{
  "id": "1306.4458",
  "version": "v2",
  "published": "2013-06-19T09:03:49.000Z",
  "updated": "2015-03-26T14:46:05.000Z",
  "title": "Stable CMC and index one minimal surfaces in conformally flat manifolds",
  "authors": [
    "Rabah Souam"
  ],
  "comment": "corrected version",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $\\Sigma$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $\\Sigma$ is homeomorphic either to a sphere or to a torus. Moreover, in case $\\Sigma$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit field normal to $\\Sigma.$ The result is sharp, we can perturb the standard metric on the 3-sphere in its conformal class to obtain metrics of nonnegative Ricci curvature admitting minimal tori which are stable as CMC surfaces. As a consequence, in any 3-sphere of positive Ricci curvature which is conformally flat, the isoperimetric domains are topologically 3-balls. This proves a special case of a conjecture of A. Ros.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-19T09:03:49.000Z",
      "abstract": "Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $\\Sigma$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $\\Sigma$ is homeomorphic either to a sphere or to a torus. Moreover, in case $\\Sigma$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit field normal to $\\Sigma.$ The result is sharp, we can perturb the standard metric on the 3-sphere in its conformal class to obtain metrics of nonnegative Ricci curvature admitting minimal tori which are stable as CMC surfaces. As a consequence, in any 3-sphere of positive Ricci curvature which is conformally flat, the isoperimetric domains are topologically 3-balls.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-26T14:46:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conformally flat manifolds",
      "minimal surface",
      "stable cmc",
      "nonnegative ricci curvature",
      "ricci curvature admitting minimal tori"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.4458S"
    }
  }
}