{
  "id": "1011.4140",
  "version": "v1",
  "published": "2010-11-18T06:27:04.000Z",
  "updated": "2010-11-18T06:27:04.000Z",
  "title": "Embeddedness of proper minimal submanifolds in homogeneous spaces",
  "authors": [
    "Sung-Hong Min"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove the three embeddedness results as follows. $({\\rm i})$ Let $\\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\\mathbb{R}^n$, where $m$ is an integer $\\geq2$. Then the total curvature of $\\Gamma_{2m+1}<2m\\pi$. In particular, the total curvature of $\\Gamma_5<4\\pi$ and thus any minimal surface $\\Sigma \\subset \\mathbb{R}^n$ bounded by $\\Gamma_5$ is embedded. Let $\\Gamma_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\\mathbb{H}^n$. Then any minimal surface $\\Sigma \\subset \\mathbb{H}^n$ bounded by $\\Gamma_5$ is embedded. If $\\Gamma_5$ is in a geodesic ball of radius $\\frac{\\pi}{4}$ in $\\mathbb{S}^n_+$, then $\\Sigma \\subset \\mathbb{S}^n_+$ is also embedded. As a consequence, $\\Gamma_5$ is an unknot in $\\mathbb{R}^3$, $\\mathbb{H}^3$ and $\\mathbb{S}^3_+$. $({\\rm ii})$ Let $\\Sigma$ be an $m$-dimensional proper minimal submanifold in $\\mathbb{H}^n$ with the ideal boundary $\\partial_{\\infty} \\Sigma = \\Gamma$ in the infinite sphere $\\mathbb{S}^{n-1}=\\partial_\\infty \\mathbb{H}^n$. If the M{\\\"o}bius volume of $\\Gamma$ $\\widetilde{\\vol}(\\Gamma) < 2\\vol(\\mathbb{S}^{m-1})$, then $\\Sigma$ is embedded. If $\\widetilde{\\vol}(\\Gamma) = 2\\vol(\\mathbb{S}^{m-1})$, then $\\Sigma$ is embedded unless it is a cone. $({\\rm iii})$ Let $\\Sigma$ be a proper minimal surface in $\\hr$. If $\\Sigma$ is vertically regular at infinity and has two ends, then $\\Sigma$ is embedded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-18T06:27:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous spaces",
      "piecewise geodesic jordan curve",
      "embeddedness",
      "dimensional proper minimal submanifold",
      "total curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.4140M"
    }
  }
}