{
  "id": "1412.1539",
  "version": "v1",
  "published": "2014-12-04T02:06:39.000Z",
  "updated": "2014-12-04T02:06:39.000Z",
  "title": "Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space",
  "authors": [
    "Yu Fu"
  ],
  "comment": "18pages,to appear in Tohoku Math. J",
  "categories": [
    "math.DG"
  ],
  "abstract": "The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\\mathbb E^m$ with at most two distinct principal curvatures ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\\delta(2)$-ideal hypersurfaces in $\\mathbb E^m$, where a $\\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\\lambda_1, \\lambda_2$ and $\\lambda_1+\\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\\times O(q)$-invariant hypersurfaces in Euclidean space $\\mathbb E^{p+q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-04T02:06:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distinct principal curvatures",
      "biharmonic hypersurfaces",
      "euclidean space",
      "chens conjecture",
      "ideal hypersurface"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.geomphys.2013.09.004",
      "journal": "Journal of Geometry and Physics",
      "year": 2014,
      "month": "Jan",
      "volume": 75,
      "pages": 113
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014JGP....75..113F"
    }
  }
}