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A COMPLEX STRUCTURE ON THE SET OF QUASICONFORMALLY EXTENDIBLE NON-OVERLAPPING MAPPINGS INTO A RIEMANN

By David Radnell and Eric Schippers

Abstract

Abstract. Let Σ be a compact Riemann surface with n distinguished points p1,...,pn. We prove that the set of n-tuples (φ1,..., φn) of univalent mappings φi from the unit disc D into Σ mapping 0 to pi, with non-overlapping images and quasiconformal extensions to a neighbourhood of D, 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 Teichmüller space. Our results are motivated by Teichmüller theory and two-dimensional conformal field theory. 1

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