Zoll Contact Forms in Contact Geometry

• Zoll contact forms are special contact forms on closed odd-dimensional manifolds whose Reeb flows are uniformly periodic with a common minimal period.
• They are rigorously classified via the Boothby–Wang construction, with results on S³ and S⁵ linking contact invariants and topological properties.
• Zoll contact forms optimize systolic ratios and spectral invariants, driving advances in symplectic capacities, systolic geometry, and dynamical systems.

A Zoll contact form is a contact form on a closed odd-dimensional manifold whose Reeb flow generates a free $S^1$-action, meaning all Reeb orbits are closed and have the same minimal period. Zoll contact forms are the contact-topological analogues of Zoll metrics in Riemannian geometry, and their paper links global geometric invariants, symplectic and contact topology, and dynamical systems theory. The characterization, rigidity, classification, and optimality properties of Zoll contact forms underpin several deep results in systolic geometry, capacity theory, and the spectral analysis of flows on manifolds.

1. Definition and Fundamental Properties

A contact form $\alpha$ on a closed manifold $M^{2n+1}$ is Zoll if its Reeb flow $\varphi^t$ generates a free circle action: every integral curve of the Reeb vector field $R_\alpha$ is periodic with the same minimal period $T_\mathrm{min}(\alpha)$. In dimension three, this is equivalent to $\alpha \wedge d\alpha > 0$ and the Reeb flow filling $M$ with circles of equal period.

Key invariants:

• Contact volume: $$\mathrm{vol}(M,\alpha) = \int_M \alpha \wedge (d\alpha)^n$$
• Systolic ratio: In dimension $2n+1$, defined as $$P_\mathrm{sys}(M,\alpha) = [T_\mathrm{min}(\alpha)]^{n+1}/\mathrm{vol}(M,\alpha)$$

All Reeb orbits are simple, periodic, and their periods coincide with $T_\mathrm{min}$.

2. Rigidity and Classification Theorems

Zoll contact forms exhibit pronounced rigidity. On spheres, the classification is strict:

• On $S^3$, all Zoll contact forms are (up to rescaling) strictly contactomorphic to the standard round contact form induced by $S^3\subset\mathbb{C}^2$.
• On $S^5$, the result of "On Zoll Contact 5-Spheres" (Chaidez et al., 11 Sep 2025) establishes that any Zoll contact form on the standard contact $5$-sphere is strictly contactomorphic to a scaling of the standard form.

Central construction: Boothby–Wang construction. A Zoll contact manifold $(Y,\alpha)$ yields a principal $S^1$-bundle $Y \to X$ over a symplectic base $(X,\Omega)$, with

• $\pi^*\Omega = (T/2\pi)d\alpha$
• Euler class $e(Y) = -[\Omega]$

Classification reduces to symplectic quotients, cohomological obstructions, and Chern classes. In the 5-sphere case, contact homology and spectral constraints force the symplectic quotient to be $(\mathbb{C}P^2, \Omega_{FS})$ with first Chern class $c_1(X,\Omega) = 3[\Omega]$, thereby establishing uniqueness.

3. Systolic Optimality, Inequalities, and Capacity Theory

Zoll contact forms locally maximize the systolic ratio $P_\mathrm{sys}$ in the $C^3$-topology:

• On $S^3$ (Abbondandolo et al., 2015, Benedetti et al., 2019), any contact form $\alpha$ sufficiently close to a Zoll contact form $\alpha_*$ satisfies $$\rho_\mathrm{sys}(\alpha) \leq \rho_\mathrm{sys}(\alpha_*)$$; equality only for Zoll forms.
• In arbitrary dimension, a precise normal form (Abbondandolo et al., 2019) allows sharp local systolic inequalities and confirms perturbative instances of Viterbo's conjecture on symplectic capacities versus volume.

The systolic ratio is unbounded globally: using plug constructions (Sağlam, 2018), one can produce contact forms with arbitrarily large systolic ratio by modifying the contact volume without shortening the minimal period.

Applications include:

• Geometric optimization: The boundary of star-shaped domains in $\mathbb{C}^n$ locally maximizes all normalized symplectic capacities if and only if it is contactomorphic to a scaling of the standard Zoll sphere (Abbondandolo et al., 2023).
• Systolic inequalities for Finsler/Riemannian metrics: Through Legendre duality and lifting, the analogous optimality holds for Riemannian/Finsler metrics close to Zoll ones.

