Periodic Reeb orbits for general contact forms

Determine whether every sufficiently C^0-small perturbation e^f\alpha of a general contact form \alpha, with f identically zero outside an open smooth domain D that is intersected by a periodic Reeb orbit of \alpha, possesses a periodic Reeb orbit intersecting D.

Background

The paper proves an existence result for periodic Reeb orbits intersecting a prescribed domain when the original contact form is Zoll, meaning that all Reeb orbits are periodic with a common minimal period. The Outlook asks whether the same localized persistence phenomenon holds without the Zoll assumption, for arbitrary contact forms admitting at least one periodic Reeb orbit through the domain.

References

As mentioned right at the beginning in Subsection \ref{subsec: Introduction - Motivation}, it makes perfect sense to ask the analogous question also for general (not necessarily Zoll) contact forms: Let $(\Sigma,\alpha)$ be a contact manifold and $D \subseteq \Sigma$ be an open smooth domain for which $\alpha$ has a periodic Reeb orbit intersecting $D$. Does every sufficiently $C0$-small perturbed contact form $ef \alpha \, $, with $f \equiv 0$ on $\Sigma \backslash D \, $, also have a periodic Reeb orbit intersecting $D \, ?$

— Periodic orbits for perturbations of Zoll contact forms  (2608.17578 - Stalljohann, 18 Aug 2026) in Section 6, Outlook