Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

TL;DR

This paper proves Grinshpan's conjecture relating the norm of the Grunsky operator to univalent functions and explores applications in Teichmüller theory, advancing understanding of quasiconformal extensions and their bounds.

Contribution

It provides a proof of Grinshpan's conjecture and investigates the analytic and geometric implications for universal Teichmüller space.

Findings

Proof of Grinshpan's conjecture established

Quantitative bounds on quasiconformal extension dilatations

Enhanced understanding of the universal Teichmüller space model

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 Teichmuller space by Grunsky coefficients.