Applications of Hofer's geometry to Hamiltonian dynamics

Abstract

We prove that for every subset A of a tame symplectic manifold (W, ω) meeting a semi-positivity condition, the π1-sensitive Hofer-Zehnder capacity of A is not greater than four times the stable displacement energy of A, cHZ°(A, W) ≤ 4e(A × S1, W × T* S1). This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in (W, ω). The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray {φFt}, t > 0, of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear. © Swiss Mathematical Society.