Hofer-Zehnder capacity and length minimizing Hamiltonian paths

Details Hofer-Zehnder capacity and length minimizing Hamiltonian paths Hofer-Zehnder capacity and length minimizing Hamiltonian paths

2001-01-10

arxiv.org/abs/math/0101085v2

Geom. Topol. 5(2001) 799-830

Mathematics

Symplectic Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper25.abs.html

Scientific paper

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.

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