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

TL;DR

This paper introduces a holomorphic and injective period mapping for genus-zero bordered Riemann surfaces with multiple boundaries, utilizing generalized Grunsky operators to embed their Teichmüller space into a space of operators.

Contribution

It defines a novel period map for genus-zero bordered surfaces using generalized Grunsky operators, extending the understanding of their Teichmüller spaces.

Findings

The period map is holomorphic.

The period map is injective.

Embedding of Teichmüller space into operator space achieved.

Abstract

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.