Periodic leaf-wise intersection points from Lagrangians

TL;DR

This paper demonstrates the existence of periodic leaf-wise intersection points on certain hypersurfaces in monotone symplectic manifolds, using Floer theory and symplectic cohomology, with applications to prequantization bundles and negative line bundles.

Contribution

It introduces new methods combining reduced symplectic cohomology with existing techniques to establish leaf-wise intersection points in broader classes of symplectic manifolds.

Findings

Periodic leaf-wise intersection points exist on Zoll hypersurfaces in monotone symplectic manifolds.

The methods apply to prequantization bundles in monotone toric negative line bundles.

Existence results extend to certain annulus subbundles in weak+-monotone negative line bundles.

Abstract

We investigate leaf-wise intersection points on hypersurfaces of contact type in monotone symplectic manifolds. We show that monotone Floer-essential Lagrangians detect periodic leaf-wise intersection points in hypersurfaces of contact type whose Reeb flow is Zoll. Examples include the prequantization bundles appearing in monotone toric negative line bundles. Generalizing, we prove the existence of leaf-wise intersection points for certain annulus subbundles in weak+-monotone negative line bundles, not necessarily toric. The proofs combine reduced symplectic cohomology with the original methods employed by Albers-Frauenfelder to prove global existence results of this kind.