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

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- by I. Beach and R. Rotman PDF
- Proc. Amer. Math. Soc.
**148**(2020), 5355-5367 Request permission

## 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 $l(M) \leq (12+3\sqrt {2})\sqrt {A}$.## References

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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
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5355-5367 - MSC (2010): Primary 53C22
- DOI: https://doi.org/10.1090/proc/15194
- MathSciNet review: 4163847