{
  "id": "1702.05118",
  "version": "v1",
  "published": "2017-02-16T19:20:36.000Z",
  "updated": "2017-02-16T19:20:36.000Z",
  "title": "Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow",
  "authors": [
    "Yongjia Zhang"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies kappa-noncollapsing on all scales. This provides an evidence for Perelman's more general assertion that on a complete ancient solution with nonnegative curvature operator, bounded entropy is equivalent to kappa-noncollapsing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-16T19:20:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gap theorem",
      "ricci flow",
      "complete ancient solution",
      "nonnegative curvature operator",
      "finite asymptotic entropy implies kappa-noncollapsing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}