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On reflections in Jordan curves


Author: Ole Jacob Broch
Journal: Conform. Geom. Dyn. 11 (2007), 12-28
MSC (2000): Primary 30C20; Secondary 30C99
DOI: https://doi.org/10.1090/S1088-4173-07-00158-0
Published electronically: March 1, 2007
MathSciNet review: 2295995
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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.


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  • 1. L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. MR 0154978 (27:4921)
  • 2. L. V. Ahlfors, Lectures on Quasiconformal Mappings, 2nd ed. University Lecture Series 38, American Mathematical Society, 2006. MR 2241787
  • 3. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142. MR 0086869 (19:258c)
  • 4. O. J. Broch, Geometry of John disks, Dr. scient.-avhandling, Doctoral Theses at NTNU 2005:21, NTNU, 2005.
  • 5. O. J. Broch, Extension of internally bilipschitz maps in John disks, Ann. Acad. Sci. Fenn. Math. 31 (2006), 13-30. MR 2210105 (2006m:30015)
  • 6. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23-48. MR 857678 (87j:30041)
  • 7. C. J. Earle and S. Nag, Conformally natural reflections in Jordan curves with applications to Teichmüller spaces, In: Holomorphic Functions and Moduli, II, Math. Sci. Res. Inst. Publ. 11, Springer, 1988, 179-193. MR 955840 (89i:30019)
  • 8. F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs 76, American Mathematical Society, 2000. MR 1730906 (2001d:32016)
  • 9. F. W. Gehring, Characteristic Properties of Quasidisks, Les presses de l'université de Montréal (1982). MR 674294 (84a:30036)
  • 10. F. W. Gehring and K. Hag, Remarks on Uniform and Quasiconformal Extension Domains, Complex Variables 9 (1987), 245-263. MR 923218 (89b:30019)
  • 11. F. W. Gehring and K. Hag, Reflections on reflections in quasidisks, Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday. Report. Univ. Jyväskylä. 83 (2001), 81-90. MR 1886615 (2003d:30058)
  • 12. F. W. Gehring and K. Hag, Sewing Homeomorphisms and Quasidisks, Comput. Methods Funct. Theory. 3 (2003), 143-150. MR 2082011 (2005e:30033)
  • 13. F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand. 65 (1989), 75-92. MR 1051825 (91i:30014)
  • 14. F. W. Gehring and W. K. Hayman, An inequality in the theory of conformal mapping, J. Math. Pures Appl. 41 (1962), 353-361. MR 0148884 (26:6381)
  • 15. F. W. Gehring and B. Palka, Quasiconformally homogeneous domains, Journal d'Analyse Math. 30 (1976), 172-199. MR 0437753 (55:10676)
  • 16. K. Hag, What is a disk? Banach Center Publications. 48 (1999), 43-53. MR 1709973 (2001d:30025)
  • 17. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer 2001. MR 1800917 (2002c:30028)
  • 18. K. Kim, Relations between certain domains in the complex plane and polynomial approximation in the domains, Bull. Korean Math. Soc. 39 (2002), 687-704. MR 1939567 (2003i:30011)
  • 19. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed. Springer-Verlag 1973. MR 0344463 (49:9202)
  • 20. J. Miller, Sector reflections in the plane, Ann. Acad. Sci. Fenn. Math. 30 (2005), 219-225. MR 2173362 (2006e:54029)
  • 21. R. Näkki and J. Väisälä, John disks, Expo. Math. 9 (1991), 3-43. MR 1101948 (92i:30021)
  • 22. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, 1992. MR 1217706 (95b:30008)
  • 23. K. Ryu, Properties of John disks, Ph.D. thesis, University of Michigan, 1991.
  • 24. P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. MR 595180 (82g:30038)
  • 25. P. Tukia and J. Väisälä, Quasiconformal extension from dimension $ n$ to $ n+1$, Ann. of Math. 115 (1982), 331-348. MR 647809 (84i:30030)

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Additional Information

Ole Jacob Broch
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Email: olejacb@math.ntnu.no

DOI: https://doi.org/10.1090/S1088-4173-07-00158-0
Keywords: John domain, reflection, hyperbolic geodesic
Received by editor(s): August 24, 2006
Published electronically: March 1, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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