Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II

Details Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II

1995-03-09

arxiv.org/abs/math/9503228v1

Mathematics

Dynamical Systems

Scientific paper

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).

Affiliated with

Also associated with

No associations

Rating

Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hofer's $L^{\infty}$-geometry: energy and stability of Hamiltonian flows, part II will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-152854