{
  "id": "1011.2728",
  "version": "v3",
  "published": "2010-11-11T18:10:16.000Z",
  "updated": "2012-05-03T17:13:34.000Z",
  "title": "The energy of a smooth metric measure space and applications",
  "authors": [
    "Jeffrey S. Case"
  ],
  "comment": "44 pages; rewritten to use the more standard language of smooth metric measure spaces, and corrected some errors in the appendix",
  "categories": [
    "math.DG"
  ],
  "abstract": "We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $\\nu$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $\\kappa$-noncollapsing property. Finally, we use the energy to prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, in the spirit of similar results for Einstein metrics and gradient Ricci solitons.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-03T17:13:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C25"
    ],
    "keywords": [
      "quasi-einstein smooth metric measure spaces",
      "compact quasi-einstein smooth metric measure",
      "applications",
      "gradient ricci solitons",
      "energy shares"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.2728C"
    }
  }
}