Quasiconformal maps and Teicmüller theory - extremal mappings, overview

M. Mateljević

In the first part we give a short review of Teicmüller theory. In the second part, we study the conditions under which unique extremality of quasiconformal mappings occurs, and we provide a broader point of view of the phenomenon of unique extremality. Furthermore, we make some contributions to what we refer to as the Teichmüller research question. In particular we report a positive answer to this question if $\mu$ is uniquely extremal on a domain $G$ and the lower oscillation of $\mu$ is less than $L^\infty$- norm of $\mu$ expect on a discrete set in $G$. Additional information is obtained by means of specialized constructions. In particular we review some results from [Ma8], in which we generalize the construction theorem in [BLMM], thus providing a more basic understanding of it. We also announce some results related extremal mappings in 3 dimensions.