The Weinstein Conjecture for Hamiltonian Fibrations

TL;DR

This paper extends the Weinstein Conjecture to non-trivial Hamiltonian fibrations over symplectically uniruled manifolds, using Gromov-Witten invariants to establish the existence of closed Reeb orbits under certain conditions.

Contribution

It generalizes previous results from trivial to non-trivial Hamiltonian fibrations over uniruled manifolds, employing a product formula for Gromov-Witten invariants.

Findings

Proves the Weinstein Conjecture for certain Hamiltonian fibrations.

Uses Gromov-Witten invariants to establish the result.

Extends known results to non-trivial fibrations.

Abstract

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type, under certain technical conditions. The proof is based on the product formula for Gromov-Witten invariants ($GW$-invariant) of Hamiltonian fibrations derived in \cite{H}.