## Institute for Mathematical Sciences

## Preprint ims95-3a

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

Abstract: Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold
$(M,\om)$ with the Hofer $L^{\infty}$-norm. A path in
$\Ham^c(M)$ will be called a geodesic if all sufficiently short
pieces of it are local minima for the Hofer length functional
$\Ll$. In this paper, we give a necessary condition for a
path $\ga$ to be a geodesic. We also develop a necessary
condition for a geodesic to be stable, that is, a local minimum
for $\Ll$. This condition is related to the existence of
periodic orbits for the linearization of the path, and so
extends Ustilovsky's work on the second variation formula.
Using it, we construct a symplectomorphism of $S^2$ which
cannot be reached from the identity by a shortest path. In
later papers in this series, we will use holomorphic methods to
prove the sufficiency of the condition given here for the
characterisation of geodesics as well as the sufficiency of the
condition for the stability of geodesics. We will also
investigate conditions under which geodesics are absolutely
length-minimizing.

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