scholarly journals SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS

2009 ◽  
Vol 87 (2) ◽  
pp. 275-288 ◽  
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Finite Type ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Closed Geodesics ◽  
Dehn Twists ◽  
Holomorphic Quadratic Differential ◽  
Simple Closed Geodesics ◽  
Disk Component ◽  
Teichmüller Mapping

AbstractLet S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

scholarly journals The growth rate of trajectories of a quadratic differential

1990 ◽  
Vol 10 (1) ◽  
pp. 151-176 ◽  
Author(s):  
Howard Masur
Keyword(s):  
Growth Rate ◽  
Riemann Surface ◽  
Quadratic Differential ◽  
Compact Riemann Surface ◽  
Saddle Connection ◽  
Asymptotic Growth ◽  
Holomorphic Quadratic Differential ◽  
Asymptotic Growth Rate

AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

SHARP POINCARÉ–SOBOLEV INEQUALITIES AND THE SHORTEST LENGTH OF SIMPLE CLOSED GEODESICS ON A TOPOLOGICAL TWO SPHERE

2004 ◽  
Vol 06 (05) ◽  
pp. 781-792 ◽  
Author(s):  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Inequalities ◽  
Square Root ◽  
Two Dimensional ◽  
Closed Geodesics ◽  
Sharp Constants ◽  
Simple Closed Geodesics

We show that the sharp constants of Poincaré–Sobolev inequalities for any smooth two dimensional Riemannian manifold are less than or equal to [Formula: see text]. For a smooth topological two sphere M2, the sharp constants are [Formula: see text] if and only if M2 is isometric to two sphere S2 with the standard metric. In the same spirit, we show that for certain special smooth topological sphere the ratio between the shortest length of simple closed geodesics and the square root of its area is less than or equals to [Formula: see text].

scholarly journals Non-simple geodesics in hyperbolic 3-manifolds

1994 ◽  
Vol 116 (2) ◽  
pp. 339-351
Author(s):  
Kerry N. Jones ◽  
Alan W. Reid
Keyword(s):  
Closed Geodesic ◽  
Closed Geodesics ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

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