On reflections in Jordan curves

Broch, Ole

doi:10.1090/S1088-4173-07-00158-0

On reflections in Jordan curves
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by Ole Jacob Broch

Conform. Geom. Dyn. 11 (2007), 12-28

DOI: https://doi.org/10.1090/S1088-4173-07-00158-0

Published electronically: March 1, 2007

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Abstract:

A purely geometric method for constructing reflections in Jordan curves on the Riemann sphere based on hyperbolic geodesics is introduced. It is then possible to investigate the relations between the geometry of a Jordan domain $D$ and the properties of the reflection by studying properties of hyperbolic geodesics. This idea is used to characterize unbounded Jordan John domains in terms of reflections satisfying a kind of Lipschitz condition.

References

Lars V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291–301. MR 154978, DOI 10.1007/BF02391816
Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360
O. J. Broch, Geometry of John disks, Dr. scient.-avhandling, Doctoral Theses at NTNU 2005:21, NTNU, 2005.
Ole Jacob Broch, Extension of internally bilipschitz maps in John disks, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 13–30. MR 2210105
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Clifford J. Earle and Subhashis Nag, Conformally natural reflections in Jordan curves with applications to Teichmüller spaces, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 179–194. MR 955840, DOI 10.1007/978-1-4613-9611-6_{1}3
Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906, DOI 10.1090/surv/076
Frederick W. Gehring, Characteristic properties of quasidisks, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 84, Presses de l’Université de Montréal, Montreal, Que., 1982. MR 674294
F. W. Gehring and K. Hag, Remarks on uniform and quasiconformal extension domains, Complex Variables Theory Appl. 9 (1987), no. 2-3, 175–188. MR 923218, DOI 10.1080/17476938708814261
F. W. Gehring and K. Hag, Reflections on reflections in quasidisks, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 81–90. MR 1886615
Frederick W. Gehring and Kari Hag, Sewing homeomorphisms and quasidisks, Comput. Methods Funct. Theory 3 (2003), no. 1-2, [On table of contents: 2004], 143–150. MR 2082011, DOI 10.1007/BF03321031
F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand. 65 (1989), no. 1, 75–92. MR 1051825, DOI 10.7146/math.scand.a-12267
F. W. Gehring and W. K. Hayman, An inequality in the theory of conformal mapping, J. Math. Pures Appl. (9) 41 (1962), 353–361. MR 148884
F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
Kari Hag, What is a disk?, Quasiconformal geometry and dynamics (Lublin, 1996) Banach Center Publ., vol. 48, Polish Acad. Sci. Inst. Math., Warsaw, 1999, pp. 43–53. MR 1709973
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Kiwon Kim, Relations between certain domains in the complex plane and polynomial approximation in the domains, Bull. Korean Math. Soc. 39 (2002), no. 4, 687–704. MR 1939567, DOI 10.4134/BKMS.2002.39.4.687
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
Jason Miller, Sector reflections in the plane, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 219–225. MR 2173362
Raimo Näkki and Jussi Väisälä, John disks, Exposition. Math. 9 (1991), no. 1, 3–43. MR 1101948
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
K. Ryu, Properties of John disks, Ph.D. thesis, University of Michigan, 1991.
P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
P. Tukia and J. Väisälä, Quasiconformal extension from dimension $n$ to $n+1$, Ann. of Math. (2) 115 (1982), no. 2, 331–348. MR 647809, DOI 10.2307/1971394