Rigid constellations of closed Reeb orbits

TL;DR

This paper employs Hamiltonian Floer theory to extend a classical rigidity theorem, demonstrating the existence of multiple closed Reeb orbits on various contact manifolds, and explores obstructions posed by fast orbits.

Contribution

It generalizes a rigidity theorem about Reeb orbits to broader classes of contact forms and manifolds, introducing new existence results and obstructions related to fast orbits.

Findings

Multiple closed Reeb orbits exist on certain contact forms close to standard forms.

Rigidity phenomena extend to prequantization spaces and general contact manifolds under nondegeneracy.

Fast closed orbits cannot be eliminated by contactomorphism-invariant conditions.

Abstract

We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekelend 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 can not be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.