Periodic orbits for perturbations of Zoll contact forms

Abstract: We consider the perturbation of a Zoll contact form on some prescribed domain of the underlying manifold by multiplying it with a positive function which is constant of value $1$ outside the domain. For every sufficiently $C<sup>0$-small such perturbation we find a periodic Reeb orbit of the perturbed contact form intersecting the domain. As an application, starting from a Zoll Riemannian manifold, we demonstrate that for every exact magnetic field, with $C<sup>0$-small magnetic potential vanishing outside some given domain, there exists a periodic magnetic geodesic intersecting this domain. The theorem is a rather direct consequence of a result which is of independent interest: For Hamiltonians sufficiently $C<sup>0$-close to a defining Hamiltonian, we show existence of a gradient flow line of the corresponding Rabinowitz action functional with a constraint on the position of the cylinder component at $(0,0) \in \mathbb{R} \times \mathbb{S}<sup>1$. This relies on a homotopy stretching argument for the Rabinowitz action functional, in the course of which we have to derive some delicate estimates to ensure compactness of the appearing moduli spaces.

• The paper proves that every C⁰-small conformal perturbation e^fα of a closed Zoll contact form, supported in a tame domain D and drawn from a C¹-bounded family, has a periodic Reeb orbit intersecting D.
• The paper develops a Rabinowitz-Floer gradient-flow existence theorem using homotopy stretching, action and energy estimates, uniform Lagrange-multiplier bounds, and compactness control for Hamiltonians C⁰-close to a defining Hamiltonian.
• The paper applies the result to localized magnetic and electromagnetic systems, producing periodic trajectories near the original Zoll period and establishing a multiplicity alternative, while identifying Morse-Bott generalizations as an open direction.

This paper establishes the existence of periodic Reeb orbits for contact forms obtained from a Zoll contact form by multiplication with a positive function that is $C^0$-small and vanishes outside a prescribed domain. The main technical contribution is an existence theorem for Rabinowitz-Floer gradient flow lines of Hamiltonians that are $C^0$-close to a defining Hamiltonian, proved via a homotopy stretching argument combined with delicate compactness estimates. The author is Tom Stalljohann (Universität Heidelberg).

Statement of the main results

The central dynamical result concerns a closed Zoll contact manifold $(\Sigma,\alpha)$, i.e. all Reeb orbits are periodic with common minimal period $\tau(\alpha)$. Given a $C^1$-bounded family $\mathscr{B} \subseteq C^\infty(\Sigma)$ and any non-empty subset $D \subseteq \Sigma$ with tame boundary (a mild condition excluding pathologies such as $\Sigma \setminus \{q\}$), there exists $\delta = \delta(\mathscr{B}) > 0$ such that for every $f \in \mathscr{B}$ with $C^0$0 and $C^0$1 on $C^0$2, the perturbed contact form $C^0$3 admits a periodic Reeb orbit intersecting $C^0$4. A refinement controls the period: for any $C^0$5 one can arrange $C^0$6. Notably, the hypothesis is only $C^0$7-smallness of $C^0$8 subject to prescribed $C^0$9 bounds — remarkable because the Reeb vector field

$(\Sigma,\alpha)$0

depends on $(\Sigma,\alpha)$1, not merely on $(\Sigma,\alpha)$2.

As an application, for a closed Zoll Riemannian manifold $(\Sigma,\alpha)$3 of dimension at least two, and for any magnetic potential $(\Sigma,\alpha)$4 that vanishes on $(\Sigma,\alpha)$5 and satisfies $(\Sigma,\alpha)$6, $(\Sigma,\alpha)$7, there exists a periodic magnetic geodesic of prescribed energy $(\Sigma,\alpha)$8 solving $(\Sigma,\alpha)$9 and intersecting $\tau(\alpha)$0. This generalizes to electromagnetic systems with a mechanic potential and metric distortion. The result is sharp in spirit regarding locality: the periodic orbit is guaranteed to visit exactly those regions where the perturbation is supported.

The Rabinowitz gradient flow line theorem

The actual core of the paper is a statement about the Rabinowski action functional $\tau(\alpha)$1 on the symplectization $\tau(\alpha)$2. For a Zoll contact manifold with SFT-type almost complex structure $\tau(\alpha)$3 and a defining Hamiltonian $\tau(\alpha)$4 (satisfying $\tau(\alpha)$5, $\tau(\alpha)$6 on $\tau(\alpha)$7, compactly supported differential), the theorem asserts:

For every $\tau(\alpha)$8 and every $\tau(\alpha)$9-datum $C^1$0 there exists $C^1$1 such that for every Hamiltonian $C^1$2 complying with $C^1$3 with $C^1$4, and for every point $C^1$5, there exists a $C^1$6-flow line $C^1$7 whose asymptotic critical points have non-zero Lagrange multipliers, whose action values remain within $C^1$8 of $C^1$9 throughout, and whose cylinder component passes through $\mathscr{B} \subseteq C^\infty(\Sigma)$0 at $\mathscr{B} \subseteq C^\infty(\Sigma)$1.

