## Institute for Mathematical Sciences

## Preprint ims93-6

** F. Lalonde & D. McDuff**
* The Geometry of Symplectic Energy*

Abstract: One of the most striking early results in symplectic topology is Gromov's "Non-Squeezing Theorem", which says that it is
impossible to embed a large ball symplectically into a thin
cylinder of the form $\R^{2n} \times B^2$, where $B^2$ is a
$2$-disc. This led to Hofer's discovery of symplectic
capacities, which give a way of measuring the size of subsets
in symplectic manifolds. Recently, Hofer found a way to
measure the size (or energy) of symplectic diffeomorphisms by
looking at the total variation of their generating
Hamiltonians. This gives rise to a bi-invariant (pseudo-)norm
on the group $\Ham(M)$ of compactly supported Hamiltonian
symplectomorphisms of the manifold $M$. The deep fact is that
this pseudo-norm is a norm; in other words, the only
symplectomorphism on $M$ with zero energy is the identity map.
Up to now, this had been proved only for sufficiently nice
symplectic manifolds, and by rather complicated analytic
arguments.
In this paper we consider a more geometric version of this
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
Non-Squeezing theorem for an arbitrary manifold $M$. We do not
need to restrict to manifolds in which the theory of
pseudo-holomorphic 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.

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