The quantitative nature of reduced Floer theory

Periodic leaf-wise intersection points from Lagrangians

“…Non-monotone line bundles can have multiple special subbundles. Although very little is understood about their Floer-essential Lagrangian tori, we showed in [19] that RFH * (Σ) is non-vanishing at a finite number of circle subbundles in weak + monotone line bundles. Precisely, the cohomological class p * c E 1 acts on the quantum cohomology of E, QH * (E), by quantum cup product.…”

“…With the reproven result, these Lemmas follow immediately from the results in [18], and we therefore omit the proofs. See also Lemma 3 in [19] for a maximum principle that is very closely related to the result we need.…”

“…REMARK 2) Theorem 5 is reminiscent of Theorem 1.2 in [4], which proved a non-vanishing result for a related homology theory termed Rabinowitz Floer homology. Theorem 5 is also the natural extension of Theorem 1 in [19]. The proof employs, with very little modification, the methods developed in [4].…”

“…Monotone line bundles have exactly one "special" circle subbundle; the hypersurface for which Theorem 1 produces a leaf-wise intersection point. This is the circle subbundle on which Floer-essential Lagrangian tori appear (in toric line bundles) and it is the only circle subbundle on which a Rabinowitz-Floer-type invariant RFH * (Σ) is non-zero [14] [19].…”

“…THEOREM 7 (VENKATESH [19]) The invariant RFH * (Σ R ) of a circle subbundle of radius R in a weak + -monotone toric negative line bundle is non-vanishing precisely when R = ev(λ i ) kπ…”