A question of Ahlfors

Samuel L. Krushkal

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2014), pp. 61-65

Résumé

In 1963, Ahlfors posed in [1] (and repeated in his book [2]) the following question which gave rise to various investigations of quasiconformal extendibility of univalent functions. Question. Let $f$ be a conformal map of the disk (or half-plane) onto a domain with quasiconformal boundary (quasicircle). How can this map be characterized? He conjectured that the characterization should be in analytic properties of the logarithmic derivative $\log f^\prime = f^{\prime\prime}/f^\prime$, and indeed, many results on quasiconformal extensions of holomorphic maps have been established using $f^{\prime\prime}/f^\prime$ and other invariants (see, e.g., the survey [9] and the references there). This question relates to another still not solved problem in geometric complex analysis: To what extent does the Riemann mapping function $f$ of a Jordan domain $D \subset \hat {\Bbb C}$ determine the geometric and conformal invariants (characteristics) of complementary domain $D^* = \hat {\Bbb C} \setminus \overline{D}$? The purpose of this paper is to provide a qualitative answer to these questions, which discovers how the inner features of biholomorphy determine the admissible bounds for quasiconformal dilatations and determine the Kobayashi distance for the corresponding points in the universal Teichmüller space.

Détail

Keywords: the Grunsky inequalities, universal Teichmüller space, Teichmüller metric, Kobayashi metric, Schwarzian derivative, Fredholm eigenvalues.
Mots-clés : Beltrami coefficient