{
  "id": "1708.01852",
  "version": "v1",
  "published": "2017-08-06T06:43:18.000Z",
  "updated": "2017-08-06T06:43:18.000Z",
  "title": "Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds",
  "authors": [
    "Jean-Marc Schlenker"
  ],
  "comment": "Notes aiming at clarifying the relations between different points of view and introducing one new notion, no real result. Not intended to be submitted at this point",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\\ext(f)$ of $f$, and an upper bound on $\\ext(f)$. We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincar\\'e-Einstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a \"constant curvature\" space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and Monge-Amp\\`ere equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-06T06:43:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasifuchsian manifolds",
      "schwarzian tensor",
      "holomorphic quadratic differential",
      "quasifuchsian hyperbolic manifold",
      "linear weingarten surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}