Rigid constellations of closed Reeb orbits

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We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby-Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.

Original languageEnglish (US)
Pages (from-to)2394-2444
Number of pages51
JournalCompositio Mathematica
Issue number11
StatePublished - Nov 1 2017


  • Floer homology
  • Reeb flows
  • periodic orbits

ASJC Scopus subject areas

  • Algebra and Number Theory


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