In this article we prove that the Weinstein conjecture holds for contact manifolds (Σ, ξ) for which Cont0(Σ, ξ) is non-orderable in the sense of Eliashberg-Polterovich [EP00]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [San13] on the existence of translated points in the non-degenerate case.

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let (M, g) denote a closed connected Riemannian manifold and σ ∈ 2 (M) a weakly exact 2-form. Let φ t : T M → T M denote the magnetic flow determined by σ , and let c(g, σ ) ∈ ‫ޒ‬ ∪ {∞} denote the Mañé critical value of the pair (g, σ ). We prove that if k > c(g, σ ), then for every nontrivial free homotopy class of loops on M there exists a closed orbit of φ t with energy k whose projection to M belongs to that free homotopy class. We also prove that for almost all k < c(g, σ ) there exists a closed orbit of φ t with energy k whose projection to M is contractible. In particular, when c(g, σ ) = ∞ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that a weakly exact magnetic flow with [σ ] = 0 on a manifold with amenable fundamental group (which implies c(g, σ ) = ∞) has contractible closed orbits on almost every energy level.

ABSTRACT. Let (M, g) be a closed connected orientable Riemannian manifold of dimension n ≥ 2. Let ω := ω0 + π * σ denote a twisted symplectic form on T * M , where σ ∈ Ω 2 (M ) is a closed 2-form and ω0 is the canonical symplectic structure dp ∧ dq on T * M . Suppose that σ is weakly exact and its pullback to the universal cover M admits a bounded primitive. Let H :, and suppose that k > c(g, σ, U ), where c(g, σ, U ) denotes the Mañé critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces.Under the stronger condition that k > c0(g, σ, U ), where c0(g, σ, U ) denotes the strict Mañé critical value, Abbondandolo and Schwarz [4] recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k > c(g, σ, U ), thus covering cases where σ is not exact.As a consequence, we deduce that the hypersurface Σ k is never (stably) displaceable for any k > c(g, σ, U ). This removes the hypothesis of negative curvature in [20, Theorem 1.3] and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in [20]. Moreover, following [6,5] we prove that for k > c(g, σ, U ), any ψ ∈ Hamc(T * M, ω) has a leaf-wise intersection point in Σ k , and that if in addition dim H * (ΛM ; 2) = ∞, dim M ≥ 2, and the metric g is chosen generically, then for a generic ψ ∈ Hamc(T * M, ω) there exist infinitely many such leaf-wise intersection points.

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser [Mos78]. Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the 'virtually contact' setting. By means of an Abbondandolo-Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.