A Hilbert manifold structure on the Weil–Petersson class Teichmüller 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 disc removed. We define a refined Teichmüller space of such Riemann surfaces 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 nonoverlapping 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.2010 Mathematics Subject Classification. Primary 30F60 ; Secondary 30C55, 30C62, 32G15, 46E20, 81T40. Key words and phrases. Refined Teichmüller space, Hilbert manifold, quasiconformal maps, moduli space of rigged Riemann surfaces, conformal field theory.Eric Schippers is partially supported by the National Sciences and Engineering Research Council. He would like to thank Nina Zorboska for several helpful conversations.

Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

“…Introduction and statement of results. In this paper, we demonstrate that the refined Teichmüller space of bordered Riemann surfaces defined by the authors in [18] possesses a convergent Weil-Petersson metric, and a simple L 2 space of Beltrami differentials models the tangent space.…”

Comparison Moduli Spaces of Riemann Surfaces

“…It is possible to extend the Weil-Petersson metric to a much wider class of surfaces, again by obtaining an L 2 theory. Radnell, Schippers and Staubach [47,48,50,51] did this for bordered surfaces of type (g, n). M. Yanagishita [79] extended the L p theory of Guo [22] and Tang [75] to surfaces satisfying "Lehner's condition", which includes bordered surfaces of type (g, n).…”

“…That is, one may view T WP (Σ) as fibred over T (Σ P ) for a compact surface with punctures Σ P , such that the fibres C −1 (p) modulo a discrete group action are biholomorphic to O qc WP (Σ P 1 ). This can be used to construct a Hausdorff, second countable topology on T WP (Σ), and a complex Hilbert manifold structure [47]. The advantage of this approach is that it is very flexible and constructive, and explicit coordinates can be given in terms of Gardiner-Schiffer variation.…”

Quasiconformal Teichmüller theory as an analytical foundation for two-dimensional conformal field theory

“…Up until recently, the only example of Teichmüller spaces with convergent Weil-Petersson pairing, aside from the finite dimensional Teichmüller spaces, was the Weil-Petersson universal Teichmüller space. In a series of papers [48,49,50,51] the authors defined a Weil-Petersson Teichmüller space of bordered surfaces of type (g, n), based on the fiber structure on Teichmüller space derived from the rigged moduli space (Theorem 4.14). M. Yanagishita independently gave a definition which includes these surfaces, based on the Bers embedding of L 2 ∩ L ∞ Beltrami differentials into an open subset of the quadratic differentials, using the Fuchsian group point of view [71].…”

“…[23,25,27] and reference therein. To our knowledge, apart from the work of the authors [44,45,48] and that of Y.-Z. Huang [25] in genus zero and K. Barron [5,6] in the genus zero super case, neither the analytic structure of these infinite-dimensional moduli spaces, nor even their point-set topology, have been rigorously defined or studied.…”