scholarly journals QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE

2006 ◽  
Vol 08 (04) ◽  
pp. 481-534 ◽  
Author(s):  
DAVID RADNELL ◽  
ERIC SCHIPPERS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Operator Algebras ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces

One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them; (2) With this complex structure, the sewing operation is holomorphic; (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

scholarly journals On augmented Schottky spaces and automorphic forms, I

1979 ◽  
Vol 75 ◽  
pp. 151-175 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Automorphic Forms ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces

With respect to Teichmüller spaces, many beautiful results are obtained by TeichmüUer, Ahlfors, Bers, Maskit, Kra, Earle, Abikoff, and others. For example, the boundary consists of b-groups, and the augmented Teichmüller space is defined by attaching a part of the boundary to the Teichmüller space. By using the augmented Teichmüller space, a compactification of the moduli space of Riemann surfaces is accomplished (cf. Abikoff [1], Bers [2]).

NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE

1998 ◽  
Vol 09 (01) ◽  
pp. 1-45 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Teichmuller Space ◽  
Open Dense Subset ◽  
Complex Structures ◽  
Thurston Boundary ◽  
Simply Connected ◽  
Singular Metric

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on ℳ(Σ), the moduli space of flat G-connections, for G any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold ℳ′(Σ), the open dense subset of smooth points of ℳ(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on ℳ′(Σ) converges to the polarization associated to (F, g). In particular we see that the Jeffrey–Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuiuty at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on ℳ′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.

scholarly journals On Domains Biholomorphic to Teichmüller Spaces

2018 ◽  
Vol 2020 (8) ◽  
pp. 2542-2560 ◽  
Author(s):  
Subhojoy Gupta ◽  
Harish Seshadri
Keyword(s):  
Boundary Point ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Teichmuller Space ◽  
Teichmüller Spaces ◽  
Strictly Convex ◽  
Teichmuller Spaces

Abstract We prove that the Teichmüller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.

scholarly journals On Teichmüller Spaces of Koebe Groups

1997 ◽  
Vol 08 (05) ◽  
pp. 611-632
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Teichmüller Space ◽  
Universal Covering ◽  
Kleinian Groups ◽  
Fuchsian Groups ◽  
Teichmuller Space ◽  
Isomorphism Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Covering Group

In this paper we study the Teichmüller space of constructible Koebe groups. These are Kleinian groups arising from planar covering of 2-orbifolds. In the first part, we parametrize the Teichmüller spaces of Koebe groups using a technique that can be applied to explicitly compute generators of these groups, maybe by programming a computer. In the second part, we study some properties of these Teichmüller spaces. More precisely, we find the covering group of these spaces (the universal covering is the Teichmüller space of the punctured surface), and prove an isomorphism theorem similar to the Bers–Greenberg theorem for Fuchsian groups.

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

2015 ◽  
Vol 17 (04) ◽  
pp. 1550016 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Banach Manifold ◽  
Holomorphic Maps ◽  
Discontinuous Group ◽  
Hilbert Manifold ◽  
Conformal Field ◽  
Compact Riemann Surfaces

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.

THE BERGMAN METRIC ON TEICHMÜLLER SPACE

2004 ◽  
Vol 15 (10) ◽  
pp. 1085-1091 ◽  
Author(s):  
BO-YONG CHEN
Keyword(s):  
Simple Proof ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Bergman Metric

We give a simple proof of a theorem of McMullen on Kähler hyperbolicity of moduli space of Riemann surfaces by using the Bergman metric on Teichmüller space.

A Remark about Components of Relative Teichmüller Spaces

1981 ◽  
Vol 24 (2) ◽  
pp. 245-246
Author(s):  
Jane Gilman
Keyword(s):  
Fixed Point ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Fixed Point Set ◽  
Teichmüller Spaces ◽  
Number Of Components ◽  
Teichmuller Spaces ◽  
Point Set

Our aim is to compute for all n > 2, ψ(n, h), the number of components of a certain quotient of the fixed point set of an involution in the "mod-n" Teichmuller space. This answers part of a question raised by Earle [2] and corrects and extends the answer due to Zarrow (See Theorem 2 of [6]).

scholarly journals Higher Complex Structures

2019 ◽  
Author(s):  
Vladimir Fock ◽  
Alexander Thomas
Keyword(s):  
Moduli Space ◽  
Geometric Structure ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Complex Structures ◽  
Punctual Hilbert Scheme

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.

Complex Geometry of the Universal Teichmüller Space. II

2007 ◽  
Vol 14 (3) ◽  
pp. 483-498
Author(s):  
Samuel Krushkal
Keyword(s):  
Complex Geometry ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Isometric Embeddings ◽  
Universal Teichmüller Space ◽  
Important Theorem ◽  
Invariant Distances

Abstract We give an alternate and simpler proof of the important theorem stating that all invariant distances on the universal Teichmüller space 𝐓 coincide, and solve for 𝐓 the problem of Kra on isometric embeddings of a disk into Teichmüller spaces.

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