Oberseminar Funktionentheorie: Prof. Eric Schippers

Abstract: The Grunsky operator is a matrix associated to a conformal map of the disk into the Riemann sphere, which under sufficient conditions on the mapping can be treated as a bounded operator. It arises classically in the theory of conformal maps, where the Grunsky inequalities give rise to a characterization of univalent maps of the disk. The Grunsky operator also surfaces in Weil-Petersson Teichmüller theory, representation theory, and conformal field theory, and is enjoying a surge of interest.

Although the Grunsky matrix has many (roughly) equivalent formulations, its interpretation is somewhat elusive.  This talk focusses on results establishing a particular geometric meaning of the Grunsky operator in terms of boundary values of functions or one-forms. We give generalizations to the case of finitely many conformal maps into a compact Riemann surface.