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 . This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general uses Liu–Tian’s construction of –invariant virtual moduli cycles. As a corollary, we find that any semifree action of on gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of . We also establish a version of the area-capacity inequality for quasicylinders.
Citation
Dusa McDuff. Jennifer Slimowitz. "Hofer–Zehnder capacity and length minimizing Hamiltonian paths." Geom. Topol. 5 (2) 799 - 830, 2001. https://doi.org/10.2140/gt.2001.5.799
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