Geometry & Topology 5 (2001) 799–830

DOI: 10.2140/gt.2001.5.799

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 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.

Keywords

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

Mathematical Subject Classification

Primary: 57R17

Secondary: 53D05, 57R57

References

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Publication

Received: 12 January 2001
Revised: 9 October 2001
Accepted: 9 November 2001
Published: 9 November 2001
Proposed: Gang Tian
Seconded: Yasha Eliashberg, Tomasz Mrowka

Authors