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


Authors: I. Beach and R. Rotman
Journal: Proc. Amer. Math. Soc. 148 (2020), 5355-5367
MSC (2010): Primary 53C22
DOI: https://doi.org/10.1090/proc/15194
Published electronically: September 24, 2020
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Abstract: In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $ l(M)$, on a complete, non-compact Riemannian surface $ M$ of finite area $ A$. We will show that $ l(M) \leq 4\sqrt {2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $ l(M) \leq 31 \sqrt {A}$. Additionally, for a surface with at least two ends we show that $ l(M) \leq 2\sqrt {2A}$, improving the prior estimate of Croke that $ l(M) \leq (12+3\sqrt {2})\sqrt {A}$.


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Additional Information

I. Beach
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
ORCID: 0000-0002-1009-1669
Email: isabel.beach@mail.utoronto.ca

R. Rotman
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
MR Author ID: 659650
Email: rina@math.toronto.edu

DOI: https://doi.org/10.1090/proc/15194
Received by editor(s): January 9, 2020
Received by editor(s) in revised form: May 17, 2020
Published electronically: September 24, 2020
Additional Notes: This research has been partially supported by the University of Toronto Work Study grant of the first author and by the NSERC Discovery Grant RGPIN-2018-04523 of the second author.
Communicated by: Jiaping Wang
Article copyright: © Copyright 2020 American Mathematical Society