Contact three-manifolds with exactly two simple Reeb orbits

Geometry & Topology, vol. 27, pp. 3801–3831

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.

Authors 4

1. Dan Cristofaro‐Gardiner

   University of California, Santa Cruz · Institute for Advanced Study · University of Maryland, College Park

   Affiliation as printed

   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. Umberto L. Hryniewicz Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen, Aachen, Germany

3. Michael Hutchings

   University of California, Berkeley

   Affiliation as printed

   Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States

4. Hui Liu

   Wuhan University

   Affiliation as printed

   School of Mathematics and Statistics, Wuhan University, Wuhan, China