The geometry of symplectic energy
Abstract
In this paper we consider a geometric variant of Hofer's symplectic energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of $M$ which had "too little" energy, one could embed a large ball into a thin cylinder $M \times B^2$. Thus there is a direct geometric relation between symplectic rigidity and energy. The second half of the paper is devoted to a proof of the NonSqueezing theorem for an arbitrary manifold $M$. We do not need to restrict to manifolds in which the theory of pseudoholomorphic curves behaves well. This is of interest since most other deep results in symplectic topology are generalised from Euclidean space to other manifolds by using this theory, and hence are still not known to be valid for arbitrary symplectic manifolds.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1993
 arXiv:
 arXiv:math/9306216
 Bibcode:
 1993math......6216L
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 Abstract shortened in migration.