Kirby Problem 3.46

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A contact form $\lambda$ on a closed $3$-manifold $Y$ is a smooth one-form with $\lambda\wedge d\lambda$ nowhere zero. Its Reeb vector field $R_\lambda$ is determined by $\lambda(R_\lambda)=1$ and $\iota_{R_\lambda}d\lambda=0$. A periodic orbit solves $\dot\gamma=R_\lambda\circ\gamma$ with some positive period. It is simple if it is not a multiple traversal of a shorter periodic orbit; simple orbits are counted by their images. A lens space $L(p,q)$, including $S^3=L(1,q)$, is the quotient of $S^3\subset\mathbb C^2$ by the free cyclic action $(z_1,z_2)\mapsto(e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2)$, where $p\ge1$ and $\gcd(p,q)=1$.

Statement recorded in the supplied source.

For every closed $3$-manifold $Y$ that is not a lens space and every contact form $\lambda$ on $Y$, must $R_\lambda$ have infinitely many geometrically distinct simple periodic orbits? No nondegeneracy hypothesis on $\lambda$ and no torsion hypothesis on the contact structure's first Chern class are imposed. [S2]

Short English statement

Must every contact form on a closed three-manifold other than a lens space have infinitely many distinct simple Reeb orbits?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2844 (KP-3.46). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.46, p. 164 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.