A question of ahlfors
Автор: Krushkal Samuel L.
Журнал: Математическая физика и компьютерное моделирование @mpcm-jvolsu
Рубрика: Математика
Статья в выпуске: 3 (22), 2014 года.
Бесплатный доступ
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 ?? be a conformal map of the disk (or halfplane) 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 ??? = ????/???, and indeed, many results on quasiconformal extensions of holomorphic maps have been established using ????/??? 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 ?? of a Jordan domain ?? ? ??C determine the geometric and conformal invariants (characteristics) of complementary domain ??* = ??C ? ??? 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 Teichmuller space.
Grunsky inequalities, beltrami coefficient, universal teichmuller space, teichmuller metric, kobayashi metric, schwarzian derivative, fredholm eigenvalues
Короткий адрес: https://sciup.org/14968755
IDR: 14968755
Список литературы A question of ahlfors
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