scholarly journals The length of a shortest closed geodesic on a surface of finite area

2020 ◽  
Vol 148 (12) ◽  
pp. 5355-5367
Author(s):  
I. Beach ◽  
R. Rotman
Keyword(s):  
Closed Geodesic ◽  
Finite Area

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals Geodesic planes in geometrically finite acylindrical -manifolds

2021 ◽  
pp. 1-40
Author(s):  
YVES BENOIST ◽  
HEE OH
Keyword(s):  
Closed Geodesic ◽  
Duke Math ◽  
Geometrically Finite ◽  
Convex Cocompact ◽  
Invent Math ◽  
Convex Core

Abstract Let M be a geometrically finite acylindrical hyperbolic $3$ -manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math.209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$ -manifold $M_0$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$ . We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

A New and Simpler Formulation for Radiative Angle Factors

1963 ◽  
Vol 85 (2) ◽  
pp. 81-87 ◽  
Author(s):  
E. M. Sparrow
Keyword(s):  
Numerical Evaluation ◽  
Energy Exchange ◽  
Analytical Calculation ◽  
Additional Benefit ◽  
Finite Area ◽  
Reduced Order ◽  
Practical Utility ◽  
New Formulation ◽  
Superposition Theorem

A new representation for diffuse angle factors has been derived which replaces the usual area integrals by more tractable contour (i.e., line) integrals. The new formulation generally simplifies analytical calculation of angle factors. The advantages of the new representation are associated with the reduced order of the integrals (i.e., double reduced to single, quadruple reduced to double) which must be evaluated to calculate the angle factor. An additional benefit of the new representation is that integrals of simpler form are encountered than in the present representation. For the numerical evaluation of angle factors, the reduction in the order of the integrals should have great practical utility. In the case of energy exchange between an infinitesimal element and a finite area, a superposition theorem has been derived which permits results for certain basic surfaces to be linearly combined to yield angle factors for surfaces at other orientations. Several illustrations of the application of the new formulation are presented.

Finite area element snow loading prediction - applications and advancements

1992 ◽  
Vol 42 (1-3) ◽  
pp. 1537-1548 ◽  
Author(s):  
Scott L. Gamble ◽  
Will W. Kochanski ◽  
Peter A. Irwin
Keyword(s):  
Finite Area ◽  
Area Element ◽  
Snow Loading

scholarly journals A Polyakov Formula for Sectors

2017 ◽  
Vol 28 (2) ◽  
pp. 1773-1839 ◽  
Author(s):  
Clara L. Aldana ◽  
Julie Rowlett
Keyword(s):  
Heat Kernel ◽  
Explicit Expression ◽  
Unit Area ◽  
Logarithmic Singularity ◽  
Conformal Factor ◽  
Opening Angle ◽  
Conformal Deformation ◽  
Finite Area ◽  
Different Parts

Abstract We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

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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