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We consider convex contact spheres Y Y all of whose Reeb orbits are closed. Any such Y Y admits a stratification by the periods of closed Reeb orbits. We show that Y Y “resembles” a contact ellipsoid: any stratum of Y Y is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of Y Y coincides with the full sequence of action values, each one repeated according to its multiplicity.
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Contact three-manifolds with exactly two simple Reeb orbits
It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.
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- PAR ID:
- 10561798
- Publisher / Repository:
- Mathematical Sciences Publishers
- Date Published:
- Journal Name:
- Geometry & Topology
- Volume:
- 27
- Issue:
- 9
- ISSN:
- 1465-3060
- Page Range / eLocation ID:
- 3801 to 3831
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.2140/gt.2023.27.3801
