#### Volume 5, issue 2 (2001)

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Hofer–Zehnder capacity and length minimizing Hamiltonian paths

### Dusa McDuff and Jennifer Slimowitz

Geometry & Topology 5 (2001) 799–830
 arXiv: math.SG/0101085
##### 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.

##### Keywords
symplectic geometry, Hamiltonian diffeomorphisms, Hofer norm, Hofer–Zehnder capacity
##### Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57R57, 53D05