# A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

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**A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces.** / Radnell, David; Schippers, Eric; Staubach, Wolfgang.

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*Communications in Contemporary Mathematics*, vol. 17, no. 04. https://doi.org/10.1142/S0219199715500169

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*Communications in Contemporary Mathematics*,

*17*(04). https://doi.org/10.1142/S0219199715500169

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TY - JOUR

T1 - A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

AU - Radnell, David

AU - Schippers, Eric

AU - Staubach, Wolfgang

PY - 2015/8

Y1 - 2015/8

N2 - 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 Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller 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 Teichmüller space by a properly discontinuous group of biholomorphisms.

AB - 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 Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller 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 Teichmüller space by a properly discontinuous group of biholomorphisms.

U2 - 10.1142/S0219199715500169

DO - 10.1142/S0219199715500169

M3 - Article

VL - 17

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 04

ER -

ID: 6931392