Tohoku Mathematical Journal

On the unique extremality of quasiconformal mappings with dilatation bounds

Yu-Liang Shen

Full-text: Open access

Abstract

Concerning the problem of extremality of quasiconformal mappings with dilatation bounds, we discuss the unique extremality of the problem and prove the if part of a conjecture on the unique extremality. To this end, we need to investigate a new extremal problem in the infinitesimal setting. In particular, we give a complete description of the unique infinitesimal extremality of partially zero Beltrami differentials.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 1 (2004), 105-123.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246383

Digital Object Identifier
doi:10.2748/tmj/1113246383

Mathematical Reviews number (MathSciNet)
MR2028920

Zentralblatt MATH identifier
1072.30034

Subjects
Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods 30F60: Teichmüller theory [See also 32G15]

Keywords
Quasiconformal mapping Beltrami differential uniquely extremal uniquely infinitesimally extremal

Citation

Shen, Yu-Liang. On the unique extremality of quasiconformal mappings with dilatation bounds. Tohoku Math. J. (2) 56 (2004), no. 1, 105--123. doi:10.2748/tmj/1113246383. https://projecteuclid.org/euclid.tmj/1113246383


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References

  • A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125--142.
  • V. Bo\u zin, N. Lakic, V. Marković and M. Mateljević, Unique extremality, J. Anal. Math. 75 (1998), 299--338.
  • V. Bo\u zin, V. Marković and M. Mateljević, Unique extremality in the tangent space of universal Teichmüller space, Integral Transform. Spec. Funct. 6 (1997), 223--227.
  • C. J. Earle, F. P. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, Inst. Hautes Études Sci. preprint 1995.
  • C. J. Earle and Z. Li, Isometrically embedded polydisks in infinite dimensional Teichmüller spaces, J. Geom. Anal. 9 (1999), 51--71.
  • R. Fehlmann, On a fundamental variational lemma for extremal quasiconformal mappings, Comment. Math. Helv. 61 (1986), 565--580.
  • R. Fehlmann and K. Sakan, On extremal quasiconformal mappings with varying dilatation bounds, Osaka J. Math. 23 (1986), 751--764.
  • F. P. Gardiner, On pratially Teichmüller Beltrami differentials, Michigan Math. J. 29 (1982), 237--247.
  • F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley-Interscience, New York, 1987.
  • F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Math. Surveys Monogr. 76, American Mathematical Society, Providence, R.I., 2000.
  • R. S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399--406.
  • S. L. Krushkal, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411--418.
  • V. Marković and M. Mateljević, The unique extremal quasiconformal mapping and uniqueness of Hahn-Banach extensions, Mat. Vesnik 48 (1996), 107--112.
  • E. Reich, Quasiconformal mappings with prescribed boundary values and a dilatation bound, Arch. Ration. Mech. Anal. 68 (1978), 99--112.
  • E. Reich, On criteria for unique extremality of Teichmüller mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 289--301.
  • E. Reich, Extremal extensions from the circle to the disk, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 321--335, A collection of papers honoring F. W. Gehring, Springer-Verlag, New York, Berlin Heidelberg, 1997.
  • E. Reich, The unique extremality counterexample, J. Anal. Math. 75 (1998), 339--347.
  • E. Reich and K. Strebel, Extremal quasiconformal mappings with prescribed boundary values, Contributions to analysis (a collection of papers dedicated to Lipman Bers), 375--391, Academic Press, New York, 1974.
  • K. Sakan, On extremal quasiconformal mappings compatible with Fuchsian groups, Tohoku Math. J. 34 (1982), 87--100.
  • K. Sakan, On quasiconformal mappings compatible with a Fuchsian group, Osaka J. Math. 19 (1982), 159--170.
  • K. Sakan, On extremal quasiconformal mappings compatible with a Fuchsian group and a dilatation bound, Tohoku Math. J. 37(1985), 79--93.
  • K. Sakan, Necessary and sufficient conditions for extremality in certain classes of quasiconformal mappings, J. Math. Kyoto Univ. 26 (1986), 31--37.
  • K. Sakan, A fundamental variational lemma for extremal quasiconformal mappings compatible with a Fuchsian group, Tohoku Math. J. 39 (1982), 105--114.
  • Y. Shen, On unique extremality, Complex Variables Theory Appl. 40 (1999), 149--162.
  • Y. Shen, On Teichmüller geometry, Complex Variables Theory Appl. 44 (2001), 73--83.
  • Y. Shen and J. Chen, Quasiconformal mappings with non-decreasable dilatations, Chinese Ann. Math. Ser. A 23 (2002), 459--466.
  • K. Strebel, On quasiconformal mappings of open Riemann surfaces, Comment. Math. Helv. 53 (1978), 301--321.
  • K. Strebel, Extremal quasiconformal mappings, Results Math. 10 (1986), 168--210.