The Length of the Shortest Closed Geodesic on a Surface of Finite Area

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