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An extremal problem of quasiconformal mappings


Authors: Zhong Li, Shengjian Wu and Zemin Zhou
Journal: Proc. Amer. Math. Soc. 132 (2004), 3283-3288
MSC (2000): Primary 30C75, 30C62
DOI: https://doi.org/10.1090/S0002-9939-04-07485-4
Published electronically: April 21, 2004
MathSciNet review: 2073303
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Abstract: In this paper, the following problem is studied. Let $\Omega _{1}$ and $\Omega _{2}$ be two domains in the complex plane with $\Omega _{1}\cap \Omega _{2}\not =\emptyset $. Suppose that $f_{j}:\Omega _{j}\to f_{j}(\Omega _{j})$ $(j=1,2)$ are two quasiconformal mappings satisfying $f_{1}\vert _{\Omega _{1}\cap \Omega _{2}} =f_{2}\vert _{\Omega _{1}\cap \Omega _{2}}$. Let $F$ be the mapping in $\Omega _{1}\cup \Omega _{2}$ defined by $F\vert _{\Omega _{j}}=f_{j}$ ($j=1,2$). If both $f_{1}$ and $f_{2}$ are uniquely extremal, is $F$ always uniquely extremal? It is shown in this paper that the answer to this problem is no.


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  • [Ah] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966. MR 34:336
  • [AH] M. Anderson and A. Hinkkanen, Quadrilaterals and extremal quasiconformal extensions, Comment. Math. Helv. 70 (1995), 455-474. MR 96g:30042
  • [BLM] V. Bozin, N. Lakic, V. Markovic and M. Mateljvic, Unique extremality, J. Anal. Math. 75 (1998), 299-338. MR 2000a:30045
  • [CS] J. Chen and Y. Shen, Oral communication.
  • [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, 1973. MR 49:9202
  • [Ma] V. Markovic, Extremal problems for quasiconformal mappings of punctured plane domains, Trans. Amer. Math. Soc. 354 (2002,) 1631-1650. MR 2002j:30074
  • [Re] E. Reich, Extremal Quasiconformal mapping of the Disk, in the book ``Handbook of Complex Analysis: Geometric function theory, Volume 1", Edited by R.Kühnau, Elsevier Science B.V., 2002, pp. 75-135. MR 2004c:30036
  • [Re1] E. Reich, An extremum problem for analytic functions with area norm, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 2 (1976), 429-445. MR 58:17102
  • [Re2] E. Reich, Uniqueness of Hahn-Banach extensions from certain spaces of analytic functions, Math. Z. 167 (1979), 81-89. MR 80j:30074
  • [RS1] E. Reich and K. Strebel, On quasiconformal mappings which keep the boundary points fixed, Trans. Amer. Math. Soc. 138 (1969), 211-222. MR 38:6059
  • [RS2] E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, in the book ``Contributions to Analysis (a collection of papers dedicated to Lipman Bers)", Academic Press, 1974, pp. 375-392. MR 50:13511
  • [RS3] E. Reich and K. Strebel, On the extremality of certain Teichmüller mappings, Comment. Math. Helv. 45 (1970), 353-362. MR 43:514
  • [Se] G. C. Sethares, The extremal property of certain Teichmüller mappings, Comment. Math. Helv. 43 (1968), 98-119. MR 37:4253
  • [St] K. Strebel, On the extremality and unique extremality of quasiconformal mappings of a parallel strip, Rev. Roumaine Math. Pures Appl. 32 (1987), 923-928. MR 89f:30042

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Additional Information

Zhong Li
Affiliation: School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, People’s Republic of China
Email: lizhong@math.pku.edu.cn

Shengjian Wu
Affiliation: School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, People’s Republic of China
Email: wusj@math.pku.edu.cn

Zemin Zhou
Affiliation: School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, People’s Republic of China
Email: zeminzhou2000@163.com

DOI: https://doi.org/10.1090/S0002-9939-04-07485-4
Received by editor(s): December 3, 2002
Received by editor(s) in revised form: July 15, 2003
Published electronically: April 21, 2004
Additional Notes: The first author was supported by the 973-Project Foundation of China (Grant TG199075105) and the second author was supported by the NNSF of China (Grants 10171003 and 10231040)
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2004 American Mathematical Society

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