Exactly n or infinitely many periodic orbits on a convex energy surface
Let $n \ge 3$ and let $\Sigma = \partial D \subset \mathbb{R}^{2n}$ be a smooth compact hypersurface bounding a strictly convex body $D$ that contains the origin, with positive definite second fundamental form. Let $R$ be the Reeb vector field of the restriction to $\Sigma$ of $$\alpha = \frac{1}{2} \sum_{j=1}^{n} (x_j\,dy_j - y_j\,dx_j),$$ so that $\alpha(R) = 1$ and $d\alpha(R, V) = 0$ for every $V \in T\Sigma$; its periodic orbits are the closed characteristics of $\Sigma$. An ellipsoid $\sum_j (x_j^2 + y_j^2)/r_j = 2$ has exactly $n$ simple periodic orbits when the ratios $r_i/r_j$, $i \ne j$, are all irrational, and infinitely many otherwise. Hofer, Wysocki and Zehnder proved that in $\mathbb{R}^4$ the number is always $2$ or infinite (1998). Çineli, Ginzburg and Gürel proved that there are always at least $n$, and exactly $n$ or infinitely many when $D$ is in addition centrally symmetric and the Reeb flow is non-degenerate (2024).
Problem. Is the number of geometrically distinct simple periodic orbits of $R$ either exactly $n$ or infinite? An orbit is simple when it is traversed with its least positive period, so repeated traversals are not counted separately.
Source: Wang, Hu and Long (2007), Conjecture 1.5, for C³ strictly convex hypersurfaces and every n ≥ 2; see also Çineli, Ginzburg and Gürel (2024), the n-or-∞ conjecture, for star-shaped domains. Stated by AIM — Open Applied Problems for smooth hypersurfaces with positive definite second fundamental form and n ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SG — Symplectic Geometry
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, closed characteristics, convex hypersurfaces, Reeb flows
References
@article{wang2007,
author = {Wang, Wei and Hu, Xijun and Long, Yiming},
title = {Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces},
journal = {Duke Math. J.},
volume = {139},
number = {3},
year = {2007},
pages = {411--462},
doi = {10.1215/S0012-7094-07-13931-0},
eprint = {math/0701608},
archivePrefix = {arXiv},
}
@misc{cineli2024,
author = {{\c{C}}ineli, Erman and Ginzburg, Viktor L. and G{\"u}rel, Ba{\c{s}}ak Z.},
title = {Closed orbits of dynamically convex {R}eeb flows: towards the {HZ}- and multiplicity conjectures},
year = {2024},
eprint = {2410.13093},
archivePrefix = {arXiv},
note = {To appear in Ann. of Math.},
}
@article{hofer1998,
author = {Hofer, H. and Wysocki, K. and Zehnder, E.},
title = {The dynamics on three-dimensional strictly convex energy surfaces},
journal = {Ann. of Math. (2)},
volume = {148},
number = {1},
year = {1998},
pages = {197--289},
doi = {10.2307/120994},
}
@misc{li2025,
author = {Li, Xiaorui and Liu, Hui and Wang, Wei},
title = {Two irrationally elliptic closed orbits of {R}eeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$},
year = {2025},
eprint = {2510.06597},
archivePrefix = {arXiv},
}
@misc{aim-convex-hamiltonian-n-or-infinity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exactly n or infinitely many periodic orbits on a convex energy surface}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/132-convex-hamiltonian-n-or-infinity.md},
note = {Entry 132 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00513, "Exactly n or infinitely many periodic orbits on a convex energy surface", clopen.solutions.