On criteria for extremality of Teichmüller mappings

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

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.