@article{96ea3a6963e9481fb5348ac999aea920, title = "A Hilbert manifold structure on the Weilâ€“Petersson class Teichm{\"u}ller space of bordered Riemann surfaces", abstract = "We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichm{\"u}ller space of such Riemann surfaces (which we refer to as the WP-class Teichm{\"u}ller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichm{\"u}ller space is a Hilbert manifold. The inclusion map from the refined Teichm{\"u}ller space into the usual Teichm{\"u}ller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichm{\"u}ller space by a properly discontinuous group of biholomorphisms.", author = "David Radnell and Eric Schippers and Wolfgang Staubach", year = "2015", month = "8", doi = "10.1142/S0219199715500169", language = "English", volume = "17", journal = "Communications in Contemporary Mathematics", issn = "0219-1997", publisher = "World Scientific Publishing Co.", number = "04", }