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On criteria for extremality of Teichmüller mappings


Author: Guowu Yao
Journal: Proc. Amer. Math. Soc. 132 (2004), 2647-2654
MSC (2000): Primary 30C75
DOI: https://doi.org/10.1090/S0002-9939-04-07420-9
Published electronically: April 21, 2004
MathSciNet review: 2054790
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Abstract: Let $f$ be a Teichmüller self-mapping of the unit disk $\Delta$corresponding to a holomorphic quadratic differential $\varphi$. If $\varphi$ satisfies the growth condition $A(r,\varphi)=\iint_{\vert z\vert<r}\vert\varphi\vert dxdy=O((1-r)^{-s})$ (as $r\to 1$), for any given $s>0$, then $f$ is extremal, and for any given $a\in (0,1)$, there exists a subsequence $\{n_k\}$ of $\mathbb{N} $ such that

\begin{displaymath}\Big\{\frac{\varphi(a^{1/2^{n_k}}z)} {\iint_\Delta\vert\varphi(a^{1/2^{n_k}}z)\vert dxdy}\Big\} \end{displaymath}

is a Hamilton sequence. In addition, it is shown that there exists $\varphi$ with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.


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

  • 1. W. K. Hayman and E. Reich, On the Teichmüller mappings of the disk, Complex Variables Theory Appl. 1 (1982), 1-12. MR 84e:30031
  • 2. X. Z. Huang, On the extremality for Teichmüller mappings, J. Math. Kyoto Univ. 35 (1995), 115-132. MR 97a:30029
  • 3. W. C. Lai and Z. M. Wu, On extremality and unique extremality of Teichmüller mappings, Chinese Ann. Math. (Ser. B) 16 (1995), 399-406. MR 97b:30027
  • 4. V. Markovic, Harmonic diffeomorphism of noncompact surfaces and Teichmüller spaces, J. London Math. Soc. 65 (2002), 103-114. MR 2002k:32015
  • 5. C. Mcmullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95-127. MR 90e:30048
  • 6. E. Reich, Construction of Hamilton sequences for certain Teichmüller mappings, Proc. Amer. Math. Soc. 103 (1988), 789-796. MR 89m:30045
  • 7. E. Reich and K. 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. 375-392. MR 50:13511
  • 8. Z. M. Wu and W. C. Lai, On criterion of the extremality and construction of Hamilton sequences for a class of Teichmüller mappings, Chinese Ann. Math. (Ser. B) 21 (2000), 339-342. MR 2002c:30031
  • 9. M. Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449-479. MR 90h:58023
  • 10. G. W. Yao, Hamilton sequences and extremality for certain Teichmüller mappings, Ann. Acad. Sci. Fenn. Math. 29 (2004), 185-194.
  • 11. G. W. Yao, A remark on Hamilton sequences of extremal Teichmüller mappings, to appear.

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Additional Information

Guowu Yao
Affiliation: School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China
Email: wallgreat@lycos.com, gwyao@mail.amss.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-04-07420-9
Keywords: Hamilton sequence, Teichm\"uller mapping, extremality
Received by editor(s): December 3, 2002
Received by editor(s) in revised form: June 5, 2003
Published electronically: April 21, 2004
Additional Notes: This research was supported by the “973” Project Foundation of China (Grant No. TG199075105) and the Foundation for Doctoral Programme
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2004 American Mathematical Society

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