Symplectic Zoll Property in Modern Geometry

• The symplectic Zoll property is defined by all closed trajectories sharing a uniform minimal period, establishing a basis for rigidity and capacity results.
• It intertwines symplectic, dynamical, and topological invariants, with applications in Zoll Riemannian metrics, convex bodies, and semiclassical quantization.
• The property underpins sharp systolic–diastolic inequalities and spectral characterizations, offering criteria for local maximization of the systolic ratio.

A symplectic form or domain is said to satisfy the "Zoll property" if all of its distinguished closed trajectories—its closed characteristics, periodic Reeb orbits, or geodesics—are closed and have a common minimal period. When formulated in the context of symplectic geometry, this property gives rise to significant rigidity phenomena, interplays with symplectic capacities and systolic inequalities, and interacts deeply with index-theoretic, dynamical, and topological concepts. Below is a technical survey of the symplectic Zoll property, its formal structures, functional invariants, and implications in modern research.

1. Zoll Structures in Symplectic and Contact Topology

Let $\Sigma$ be a closed, oriented manifold of odd dimension $2n+1$. A Zoll odd-symplectic form $\Omega \in \Omega^2(\Sigma)$ is a closed 2-form whose kernel is a one-dimensional cooriented distribution,

$$\ker\,\Omega = \{ v \in T\Sigma : \Omega(v,\,\cdot\,) = 0 \}$$

with the further requirement that the integral curves of this line field generate a free $S^1$-action on $\Sigma$. In explicit terms, $\Omega$ is Zoll if and only if there exists a principal $S^1$-bundle $p:\Sigma\to M$, with Euler class $e \in H^2(M;\mathbb{R})$ and a symplectic form $\omega$ on $M$, such that $\Omega = p^*\omega$ and the leaves of $\ker \Omega$ are the fibers of $p$ (Benedetti et al., 2019).

For contact manifolds, a contact form $\alpha$ on a closed $(2n-1)$-manifold is called Zoll if its Reeb flow generates the orbits of a free $S^1$-action—that is, every Reeb orbit is periodic and has the same minimal period (Benedetti et al., 2018).

For smooth, strictly convex domains $K\subset\mathbb{R}^{2n}$ with boundary $\partial K$, the symplectic Zoll property stipulates that $\partial K$ is foliated by closed (generalized) characteristics, each with action exactly equal to the Ekeland–Hofer–Zehnder capacity $c_{EHZ}(K)$ (Haim-Kislev, 20 Nov 2025).

2. Action and Volume Functionals

Given a reference odd-symplectic form $\Omega_0$ and a perturbation by $\alpha \in \Omega^1(\Sigma)$, set $\Omega_\alpha = \Omega_0 + d\alpha$. The volume functional is

$$\mathrm{Vol}(\alpha) = \sum_{j=0}^n \frac{1}{j+1} \binom{n}{j} \int_\Sigma \alpha \wedge (d\alpha)^j \wedge \Omega_0^{n-j}$$

with normalization choices as needed. This generalizes classical contact and symplectic volumes (Benedetti et al., 2019).

For a closed characteristic $\gamma$ tangent to $\ker \Omega$, the action is

$\mathcal{A}(\gamma) = \int_{S^1} \gamma^*\alpha$

where $\alpha$ is chosen with $\ker\alpha = \ker\Omega$ and $\alpha(R) = 1$ for the generator $R$ of the $S^1$-action (Benedetti et al., 2018). In particular, in the strictly Zoll case (all orbits of the $S^1$-action are minimal closed characteristics), there is a polynomial relation linking the volume and action:

$\mathrm{Vol}(\Omega) = P(\mathcal{A}(\Omega))$

for a suitable homogeneous polynomial $P$ determined by the topology of the fibration (Benedetti et al., 2019). In dimension three, this specializes to $$\mathrm{Vol}(\alpha) = t_\Sigma (\min\text{ period})^2$$ for contact forms (Benedetti et al., 2018).

3. Systolic–Diastolic Inequalities and Local Rigidity

The symplectic Zoll property anchors sharp systolic–diastolic inequalities: in a $C^k$-neighborhood $\mathcal{U}$ of a Zoll form $\Omega_*$, every $\Omega\in \mathcal{U}$ satisfies

$$P(\mathcal{A}_{\min}(\Omega)) \leq \mathrm{Vol}(\Omega) \leq P(\mathcal{A}_{\max}(\Omega))$$

where $\mathcal{A}_{\min}(\Omega)$ and $\mathcal{A}_{\max}(\Omega)$ denote the minimal and maximal action of closed characteristics, and equality holds if and only if $\Omega$ is Zoll (Benedetti et al., 2019, Benedetti et al., 2018).

The systolic ratio for a contact form $\alpha$ is defined as

$$\rho(\alpha) = \frac{T_{\min}(\alpha)^n}{\mathrm{Vol}(\alpha)}$$

where $T_{\min}$ is the minimal period of the Reeb flow. Zoll forms strictly locally maximize this ratio: any sufficiently small perturbation in the space of contact forms reduces $\rho$ unless it preserves the Zoll property (Abbondandolo et al., 2019).

