Minimal length of nonsimple closed geodesics on hyperbolic surfaces

TL;DR

This paper establishes that the shortest closed geodesics with self-intersections on hyperbolic surfaces grow logarithmically with the number of intersections, providing a precise asymptotic order.

Contribution

It proves that the minimal length of such geodesics scales as 2 log k for large k, revealing a fundamental geometric relationship.

Findings

Minimal length scales as 2 log k for large k

Provides asymptotic order of geodesic lengths with self-intersections

Enhances understanding of hyperbolic surface geometry

Abstract

In the present paper, we show that the minimal length of closed geodesics on finite-type hyperbolic surfaces with self-intersection number $k$ has order $2\log k$ as $k$ gets large.