The Grunsky norm of univalent functions and abelian holomorphic differentials

We establish that the Grunsky norm of any normalized univalent function on the disk is completely determined by the squares of holomorphic abelian differentials (in contrast to the Teichmuller norm, which relates to all integrable holomorphic quadratic differentials). This result has important interesting applications. In particular, it provides an explicit representation of Fredholm eigevalues of all quasiconformal curves.