An orthogonal geodesic chord (OGC) of a Riemannian domain is a nonconstant geodesic that meets the boundary orthogonally at both endpoints; via the Jacobi (Maupertuis) metric, brake orbits of a natural Hamiltonian system at fixed energy correspond to OGCs of a strongly concave Riemannian disk. All known multiplicity theorems for OGCs on nonconvex disks — including the theorem of Giambò–Giannoni–Piccione (2018, no O–T chords) and their 2020 preprint proving Seifert's conjecture (strongly concave disks plus an auxiliary chord-avoidance assumption (3.13)) — require a chord-exclusion hypothesis that the authors themselves describe as possibly inessential, and genericity of multiplicity among smooth metrics is known only for convex boundary. The open problem is the completion of this multiplicity theory: (1) prove that every strongly concave Riemannian N-disk carries at least N geometrically distinct OGCs with no chord-avoidance hypothesis, or exhibit a counterexample; (2) settle whether multiplicity is generic (Baire-residual or open-dense) among smooth metrics on the disk without boundary convexity, and correspondingly robust for brake orbits among potentials.

Background

A natural (classical-type) Hamiltonian on a configuration manifold is of the form H(q,p)=12⟨p,p⟩g+V(q)H(q,p)=\frac{1}{2}\langle p,p\rangle_{g}+V(q), with Riemannian metric gg and potential VV; equivalently one writes the Lagrangian system Ddtx˙=−∇V(x)\frac{D}{dt}\dot{x}=-\nabla V(x). A brake orbit is a periodic solution of such a system that oscillates between two turning points where the velocity (momentum) vanishes, both lying on the level set V−1(E)V^{-1}(E) at fixed energy EE. If EE is a regular value of VV and the potential well V−1((−∞,E])V^{-1}((-\infty,E]) is homeomorphic to an N-dimensional disk, Seifert conjectured in 1948 that the system admits at least NN geometrically distinct brake orbits; this bound is sharp for anisotropic quadratic potentials. By Maupertuis' principle, motions of energy EE are reparametrized geodesics of the conformal Jacobi Metric (E−V) g(E-V)\,g on the well, and brake orbits correspond to orthogonal geodesic chords (OGCs) — nonconstant geodesic segments γ:[a,b]→Ωˉ\gamma:[a,b]\to\bar{\Omega} of a domain Ω\Omega with interior in Ω\Omega that start and arrive orthogonally to ∂Ω\partial\Omega. The Jacobi Metric makes the relevant sublevel domains strongly concave: the second fundamental form of the boundary is positive definite in the outward normal direction; equivalently, geodesics starting tangentially to ∂Ω\partial\Omega move locally into the interior, and geodesics with both endpoints on ∂Ω\partial\Omega cannot remain uniformly close to the boundary.

For convex domains the classical Lusternik–Schnirelman theorem and Bos's 1963 generalization give at least NN distinct OGCs by curve-shortening. Beyond convexity these methods fail: geodesics may be tangent to the boundary, and arbitrarily small shortening variations of such curves leave the domain, so gradient flows of the geodesic action functional are not future-complete. Bos already observed that nonconvex plane regions may admit no OGC at all, so some hypothesis beyond disk topology is necessary in the fully general nonconvex class. The relevant partial-regularity substitute for critical points is the theory of geodesics with obstacle (Marino–Scolozzi): curves that alternate between boundary arcs and interior geodesic segments; an O–T chord is a geodesic with both endpoints on ∂Ω\partial\Omega which is orthogonal to the boundary at one endpoint and tangent at the other; a WOGC (weak orthogonal geodesic chord) is a chord meeting the boundary orthogonally at the endpoints but touching it also at an interior time.

