HOFERS L(INFINITY)-GEOMETRY - ENERGY AND STABILITY OF HAMILTONIAN FLOWS .2.

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 sym plectomorphisms. This applies with no restriction on M. We then discus s conditions which guarantee that such a path minimizes the Hofer leng th. Our argument relies on a general geometric construction (the gluin g of monodromies) and on an extension of Gromov's non-squeezing theore m both to more general manifolds and to more general capacities. The m anifolds we consider are quasi-cylinders, that is spaces homeomorphic to M x D-2 which are symplectically ruled over D-2. When we work with the usual capacity (derived from embedded balls), we can prove the exi stence of paths which minimize the length among all homotopic paths, p rovided that M is semi-monotone. (This restriction occurs because of t he well-known difficulty with the theory of J-holomorphic curves in ar bitrary M.) However, we can only grove the existence of length-minimiz ing paths (i.e. paths which minimize length amongst all paths, not onl y the homotopic ones) under even more restrictive conditions on M, for example when M is exact and convex or of dimension 2. The new difficu lty is caused by the possibility that there are non-trivial and very s hort loops in Ham(c)(M). When such length-minimizing paths do exist, w e can extend the Bialy-Polterovich calculation of the Kofer 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 cylind ers extends more easily to quasi-cylinders in this case. As applicatio ns, we generalise Hofer's estimate of the time for which an autonomous flow is length-minimizing to some manifolds other than R(2n), and der ive new results such as the unboundedness of Hofer's metric on some cl osed manifolds, and a linear rigidity result.