scholarly journals Fenchel–Nielsen coordinates on upper bounded pants decompositions

2015 ◽  
Vol 158 (3) ◽  
pp. 385-397 ◽  
Author(s):  
DRAGOMIR ŠARIĆ
Keyword(s):  
Teichmüller Space ◽  
Hyperbolic Surface ◽  
Length Spectrum ◽  
Teichmuller Space ◽  
Closed Geodesics ◽  
Infinite Type ◽  
Pants Decomposition ◽  
Length Spectrum Metric ◽  
Path Connected ◽  
Simple Closed Geodesics

AbstractLet X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

scholarly journals Geodesically Complete Hyperbolic Structures

2017 ◽  
Vol 166 (2) ◽  
pp. 219-242
Author(s):  
ARA BASMAJIAN ◽  
DRAGOMIR ŠARIĆ
Keyword(s):  
Teichmüller Space ◽  
Hyperbolic Surface ◽  
Teichmuller Space ◽  
Geometric Proof ◽  
Hyperbolic Structures ◽  
Infinite Type ◽  
Length Spectrum Metric ◽  
Geodesically Complete ◽  
The Relationship ◽  
Convex Core

AbstractIn the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

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.

SYSTOLIC FILLINGS OF SURFACES

2018 ◽  
Vol 98 (3) ◽  
pp. 502-511
Author(s):  
BIDYUT SANKI
Keyword(s):  
Closed Geodesic ◽  
Disjoint Union ◽  
Hyperbolic Surface ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Simple Closed Geodesics

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

scholarly journals On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space

2015 ◽  
Vol 179 (2) ◽  
pp. 165-189 ◽  
Author(s):  
D. Alessandrini ◽  
L. Liu ◽  
A. Papadopoulos ◽  
W. Su
Keyword(s):  
Teichmüller Space ◽  
Length Spectrum ◽  
Teichmuller Space

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.

Length spectrum characterization of asymptotic Teichmüller space

2018 ◽  
Vol 186 (1) ◽  
pp. 73-91
Author(s):  
Jun Hu ◽  
Francisco G. Jimenez-Lopez
Keyword(s):  
Teichmüller Space ◽  
Length Spectrum ◽  
Teichmuller Space

Teichmuller Space

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hyperbolic Structure ◽  
Closed Curves ◽  
Pants Decomposition

This chapter deals with Teichmüller space Teich(S) of a surface S. It first defines Teichmüller space and a topology on Teich(S) before giving two heuristic counts of its dimension. It then describes explicit coordinates on Teich(Sɡ) coming from certain length and twist parameters for curves in a pair of pants decomposition of Sɡ; these are the Fenchel–Nielsen coordinates on Teich(Sɡ). The chapter also considers the Teichmüller space of the torus and concludes by proving the 9g – 9 theorem, which states that a hyperbolic structure on Sɡ is completely determined by the lengths assigned to 9g – 9 isotopy classes of simple closed curves in Sɡ.

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