A local contact systolic inequality in dimension three
Gabriele Benedetti, Jungsoo Kang
备注：
42 pages, a revised version of Part I in "A local systolic-diastolic inequality in contact and symplectic geometry" arXiv:1801.00539 (now withdrawn), accepted for publication in JEMS
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 contact forms such that, for every $α\in\mathcal U$, there holds $ρ_{\mathrm{sys}}(α)\leq ρ_{\mathrm{sys}}(α_*)$ with equality if and only if $α$ is Zoll.
We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close to a Zoll one or large enough.
The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena.
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…
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 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.
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…
We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct a sequence of tight contact forms $α_n$, $n\geq 2$, with systolic ratio arbitrarily close to $n$ and suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.
If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a contact form with arbitrarily large systolic ratio. This statement generalizes the recent result of Abbondandolo et al. in dimension three to higher dimensions. We extend the plug construction of Abbondandolo et. al. to any dimension, by means of generalizing the hamiltonian disc m…
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{…
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
We show that any co-oriented closed contact manifold of dimension at least five admits a contact form such that the contact volume is arbitrarily small but the Reeb flow admits a global hypersurface of section with the property that the minimal period on the boundary of the hypersurface and the first return time in the interior of the hypersurface are bounded below. An immediate consequence of this statement is that every co-oriented contact structure on any closed manifold admits a contact form with arbitrarily large systolic ratio. This generalizes the result of Abbondandolo et al. in dimens…
Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Ab…
We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy…
Let $(M^5,α,g_α,J)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $α$, associated metric $g_α$ and almost complex structure $J$ and $L$ a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the following equation \begin{eqnarray}\label{equ} -Δ^νH+(K-1)H=0, \end{eqnarray} where $Δ^ν$ is the normal Laplacian w.r.t the metric $g$ on $L$ induced from $g_α$ and $K$ is the Gauss curvature of $(L,g)$. Using equation \eqref{equ} and a new Simons' type inequality for Legendrian surfaces in the standard unit sphere $\mathbb{S}^5$, we prove an integral ine…
We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a ``large enough" tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborh…