Giambò–Giannoni–Piccione (2018) proved, using nonsmooth critical point theory — the weak-slope deformation method, a nonsmooth analogue of classical gradient-flow deformation arguments — plus the obstacle-geodesic regularity theory, that a Riemannian N-disk without O–T chords has at least NN distinct OGCs, and they noted that the no-O–T condition is C1C^{1}-open near rotationally symmetric metrics while its failure has nonempty C2C^{2}-interior, and also observed — informally, by the same transversality reasoning — that the existence of multiple OGCs is itself not a generic property for metrics on the disk, suggesting that generic multiplicity may fail outside restricted classes. Their preprint of 2020 (v3 2022) strengthens this: a strongly concave Riemannian N-disk satisfying the auxiliary assumption (3.13) — the existence of a point x0∈Ωx_{0}\in\Omega and M0>L(2)(Ω)M_{0}>L^{(2)}(\Omega) such that x0x_{0} lies on no geodesic of length at most M0M_{0} joining boundary to boundary that is tangent at an endpoint — carry at least NN geometrically distinct OGCs. Since suitable Jacobi sublevel domains satisfy (3.13) whenever the number of brake orbits is finite, the authors derive the full Seifert conjecture. However, this 2020 preprint has no published journal version, its unconditional result is still only announced rather than proved, and the authors themselves flag (3.13) as "possibly inessential". On the genericity side, Rademacher (published 2026) proved, combining Bettiol–Giambò genericity machinery with the GGP existence theory, that for an open dense set of metrics on an n-disk (n≥3n\ge 3) with strictly convex boundary there are nn geometrically distinct OGCs without self-intersections, and that for generic potentials the resulting prime brake orbits are nondegenerate, simple, and pairwise disjoint; nothing comparable is known for nonconvex or strongly concave boundaries. Earlier landmark partial results on Seifert's conjecture include the even-convex case (Liu–Zhang 2014), dimension N=2N=2 (Giambò–Giannoni–Piccione 2016), and symmetric convex domains (Long–Zhang–Zhu 2006; at least two brake orbits in bounded symmetric convex domains, later refined to nn in several subcases).

Problem Statement

The problem is the completion of the multiplicity theory of orthogonal geodesic chords on nonconvex Riemannian disks — the two questions left open by the hypothesis-conditional theorems of Giambò–Giannoni–Piccione (2018, 2020). Recall the terminology: an orthogonal geodesic chord (OGC) of a Riemannian domain Ω\Omega is a nonconstant geodesic segment with both endpoints on ∂Ω\partial\Omega and image otherwise contained in Ω\Omega, meeting ∂Ω\partial\Omega orthogonally at both endpoints; two OGCs are geometrically distinct if their images are different sets; a Riemannian N-disk is a smooth Riemannian manifold with boundary whose underlying space is homeomorphic to the closed NN-dimensional ball; and its boundary is strongly concave when the second fundamental form of ∂Ω\partial\Omega, computed with respect to the outward-pointing unit normal, is positive definite at every boundary point. The two components share one objective: a definitive, hypothesis-free (within the strongly concave class) and robust lower-bound theorem for OGCs, which is exactly what the brake-orbit (Seifert) program still requires beyond the current claimed proof.

(1) Removal of the chord-exclusion hypothesis. Let (Ω,g)(\Omega,g) be a smooth Riemannian N-disk with strongly concave boundary. Prove, without any auxiliary chord-avoidance assumption — such as the absence of O–T chords (geodesic chords with both endpoints on ∂Ω\partial\Omega that are orthogonal to ∂Ω\partial\Omega at one endpoint and tangent at the other) in the 2018 theorem, or assumption (3.13) of arXiv:2002.09687 in the 2020 preprint (the existence of a point of Ω\Omega and a constant M0M_{0} such that no geodesic of length at most M0M_{0} joining ∂Ω\partial\Omega to itself and tangent to ∂Ω\partial\Omega at an endpoint passes through that point) — that Ω\Omega contains at least NN geometrically distinct orthogonal geodesic chords; or, alternatively, construct a strongly concave Riemannian N-disk admitting fewer than NN geometrically distinct OGCs, thereby showing the auxiliary hypothesis is essential.

(2) Generic multiplicity. Settle whether multiplicity is generic without boundary convexity: does there exist a Baire-residual (or at least open and dense) subset of the space of smooth Riemannian metrics on the N-disk, with no convexity condition on the boundary, such that every metric in it admits at least NN geometrically distinct OGCs — or, on the contrary, does multiplicity fail on a residual or nonempty-open set of metrics, as the authors of the 2018 paper suggest by transversality heuristics? Establish the corresponding robust statement for brake orbits: for a generic smooth potential (within the class whose energy sublevels are disks, at a regular energy value), are there at least NN geometrically distinct brake orbits that persist under small smooth perturbations of the data? The known generic-multiplicity results of Rademacher cover only strictly convex boundary (n≥3n\ge 3) and generic potentials in that setting; the nonconvex and strongly concave cases are open.

