Contact three-manifolds with exactly two simple Reeb orbits
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 struture 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.
Cite
@article{arxiv.2102.04970,
title = {Contact three-manifolds with exactly two simple Reeb orbits},
author = {Dan Cristofaro-Gardiner and Umberto Hryniewicz and Michael Hutchings and Hui Liu},
journal= {arXiv preprint arXiv:2102.04970},
year = {2023}
}
Comments
29 pages; v4 has minor edits, to appear in Geometry and Topology