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

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.