References

  1. Roberto Giambò, Fabio Giannoni, Paolo Piccione, "Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles", Calculus of Variations and Partial Differential Equations 57 (2018), article 117. DOI: 10.1007/s00526-018-1394-y, https://doi.org/10.1007/s00526-018-1394-y (preprint arXiv:1807.00887, https://arxiv.org/abs/1807.00887)
  2. Roberto Giambò, Fabio Giannoni, Paolo Piccione, "Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits", preprint (2020, revised 2022). arXiv:2002.09687, https://arxiv.org/abs/2002.09687 (no published journal version)
  3. Herbert Seifert, "Periodische Bewegungen mechanischer Systeme", Mathematische Zeitschrift 51 (1948), 197–216. DOI: 10.1007/BF01291002, https://doi.org/10.1007/BF01291002
  4. W. Bos, "Kritische Sehnen auf Riemannschen Elementarraumstücken", Mathematische Annalen 151 (1963), 431–451. DOI: 10.1007/BF01439453, https://doi.org/10.1007/BF01439453
  5. Roberto Giambò, Fabio Giannoni, Paolo Piccione, "Multiple brake orbits in m-dimensional disks", Calculus of Variations and Partial Differential Equations 54 (2015), 2553–2580. DOI: 10.1007/s00526-015-0875-5, https://doi.org/10.1007/s00526-015-0875-5
  6. Roberto Giambò, Fabio Giannoni, Paolo Piccione, "Functions on the sphere with critical points in pairs and orthogonal geodesic chords", Journal of Differential Equations 260 (2016), 8261–8275. DOI: 10.1016/j.jde.2016.02.018, https://doi.org/10.1016/j.jde.2016.02.018 (preprint arXiv:1503.06176, https://arxiv.org/abs/1503.06176)
  7. Chungen Liu, Duanzhi Zhang, "Seifert conjecture in the even convex case", Communications on Pure and Applied Mathematics 67 (2014), 1563–1604. DOI: 10.1002/cpa.21525, https://doi.org/10.1002/cpa.21525
  8. Yiming Long, Duanzhi Zhang, Chaofeng Zhu, "Multiple brake orbits in bounded convex symmetric domains", Advances in Mathematics 203 (2006), 568–635. DOI: 10.1016/j.aim.2005.05.005, https://doi.org/10.1016/j.aim.2005.05.005
  9. Dario Corona, "A multiplicity result for orthogonal geodesic chords in Finsler disks", Discrete and Continuous Dynamical Systems 41 (2021), 5329–5357. DOI: 10.3934/dcds.2021079, https://doi.org/10.3934/dcds.2021079 (preprint arXiv:2010.05604, https://arxiv.org/abs/2010.05604)
  10. Dario Corona, Fabio Giannoni, "Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics", Advances in Nonlinear Analysis 11 (2022), 1223–1248. DOI: 10.1515/anona-2022-0222, https://doi.org/10.1515/anona-2022-0222 (preprint arXiv:2111.06218, https://arxiv.org/abs/2111.06218, circulated under the preprint title "Brake orbits for Hamiltonian systems of classical type via Finsler geodesics")
  11. Hans-Bert Rademacher, "Geodesic loops and orthogonal geodesic chords without self-intersections", Nonlinear Analysis 263 (2026), article 113952. DOI: 10.1016/j.na.2025.113952, https://doi.org/10.1016/j.na.2025.113952 (preprint arXiv:2407.02905, https://arxiv.org/abs/2407.02905)
  12. Hans-Bert Rademacher, "Periodic orbits of reversible Lagrangian systems without self-intersections and Mañé genericity", preprint (2026). arXiv:2602.17833, https://arxiv.org/abs/2602.17833
  13. Isabel Beach, "Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary", preprint (2025). arXiv:2507.07797, https://arxiv.org/abs/2507.07797
  14. Isabel Beach, Haydeé Contreras-Peruyero, Erin Griffin, Regina Rotman, Catherine Searle, "Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds", preprint (2025). arXiv:2509.19620, https://arxiv.org/abs/2509.19620