Geometric filling curves on surfaces

TL;DR

This paper investigates the density of closed geodesics on hyperbolic surfaces, providing bounds on the shortest geodesic length that densely covers the surface within a specified approximation.

Contribution

It introduces upper bounds on the length of the shortest closed geodesic that $ ext{ extepsilon}$-fills a hyperbolic surface, advancing understanding of geodesic density.

Findings

Established upper bounds on geodesic lengths for $ ext{ extepsilon}$-filling curves

Quantitative measures of geodesic density on hyperbolic surfaces

Insights into the geometric structure of closed geodesics

Abstract

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.