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 Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650025 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Tangent Space ◽  
Riemann Surfaces ◽  
Tangent Vector ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Holomorphic Curve ◽  
Hilbert Manifold ◽  
Direct Sum ◽  
Genus Formula

Consider a Riemann surface of genus [Formula: see text] bordered by [Formula: see text] curves homeomorphic to the unit circle, and assume that [Formula: see text]. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in [Formula: see text] and [Formula: see text], and furthermore that the space of [Formula: see text] differentials in [Formula: see text] decomposes as a direct sum of infinitesimally trivial differentials and [Formula: see text] harmonic [Formula: see text] differentials. Thus the tangent space of this Teichmüller space is given by [Formula: see text] harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of [Formula: see text] harmonic Beltrami differentials.

scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

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 ON STABILIZERS OF SOME TEICHMÜLLER DISKS IN POINTED MAPPING CLASS GROUPS

2010 ◽  
Vol 52 (3) ◽  
pp. 593-604
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Mapping Class Group ◽  
Isometric Embedding ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class Groups ◽  
Natural Projection ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Groups

AbstractWe prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

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