Applications of Hofer's geometry to Hamiltonian dynamics

Abstract

We prove that for every subset $A$ of a tame symplectic manifold $(W,\omega)$ meeting a semi-positivity condition, the $\pi_1$-sensitive Hofer--Zehnder capacity of $A$ is not greater than four times the stable displacement energy of $A$, $$ c_{HZ}^\circ(A,W)\le 4e (A\times S^1, W\times T^*S^1). $$ This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: $\bullet$ The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in $(W,\omega)$. $\bullet$ 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 $\{\varphi_F^t \}$, $t \ge 0$, 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