{
  "id": "1209.3651",
  "version": "v1",
  "published": "2012-09-17T13:50:11.000Z",
  "updated": "2012-09-17T13:50:11.000Z",
  "title": "Rotational Surfaces in S^3 with constant mean curvature",
  "authors": [
    "Oscar Perdomo"
  ],
  "comment": "13 pages, 8 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\\sqrt{1+H^2}) there is an axially symmetry surface \\Sigma_{H,C} with mean curvature H. In 2010, Perdomo showed that for every H between cot(\\pi/m) and (m^2-2)/(2(m^2-1)^1/2), there exists an embedded axially symmetric example with non constant principal curvatures that is invariant under the ciclic group Z_m. Andrews and Li, showed that these examples are the only non-isoparametric embedded examples in the family when H>0. In this paper we study those examples in the family with H<0. We prove that there are no embedded examples in the family when H<0 and we also prove that for every integer m>2 there is a properly immersed example in this family that contains a great circle and is invariant under the ciclic group Z_m. We will say that these examples contain the axis of symmetry. Finally we show that every non-isoparametric surface \\Sigma_{H,C} is either properly immersed invariant under the ciclic group Z_m for some integer m>1 or it is dense in the region bounded by two isoparametric tori if the surface \\Sigma_{H,C} does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface \\Sigma_{H,C} contains the axis of symmetry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-17T13:50:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53C10"
    ],
    "keywords": [
      "constant mean curvature",
      "rotational surfaces",
      "ciclic group",
      "non constant principal curvatures",
      "axially symmetric cmc surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.3651P"
    }
  }
}