{
  "id": "1201.1266",
  "version": "v2",
  "published": "2012-01-05T19:08:44.000Z",
  "updated": "2013-01-28T21:03:33.000Z",
  "title": "Collapsing in the $L^2$ curvature flow",
  "authors": [
    "Jeff Streets"
  ],
  "comment": "to appear in Comm. PDE",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We show some results for the $L^2$ curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for $SO(3)$-invariant initial data on $S^3$, as well as a long time existence and convergence statement for three-manifolds with initial $L^2$ norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension $n = 4$ we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension $n \\geq 5$, and show examples of finite time singularities in dimension $n \\geq 6$ which are collapsed on the scale of curvature.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-28T21:03:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "curvature flow",
      "long time existence",
      "finite time singularities",
      "curvature chosen small",
      "related low-energy convergence statement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}