Reeb orbits frequently intersecting a symplectic surface
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910962591727616 |
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| author | Hutchings, Michael |
| author_facet | Hutchings, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reeb orbits frequently intersecting a symplectic surface Hutchings, Michael Symplectic Geometry Dynamical Systems 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. |
| title | Reeb orbits frequently intersecting a symplectic surface |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2504.19332 |