## Institute for Mathematical Sciences

## Preprint ims95-3b

** F. Lalonde and D. McDuff**
* Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II*

Abstract: In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition
for a path to be a geodesic in the group $\Ham^c(M)$ of
compactly supported Hamiltonian symplectomorphisms. This
applies with no restriction on $M$. We then discuss conditions
which guarantee that such a path minimizes the Hofer length.
Our argument relies on a general geometric construction (the
gluing of monodromies) and on an extension of Gromov's
non-squeezing theorem both to more general manifolds and to
more general capacities. The manifolds we consider are
quasi-cylinders, that is spaces homeomorphic to $M \times D^2$
which are symplectically ruled over $D^2$. When we work with
the usual capacity (derived from embedded balls), we can prove
the existence of paths which minimize the length among all
homotopic paths, provided that $M$ is semi-monotone. (This
restriction occurs because of the well-known difficulty with
the theory of $J$-holomorphic curves in arbitrary $M$.)
However, we can only prove the existence of length-minimizing
paths (i.e. paths which minimize length amongst {\it all}
paths, not only the homotopic ones) under even more restrictive
conditions on $M$, for example when $M$ is exact and convex or
of dimension $2$. The new difficulty is caused by the
possibility that there are non-trivial and very short loops in
$\Ham^c(M)$. When such length-minimizing paths do exist, we
can extend the Bialy--Polterovich calculation of the Hofer norm
on a neighbourhood of the identity ($C^1$-flatness).
Although it applies to a more restricted class of
manifolds, the Hofer-Zehnder capacity seems to be better
adapted to the problem at hand, giving sharper estimates in
many situations. Also the capacity-area inequality for split
cylinders extends more easily to quasi-cylinders in this case.
As applications, we generalise Hofer's estimate of the time for
which an autonomous flow is length-minimizing to some manifolds
other than $\R^{2n}$, and derive new results such as the
unboundedness of Hofer's metric on some closed manifolds, and a
linear rigidity result.

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