Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric

Herron, David; Ma, William; Minda, David

doi:10.1090/S1088-4173-08-00178-1

Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric
HTML articles powered by AMS MathViewer

by David A. Herron, William Ma and David Minda PDF

Conform. Geom. Dyn. 12 (2008), 67-96 Request permission

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

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0054016
Dimitrios Betsakos, Estimation of the hyperbolic metric by using the punctured plane, Math. Z. 259 (2008), no. 1, 187–196. MR 2377748, DOI 10.1007/s00209-007-0218-0
A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), no. 3, 475–483. MR 518232, DOI 10.1112/jlms/s2-18.3.475
C. Carathéodory, Theory of functions of a complex variable, $2^\textrm {nd}$ English ed., vol. 2, Chelsea Publ. Co., New York, 1960.
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
Frederick P. Gardiner and Nikola Lakic, Comparing Poincaré densities, Ann. of Math. (2) 154 (2001), no. 2, 245–267. MR 1865971, DOI 10.2307/3062097
Maurice Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60. MR 143901, DOI 10.1017/S002776300002376X
Dennis A. Hejhal, Universal covering maps for variable regions, Math. Z. 137 (1974), 7–20. MR 349989, DOI 10.1007/BF01213931
Joachim A. Hempel, The Poincaré metric on the twice punctured plane and the theorems of Landau and Schottky, J. London Math. Soc. (2) 20 (1979), no. 3, 435–445. MR 561135, DOI 10.1112/jlms/s2-20.3.435
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.
James A. Jenkins, On explicit bounds in Landau’s theorem. II, Canadian J. Math. 33 (1981), no. 3, 559–562. MR 627642, DOI 10.4153/CJM-1981-045-1
David Minda, Lower bounds for the hyperbolic metric in convex regions, Rocky Mountain J. Math. 13 (1983), no. 1, 61–69. MR 692577, DOI 10.1216/RMJ-1983-13-1-61
David Minda, Inequalities for the hyperbolic metric and applications to geometric function theory, Complex analysis, I (College Park, Md., 1985–86) Lecture Notes in Math., vol. 1275, Springer, Berlin, 1987, pp. 235–252. MR 922304, DOI 10.1007/BFb0078356
Zeev Nehari, Conformal mapping, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031
Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192–199. MR 534933, DOI 10.1007/BF01238490
Ch. Pommerenke, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), no. 3-4, 299–321. MR 780609, DOI 10.1524/anly.1984.4.34.299
A. Yu. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), no. 6, 148–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015–1038. MR 1458141
A. Yu. Solynin, Ordering of sets, hyperbolic metric, and harmonic measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 129–147, 230 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2256–2266. MR 1691288, DOI 10.1007/BF02172470
A. Yu. Solynin, Radial projection and the Poincaré metric, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 148–160, 230 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2267–2275. MR 1691289, DOI 10.1007/BF02172471
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, DOI 10.1007/BF02772606
Toshiyuki Sugawa and Matti Vuorinen, Some inequalities for the Poincaré metric of plane domains, Math. Z. 250 (2005), no. 4, 885–906. MR 2180380, DOI 10.1007/s00209-005-0782-0