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arXiv · 2504.19332

Reeb orbits frequently intersecting a symplectic surface

Michael Hutchings
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Abstract

Consider a symplectic surface in a three-dimensional contact manifold with boundary on Reeb orbits (periodic orbits of the Reeb vector field). We assume that the rotation numbers of the boundary Reeb orbits satisfy a certain inequality, and we also make a technical assumption that the Reeb vector field has a particular ``nice'' form near the boundary of the surface. We then show that there exist Reeb orbits which intersect the interior of the surface, with a lower bound on the frequency of these intersections in terms of the symplectic area of the surface and the contact volume of the three-manifold. No genericity of the contact form is assumed. As a corollary of the main result, we obtain a generalization of various recent results relating the mean action of periodic orbits to the Calabi invariant for area-preserving surface diffeomorphisms.

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Michael Hutchings. 2025-04-27. Reeb orbits frequently intersecting a symplectic surface. https://arxiv.org/abs/2504.19332

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