Giambò, Giannoni, and Piccione (arXiv:2002.09687, v3 2022) proved that an open subset of a Riemannian manifold whose boundary is strongly concave and which is homeomorphic to an N-dimensional disk contains at least N geometrically distinct orthogonal geodesic chords (geodesics meeting the boundary orthogonally at both endpoints), provided an auxiliary geometric condition holds: there is an interior point and an energy level above the natural minimax threshold such that no 'special' geodesic (tangent to the boundary at both endpoints, or tangent at one and orthogonal at the other) of energy below that level passes through the point. The authors state that this exclusion condition 'has a technical nature' and is 'possibly inessential' for the theorem, but their proof uses it essentially. The problem is to determine whether the condition is dispensable — i.e., whether every strongly concave Riemannian N-disk (of dimension N≥2N\ge 2, per the source framework's standing assumption dim⁡M=N≥2\dim M=N\ge 2) carries at least N geometrically distinct orthogonal geodesic chords with no exclusion hypothesis — or essential, by exhibiting a strongly concave Riemannian N-disk with fewer than N geometrically distinct orthogonal geodesic chords.

Background

A natural Hamiltonian (or Lagrangian) system at fixed energy in a potential well can be reparametrized, via the Jacobi metric (the conformal rescaling gE=(E−V)gg_E=(E-V)g that turns trajectories of energy EE into geodesics), into a geodesic problem on the energy sublevel of the potential. Brake orbits — periodic solutions whose velocities vanish at two instants, so the orbit 'bounces' perpendicularly off the boundary of the sublevel — then correspond to orthogonal geodesic chords (OGCs): geodesic segments in the closure of a domain Ω\Omega whose endpoints lie on ∂Ω\partial\Omega and which meet ∂Ω\partial\Omega orthogonally at both endpoints. A classical conjecture of Seifert (1948) predicts that if the energy sublevel is homeomorphic to an NN-dimensional disk, the system has at least NN geometrically distinct brake orbits; via the Jacobi metric this becomes a question about the number of OGCs in a Riemannian domain, and for the Jacobi metric of a potential well the domain has strongly concave boundary in the sense below. Two OGCs are called geometrically distinct if their images are different sets.

Giambò, Giannoni, and Piccione studied the OGC problem directly with nonsmooth critical point theory (minimax methods applied to the geodesic action functional F(x)=∫01g(x˙,x˙) dsF(x)=\int_0^1 g(\dot x,\dot x)\,ds on a space of H1H^1-curves, with admissible variations constrained at the boundary). Their central result, Theorem 3.6 of arXiv:2002.09687, states: if Ω\Omega is an open subset of a Riemannian manifold with C2C^2 boundary, is strongly concave — meaning that for a smooth defining function ϕ\phi of ∂Ω\partial\Omega (with Ω={ϕ<0}\Omega=\{\phi<0\}) the Hessian satisfies Hϕ(x)(v,v)<0H\phi(x)(v,v)<0 for every x∈∂Ωx\in\partial\Omega and every nonzero v∈Tx∂Ωv\in T_x\partial\Omega, so that geodesics tangent to the boundary locally move inside Ω\Omega — and is homeomorphic to an NN-disk (the paper's standing framework takes N=dim⁡M≥2N=\dim M\ge 2, and the homeomorphism to the unit disk is used up to the boundary, so that Ω‾\overline{\Omega} is homeomorphic to the closed NN-disk), then, under one additional hypothesis, Ω\Omega contains at least NN geometrically distinct OGCs. The additional hypothesis, condition (3.13), is an exclusion condition on 'special' geodesics: a geodesic x:[0,1]→Ω‾x:[0,1]\to\overline{\Omega} with x(]0,1[)⊂Ωx(]0,1[)\subset\Omega and both endpoints x(0),x(1)∈∂Ωx(0),x(1)\in\partial\Omega is special if it is tangent to ∂Ω\partial\Omega at both endpoints, or tangent at one endpoint and orthogonal at the other; writing GLG_L for the set of special geodesics with 0<F≤L20<F\le L^2 and L(2)(Ω)L^{(2)}(\Omega) for the minimax value inf⁡γsup⁡A,B∈∂ΩF(γA,B)\inf_\gamma\sup_{A,B\in\partial\Omega}F(\gamma_{A,B}) over symmetric boundary-to-boundary sweep-outs γ\gamma (families of H1H^1 curves joining boundary points, constant on the diagonal, and symmetric under reversal of the parameter), condition (3.13) requires that there exist M02>L(2)(Ω)M_0^2>L^{(2)}(\Omega) and a point x0∈Ωx_0\in\Omega such that x0x_0 does not lie on the image of any geodesic in GM0G_{M_0}.

The assumption enters the proof at a precise place. The minimax argument must handle V−V_--critical curves — curves stationary only under variations that point inward at the boundary, which are 'geodesics with obstacle' — and the critical sub-curves of these that fail to be OGCs are exactly special geodesics of energy at most M02M_0^2 (Remark 4.9 of the paper). Condition (3.13) then produces a ball Bρ0(x0)B_{\rho_0}(x_0) avoided by all such special geodesics, which is what allows the construction of an invariant set Λ∗\Lambda^* for the pseudo-gradient flow and its entrance-set analysis, on which the deformation lemmas powering the minimax count are built. The authors note in the introduction that the assumption 'has a technical nature' and is 'possibly inessential for the validity of the result of Theorem 3.6', but they do not remove it.

