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On criteria for extremality of Teichmüller mappings

Authors:	Guowu Yao

Affiliation:	School of Mathematical Sciences, Peking University, Beijing, 100871, People's Republic of China

Abstract:	Let be a Teichmüller self-mapping of the unit disk corresponding to a holomorphic quadratic differential . If satisfies the growth condition $A(r,varphi)=iint_{vert zvert<r}vertvarphivert dxdy=O((1-r)^{-s})$ (as ), for any given , then is extremal, and for any given , there exists a subsequence ${n_k}$ of such that $begin{displaymath}Big{frac{varphi(a^{1/2^{n_k}}z)} {iint_Deltavertvarphi(a^{1/2^{n_k}}z)vert dxdy}Big} end{displaymath}$ is a Hamilton sequence. In addition, it is shown that there exists 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.

Keywords:	Hamilton sequence, Teichm" uller mapping, extremality

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