The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

  • Daniel Cristofaro-Gardiner

    University of California at Santa Cruz, USA
  • Marco Mazzucchelli

    École Normale Supérieure de Lyon, France and Math. Sciences Research Institute, Berkeley, USA
The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed cover

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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 this article

Daniel Cristofaro-Gardiner, Marco Mazzucchelli, The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed. Comment. Math. Helv. 95 (2020), no. 3, pp. 461–481

DOI 10.4171/CMH/493