4. Spectral Characterization via Min–Max and ECH Invariants

Spectral invariants characterize Zoll contact forms using pseudoholomorphic curve theory and embedded contact homology (ECH):

• Elementary (max–min) spectral invariants $c_k(M,\alpha)$, defined via energies of pseudoholomorphic curves, satisfy $$c_1(S^3,\alpha) = c_2(S^3,\alpha) = \mathcal{A}_\mathrm{min}(S^3,\alpha)$$ iff $\alpha$ is Zoll (Fernandes et al., 7 Oct 2025).
• Spectral gap closing: If the difference $c_k - c_{k-1}$ vanishes, then every point lies on a closed Reeb orbit of action $\mathcal{A}_\mathrm{min}$.
• On lens spaces $L(p,1)$, the analogous ECH invariants must satisfy $$c_\sigma^{ECH}(L(p,1),\lambda) = c_{U\sigma}^{ECH}(L(p,1),\lambda) = \mathcal{A}_\mathrm{min}(L(p,1),\lambda)$$ for Zoll forms, with distinct behavior for $p>1$.

In convex or restricted contact type settings, S$^1$–equivariant spectral invariants $c_i$ and their collapse (e.g., $c_i = c_{i+n-1}$) force Besse or Zoll properties (Ginzburg et al., 2019).

The spectral invariants also determine the Banach–Mazur pseudo–metric for contact forms close to Zoll ones and the existence of minimizing geodesics in the corresponding moduli spaces (Abbondandolo et al., 2023).

5. Normal Forms and Deviation Analysis

Contact forms $C^2$–close to a Zoll form admit explicit normal forms:

• After a diffeomorphism $u$, such forms can be expressed as $u^*\alpha = S\alpha_0 + \nu + df$ (Abbondandolo et al., 2019), with $S$ a positive function invariant under the Reeb flow of $\alpha_0$ and $\nu$ orthogonal to the Reeb direction.
• A further refinement (Abbondandolo et al., 2023) shows that $y^*\alpha = T\,e^h\,\alpha_0$, with $T$ S$^1$–invariant and $h$ vanishing (with $dh$) on critical points of $T$.
• The minimal/maximal periods of short closed Reeb orbits correspond directly to $\sys(\alpha_0)\cdot\min T$ and $\sys(\alpha_0)\cdot\max T$.

These normal forms underpin the proof of local optimality and control capacity comparisons, projecting symplectic invariants and the oscillation pseudo–metric directly onto dynamical data.

6. Related Structures and Generalizations

Zoll properties extend to Lorentzian geometry, odd-symplectic forms, and magnetic flows:

• Lorentzian Zoll surfaces admit explicit classification, integral conditions, and conformal rigidity phenomena (Mounoud et al., 2014). The set of spacelike Zoll surfaces is richer than the Riemannian case.
• Odd-symplectic forms (Hamiltonian structures) on S$^1$–bundles satisfy volume–action polynomial equations if they are Zoll (Benedetti et al., 2019). The local systolic–diastolic inequalities generalize both contact and Calabi-type invariants.
• Magnetic flows: Zoll flows in strong magnetic systems require both the magnetic function and the underlying metric to be constant (on S$^2$ for vanishing magnetic term) (Asselle et al., 2020). Exotic Zoll flows may exist on tori for special parameter values.

Highly symmetric contact forms (e.g. those invariant under SO$(2p)$ on SL$(2p)$ (Remm, 21 Jun 2024)) may not always be Zoll, though maximal symmetry often suggests regular periodic flows as for classical Boothby–Wang fibrations.

7. Historical Context and Key References

Foundational results date to Martinet's classification of contact forms [19], the Boothby–Wang construction [BW58], Besse's monograph "Manifolds all of whose geodesics are closed," and works of Epstein, Wadsley, and Giroux. Modern systolic inequalities on contact manifolds, the introduction of spectral invariants via pseudoholomorphic curve counts (ECH), and rigidity phenomena in higher dimensions are areas of active research (Abbondandolo et al., 2015, Benedetti et al., 2019, Abbondandolo et al., 2019, Chaidez et al., 11 Sep 2025, Fernandes et al., 7 Oct 2025).

Recent works reveal deep links between contact geometry, dynamical systems, and symplectic topology—especially in the context of systolic optimization, spectral rigidity, and classification of periodic flows.

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In summary, Zoll contact forms represent a central paradigm of global rigidity in contact geometry, connecting dynamics (periodic Reeb flows), topological classifications (via Boothby–Wang), geometric optimization (systolic inequalities, symplectic volumes, capacities), and spectral invariants (ECH, min–max energies). Their analytic and topological characterizations continue to inform the broader theory of contact, symplectic, and dynamical systems.