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A. Grinshpan posed a deep conjecture on the norm of the Grunsky operatorgenerated by univalent functions in the disk. It gives a quantitative answer interms of the Grunsky coefficients, to which extent a univalent functiondetermines the bound of dilatations of its quasiconformal extensions. Weprovide the proof of this conjecture and its various analytic, geometric andpotential applications.  Another result concerns the model of universal Teichmuller space by Grunskycoefficients.

Grunsky operator, Grinshpan's conjecture and universal Teichmuller space