Leaf-Wise Intersections and Rabinowitz Floer Homology

Abstract

In this article we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [Mos78].2000 Mathematics Subject Classification. 53D40, 37J10, 58J05. Key words and phrases. Leaf-wise Intersections, Rabinowitz Floer homology, stretching the neck, local homology. A perturbation of the Rabinowitz action functionalWe recall that Σ ⊂ (M, ω = dλ) is a closed hypersurface in an exact symplectic manifold such that (Σ, α = λ |Σ ) is a contact manifold. Moreover, Σ is assumed to bound a compact region in M . We denote by R the Reeb vector field of α. Moreover, we define the vector field Y by dλ(Y, ·) = λ(·).Lemma 2.1. The vector field Y is a Liouville vector field for (Σ, α), that is, L Y ω = ω and Y ⋔ Σ. In particular, (Σ, α) is of restricted contact type.

Floer Homologies, with Applications

“…The Floer homology RFH of this functional was constructed by Cieliebak and Frauenfelder in [43]. In Theorem 6.3 one leafwise intersection point and assertion (ii) was established in [12], while the cuplength estimate was proven in [14]. RFH was used in [45] to study the dynamics of a charged particle in a magnetic field at different energy levels.…”

The Omega limit set of a family of chords

“…On this symplectic manifold we can choose a 1-parameter family of Hamiltonians with Σ µ ∼ = ST * S g , where the dynamics change from geodesic flow into the horocycle flow. The latter has no periodic orbits, showing that the family can simply stop in such a case without ending up in either option (1) or (2).…”

Bounds for tentacular Hamiltonians

“…In order to define the boundary operator, one first has to endow C ∞ (S 1 , M ) with the L 2 -Riemannian structure induced by an ω-compatible 2-parameter family of almost complex structures {J t (·, η)} (t,η)∈S 1 ×R on M . 3 Then we can calculate the gradient of the Rabinowitz action functional with respect to that metric:…”