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On the extremality of quasiconformal mappings
and quasiconformal deformations


Author: Shen Yu-Liang
Journal: Proc. Amer. Math. Soc. 128 (2000), 135-139
MSC (1991): Primary 30C70, 30C62
DOI: https://doi.org/10.1090/S0002-9939-99-04980-1
Published electronically: June 30, 1999
MathSciNet review: 1616613
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a family of quasiconformal deformations $F(w, t)$ such that $\overline{\partial }F$ has a uniform bound $M$, the solution $f(z, t) ( f(z, 0)=z ) $ of the Löwner-type differential equation

\begin{equation*}\frac{dw}{dt}=F(w, t)\end{equation*}

is an $e^{2Mt}$-quasiconformal mapping. An open question is to determine, for each fixed $t>0$, whether the extremality of $f(z, t)$ is equivalent to that of $F(w, t)$. The note gives this a negative approach in both directions.


References [Enhancements On Off] (What's this?)

  • 1. L. V. Ahlfors, Quasiconformal deformations and mapings in $R^{n}$, J. d'Analyse Math. 30 (1976), 74-97. MR 58:11384
  • 2. A. Beurling & L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta. Math. 96 (1956), 125-142. MR 19:258c
  • 3. F. P. Gardiner & D. P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683-736. MR 95h:30020
  • 4. R. S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Tran. Amer. Math. Soc. 138 (1969), 399-406. MR 39:7093
  • 5. S. L. Krushkal, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411-418.
  • 6. E. Reich, A quasiconformal extension using the parametric representation, J. d'Analyse Math. 54 (1990), 246-258. MR 91c:30035
  • 7. E. Reich & J. X. Chen, Extensions with bounded $\overline{\partial }$-derivative, Ann. Acad. Sci. Fenn. A I Math. 16 (1991), 377-389. MR 93b:30018
  • 8. E. Reich & K. Strebel, Extremal quasiconformal mappings with prescribed boundary values, Contributions to Analysis, A collection of papers dedicated to Lipman Bers. Academic Press, New York (1974), 375-391. MR 50:13511
  • 9. A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47-76. MR 7:60b

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Additional Information

Shen Yu-Liang
Affiliation: Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China
Email: ylshen@suda.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-99-04980-1
Keywords: Quasiconformal mapping, quasiconformal deformation, extremality
Received by editor(s): December 23, 1997
Received by editor(s) in revised form: March 10, 1998
Published electronically: June 30, 1999
Additional Notes: Project supported by the National Natural Science Foundation of China.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society

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