A characterization of biholomorphic automorphisms of Teichmüller space

2012 ◽  
Vol 154 (1) ◽  
pp. 71-83
Author(s):  
RYOSUKE MINEYAMA ◽  
HIDEKI MIYACHI
Keyword(s):  
Riemann Surface ◽  
Finite Type ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Alternative Approach

AbstractIn this paper, we give an alternative approach to Royden–Earle–Kra–Markovic's characterization of biholomorphic automorphisms of Teichmüller space of Riemann surface of analytically finite type.

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 boundaries of Schottky spaces

1976 ◽  
Vol 62 ◽  
pp. 97-124 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Compact Riemann Surface ◽  
Teichmüller Space ◽  
Schottky Group ◽  
Teichmuller Space ◽  
Upper Half Plane

Let S be a compact Riemann surface and let Sn be the surface obtained from S in the course of a pinching deformation. We denote by Γn the quasi-Fuchsian group representing Sn in the Teichmüller space T(Γ), where Γ is a Fuchsian group with U/Γ = S (U: the upper half plane). Then in the previous paper [7] we showed that the limit of the sequence of Γn is a cusp on the boundary ∂T(Γ). In this paper we will consider the case of Schottky space . Let Gn be a Schottky group with Ω(Gn)/Gn = Sn. Then the purpose of this paper is to show what the limit of Gn is.

scholarly journals The geometry of measured geodesic laminations and measured train tracks

1989 ◽  
Vol 9 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Howard Weiss
Keyword(s):  
Riemann Surface ◽  
Linear Model ◽  
Tangent Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hölder Continuous ◽  
Geodesic Laminations ◽  
Non Linear ◽  
Train Tracks ◽  
Hyperbolic Riemann Surface

AbstractThurston and Kerckhoff have shown that the space of measured geodesic laminations on a hyperbolic Riemann surface serves as a non-linear model of the tangent space to Teichmüller space at the surface. In this paper we show that the natural map between these manifolds has stronger than Hölder continuous regularity.

scholarly journals On the classification of mapping class actions on Thurston's asymmetric metric

2013 ◽  
Vol 155 (3) ◽  
pp. 499-515 ◽  
Author(s):  
L. LIU ◽  
A. PAPADOPOULOS ◽  
W. SU ◽  
G. THÉRET
Keyword(s):  
Finite Type ◽  
Mapping Class Group ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Actions ◽  
The One

AbstractWe study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil–Petersson metric.

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