1999), no. 3, 2267-2275. MR 1691289 (2000c:30086)
  • [SV01] A.Yu. Solynin and M. Vuorinen, Estimates for the hyperbolic metric of the punctured plane and applications, Israel J. Math. 124 (2001), 29-60. MR 1856503 (2002j:30071)
  • [SV05] T. Sugawa and M. Vuorinen, Some inequalities for the Poincaré metric of plane domains, Mathematische Zeitschrift 250 (2005), no. 4, 885-906. MR 2180380 (2006g:30075)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F45, :, 30C55, 30F30

Retrieve articles in all journals with MSC (2000): 30F45, :, 30C55, 30F30


Additional Information

David A. Herron
Affiliation: Department of Mathematical Sciences, 839 Old Chemistry Building, P.O. Box 210025, Cincinnati, Ohio 45221-0025
Email: David.Herron@math.UC.edu

William Ma
Affiliation: School of Integrated Studies, Pennsylvania College of Technology, Williamsport, Pennsylvania 17701
Email: wma@pct.edu

David Minda
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221
Email: david.minda@math.uc.edu

DOI: https://doi.org/10.1090/S1088-4173-08-00178-1
Keywords: M\"obius metrics, Poincar\'e hyperbolic metric, uniformly perfect
Received by editor(s): November 30, 2007
Published electronically: June 10, 2008
Additional Notes: The first and third authors were supported by the Charles Phelps Taft Research Center.
Dedicated: Dedicated to Roger Barnard on the occasion of his $65^{th}$ birthday.
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society