The Seifert conjecture application in the same paper does not depend on removing (3.13): Proposition 3.8 proves that if the underlying system has only finitely many brake orbits, then a suitable truncated domain Ωδ\Omega_\delta (on which the Jacobi metric is considered) does satisfy (3.13), and Theorem 3.6 is applied to that domain. So the Seifert corollary verifies the exclusion condition rather than needing its removal, and the genuinely open issue is intrinsic to the Riemannian-geometric theorem: whether the multiplicity of OGCs in strongly concave Riemannian disks holds without any exclusion hypothesis on special geodesics.

Problem Statement

Let (M,g)(M,g) be a Riemannian manifold of class C3C^3 whose metric gg is of class C2C^2, with N=dim⁡M≥2N=\dim M\ge 2 (the source framework's standing assumptions; this regularity guarantees uniqueness of solutions for the geodesic Cauchy problem), and let Ω⊂M\Omega\subset M be an open subset whose boundary ∂Ω\partial\Omega is of class C2C^2, such that Ω\Omega is strongly concave (for a defining function ϕ\phi with Ω={ϕ<0}\Omega=\{\phi<0\}, Hϕ(x)(v,v)<0H\phi(x)(v,v)<0 for all x∈∂Ωx\in\partial\Omega and nonzero v∈Tx∂Ωv\in T_x\partial\Omega) and Ω‾\overline{\Omega} is homeomorphic to the closed NN-dimensional disk. Recall that an orthogonal geodesic chord (OGC) is a geodesic x:[0,1]→Ω‾x:[0,1]\to\overline{\Omega} with x(0),x(1)∈∂Ωx(0),x(1)\in\partial\Omega and x˙\dot x orthogonal to ∂Ω\partial\Omega at both endpoints, and that OGCs are geometrically distinct when their images differ.

Giambò, Giannoni, and Piccione proved (Theorem 3.6 of arXiv:2002.09687) that such an Ω\Omega contains at least NN geometrically distinct OGCs under the additional exclusion condition (3.13): there exist a constant M02>L(2)(Ω)M_0^2>L^{(2)}(\Omega) (where L(2)(Ω)L^{(2)}(\Omega) is the minimax value of the source framework: the infimum, over the symmetric boundary-to-boundary sweep-outs of Lemma 3.4 of the source — families of H1H^1 curves joining boundary points, constant on the diagonal, and symmetric under parameter reversal — of the supremum of the geodesic action FF on the sweep-out) and a point x0∈Ωx_0\in\Omega such that x0x_0 does not belong to the image of any special geodesic of energy ≤M02\le M_0^2, a special geodesic being one x:[0,1]→Ω‾x:[0,1]\to\overline{\Omega} with x(]0,1[)⊂Ωx(]0,1[)\subset\Omega and both endpoints on ∂Ω\partial\Omega that is tangent to ∂Ω\partial\Omega at both endpoints or tangent at one endpoint and orthogonal at the other. The authors state that this condition is 'possibly inessential' but their proof — a pseudo-gradient minimax argument in which the excluded ball Bρ0(x0)B_{\rho_0}(x_0) is needed to construct the invariant set for the deformation lemmas — uses it essentially.

The problem is to settle, by one of the two routes below, whether the exclusion condition is dispensable in this multiplicity theorem:

  1. Prove that every strongly concave Riemannian N-disk as above contains at least NN geometrically distinct orthogonal geodesic chords, with no hypothesis of the form (3.13) — in particular, the argument must handle domains in which every interior point lies on the image of some special geodesic of energy ≤M02\le M_0^2 for every M02>L(2)(Ω)M_0^2>L^{(2)}(\Omega); or

  2. Refute the general statement by exhibiting a strongly concave Riemannian N-disk with N≥2N\ge 2 (with C2C^2 boundary, whose closure is homeomorphic to the closed NN-disk) that has fewer than NN geometrically distinct orthogonal geodesic chords, thereby showing that condition (3.13) is essential.

References

  1. Roberto Giambò, Fabio Giannoni, Paolo Piccione, Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits, arXiv:2002.09687 (v3, 2022), DOI 10.48550/arXiv.2002.09687, https://arxiv.org/abs/2002.09687
  2. Dario Corona, Fabio Giannoni, Brake orbits for Hamiltonian systems of classical type via Finsler geodesics, arXiv:2111.06218 (2021), DOI 10.48550/arXiv.2111.06218, https://arxiv.org/abs/2111.06218
  3. Dario Corona, A multiplicity result for orthogonal geodesic chords in Finsler disks, Discrete and Continuous Dynamical Systems 41(11) (2021), 5329-5357, DOI 10.3934/dcds.2021079, https://arxiv.org/abs/2010.05604
  4. Isabel Beach, Haydée Contreras-Peruyero, Erin Griffin, Regina Rotman, Catherine Searle, Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds, arXiv:2509.19620 (2025), DOI 10.48550/arXiv.2509.19620, https://arxiv.org/abs/2509.19620
  5. Isabel Beach, Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary, arXiv:2507.07797 (2025), DOI 10.48550/arXiv.2507.07797, https://arxiv.org/abs/2507.07797
  6. R. Giambò, F. Giannoni, P. Piccione, WITHDRAWN: Potential wells with a unique brake orbit. Counterexamples to a conjecture by H. Seifert, arXiv:1203.5198 (2012), DOI 10.48550/arXiv.1203.5198, https://arxiv.org/abs/1203.5198