For convex bodies, local maximizers of the symplectic systolic ratio

$$\rho_{\mathrm{sys}}(K) = \frac{c_{EHZ}(K)^n}{n!\,\mathrm{Vol}(K)}$$

are precisely the symplectic Zoll bodies among smooth convex domains. The property is characterized in the nonsmooth context by "cut additivity" of the capacity $c_{EHZ}$ under hyperplane splits (Haim-Kislev, 20 Nov 2025).

4. Index-Theoretic and Capacity Characterizations

The systolic $S^1$-index, $\mathrm{ind}(C)$, of a convex body $C\subset\mathbb{R}^{2n}$ (not necessarily smooth) is the Fadell–Rabinowitz index of the $S^1$-space of centralized generalized systoles with minimal action. This is a symplectic invariant:

$$\mathrm{ind}(C) = \max\{k \mid c^{GH}_k(C) = c^{GH}_1(C)\}$$

where $c^{GH}_k$ are the Gutt–Hutchings capacities (equal to Ekeland–Hofer capacities on convex bodies). The body $C$ is generalized Zoll if $\mathrm{ind}(C) \geq n$, equivalently $c^{GH}_n(C) = c^{GH}_1(C)$. When $\partial C$ is smooth, being generalized Zoll coincides with all Reeb orbits being closed with common minimal period—the classical Zoll property (Matijević, 23 Jan 2025).

For contact forms on $Y$, the $S^1$-equivariant spectral invariants $c_k^{S^1}(Y,\alpha)$ admit a spectral characterization:

• $\alpha$ is Zoll of minimal period $\tau$ if and only if $c_k^{S^1}(Y,\alpha) = k\,\tau$ for all $k\geq 0$ (Ginzburg et al., 2019).
• Equality $c_i=c_{i+n-1}$ for some $i$ implies the Besse property (all orbits close), and $c_0=c_{n-1}$ characterizes strict Zoll.

5. Non-Smooth and Dynamical Extensions

The symplectic Zoll property extends dynamically to non-smooth convex bodies via cuts additivity: a convex body $K$ is called "cuts additive" if every hyperplane splitting $K$ into $K_1$ and $K_2$ satisfies

$c_{EHZ}(K) = c_{EHZ}(K_1) + c_{EHZ}(K_2)$

This is equivalent (under mild hypotheses) to the generalized Zoll property defined via the Fadell–Rabinowitz index of minimizing closed characteristics:

$\mathrm{ind}_{FR}(\mathrm{Sys}(K)) \geq n$

(Haim-Kislev, 20 Nov 2025).

Action-minimizing closed characteristics in the nonsmooth case exhibit three behaviors: (i) extreme-ray motion, (ii) coisotropic face sliding, (iii) more pathological isotropic gliding, with $H^1$-compactness of the quotient space of generalized systoles except in the presence of (iii). This structure ensures that local maximizers of the systolic ratio among (possibly nonsmooth) convex bodies are detectable by the same dynamical and topological criteria as in the smooth setting.

6. Examples and Applications

• Unit cotangent sphere bundles $S^*_gM$ for Zoll Riemannian metrics $g$ (e.g., spheres and rank-one symmetric spaces) are paradigmatic Zoll domains; their Reeb/Geodesic flow is $S^1$-periodic (Shelukhin, 2018).
• The standard ball or an ellipsoid $E(a_1,\ldots,a_n)$ is Zoll if and only if all radii $a_i$ are equal; otherwise, Besse but not Zoll (Ginzburg et al., 2019).
• Zoll magnetic systems on the two-torus and Zoll deformations of the Kepler problem furnish explicit integrable systems with the symplectic Zoll property at selected energy levels, including infinite-dimensional deformation families and sharp area-period inequalities (Asselle et al., 2019, Asselle et al., 2023, Luca et al., 26 Aug 2024).
• Unit disk bundles in $T^*M$ for manifolds with all geodesics closed yield Zoll-type domains whose boundaries are contact type, and under Bohr–Sommerfeld quantization, fit into the semiclassical quantization formalism (Hernández-Dueñas et al., 2011).

7. Implications and Open Problems

The symplectic Zoll property underpins several rigidity and extremality results: it detects strict local maximizers for systolic ratios, provides sharp bounds for symplectic capacities, and characterizes (locally) equality cases in Viterbo's conjecture and non-squeezing inequalities (Abbondandolo et al., 2019, Matijević, 23 Jan 2025). Open questions persist regarding global maximality for these ratios, the role of bott–Morse closed orbit families, the behavior of the spectral invariants under infinite-order tangency, and the classification of non-smooth dynamical behaviors (Abbondandolo et al., 2019, Haim-Kislev, 20 Nov 2025).

This unification of dynamical, spectral, and topological approaches to the symplectic Zoll property continues to frame key advances in contact and symplectic topology, with index-theoretic, capacity-theoretic, and operator-theoretic invariants as its central analytic tools (Benedetti et al., 2019, Matijević, 23 Jan 2025, Haim-Kislev, 20 Nov 2025, Benedetti et al., 2018).