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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
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.

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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

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