{
  "id": "math/0609494",
  "version": "v1",
  "published": "2006-09-18T12:12:42.000Z",
  "updated": "2006-09-18T12:12:42.000Z",
  "title": "A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space",
  "authors": [
    "Bruno Colbois",
    "Jean-Francois Grosjean"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we give pinching Theorems for the first nonzero eigenvalue $\\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $\\epsilon>0$, there exists a constant $C\\_{\\epsilon}$ depending on the dimension $n$ of $M$ and the $L\\_{\\infty}$-norm of the mean curvature $H$, so that if the $L\\_{2p}$-norm $\\|H\\|\\_{2p}$ ($p\\geq 2$) of $H$ satisfies $n\\|H\\|\\_{2p}-C\\_{\\epsilon}<\\lambda$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/\\lambda)^{1/2}$ is smaller than $\\epsilon$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L\\_{\\infty}$-norm of the second fundamental form, then the pinching condition $n\\|H\\|\\_{2p}-C<\\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-18T12:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A07",
      "53C21"
    ],
    "keywords": [
      "euclidean space",
      "pinching theorem",
      "first eigenvalue",
      "first nonzero eigenvalue",
      "second fundamental form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9494C"
    }
  }
}