{
  "id": "1409.4170",
  "version": "v1",
  "published": "2014-09-15T08:07:00.000Z",
  "updated": "2014-09-15T08:07:00.000Z",
  "title": "Minimal Lagrangian submanifolds via the geodesic Gauss map",
  "authors": [
    "Chris Draper",
    "Ian McIntosh"
  ],
  "comment": "15 pages",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "abstract": "For an oriented isometric immersion $f:M\\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\\gamma$ projects this into the manifold of oriented geodesics in $S^n$ (the Grassmannian of oriented 2-planes in $\\mathbb{R}^{n+1}$), giving a Lagrangian immersion of $UM^\\perp$ into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of $f$, and show that when $f$ has conformal shape form this depends only on the mean curvature of $f$. We deduce conditions under which $\\gamma$ is a minimal Lagrangian immersion. We give simple proofs that: deformations of $f$ always correspond to Hamiltonian deformations of $\\gamma$; the mean curvature vector of $\\gamma$ is always a Hamiltonian vector field. This extends work of Palmer on the case when $M$ is a hypersurface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-15T08:07:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53D12"
    ],
    "keywords": [
      "geodesic gauss map",
      "minimal lagrangian submanifolds",
      "mean curvature vector",
      "second fundamental form",
      "conformal shape form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4170D"
    }
  }
}