Hofer-Zehnder capacity and length minimizing Hamiltonian paths
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.
Cite
@article{arxiv.math/0101085,
title = {Hofer-Zehnder capacity and length minimizing Hamiltonian paths},
author = {Dusa McDuff and Jennifer Slimowitz},
journal= {arXiv preprint arXiv:math/0101085},
year = {2014}
}
Comments
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper25.abs.html