Hofer–Zehnder capacity and length minimizing Hamiltonian paths

Abstract

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 $S^1$–invariant virtual moduli cycles. As a corollary, we find that any semifree action of $S^1$ on $M$ gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of $M$. We also establish a version of the area-capacity inequality for quasicylinders.