Krushkal S.L. A question of Ahlfors

Krushkal Samuel L.

Doctor of Physical and Mathematical Sciences, Professor, Department of Mathematics Bar-Ilan University and University of Virginia
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5290002 Ramat-Gan, Israel and Charlottesville, VA 22904-4137, USA

Abstract. 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 logf′= f′′/f′, and indeed, many results on quasiconformal extensions of holomorphic maps have been established using f′′/f′ 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 ⊂C^ determine the geometric and conformal invariants (characteristics) of complementary domain D*=C^ ∖ 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.

Key words: the Grunsky inequalities, Beltrami coefficient, universal Teichmüller space, Teichmüller metric, Kobayashi metric, Schwarzian derivative, Fredholm eigenvalues.

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A Question of Ahlfors by Krushkal S.L. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №3 (22) 2014 pp. 61-65

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