  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.
