Скачать или смотреть Eric Schippers: A geometric interpretation of the Grunsky operator

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 potential theory and extremal problems for conformal maps, for example. The Grunsky operator also surfaces in Weil-Petersson Teichmüller theory, representation theory, and conformal field theory, and is enjoying a surge of interest. It has many (roughly) equivalent formulations, but 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 and polarizations. We give generalizations to the case of finitely many conformal maps into a compact Riemann surface.