Hofer–Zehnder capacity and length minimizing Hamiltonian paths

We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M. This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses Liu–Tian’s construction of S1-invariant virtual moduli cycles. As a corollary, we find that any semifree action of S1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group ofM. We also establish a version of the area–capacity inequality for quasicylinders