Geometry & Topology, Vol. 5 (2001) Paper no. 25, pages 799--830.

Hofer-Zehnder capacity and length minimizing Hamiltonian paths

Dusa McDuff, Jennifer Slimowitz

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

AMS subject classification. Primary: 57R17. Secondary: 57R57, 53D05.

DOI: 10.2140/gt.2001.5.799

E-print: arXiv:math.SG/0101085

Submitted to GT on 12 January 2001. (Revised 9 October 2001.) Paper accepted 9 November 2001. Paper published 9 November 2001.

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Dusa McDuff, Jennifer Slimowitz

Department of Mathematics, State University of New York
Stony Brook, NY 11794-3651, USA
and
Department of Mathematics - MS 136, Rice University
Houston, TX 77005, USA

Email: dusa@math.sunysb.edu, jslimow@rice.edu

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