{
  "id": "math/0111136",
  "version": "v5",
  "published": "2001-11-12T16:40:25.000Z",
  "updated": "2002-09-30T09:24:39.000Z",
  "title": "Hyperbolic manifolds with polyhedral boundary",
  "authors": [
    "Jean-Marc Schlenker"
  ],
  "comment": "Updated version on http://picard.ups-tlse.fr/~schlenker/texts/papers.html New version: several typos corrected, a few remarks added",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let $(M, \\partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\\dr M$ is smooth and strictly convex, the induced metric on $\\dr M$ has curvature $K>-1$, and each such metric on $\\dr M$ is obtained for a unique choice of $g$. A dual statement is that, for each $g$ as above, the third fundamental form of $\\dr M$ has curvature $K<1$, and its closed geodesics which are contractible in $M$ have length $L>2\\pi$. Conversely, any such metric on $\\dr M$ is obtained for a unique choice of $g$. We are interested here in the similar situation where $\\partial M$ is not smooth, but rather looks locally like an ideal polyhedron in $H^3$. We can give a fairly complete answer to the question on the third fundamental form -- which in this case concerns the dihedral angles -- and some partial results about the induced metric. This has some by-products, like an affine piecewise flat structure on the Teichmueller space of a surface with some marked points, or an extension of the Koebe circle packing theorem to many 3-manifolds with boundary.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2002-09-30T09:24:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C45"
    ],
    "keywords": [
      "hyperbolic manifolds",
      "polyhedral boundary",
      "third fundamental form",
      "unique choice",
      "convex co-compact hyperbolic metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....11136S"
    }
  }
}