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

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.