Contact three-manifolds with exactly two simple Reeb orbits
Daniel Cristofaro‐Gardiner, Umberto L. Hryniewicz, Michael Hutchings, Hui Liu
Abstract
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.