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.

In: Communications in Contemporary Mathematics, Vol. 17, No. 04, 08.2015.

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@article{96ea3a6963e9481fb5348ac999aea920,
title = "A Hilbert manifold structure on the Weil–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",

}

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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