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

2008/03/21 by David Radnell, Eric Schippers, Radnell, David +1
Mathematics · Physics and Astronomy · #30C55 #30C62 #30F60 (Primary) 81T40 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CV #math.MP #msc:30C55 #msc:30C62 #msc:30F60 #msc:81T40

paper · pdf · doi:10.48550/arxiv.0803.3211

12 pages. Minor corrections made. To appear in Journal d'Analyse Mathematique

arxiv created 2008/07/18 · arxiv updated 2009/12/01

Abstract

Let Σbe a compact Riemann surface with n distinguished points p₁,...,p_n. We prove that the set of n-tuples (ϕ₁,...,ϕ_n) of univalent mappings ϕ_i from the open unit disc into Σmapping 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.

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