  We consider Riemann surfaces $\Sigma$ 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.
