Kirby Problem 3.47

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. On $\mathbb R^4=\mathbb C^2$ put $\lambda_0=\tfrac12\sum_{j=1}^2(x_j\,dy_j-y_j\,dx_j)$. The standard contact structure on $S^3$ is $\xi_{\rm std}=\ker(\lambda_0|_{S^3})$. A periodic Reeb orbit is elliptic if every eigenvalue of the linearized Poincaré return map on the transverse contact plane has modulus one; this includes degenerate unit eigenvalues. For a smooth bounded strictly convex domain $D\subset\mathbb R^4$, translate an interior point to the origin and use the contact form $\lambda_0|_{\partial D}$; strict convexity means that for any distinct $x,y\in\overline D$ and every $0<t<1$, $(1-t)x+ty\in D$. No positive-curvature condition on the boundary is added. [S3]

Statement recorded in the supplied source.

(a) For every contact form $\lambda$ on $S^3$ defining $\xi_{\rm std}$, does its Reeb flow have an elliptic periodic orbit? (b) Does every Reeb flow obtained from the boundary of a smooth strictly convex domain in $\mathbb R^4$ have an elliptic periodic orbit? Both are universal existence questions, without genericity or pinching assumptions. [S2]

Short English statement

Does every standard-contact Reeb flow on the three-sphere have an elliptic closed orbit, and does this hold for every strictly convex hypersurface in four-dimensional Euclidean space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2845 (KP-3.47). 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 3.47, p. 164 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Miguel Abreu and Leonardo Macarini, Dynamical convexity and elliptic periodic orbits for Reeb flows, Mathematische Annalen 369 (2017), 331--386; arXiv:1411.2543v3, introduction, p. 1 and footnote 1. https://arxiv.org/abs/1411.2543v3.