scholarly journals Connected components of the strata of quadratic differentials over the Teichmüller space

2009 ◽  
Vol 142 (1) ◽  
pp. 47-60 ◽  
Author(s):  
Katharine C. Walker
Keyword(s):  
Teichmüller Space ◽  
Connected Components ◽  
Teichmuller Space ◽  
Quadratic Differentials

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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document