Curvature-free linear length bounds on geodesics in closed Riemannian surfaces

TL;DR

This paper establishes new upper bounds on the lengths of the shortest geodesics in closed Riemannian surfaces, improving previous estimates and providing explicit bounds based on the surface's diameter and the geodesic's order.

Contribution

It introduces tighter, curvature-free bounds for the lengths of the $k$th shortest geodesics in closed Riemannian surfaces, refining earlier results by Nabutovsky and Rotman.

Findings

Length of the $k$th shortest geodesic is at most 8kd.

If $p = q$, the bound improves to 6kd.

Bounds are independent of curvature, depending only on diameter and geodesic order.

Abstract

This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k^\text{th}$-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$. This improves prior estimates by A. Nabutovsky and R. Rotman.