A Model of the Teichm\"uller space of genus-zero bordered surfaces by period maps

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