Narrow equidistribution and counting of closed geodesics on noncompact manifolds

Filling words in fundamental groups of Riemann surfaces

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.1.1227 ◽

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.

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A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture

Advances in Mathematics ◽

10.1016/j.aim.2019.106852 ◽

2019 ◽

Vol 358 ◽

pp. 106852

Author(s):

Ilya Gekhtman ◽

Samuel J. Taylor ◽

Giulio Tiozzo

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Closed Geodesics

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Angular self-intersections for closed geodesics on surfaces

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-05-08382-6 ◽

2005 ◽

Vol 134 (02) ◽

pp. 419-426 ◽

Cited By ~ 2

Author(s):

Mark Pollicott ◽

Richard Sharp

Keyword(s):

Closed Geodesics

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WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000232 ◽

2021 ◽

pp. 1-38

Author(s):

Xianzhe Dai ◽

Junrong Yan

Keyword(s):

Essential Role ◽

Morse Function ◽

Noncompact Manifold ◽

Morse Inequalities ◽

Bounded Geometry ◽

Noncompact Manifolds ◽

Witten Laplacian ◽

Relative Cohomology ◽

Witten Deformation ◽

Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

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