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Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

Dan Cristofaro-Gardiner, Michael Hutchings and Daniel Pomerleano

Geometry & Topology 23 (2019) 3601–3645

We prove that every nondegenerate contact form on a closed connected three-manifold such that the associated contact structure has torsion first Chern class has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for nontorsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.

Reeb dynamics, Weinstein conjecture, embedded contact homology
Mathematical Subject Classification 2010
Primary: 53D10
Secondary: 53D42
Received: 21 June 2018
Accepted: 24 March 2019
Published: 30 December 2019
Proposed: Yasha Eliashberg
Seconded: Tomasz Mrowka, Leonid Polterovich
Dan Cristofaro-Gardiner
Mathematics Department
University of California, Santa Cruz
Santa Cruz, CA
United States
Michael Hutchings
Mathematics Department
University of California, Berkeley
Berkeley, CA
United States
Daniel Pomerleano
Mathematics Department
University of Massachusetts, Boston
Boston, MA
United States