Fiber Structure and Local Coordinates for the Teichmueller Space of a Bordered Riemann Surface

TL;DR

This paper demonstrates that the Teichmueller space of a bordered Riemann surface can be viewed as a holomorphic fiber space over the space of punctured surfaces, with fibers modeled on complex Banach manifolds, and introduces new local coordinates.

Contribution

It establishes a fiber space structure for the infinite-dimensional Teichmueller space of bordered Riemann surfaces and introduces novel holomorphic local coordinates using internal Schiffer variation.

Findings

Teichmueller space is a holomorphic fiber space over punctured surface space.

Fibers are complex Banach manifolds modeled on extended universal Teichmueller space.

New holomorphic local coordinates are provided for the infinite-dimensional space.

Abstract

We show that the infinite-dimensional Teichmueller space of a Riemann surface whose boundary consists of n closed curves is a holomorphic fiber space over the Teichmueller space of n-punctured surfaces. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichmueller space. The local model of the fiber, together with the coordinates from internal Schiffer variation, provides new holomorphic local coordinates for the infinite-dimensional Teichmueller space.