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

TL;DR

This paper establishes improved upper bounds for the length of the shortest closed geodesic on finite-area non-compact surfaces, refining previous estimates and providing tighter bounds based on the number of ends.

Contribution

The paper introduces new, sharper upper bounds for the shortest closed geodesic length on surfaces with finite area, depending on the number of ends, improving prior estimates by Croke.

Findings

For surfaces with one end, shortest geodesic length ≤ 4√(2A).

For surfaces with at least two ends, shortest geodesic length ≤ 2√(2A).

Improved bounds are tighter than previous estimates by Croke.

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