{
  "id": "1709.02078",
  "version": "v1",
  "published": "2017-09-07T05:44:36.000Z",
  "updated": "2017-09-07T05:44:36.000Z",
  "title": "Density of a minimal submanifold and total curvature of its boundary",
  "authors": [
    "Jaigyoung Choe",
    "Robert Gulliver"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Given a piecewise smooth submanifold $\\Gamma^{n-1} \\subset \\R^m$ and $p \\in \\R^m$, we define the {\\em vision angle} $\\Pi_p(\\Gamma)$ to be the $(n-1)$-dimensional volume of the radial projection of $\\Gamma$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $\\Sigma \\subset \\R^m$ with boundary $\\Gamma$, then we show the density of $\\Sigma$ at $p$ is $\\leq$ the density at its vertex $p$ of the cone over $\\Gamma$. It follows that if $\\Pi_p(\\Gamma)$ is less than twice the volume of $S^{n-1}$, for all $p \\in \\Gamma$, then $\\Sigma$ is an embedded submanifold. As a consequence, we prove that given two $n$-planes $R^n_1, R^n_2$ in $\\R^m$ and two compact convex hypersurfaces $\\Gamma_i$ of $R^n_i, i=1,2$, a nonflat minimal submanifold spanned by $\\Gamma:=\\Gamma_1\\cup\\Gamma_2$ is embedded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-07T05:44:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10"
    ],
    "keywords": [
      "total curvature",
      "compact convex hypersurfaces",
      "nonflat minimal submanifold",
      "piecewise smooth submanifold",
      "dimensional volume"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}