{
  "id": "math/9503228",
  "version": "v1",
  "published": "1995-03-09T00:00:00.000Z",
  "updated": "1995-03-09T00:00:00.000Z",
  "title": "Hofer's $L^{\\infty}$-geometry: energy and stability of Hamiltonian flows, part II",
  "authors": [
    "François Lalonde",
    "Dusa McDuff"
  ],
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "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).",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-03-09T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamiltonian flows",
      "length-minimizing paths",
      "general geometric construction",
      "hofer norm",
      "holomorphic curves"
    ],
    "publication": {
      "doi": "10.1007/BF01232394",
      "journal": "Inventiones Mathematicae",
      "year": 1996,
      "month": "Dec",
      "volume": 123,
      "number": 1,
      "pages": 613
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996InMat.123R.613L"
    }
  }
}