Applications of Hofer’s Geometry to Hamiltonian Dynamics

We prove the following three results in Hamiltonian dynamics. • The Weinstein conjecture holds true for every displaceable hypersurface of contact type. • Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. • Every closed Lagrangian submanifold whose fundamental group injects and which admits a Riemannian metric without closed geodesics has the intersection property. The proofs all rely on the following creation mechanism for closed orbits: If the ray {φ H }, t ≥ 0, of Hamiltonian diffeomorphisms generated by a sufficiently nice compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible closed orbit must appear.