Geometry & Topology
- Geom. Topol.
- Volume 5, Number 2 (2001), 799-830.
Hofer–Zehnder capacity and length minimizing Hamiltonian paths
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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.gt/1513883044