{
  "id": "math/0601582",
  "version": "v1",
  "published": "2006-01-24T12:47:43.000Z",
  "updated": "2006-01-24T12:47:43.000Z",
  "title": "Complete subamanifolds of $\\mathbb{R}^{n}$ with finite topology",
  "authors": [
    "G. Pacelli Bessa",
    "L. Jorge",
    "J. Fabio Montenegro"
  ],
  "comment": "8 pages",
  "journal": "Comm. Anal. Geom. vol. 15, n.4 (2007) 725-732",
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that a complete $m$-dimensional immersed submanifold $M$ of $\\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\\in [0,\\infty]$ is an scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifolds $M$ with $a(M)<1$ contains all complete minimal surfaces in $\\mathbb{R}^{n}$ with finite total curvature, all $m$-dimensional minimal submanifolds $M $ of $ \\mathbb{R}^{n}$ with finite total scalar curvature $\\smallint_{M}| \\alpha |^{m} dV<\\infty $ and all complete 2-dimensional complete surfaces with $\\smallint_{M}| \\alpha |^{2} dV<\\infty $ and nonpositive curvature with respect to every normal direction, since $a(M)=0$ for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-24T12:47:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42"
    ],
    "keywords": [
      "finite topology",
      "complete subamanifolds",
      "finite total scalar curvature",
      "second fundamental form decays",
      "dimensional minimal submanifolds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1582P"
    }
  }
}