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

2019/12/16 by Isabel Beach, Beach, I., Regina Rotman +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1912.07711

openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) ≤ 4√(2A) on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that l(M) ≤ 31 √(A). Additionally, for a surface with at least two ends we show that l(M) ≤ 2√(2A), improving the prior estimate of Croke that l(M) ≤ (12+3√(2))√(A).

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