A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

Abstract

We consider Riemann surfaces Σ \Sigma with n n borders homeomorphic to S 1 \mathbb {S}^1 and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an n n -fold direct sum of Bergman spaces of the disk. We show that this period mapping is holomorphic and injective.