Abstract
The main results of this paper are characterizations of John disks-the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain’s hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property.
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References
Beardon A F, The Apollonian metric of a domain in ℝn, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) pp. 91–108 (New York: Springer) (1998)
Broch O J, Geometry of John disks, Ph.D. Thesis, NTNU (2004)
Broch O J, On reflections in Jordan curves, Conform. Geom. Dyn. 11 (2007) 12–28
Gehring F W, Characterizations of quasidisks, quasiconformal geometry and dynamics (Lublin, 1996) pp. 11–41, Banach Center Publ., 48 (Warsaw: Polish Acad. Sci.) (1999)
Gehring F W and Hag K, Remarks on uniform and quasiconformal extension domains, Complex Variables Theory Appl. 9(2–3) (1987) 175–188
Gehring F W and Hag K, The Apollonian metric and quasiconformal mappings, in the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) pp. 143–163, Contemp. Math. 256 (RI: Amer. Math. Soc., Providence) (2000)
Gehring F W, Hag K and Martio O, Quasihyperbolic geodesic in John domains, Math. Scand. 65(1) (1989) 75–92
Gehring F W and Osgood B O, Uniform domains and the quasi-hyperbolic metric, J. Analyse Math. 36 (1979) 50–74
Gehring F W and Palka B P, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976) 172–199
Herron D A, John domains and the quasihyperbolic metric, Complex Variables Theory Appl. 39(4) (1999) 327–334
John F, Rotation and strain, Comm. Pure Appl. Math. 14 (1961) 391–413
Kim K and Langmeyer N, Harmonic measure and hyperbolic distance in John disks, Math. Scand. 83(2) (1998) 283–299
Martio O and Sarvas J Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4(2) (1979) 383–401
Näkki R and Väisälä J, John disks, Exposition. Math. 9(1) (1991) 3–43
Pommerenke Ch., Boundary behaviour of conformal Maps. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 299 (Berlin: Springer-Verlag) (1992) x+300 pp.
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Dedicated to Matti Vuorinen on the occasion of his 60th birthday
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Huang, M., Ponnusamy, S. & Wang, X. John disks, the Apollonian metric, and min-max properties. Proc Math Sci 120, 83–96 (2010). https://doi.org/10.1007/s12044-010-0002-7
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DOI: https://doi.org/10.1007/s12044-010-0002-7