A Note on Collars of Simple Closed Geodesics

2005 ◽  
Vol 112 (1) ◽  
pp. 165-168 ◽  
Author(s):  
Hugo Parlier
Keyword(s):  
Closed Geodesics ◽  
Simple Closed Geodesics

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.

scholarly journals Simple closed geodesics on convex surfaces

1992 ◽  
Vol 36 (3) ◽  
pp. 517-549 ◽  
Author(s):  
Eugenio Calabi ◽  
Jian Guo Cao
Keyword(s):  
Closed Geodesics ◽  
Convex Surfaces ◽  
Simple Closed Geodesics

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}.

scholarly journals Parametrizing simple closed geodesy on Γ3\ℋ

2003 ◽  
Vol 74 (1) ◽  
pp. 43-60 ◽  
Author(s):  
Thomas A. Schmidt ◽  
Mark Sheingorn
Keyword(s):  
Lower Bound ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Closed Curves ◽  
Bounded Interval ◽  
Fundamental Domains ◽  
Simple Closed Geodesics

AbstractWe exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3with Markoff triples.The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

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