scholarly journals On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Yan Wu ◽  
Yi Qi ◽  
Zunwei Fu
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Dual Space ◽  
Teichmüller Space ◽  
Geodesic Segment ◽  
Teichmuller Space ◽  
Quadratic Differentials ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Infinite Type

LetAZ(R)be the infinitesimal asymptotic Teichmüller space of a Riemann surfaceRof infinite type. It is known thatAZ(R)is the quotient Banach space of the infinitesimal Teichmüller spaceZ(R), whereZ(R)is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points inAZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller spaceAZ(D)by constructing a special degenerating sequence.

scholarly journals Harmonic maps and wild Teichmüller spaces

2019 ◽  
pp. 1-45
Author(s):  
Subhojoy Gupta
Keyword(s):  
Riemann Surface ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Hyperbolic Surface ◽  
Teichmuller Space ◽  
Quadratic Differentials ◽  
Hopf Differential ◽  
Principal Parts

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

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 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 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.

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.

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]).

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