On the extremality of quasiconformal mappings and quasiconformal deformations
HTML articles powered by AMS MathViewer
- by Shen Yu-Liang PDF
- Proc. Amer. Math. Soc. 128 (2000), 135-139 Request permission
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
- Lars V. Ahlfors, Quasiconformal deformations and mappings in $\textbf {R}^{n}$, J. Analyse Math. 30 (1976), 74–97. MR 492238, DOI 10.1007/BF02786705
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683–736. MR 1175689, DOI 10.2307/2374795
- Richard S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399–406. MR 245787, DOI 10.1090/S0002-9947-1969-0245787-3
- S. L. Krushkal, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411-418.
- Edgar Reich, A quasiconformal extension using the parametric representation, J. Analyse Math. 54 (1990), 246–258. MR 1041184, DOI 10.1007/BF02796151
- Edgar Reich and Jixiu Chen, Extensions with bounded $\overline \partial$-derivative, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 377–389. MR 1139804, DOI 10.5186/aasfm.1991.1623
- Edgar Reich and Kurt Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 375–391. MR 0361065
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Additional Information
- Shen Yu-Liang
- Affiliation: Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China
- MR Author ID: 360822
- Email: ylshen@suda.edu.cn
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1616613