Tohoku Mathematical Journal

Conformal invariants of QED domains

Yu-Liang Shen

Full-text: Open access

Abstract

Given a Jordan domain $\Omega$ in the extended complex plane $\overline{\kern-1.5pt\Bbb C}$, denote by $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ the boundary quasiextremal distance constant, quasiextremal distance constant and quasiconformal reflection constant of $\Omega$, respectively. It is known that $M_b(\Omega)\le M(\Omega)\le R(\Omega)+1$. In this paper, we will give some further relations among $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ by introducing and studying some other closely related constants. Particularly, we will give a necessary and sufficient condition for $M_b(\Omega)=R(\Omega)+1$ and show that $M(\Omega)<R(\Omega)+1$ for all asymptotically conformal extension domains other than disks. This gives an affirmative answer to a question asked by Yang, showing that the conjecture $M(\Omega)=R(\Omega)+1$ by Garnett and Yang is not true for all asymptotically conformal extension domains other than disks. Our discussion relies heavily on the theory of extremal quasiconformal mappings, which in turn gives some interesting results in the extremal quasiconformal mapping theory as well.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 3 (2004), 445-466.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246678

Digital Object Identifier
doi:10.2748/tmj/1113246678

Mathematical Reviews number (MathSciNet)
MR2075777

Zentralblatt MATH identifier
1063.30017

Subjects
Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods

Keywords
Boundary quasiextremal distance constant quasiextremal distance constant quasiconformal reflection constant quasisymmetric homeomorphism QED domain

Citation

Shen, Yu-Liang. Conformal invariants of QED domains. Tohoku Math. J. (2) 56 (2004), no. 3, 445--466. doi:10.2748/tmj/1113246678. https://projecteuclid.org/euclid.tmj/1113246678


Export citation

References

  • L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.
  • J. M. Anderson and A. Hinkkanen, Quadrilaterals and extremal quasiconformal extensions, Comment. Math. Helv. 70 (1995), 455--474.
  • T. Bagby, The modulus of a plane condenser, J. Math. Mech. 17 (1967), 315--329.
  • A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125--142.
  • J. Chen and Z. Chen, A remark on ʽʽ An approximation condition and extremal quasiconformal extensionsʼʼ, Chinese Sci. Bull. 42 (1997), 1765--1767.
  • C. J. Earle and Li Zhong, Isometrically embedded polydisks in infinite dimensional Teichmüller spaces, J. Geom. Anal. 9 (1999), 51--71.
  • F. P. Gardiner and D. P. Sullivan, Symmetric and quasisymmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683--736.
  • J. B. Garnett and S. Yang, Quasiextremal distance domains and integrability of derivatives of conformal mappings, Michigan Math. J. 41 (1994), 389--406.
  • F. W. Gehring, Quasiconformal mappings, Complex Analysis and Its Applications (Lectures, Internat. Sem., Trieste, 1975) vol. 2, 213--268, Internat. Atomic Energy Agency, Vienna, 1976.
  • F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. d'Analyse Math. 45 (1985), 181--206.
  • J. G. Krzyz and D. Partyka, Generalized Neumann-Poincaré operator, chord-arc curves and Fredholm eigenvalues, Complex Variables Theory Appl. 21 (1993), 253--263.
  • N. Lakic, Strebel points, Contemp. Math. 211 (1997), 417--431.
  • O. Lehto, Univalent Functions and Teichmüller Spaces, Springer-Verlag, New York, 1986.
  • S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Wiley-Interscience, 1988.
  • S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^1/2$ space on the circle, Osaka J. Math. 32(1995), 1--34.
  • D. Partyka, Generalized harmonic conjugation operator, Ber. Univ. Jyväskylä Math. Inst. 55 (1992), 143--155, Proc. of the Fourth Finnish-Polish Summer School in Complex Analysis at Jyväskylä, 1993.
  • D. Partyka, Spectral values and eigenvalues of a quasicircle, Ann. Univ. Mariae Curie-Sklodowska Sect. A 46(1993), 81--98.
  • D. Partyka, The smallest positive eigenvalue of a quasisymmetric automorphisms of the unit circle, Topics in Complex Analysis (Warsaw, 1992), 303--310, Banach Center Publ. 31, Polish Acad. Sci., Warszawa, 1995.
  • D. Partyka, Some extremal problems concerning the operator $B_\gamma$, Ann. Univ. Mariae Curie-Sklodowska Sect. A 49 (1996), 163--184.
  • D. Partyka, The generalized Neumann-Poincaré operator and its spectrum, Dissertations Math. No. 484, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997.
  • E. Reich, An approximation condition and extremal quasiconformal extensions, Proc. Amer. Math. Soc. 125 (1997), 1479--1481.
  • G. Schober, Continuity of curve functionals and a technique involving quasiconformal mappings, Arch. Ration. Mech. Anal. 29 (1968), 378--389.
  • Y. Shen, Quasiconformal mappings and harmonic functions, Adv. in Math. (Beijing) 28 (1999), 347--357.
  • Y. Shen, A counterexample theorem in quasiconformal mapping theory, Sci. China Ser. A 43 (2000), 929--936.
  • Y. Shen, Pull-back operators by quasisymmetric functions and invariant metrics on Teichmüller spaces, Complex Variables Theory Appl. 42 (2000), 289--307.
  • Y. Shen, Notes on pull-back operators by quasisymmetric homeomorphisms with applications to Schober's functionals, Chinese Annals of Math. 24A (2003), 209--218; Chinese J. of Contemporary Math. 24 (2003), 187--196.
  • H. Shiga and H. Tanigawa, Grunsky's inequality and its applications to Teichmüller spaces, Kodai Math. J. 16 (1993), 361--378.
  • K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises, Comment. Math. Helv. 36 (1962), 306--323.
  • K. Strebel, On the existence of extremal Teichmüller mappings, J. Anal. Math. 30 (1976), 464--480.
  • K. Strebel, Extremal quasiconformal mappings, Result. Math. 10 (1986), 169--209.
  • K. Strebel, On the dilatation of extremal quasiconformal mappings of polygons, Comment. Math. Helv. 74 (1999), 143--149.
  • J. Väisälä, Lectures on $n$-dimensional Quasiconformal Mappings, Lecture Notes in Math. 229, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
  • S. Wu, Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions, Comment. Math. Helv. 72 (1997), 593--604.
  • S. Wu and S. Yang, On symmetric quasicircles, J. Austr. Math. Soc. Ser. A 68 (2000), 131--144.
  • S. Yang, QED domains and NED sets in $\overline R^n$, Trans. Amer. Math. Soc. 334 (1992), 97--120.
  • S. Yang, On dilatations and substantial boundary points of homeomorphisms of Jordan curves, Result. Math. 31 (1997), 180--188.
  • S. Yang, Conformal invariants of smooth domains and extremal quasiconformal mappings of ellipses, Illinois J. Math. 41 (1997), 438--452.
  • S. Yang, A modulus inequality for condensers and conformal invariants of smooth domains, J. Anal. Math. 75 (1998), 173--183.