Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles

TL;DR

This paper proves the existence of multiple orthogonal geodesic chords in nonconvex Riemannian disks with boundary, using nonsmooth critical point theory and obstacle geodesics, advancing the understanding of brake orbits in Hamiltonian systems.

Contribution

It introduces a novel multiplicity result for orthogonal geodesic chords in nonconvex Riemannian disks, linking geometric analysis with Hamiltonian dynamics.

Findings

Multiple orthogonal geodesic chords exist in nonconvex Riemannian disks.

The results apply to brake orbits in potential wells of Hamiltonian systems.

Progress towards Seifert's conjecture on brake orbits.

Abstract

We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an $N$-disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert.