A Multiplicity Result for Orthogonal Geodesic Chords in Finsler disks

TL;DR

This paper proves the existence of multiple orthogonal Finsler geodesic chords in disk-shaped manifolds under weaker conditions than convexity, with results depending on the reversibility of the Finsler metric.

Contribution

It establishes new multiplicity results for orthogonal Finsler geodesic chords under relaxed assumptions, extending previous work beyond convexity.

Findings

At least N orthogonal geodesic chords for reversible metrics

At least two orthogonal geodesic chords with different energies without reversibility

Results apply to manifolds homeomorphic to a disk

Abstract

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a N-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least N orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.