Strengthened Moser's conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

被引:14
作者
Krushkal, Samuel [1 ,2 ]
机构
[1] Bar Ilan Univ, Dept Math, IL-52900 Ramat Gan, Israel
[2] Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA
来源
CENTRAL EUROPEAN JOURNAL OF MATHEMATICS | 2007年 / 5卷 / 03期
关键词
quasiconformal; univalent function; Grunsky coefficient inequalities; universal Teichmuller space; subharmonic function; Strebel's point; Kobayashi metric; generalized Gaussian curvature; holomorphic curvature; Fredholm eigenvalues;
D O I
10.2478/s11533-007-0013-5
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The Grunsky and Teichmuller norms chi(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D is an element of infinity with quasiconformal extension to (C) over cap are related by chi(f) <= k(f). In 1985, Jurgen Moser conjectured that any univalent function in the disk Delta* = {z : |z| > 1} can be approximated locally uniformly by functions with chi(f) < k(f). This conjecture has been recently proved by R. Kuhnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kuhnau problem on an exact lower bound in the inverse inequality estimating k(f) by chi(f), and in the related Ahlfors inequality. (C) Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.
引用
收藏
页码:551 / 580
页数:30
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