Giambò–Giannoni–Piccione (J. Differential Equations 260 (2016) 8261–8275) proved that a compact Riemannian manifold diffeomorphic to an nn-ball carries at least nn orthogonal geodesic chords — geodesic segments meeting the boundary orthogonally at both endpoints — for a generic set of regular metrics that are non-focal (free of boundary focal points) and have nondegenerate chords, and deduced the lower bound conjectured by H. Seifert for brake orbits of potential wells under corresponding hypotheses. The non-focal hypothesis is essential to their method and cannot be removed by perturbation, since the non-focal class is not dense. This problem asks for the multiplicity theory of orthogonal geodesic chords on ball-diffeomorphic Riemannian manifolds beyond the non-focal class: whether the nn-chord lower bound persists, and how it must be modified, when boundary focal points are present. The strongly concave instance of the same question is Seifert's 1948 conjecture on brake orbits, whose only claimed full resolution exists as an unpublished preprint.

Background

A brake orbit is a periodic solution of a natural Hamiltonian or Lagrangian system — one whose energy is the sum of a kinetic and a potential term — that oscillates like a pendulum: it leaves a level set of the potential energy with zero velocity and returns to it with zero velocity. H. Seifert proved in 1948 that an analytic natural system whose fixed-energy sublevel {q:V(q)≤E}\{q:V(q)\le E\} is homeomorphic to a ball admits at least one brake orbit, and conjectured in a footnote that if the sublevel is homeomorphic to an NN-dimensional ball then there are at least NN geometrically distinct brake orbits. The bound is sharp: the quadratic potential V=∑iλi2xi2V=\sum_i\lambda_i^2x_i^2 with rationally independent ratios λi2/λj2\lambda_i^2/\lambda_j^2 has exactly NN brake orbits. Under the Jacobi metric — a conformal rescaling of the ambient metric, depending on the fixed energy, for which trajectories of the system become geodesics — brake orbits correspond to orthogonal geodesic chords (OGCs): geodesic segments in a compact domain that meet the boundary orthogonally at both endpoints; for a potential well the boundary of the Jacobi domain is strongly concave, i.e. its second fundamental form is definite with the sign for which the domain lies on the concave side of every boundary tangent plane. The dynamical multiplicity question thus transfers to a Riemannian-geometric one: how many OGCs must a Riemannian manifold diffeomorphic to a ball contain?

The geometric question is subtle. W. Bos (1963) exhibited planar regions homeomorphic to a disk, with non-convex pieces of boundary, containing no OGC at all, so hypotheses are necessary. For strongly concave boundaries, existence of OGCs on disks is known and on surfaces at least two chords exist; for even, convex potentials in Euclidean space the full NN-bound was proved by Liu and Zhang (2014). The source of the present problem, Giambò–Giannoni–Piccione (2016), obtained the first multiplicity result with no convexity or concavity assumption: for a generic set of metrics in the class Reg∗(Bm+1)\mathrm{Reg}_*(B^{m+1}) of regular non-focal metrics on the ball there are at least m+1m+1, i.e. at least dim⁡M\dim M, distinct OGCs (their Theorem B), and consequently Seifert's conjectured brake-orbit bound holds under their hypotheses and generically for potentials in a C2C^2-open neighborhood of rotationally symmetric ones. Here a metric is regular if the crossing time function sg:∂M→(0,∞)s_g:\partial M\to(0,\infty) is well defined, meaning every geodesic leaving the boundary orthogonally meets ∂M\partial M again transversally, after time sg(p)s_g(p); a point qq is a ∂M\partial M-focal point if it is a critical value of the normal exponential map of the boundary (its differential is singular there, so infinitesimally nearby normal geodesics from the boundary fail to form a local diffeomorphism around qq); and a metric is non-focal if no ∂M\partial M-focal point lies on ∂M\partial M. For regular non-focal metrics whose OGCs are nondegenerate — meaning that the corresponding critical points of sgs_g have nondegenerate Hessian — sgs_g is an even-Morse function on ∂M≅Sm\partial M\cong S^m, i.e. a Morse function whose critical points come in pairs with equal Morse indices (the index being the number of negative eigenvalues of the Hessian at the critical point), and a topological lower estimate on critical points of even-Morse functions on spheres then produces the m+1m+1 chords.

The non-focal hypothesis is exactly where this method breaks, and the source paper shows it cannot be removed by perturbation. First, by the Morse index theorem for geodesics between submanifolds, the Morse indices of the two orientations of a chord differ by the number of ∂M\partial M-focal points encountered, so when focal points are present sgs_g can be a Morse function without being even-Morse (their Figure 2). Second, for metrics that are not non-focal, critical points of sgs_g need not correspond to OGCs at all (their Figure 1 and Remark 2). Third, the class of non-focal metrics, while C2C^2-open and containing all rotationally symmetric metrics, is not dense: there are domains in which every sufficiently small C1C^1-perturbation of the flat metric produces ∂M\partial M-focal points (their Example 1 and Section 3.2), so focal points cannot be generically perturbed away; the only general membership test is a sufficient quantitative inequality (3.2) bounding sectional curvature and the shape operator of the boundary — the derivative of the boundary's outward unit normal along the boundary, encoding how the boundary curves in the ambient space — in terms of the maximal crossing time. Finally, the multiplicity conclusion genuinely fails without boundary connectedness: an annular region between two non-concentric spheres is regular and non-focal but admits only two OGCs regardless of dimension (their Figure 3).

Problem Statement

Let (M,g)(M,g) be a compact Riemannian manifold of dimension n≥2n\ge 2 whose boundary ∂M≅Sn−1\partial M\cong S^{n-1} is smooth and connected, with MM diffeomorphic to the closed nn-ball, and let gg be a regular metric. Determine the multiplicity of orthogonal geodesic chords in this setting when the metric is allowed to have ∂M\partial M-focal points, i.e. beyond the class of regular non-focal metrics in which the nn-chord lower bound is known. Concretely:

Focal multiplicity. Does the generic multiplicity theorem persist when the non-focal hypothesis is dropped — for a suitable generic (e.g. residual) class of regular metrics on MM that includes metrics with ∂M\partial M-focal points, are there at least nn geometrically distinct OGCs? If not, establish the correct sharp lower bound and make the role of focal points explicit.

The objective is a multiplicity theory of OGCs on ball-diffeomorphic Riemannian manifolds that covers metrics with boundary focal points and assumes no convexity or concavity of the boundary. The strongly concave case of the same OGC problem is, via the Jacobi metric, precisely Seifert's 1948 conjecture that a potential well whose sublevel is homeomorphic to an NN-ball has at least NN geometrically distinct brake orbits; that named conjecture is a special instance of the framework.

References

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