Geodesic
Quasiconformal maps and Teicmüller theory - extremal mappings, overview
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 38 (2013), p. 129 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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.
Classification : 30C75 30C70 30H05 46A22
Keywords: unique extremality in Teicmüller and infinitesimal class, nonconstant uniquely extremal complex dilatation, complex dilatation of Teicmüller type, Hahn-Banach unique extension, Hamilton condition, Runge theorem 
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     author = {M. Mateljevi\'c},
     title = {Quasiconformal maps and {Teicm\"uller} theory - extremal mappings, overview},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {129 },
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M. Mateljević. Quasiconformal maps and Teicmüller theory - extremal mappings, overview. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 38 (2013), p. 129 . http://geodesic.mathdoc.fr/item/BASS_2013_38_a7/