{
  "id": "math/0608067",
  "version": "v1",
  "published": "2006-08-02T20:37:41.000Z",
  "updated": "2006-08-02T20:37:41.000Z",
  "title": "Area-stationary surfaces inside the sub-Riemannian three-sphere",
  "authors": [
    "Ana Hurtado",
    "César Rosales"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "abstract": "We consider the sub-Riemannian metric $g_{h}$ on $\\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carath\\'eodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in $(\\mathbb{S}^3,g_{h})$. By following the ideas and techniques in [RR] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carath\\'eodory geodesics and the ruling property of constant mean curvature surfaces to show that the only $C^2$ compact, connected, embedded surfaces in $(\\mathbb{S}^3,g_{h})$ with empty singular set and constant mean curvature $H$ such that $H/\\sqrt{1+H^2}$ is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in $(\\mathbb{S}^3,g_{h})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-02T20:37:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C17",
      "49Q20"
    ],
    "keywords": [
      "area-stationary surfaces inside",
      "sub-riemannian three-sphere",
      "constant mean curvature surfaces",
      "clifford torus",
      "hopf vector field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8067H"
    }
  }
}