15 problems

Let $v$ be a non-vanishing (traversing ?) vector field on a compact $(2n+1)$-dimensional manifold $X$, and let $\xi$ be a $(2n)$-dimensional distribution normal to $v$ that admits…

Let $\Sigma^{2n-1}$ bound a strictly convex domain in $\mathbb{R}^{2n}$. Elliptic orbit conjecture. There exists at least one elliptic closed Reeb orbit on $\Sigma$. The source des…

Let $\Sigma^{2n-1}$ bound a convex domain in $\mathbb{R}^{2n}$, with $n\geq 2$, and let $\mathcal{T}(\Sigma)$ denote the set of prime closed Reeb orbits on $\Sigma$. A closed Reeb…

Let $\Sigma^{2n-1}$ bound a star-shaped domain in $\mathbb{R}^{2n}$. Higher-dimensional two-or-infinity conjecture. For each such $\Sigma$, … The dimension-four case is solved by t…

Let $\Sigma^{2n-1}$ be the boundary of a star-shaped domain in $\mathbb{R}^{2n}$, and let $\mathcal{T}(\Sigma)$ denote the set of prime closed Reeb orbits on $\Sigma$. Multiplicity…

Two-or-infinitely-many periodic Reeb orbits conjecture. The associated Reeb flow has either exactly two or infinitely many periodic orbits.

Reeb 1-curve conjecture. The manifold $M$ has a Reeb 1-curve.

Let $(Y,\xi)$ be a closed contact manifold with contact form $\alpha$. Write $\mathfrak{gap}_\sigma(Y,\alpha)$ for the spectral gap associated with a class $\sigma\in CH(Y,\xi)$. P…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$ of period $T$. Let $CH(Y,\xi)$ denote contact homology, $\mathfrak{s}_\sigma(Y,\alpha)$ the spectra…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$. The closed Reeb orbits form components $S$ of the orbit orbifolds, and to each component associate…

Let $E \subset \mathbb{C}^n$ be an ellipsoid and let $(\partial E,\lambda|_{\partial E})$ be its contact boundary, where $\lambda$ is the standard Liouville form. Irie's strong clo…

Viterbo's weak conjecture. The systolic ratio satisfies

Let $M$ be a closed $3$-manifold and let $R$ be a nondegenerate Reeb vector field on $M$. A Birkhoff section is a surface with boundary whose interior is embedded and transverse to…

Let $(Y,\xi)$ be a closed contact manifold equipped with a selected contact form $\alpha$ such that $\ker\alpha=\xi$. Let $\phi$ be a contactomorphism of $Y$ that preserves…

Let $M$ be a closed hyperbolic $3$-manifold with a universally tight contact structure $\xi$, and let $\alpha$ be a non-degenerate contact form defining $\xi$. Denote by…