The pointwise constraint $\mathscr{B} \subseteq C^\infty(\Sigma)$2 is what converts the abstract Floer-theoretic existence into the localized Reeb orbit statement: if no periodic orbit intersects $\mathscr{B} \subseteq C^\infty(\Sigma)$3 exists, then all relevant critical points lie over $\mathscr{B} \subseteq C^\infty(\Sigma)$4, where $\mathscr{B} \subseteq C^\infty(\Sigma)$5 agrees with $\mathscr{B} \subseteq C^\infty(\Sigma)$6 in value and differential; the action spectrum gap then forces a contradiction via the cobordism argument. The same mechanism yields a multiplicity alternative: either $\mathscr{B} \subseteq C^\infty(\Sigma)$7 has two geometrically distinct periodic orbits, or it has an orbit of period at most $\mathscr{B} \subseteq C^\infty(\Sigma)$8.

Proof strategy: homotopy stretching

The proof interpolates between $\mathscr{B} \subseteq C^\infty(\Sigma)$9 and $D \subseteq \Sigma$0 using bump functions $D \subseteq \Sigma$1 supported near $D \subseteq \Sigma$2, producing functionals $D \subseteq \Sigma$3. Moduli spaces $D \subseteq \Sigma$4 of finite-energy gradient flow lines asymptotic to the Morse-Bott critical manifold $D \subseteq \Sigma$5 (with $D \subseteq \Sigma$6) are shown to be non-empty for each fixed $D \subseteq \Sigma$7 by applying an abstract perturbation theorem for Fredholm sections to Banach bundle data built on weighted Sobolev spaces $D \subseteq \Sigma$8. Stretching $D \subseteq \Sigma$9 and extracting a diagonal subsequence produces the desired gradient flow line of $\Sigma \setminus \{q\}$0 itself.

Three uniform estimates make this work, and they constitute the paper's analytic substance:

1. Action and energy bounds: $\Sigma \setminus \{q\}$1 and $\Sigma \setminus \{q\}$2, where $\Sigma \setminus \{q\}$3 along the trajectory.
2. Uniform Lagrange multiplier bound: $\Sigma \setminus \{q\}$4, adapting Cieliebak–Frauenfelder's parabola argument to interpolated functionals. Crucially, compliance with the $\Sigma \setminus \{q\}$5-datum (a bound on $\Sigma \setminus \{q\}$6 over $\Sigma \setminus \{q\}$7) enters here — which is precisely why the theorem requires $\Sigma \setminus \{q\}$8-bounded families rather than arbitrary small functions.
3. Confinement of cylinders: $\Sigma \setminus \{q\}$9, proved by a maximum principle for the upper bound and, for the lower bound, by combining a Hofer energy estimate with Albers–Fuchs–Merry's result that low Hofer energy forces short Reeb orbits — contradicting minimality of $\delta = \delta(\mathscr{B}) > 0$0 when the energy drops below $\delta = \delta(\mathscr{B}) > 0$1.

With these bounds, Gromov compactness applies; breaking is excluded because the energy bound $\delta = \delta(\mathscr{B}) > 0$2 is smaller than the spectral gap of the component $\delta = \delta(\mathscr{B}) > 0$3, and convergence in the weighted topology follows from a uniform exponential decay estimate for gradient flow lines converging to $\delta = \delta(\mathscr{B}) > 0$4. The Fredholm analysis computes the index of $\delta = \delta(\mathscr{B}) > 0$5 as $\delta = \delta(\mathscr{B}) > 0$6 via spectral flow, propagates constancy of the index across the connected manifold $\delta = \delta(\mathscr{B}) > 0$7, and proves surjectivity at $\delta = \delta(\mathscr{B}) > 0$8 by a semigroup/kernel argument exploiting Morse-Bott nondegeneracy transverse to $\delta = \delta(\mathscr{B}) > 0$9.

The author also notes a correction to the literature: the exponential decay result stated in Fauck's thesis is believed flawed, since its proof differentiates the operator family in the wrong direction; the present paper supplies a corrected adaptation.

Limitations and open questions

Several restrictions are explicit. First, everything relies on the Zoll hypothesis; for general contact manifolds the analogous question remains open. The author identifies a plausible route under the weaker Morse-Bott condition (MB$f \in \mathscr{B}$0) on $f \in \mathscr{B}$1-periodic orbits, but the final step requires a "slice" $f \in \mathscr{B}$2 meeting the orbit manifold $f \in \mathscr{B}$3 transversally in an odd number of points, which imposes nontrivial homological conditions ($f \in \mathscr{B}$4 mod 2), so the generalization is conjectural rather than established. Second, the multiplicity result is explicitly not claimed optimal. Third, in the electromagnetic application the requirement of $f \in \mathscr{B}$5-bounds is admittedly not optimal: a purely Lagrangian argument via the Mañé critical value yields periodic orbits above $f \in \mathscr{B}$6 without any derivative control — though without the localization conclusion. Finally, whether the exponential decay constant $f \in \mathscr{B}$7 can be taken up to the Hessian spectral minimum is left unproven.

Conclusion

The paper reduces a concrete question in contact dynamics — persistence of periodic Reeb orbits under localized $f \in \mathscr{B}$8-small conformal rescalings of Zoll contact forms, with quantitative localization of the resulting orbit — to a robust existence principle for Rabinowitz-Floer gradient flow lines of nearby Hamiltonians, valid uniformly over $f \in \mathscr{B}$9-small perturbations within a $C^0$00-bounded class. The combination of spectral-gap-controlled stretching, weighted Sobolev Fredholm theory, and the abstract cobordism/perturbation framework provides a template likely to extend beyond the Zoll setting whenever suitable slice geometry is available.