{
  "id": "1506.06576",
  "version": "v1",
  "published": "2015-06-22T12:52:40.000Z",
  "updated": "2015-06-22T12:52:40.000Z",
  "title": "Derivatives of length functions and shearing coordinates on teichm{ü}ller spaces",
  "authors": [
    "Matthieu Gendulphe"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\\teich(S)$ its Teichm\\\"uller space. For any isotopy class of closed curves $\\g$, we compute the first three derivatives of the length function $\\ell\\_\\g:\\teich(S)\\rightarrow\\R\\_+$ in the shearing coordinates associated to a maximal geodesic lamination $\\l$. We show that if $\\g$ intersects any leaf of $\\l$, then the Hessian of $\\ell\\_\\g$ is positive-definite. We extend this result to length functions of measured laminations. We also provide a method to compute higher derivatives of length functions of geodesics. We use Bonahon's theory of transverse H\\\"older distributions and shearing coordinates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-22T12:52:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "length function",
      "shearing coordinates",
      "ller spaces",
      "maximal geodesic lamination",
      "isotopy class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606576G"
    }
  }
}