Hofer'sL ^∞-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 onM. 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 toM×D ² which are symplectically ruled overD 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 thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM 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 lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (C ^l-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 thanR ²ⁿ, and derive new results such as the unboundedness of Hofer's metric on some closed manifolds, and a linear rigidity result.