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 $M$ . 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

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57R57, 53D05

References

Forward citations

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

Volume 5, issue 2 (2001)

Dusa McDuff and Jennifer Slimowitz