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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 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 2000
Primary: 57R17
Secondary: 57R57, 53D05
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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
Dusa McDuff
Department of Mathematics
State University of New York
Stony Brook
New York 11794-3651
USA
Jennifer Slimowitz
Department of Mathematics
Rice University
Houston
Texas 77005
USA