A complex structure on the set of quasiconformally extendible non-overlapping mappings into a Riemann surface

TL;DR

This paper establishes a natural complex Banach manifold structure on the set of non-overlapping, quasiconformally extendible univalent mappings into a Riemann surface, linking Teichmüller theory and conformal field theory.

Contribution

It introduces a new complex structure on mappings into Riemann surfaces, modeled on an extension of the universal Teichmüller space, advancing the understanding of their geometric properties.

Findings

The set of such mappings forms a complex Banach manifold.

The structure is locally modeled on a product of extended Teichmüller spaces.

Results connect Teichmüller theory with conformal field theory.

Abstract

Let \Sigma be a compact Riemann surface with n distinguished points p_1,...,p_n. We prove that the set of n-tuples (\phi_1,...,\phi_n) of univalent mappings \phi_i from the open unit disc into \Sigma mapping 0 to p_i, with non-overlapping images and quasiconformal extensions to a neighbourhood of the closed unit disk, carries a natural complex Banach manifold structure. This complex structure is locally modelled on the n-fold product of a two complex-dimensional extension of the universal Teichmueller space. Our results are motivated by Teichmueller theory and two-dimensional conformal field theory.