{
  "id": "1304.6356",
  "version": "v2",
  "published": "2013-04-23T17:28:17.000Z",
  "updated": "2015-02-12T13:43:11.000Z",
  "title": "Rigidity of generic singularities of mean curvature flow",
  "authors": [
    "Tobias Holck Colding",
    "Tom Ilmanen",
    "William P. Minicozzi II"
  ],
  "comment": "revised after acceptance for Publications IHES",
  "categories": [
    "math.DG",
    "math.AP",
    "math.GT"
  ],
  "abstract": "Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, [CM1] showed that the only generic are round cylinders $\\SS^k\\times \\RR^{n-k}$. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows. Our results hold in all dimensions and do not require any a priori smoothness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-23T17:28:17.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-12T13:43:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean curvature flow",
      "generic singularities",
      "round cylinder",
      "first general rigidity theorem",
      "nonlinear geometric flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.6356H"
    }
  }
}