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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: June 30, 1999
MathSciNet review: 1616613
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-99-04980-1
PII: S 0002-9939(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