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Short closed geodesics with self-intersections

Part of: Riemann surfaces Deformations of analytic structures Global differential geometry

Published online by Cambridge University Press:  24 January 2020

VIVEKA ERLANDSSON  and
HUGO PARLIER
VIVEKA ERLANDSSON
Affiliation:
School of Mathematics, University of Bristol, e-mail: v.erlandsson@bristol.ac.uk
HUGO PARLIER
Affiliation:
Department of Mathematics, University of Luxembourg, e-mail: hugo.parlier@uni.lu
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Abstract

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

MSC classification

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory
Secondary: 30F10: Compact Riemann surfaces and uniformization 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 53C22: Geodesics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 623 - 638
Copyright
© Cambridge Philosophical Society 2020

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