{
  "id": "1009.3573",
  "version": "v3",
  "published": "2010-09-18T17:34:47.000Z",
  "updated": "2010-11-17T16:15:47.000Z",
  "title": "Lower bounds on the Hausdorff measure of nodal sets",
  "authors": [
    "Christopher D. Sogge",
    "Steve Zelditch"
  ],
  "comment": "Added detail to exposition (especially Proposition 1) and added references to recent results of Colding-Minicozzi and of Mangoubi. To appear in MRL",
  "journal": "Math. Res. Lett. 18 (2011), no. 1, 25-37",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\ncal_{\\phi_{\\lambda}}$ be the nodal hypersurface of a $\\Delta$-eigenfunction $\\phi_{\\lambda}$ of eigenvalue $\\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\\hcal^{n-1}(\\ncal_{\\phi_{\\lambda}}) \\geq C \\lambda^{\\frac74-\\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \\lambda}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-11-17T16:15:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nodal sets",
      "hausdorff measure",
      "best prior lower bound appears",
      "smooth riemannian manifold",
      "nodal hypersurface"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.3573S"
    }
  }
}