{
  "id": "1608.02156",
  "version": "v1",
  "published": "2016-08-06T21:26:15.000Z",
  "updated": "2016-08-06T21:26:15.000Z",
  "title": "Stability of Catenoids and Helicoids in Hyperbolic Space",
  "authors": [
    "Biao Wang"
  ],
  "comment": "This paper replaces both arXiv:1204.4943 and arXiv:1502.04764",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\\{\\mathcal{C}_a\\}_{a>0}$ in $\\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that $\\bullet$ $\\mathcal{C}_a$ is an unstable minimal surface with index one if $a<a_c$, $\\bullet$ $\\mathcal{C}_a$ is a globally stable minimal surface if $a\\geq{}a_c$, and $\\bullet$ $\\mathcal{C}_a$ is a least area minimal surface in the sense of Meeks and Yau if $a\\geq{}a_l$. (2) For a family of minimal helicoids $\\{\\mathcal{H}_{\\bar{a}}\\}_{\\bar{a}\\geq{}0}$ in $\\mathbb{H}^3$, there exists a constant $\\bar{a}_c=\\coth(a_c)$ such that $\\bullet$ $\\mathcal{H}_{\\bar{a}}$ is a globally stable minimal surface if $0\\leq\\bar{a}\\leq\\bar{a}_c$, and $\\bullet$ $\\mathcal{H}_{\\bar{a}}$ is an unstable minimal surface with index infinity if $\\bar{a}>\\bar{a}_c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-06T21:26:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10"
    ],
    "keywords": [
      "hyperbolic space",
      "globally stable minimal surface",
      "unstable minimal surface",
      "area minimal surface",
      "index infinity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}