Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary

TL;DR

This paper proves the existence of multiple short orthogonal geodesic chords on convex 2-disks, with bounds depending on geometric properties, and establishes the existence of at least one such chord on any Riemannian 2-disk.

Contribution

It introduces new existence results for short orthogonal geodesic chords on convex and general Riemannian 2-disks, with explicit length bounds.

Findings

At least two distinct short orthogonal geodesic chords exist on convex 2-disks.

Length bounds depend on boundary length, diameter, and area of the disk.

Existence of a short geodesic chord on any Riemannian 2-disk is established.

Abstract

We prove the existence of at least two distinct short, simple orthogonal geodesic chords on a Riemannian 2-disk $M$ with convex boundary. The lengths of these curves are bounded in terms of the length of $\partial M$, the diameter of $M$, and the area of $M$. We also prove the existence of a short, simple geodesic chord on any Riemannian 2-disk.