{
  "id": "1602.01939",
  "version": "v1",
  "published": "2016-02-05T07:43:04.000Z",
  "updated": "2016-02-05T07:43:04.000Z",
  "title": "Compactness and Shi-type estimate of the Ricci flow based on Ricci curvature",
  "authors": [
    "Chih-Wei Chen"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that a uniform local bound for the curvature operator can be derived from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. As a consequence, we obtain new compactness theorems for the Ricci flow and Ricci soliton without assuming any bounds on the curvature operator. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, a Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-05T07:43:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "58J05"
    ],
    "keywords": [
      "ricci curvature",
      "shi-type estimate",
      "curvature operator",
      "injectivity radius",
      "uniform local bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160201939C"
    }
  }
}