{
  "id": "1605.06602",
  "version": "v1",
  "published": "2016-05-21T07:34:31.000Z",
  "updated": "2016-05-21T07:34:31.000Z",
  "title": "Mean curvature in manifolds with Ricci curvature bounded from below",
  "authors": [
    "Jaigyoung Choe",
    "Ailana Fraser"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\\Sigma$ does not separate $M$ then $\\Sigma$ is totally geodesic and $M\\setminus\\Sigma$ is isometric to the Riemannian product $\\Sigma\\times(a,b)$, and if $\\Sigma$ separates $M$ then the map $i_*:\\pi_1(\\Sigma)\\rightarrow \\pi_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\\geq(n-1)\\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\\geq-(n-1)k,k>0$, and for a free boundary minimal hypersurface in a manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $n$-dimensional manifold $N$ with the number of generators of $\\pi_1(N)<n$ cannot be minimally embedded in the flat torus $T^{n+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-21T07:34:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20"
    ],
    "keywords": [
      "mean curvature",
      "nonnegative ricci curvature",
      "free boundary minimal hypersurface",
      "compact riemannian manifold",
      "nonempty strictly convex boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}