{
  "id": "2303.12449",
  "version": "v1",
  "published": "2023-03-22T10:36:04.000Z",
  "updated": "2023-03-22T10:36:04.000Z",
  "title": "Hyperbolic Plane in $\\mathbb{E}^3$",
  "authors": [
    "Vincent Borrelli",
    "Roland Denis",
    "Francis Lazarus",
    "Mélanie Theillière",
    "Boris Thibert"
  ],
  "comment": "49 pages, 8 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We build an explicit $C^1$ isometric embedding $f_{\\infty}:\\mathbb{H}^2\\to\\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. In parallel, we introduce a formal corrugation process leading to a limit map $\\Phi_{\\infty}:\\mathbb{H}^2\\to \\mathcal{L}(\\mathbb{R}^2,\\mathbb{R}^3)$ that we call the formal analogue of $df_{\\infty}$. We show that $\\Phi_{\\infty}$ approximates $df_{\\infty}$. In fact, when the sequence of corrugation numbers $(N_{k})_{k}$ grows to infinity the map $\\Phi_{\\infty}$ encodes the asymptotic behavior of $df_{\\infty}$. Moreover, analysing the geometry of $\\Phi_{\\infty}$ appears much easier than studying $f_{\\infty}$. In particular, we are able to exhibit a form of self-similarity of $\\Phi_{\\infty}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-22T10:36:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53C21",
      "30F45"
    ],
    "keywords": [
      "hyperbolic plane",
      "limit map",
      "limit set",
      "asymptotic behavior",
      "formal corrugation process leading"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}