{
  "id": "0910.5373",
  "version": "v2",
  "published": "2009-10-28T13:36:25.000Z",
  "updated": "2010-04-22T11:33:59.000Z",
  "title": "Parabolic stable surfaces with constant mean curvature",
  "authors": [
    "Jose M. Manzano",
    "Joaquin Perez",
    "M. Magdalena Rodriguez"
  ],
  "comment": "15 pages, 1 figure. Minor changes have been incorporated (exchange finite capacity by parabolicity, and simplify the proof of Theorem 1).",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of such functions is 1-dimensional. We obtain consequences for orientable, complete stable surfaces with constant mean curvature $H\\in\\mathbb{R}$ in homogeneous spaces $\\mathbb{E}(\\kappa,\\tau)$ with four dimensional isometry group. For instance, if M is an orientable, parabolic, complete immersed surface with constant mean curvature H in $\\mathbb{H}^2\\times\\mathbb{R}$, then $|H|\\leq 1/2$ and if equality holds, then M is either an entire graph or a vertical horocylinder.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-04-22T11:33:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "49Q05",
      "53C42"
    ],
    "keywords": [
      "constant mean curvature",
      "parabolic stable surfaces",
      "parabolic riemannian manifold",
      "dimensional isometry group",
      "schrodinger operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.5373M"
    }
  }
}