Contact three-manifolds with exactly two simple Reeb orbits

Cristofaro-Gardiner, Daniel ¹   ; Hryniewicz, Umberto ²   ; Hutchings, Michael ³   ; Liu, Hui ⁴

¹ Department of Mathematics, University of California, Santa Cruz, Santa Cruz, CA, United States, School of Mathematics, Institute for Advanced Study, Princeton, NJ, United States, Department of Mathematics, University of Maryland, College Park, MD, United States
² RWTH Aachen, Aachen, Germany
³ Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States
⁴ School of Mathematics and Statistics, Wuhan University, Wuhan, China

Geometry & topology, Tome 27 (2023) no. 9, pp. 3801-3831

Résumé

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.

Détail

DOI : 10.2140/gt.2023.27.3801

Keywords: Reeb orbit, embedded contact homology, lens space, pseudorotation

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