Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

Abstract

A. Grinshpan posed a deep conjecture on the norm of the Grunsky operator generated by univalent functions in the disk. It gives a quantitative answer in terms of the Grunsky coefficients, to which extent a univalent function determines the bound of dilatations of its quasiconformal extensions. We provide the proof of this conjecture and its various analytic, geometric and potential applications.Another result concerns the model of universal Teichmüller space by Grunsky coefficients.