[Paper Review] Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits

Roberto Giambò, Fabio Giannoni+1 more|arXiv (Cornell University)|Feb 22, 2020

Geometric Analysis and Curvature Flows15 references4 citations

TL;DR

This paper proves Seifert's conjecture by establishing the existence of at least $N$ geometrically distinct orthogonal geodesic chords (OGCs) in a strongly concave $N$-dimensional Riemannian disk using a pseudo-gradient flow and minimax theory. The result confirms that a natural Lagrangian/Hamiltonian system in a potential well has at least $N$ distinct brake orbits, resolving a long-standing conjecture in dynamical systems and geometric analysis.

ABSTRACT

Using nonsmooth critical point theory, we prove the existence of at least N orthogonal geodesic chords in a class of Riemannian N-disk with strongly concave boundary. This yields a proof of a celebrated conjecture by H.Seifert on the number of brake orbits in a potential well of a natural Lagrangian/Hamiltonian system.

Motivation & Objective

• To resolve Seifert's conjecture on the minimal number of brake orbits in a potential well of a natural Lagrangian system.
• To establish the existence of at least $N$ geometrically distinct orthogonal geodesic chords (OGCs) in a strongly concave Riemannian $N$-disk.
• To extend the classical result of Bos on convex domains to the non-convex, strongly concave setting using new variational techniques.
• To develop a minimax approach based on relative category and pseudo-gradient flows to overcome failure of standard gradient flow methods in non-convex settings.
• To rigorously link the geometric problem of OGCs to the dynamical problem of brake orbits via Maupertuis' principle and conformal metrics.

Proposed method

• Employ a pseudo-gradient flow approach to construct a deformation retract that preserves the relative category structure in the space of paths.
• Use minimax theory via the relative Lusternik-Schnirelman category to extract critical points of the energy functional on the space of $\mathbb{Z}_2$-symmetric paths.
• Define a $\mathcal{V}^{-}$- and $\mathcal{V}^{+}$-vector field framework to analyze Palais-Smale sequences and ensure convergence to OGCs.
• Apply a deformation lemma to show that the functional $\mathcal{F}$ satisfies the necessary compactness and homotopy properties for critical point extraction.
• Use the quotient space $\widetilde{\mathfrak{C}}$ under $\mathbb{Z}_2$-action to compute $\operatorname{cat}_{\widetilde{\mathfrak{C}},\widetilde{\mathfrak{C}}_0}(\widetilde{\mathfrak{C}}) = N$, leveraging the topology of real projective space.
• Establish that distinct critical values $c_1 < c_2 < \cdots < c_N$ correspond to $N$ geometrically distinct OGCs via transversality and energy uniqueness.

Experimental results

Research questions

• RQ1 Does a strongly concave Riemannian $N$-disk admit at least $N$ geometrically distinct orthogonal geodesic chords?
• RQ2 Can the classical result on OGCs in convex domains be extended to non-convex, strongly concave domains?
• RQ3 Is Seifert's conjecture on the minimal number of brake orbits in a potential well true for natural Lagrangian systems?
• RQ4 Can a pseudo-gradient flow and minimax method overcome the failure of standard gradient flow techniques in non-convex settings?
• RQ5 What is the role of relative category and $\mathbb{Z}_2$-symmetry in extracting multiple critical points for the energy functional?

Key findings

• The paper proves that any $N$-dimensional Riemannian disk with strongly concave boundary contains at least $N$ geometrically distinct orthogonal geodesic chords.
• The existence of $N$ distinct OGCs is established via a minimax scheme based on the relative $\mathbb{Z}_2$-equivariant category, yielding $N$ distinct critical values $c_1 < c_2 < \cdots < c_N$ for the energy functional.
• The critical values $c_i$ correspond to $N$ distinct OGCs, with $c_1 \geq \delta_1^2 / K_0^2 > 0$, ensuring non-degeneracy and positivity.
• The proof establishes that the number of brake orbits in a potential well of a natural Lagrangian system is at least $N$, confirming Seifert's conjecture.
• The method successfully handles non-convexity by avoiding curve-shortening flows and instead using a pseudo-gradient construction that preserves domain invariance.
• The result holds under the assumption of strong concavity of the boundary, which ensures the existence of a suitable conformal metric linking OGCs to brake orbits via Maupertuis' principle.