scholarly journals Some metric properties of the Teichmüller space of a closed set in the Riemann sphere

2017 ◽  
Vol 42 ◽  
pp. 755-770
Author(s):  
Nishan Chatterjee
Keyword(s):  
Riemann Sphere ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Set ◽  
Metric Properties

scholarly journals Teichmüller space of a countable set of points on the Riemann sphere

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 45-51
Author(s):  
Masahiko Taniguchi
Keyword(s):  
Configuration Space ◽  
Infinite Number ◽  
Riemann Sphere ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Limit Set ◽  
Finitely Generated ◽  
Forward Limit ◽  
Set Of Points ◽  
Totally Disconnected

We introduce the Teichm?ller space T(E) of an ordered countable set E of infinite number of distinct points on the Riemann sphere. We discuss the relation between the Teichm?ller distance on T(E) and a natural one on the configuration space for E. Also we give a system of global holomorphic coordinates for T(E) when E is determined from a finitely generated semigroup consisting of M?bius transformations with the totally disconnected forward limit set.

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.

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