Hofer–Zehnder capacity and length minimizing Hamiltonian paths

McDuff, Dusa¹ ; Slimowitz, Jennifer²

¹ Department of Mathematics, State University of New York, Stony Brook, New York 11794-3651, USA
² Department of Mathematics, Rice University, Houston, Texas 77005, USA

Geometry & topology, Tome 5 (2001) no. 2, pp. 799-830.

Résumé

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.

Détail

DOI : 10.2140/gt.2001.5.799

Keywords: symplectic geometry, Hamiltonian diffeomorphisms, Hofer norm, Hofer–Zehnder capacity

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