On criteria for extremality of Teichmüller mappings

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 $A(r,\varphi )=\iint _{|z|<r}|\varphi |dxdy=O((1-r)^{-s})$ (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{equation*} \Big \{\frac {\varphi (a^{1/2^{n_k}}z)} {\iint _\Delta |\varphi (a^{1/2^{n_k}}z)|dxdy}\Big \} \end{equation*} 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.