On the spectral characterization of Besse and Zoll Reeb flows

Sharp systolic inequalities for Reeb flows on the three-sphere

The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $ρ_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a…

关于 Zoll 切触 5-球面

On Zoll Contact 5-Spheres

本文证明了标准切触5-球面上具有Zoll Reeb流的任意切触形式，均严格切触同构于标准Zoll切触形式的伸缩。

伪全纯曲线的极大极小能量与三维 Reeb 流

Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$

本文利用伪全纯曲线的极大极小能量谱不变量，刻画了三维 Reeb 流中 Zoll 接触形式，并推广至 Lens 空间及 Besse 情形。

A local systolic-diastolic inequality in contact and symplectic geometry

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $α$ respectively. We say that $α$ is Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that every Zoll contact form $α_*$ on $Σ$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that \[ t_ΣT_{\min}(α)^2\leq \mathrm{…

A local contact systolic inequality in dimension three

Let $α$ be a contact form on a connected closed three-manifold $Σ$. The systolic ratio of $α$ is defined as $ρ_{\mathrm{sys}}(α):=\tfrac{1}{\mathrm{Vol}(α)}T_{\min}(α)^2$, where $T_{\min}(α)$ and $\mathrm{Vol}(α)$ denote the minimal period of periodic Reeb orbits and the contact volume. The form $α$ is said to be Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that the set of Zoll contact forms on $Σ$ locally maximises the systolic ratio in the $C^3$-topology. More precisely, we show that every Zoll form $α_*$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of conta…

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.