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

Samuel Krushkal

Open Mathematics (2007)

  • Volume: 5, Issue: 3, page 551-580
  • ISSN: 2391-5455

Abstract

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The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to ℂ ^ are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau 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 Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.

How to cite

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Samuel Krushkal. "Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues." Open Mathematics 5.3 (2007): 551-580. <http://eudml.org/doc/269263>.

@article{SamuelKrushkal2007,
abstract = {The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \[\widehat\{\mathbb \{C\}\}\] are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau 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 Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.},
author = {Samuel Krushkal},
journal = {Open Mathematics},
keywords = {quasiconformal; univalent function; Grunsky coefficient inequalities; universal Teichmüller space; subharmonic function; Strebel’s point; Kobayashi metric; generalized Gaussian curvature; holomorphic curvature; Fredholm eigenvalues; univalent functions; quasiconformal mappings; Strebel's point},
language = {eng},
number = {3},
pages = {551-580},
title = {Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues},
url = {http://eudml.org/doc/269263},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Samuel Krushkal
TI - Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues
JO - Open Mathematics
PY - 2007
VL - 5
IS - 3
SP - 551
EP - 580
AB - The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \[\widehat{\mathbb {C}}\] are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau 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 Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.
LA - eng
KW - quasiconformal; univalent function; Grunsky coefficient inequalities; universal Teichmüller space; subharmonic function; Strebel’s point; Kobayashi metric; generalized Gaussian curvature; holomorphic curvature; Fredholm eigenvalues; univalent functions; quasiconformal mappings; Strebel's point
UR - http://eudml.org/doc/269263
ER -

References

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