  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.
