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 LiuTian'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.

Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds

“…This was later extended to the case when (M, ω) is rational by Hofer and Viterbo in [36]. In the context of studying length minimizing Hamiltonian paths, McDuff and Slimowitz also proved more general results for slightly different capacities in [53]. The result for general symplectic manifolds (M, ω) was proved recently by G. Lu in [49] using the theory of Gromov-Witten invariants and, in particular, Liu and Tian's construction of an equivariant virtual moduli cycle from [47].…”

“…These sets can be symplectically embedded into a trivial symplectic tubular neighborhood W × B 2 (R) whose capacity is known to be finite in various cases by [36,48,49,53]. When applied to U R \ M this yields bounds on certain relative capacities and allows Macarini to further improve the existence results from [12,26] by relaxing the assumption that the ambient manifold is symplectically aspherical.…”

“…The first of these methods was introduced in [20] and developed in [36] (see also [47,48,49,53]). This method utilizes Floer's version of Gromov's compactness theorem for perturbed J -holomorphic spheres.…”

“…We only mention here that at the center of the argument lies the main result from [53] which gives a sufficient condition, in terms of periodic orbits, for a Hamiltonian H to generate a length minimizing path with respect to Hofer's norm. In turn, the main result in [53] is implied by an upper bound for a particular Hofer-Zehnder capacity of trivial symplectic tubular neighborhoods [44]. It is a remarkable fact that all the finiteness theorems for capacities implied by Schlenk's Theorem 1.6 can be traced to this one bound for the capacity of trivial symplectic tubular neighborhoods.…”

The Sharp Energy-Capacity Inequality

“…However, the properties of ρ(H; 1) are now fairly well-understood when H is nondegenerate, and (even though the Hamiltonians involved in the definition of c HZ are quite degenerate) we are able to apply this understanding to push through enough of the arguments of [3] with the hypotheses that Frauenfelder, Ginzburg, and Schlenk impose on σ replaced by known properties of ρ(·; 1). Incidentally, as we explain in, respectively, Remarks 2.2 and 4.2, as byproducts of the proof of Theorem 1.1 we obtain new proofs of the nondegeneracy of Oh's spectral norm (originally proven in [20]) and of Conjecture 1.2 of [16] (originally proven in [25]). …”