{
  "id": "math/0701548",
  "version": "v2",
  "published": "2007-01-19T16:59:39.000Z",
  "updated": "2007-01-26T16:16:42.000Z",
  "title": "Eigenvalues and energy functionals with monotonicity formulae under Ricci flow",
  "authors": [
    "Junfang Li"
  ],
  "comment": "19 pages, one reference added, to appear in Mathematische Annalen",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In this note, we construct families of functionals of the type of $\\mathcal{F}$-functional and $\\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend X. Cao's methods of eigenvalues\\cite{C} and improve their results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-26T16:16:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "35K55"
    ],
    "keywords": [
      "ricci flow",
      "energy functionals",
      "monotonicity formulae",
      "compact steady ricci breathers",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1548L"
    }
  }
}