{
  "id": "1006.0540",
  "version": "v1",
  "published": "2010-06-03T02:23:26.000Z",
  "updated": "2010-06-03T02:23:26.000Z",
  "title": "The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow",
  "authors": [
    "Xiaodong Cao",
    "Qi S. Zhang"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $\\kappa$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $\\kappa$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-03T02:23:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44"
    ],
    "keywords": [
      "conjugate heat equation",
      "ricci flow",
      "ancient solutions",
      "fundamental solution",
      "non-flat gradient shrinking ricci soliton"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.0540C"
    }
  }
}