On the extremality of quasiconformal mappings and quasiconformal deformations

Abstract:

Given a family of quasiconformal deformations $F(w, t)$ such that $\overline {\partial }F$ has a uniform bound $M$, the solution $f(z, t) ( f(z, 0)=z )$ of the Löwner-type differential equation \begin{equation*}\frac {dw}{dt}=F(w, t)\end{equation*} is an $e^{2Mt}$-quasiconformal mapping. An open question is to determine, for each fixed $t>0$, whether the extremality of $f(z, t)$ is equivalent to that of $F(w, t)$. The note gives this a negative approach in both directions.