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

2025/07/10 by Isabel Beach, Beach, Isabel · 1 citation
Mathematics · #53C22 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2507.07797

openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∂ 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.

Citations

• Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary
• Multiple orthogonal geodesic chords and a proof of Seifert's conjecture\n on brake orbits
• Constructing monotone homotopies and sweepouts
• Lengths of three simple periodic geodesics on a Riemannian 2-sphere
• Monotone homotopies and contracting discs on Riemannian surfaces
• Converting homotopies to isotopies and dividing homotopies in half in an\n effective way
• Contracting the boundary of a Riemannian 2-disc
• Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds

Cited by

• Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds

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• Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds
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• Multiple orthogonal geodesic chords and a proof of Seifert's conjecture\n on brake orbits
• Riemannian 3-spheres that are hard to sweep out by short curves
• Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary