On a systolic inequality for closed magnetic geodesics on surfaces

Infinitesimal Systolic Rigidity of Metrics all of whose Geodesics are Closed and of the same Length

The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.

A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and s…

On a local systolic inequality for odd-symplectic forms

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $Ω$ be an odd-symplectic form on an oriented closed manifold $Σ$ of odd dimension. We say that $Ω$ is Zoll if the trajectories of the flow given by $Ω$ are the orbits of a free $S^1$-action. After defining the volume of $Ω$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $Ω$ is Zoll. This builds the equality case of a conjectural systolic inequality fo…

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…

三维中S¹不变切触形式的收缩不等式

Systolic inequalities for S1-invariant contact forms in dimension three

该研究在切触几何领域，针对欧拉数非零的Seifert丛，证明了适用于所有底层圆周作用不变切触形式的通用收缩不等式，即用切触体积一致上界闭Reeb轨道的最短周期。

Positive loops and $L^{\infty}$-contact systolic inequalities

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16]. As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers of the $L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic inequalities in the…