The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed
Abstract
A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.
Cite
@article{arxiv.1901.10617,
title = {The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed},
author = {Daniel Cristofaro-Gardiner and Marco Mazzucchelli},
journal= {arXiv preprint arXiv:1901.10617},
year = {2020}
}
Comments
18 pages; version 3: we specified that the contact manifolds are required to be connected. To appear in Commentarii Mathematici Helvetici