Geometry & Topology

Hofer–Zehnder capacity and length minimizing Hamiltonian paths

Dusa McDuff and Jennifer Slimowitz

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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 S1–invariant virtual moduli cycles. As a corollary, we find that any semifree action of S1 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.

Article information

Geom. Topol., Volume 5, Number 2 (2001), 799-830.

Received: 12 January 2001
Revised: 9 October 2001
Accepted: 9 November 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX] 53D05: Symplectic manifolds, general

symplectic geometry Hamiltonian diffeomorphisms Hofer norm Hofer–Zehnder capacity


McDuff, Dusa; Slimowitz, Jennifer. Hofer–Zehnder capacity and length minimizing Hamiltonian paths. Geom. Topol. 5 (2001), no. 2, 799--830. doi:10.2140/gt.2001.5.799.

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