{
  "id": "math/0404326",
  "version": "v2",
  "published": "2004-04-19T03:03:45.000Z",
  "updated": "2010-02-08T04:45:01.000Z",
  "title": "Convex solutions to the mean curvature flow",
  "authors": [
    "Xu-Jia Wang"
  ],
  "comment": "47 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\\R^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\\le k\\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\\times \\R^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist non-rotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is $k$-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space $\\R^{n+1}$ is a shrinking sphere or cylinder.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-08T04:45:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "35J60"
    ],
    "keywords": [
      "mean curvature flow",
      "entire convex translating solution",
      "ancient convex solution",
      "rotationally symmetric",
      "high dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4326W"
    }
  }
}