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Reeb orbits frequently intersecting a symplectic surface

Michael Hutchings

TL;DR

This work proves that in a closed 3D contact manifold with a symplectic surface boundary on Reeb orbits, and under a niceness condition near the boundary, there exists a Reeb orbit intersecting the surface with a frequency lower bound given by $\operatorname{Area}(\Sigma,d\lambda)/\operatorname{vol}(Y,\lambda)$, independent of genericity assumptions. The authors’ approach hinges on inflating the contact form near the surface and leveraging Weyl laws for elementary spectral invariants and their interaction with alternative ECH capacities to force the bound. The result yields a broad generalization relating mean action and the Calabi invariant for area-preserving diffeomorphisms and illustrates a robust link between geometric intersection data and spectral invariants in contact geometry. The paper also outlines open questions, including removing the niceness hypothesis, achieving reverse inequalities, and extending the framework to higher dimensions, with potential implications for geodesic and Hamiltonian dynamics beyond the current setting.

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.

Reeb orbits frequently intersecting a symplectic surface

TL;DR

This work proves that in a closed 3D contact manifold with a symplectic surface boundary on Reeb orbits, and under a niceness condition near the boundary, there exists a Reeb orbit intersecting the surface with a frequency lower bound given by , independent of genericity assumptions. The authors’ approach hinges on inflating the contact form near the surface and leveraging Weyl laws for elementary spectral invariants and their interaction with alternative ECH capacities to force the bound. The result yields a broad generalization relating mean action and the Calabi invariant for area-preserving diffeomorphisms and illustrates a robust link between geometric intersection data and spectral invariants in contact geometry. The paper also outlines open questions, including removing the niceness hypothesis, achieving reverse inequalities, and extending the framework to higher dimensions, with potential implications for geodesic and Hamiltonian dynamics beyond the current setting.

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.
Paper Structure (15 sections, 15 theorems, 158 equations)

This paper contains 15 sections, 15 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Idea of the proof
  4. Questions for further research
  5. The rest of the paper.
  6. Acknowledgments.
  7. Application to mean action and the Calabi invariant
  8. Spectral invariants
  9. Alternative ECH capacities
  10. Elementary spectral invariants
  11. Proof of the main theorem
  12. Reduction to the very nice case
  13. Inflating the contact form in a slab
  14. Behavior of the spectral invariants under inflation
  15. Proof of Theorem \ref{['thm:main1']}

Key Result

Theorem 1.10

Let $Y$ be a closed connected three-manifold, let $\lambda$ be a contact form on $Y$, and let $u:\Sigma\to Y$ be an admissible symplectic surface. Suppose that $(Y,\lambda,u)$ is nice. Then

Theorems & Definitions (56)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 46 more