{
  "id": "math/0603525",
  "version": "v2",
  "published": "2006-03-22T00:35:47.000Z",
  "updated": "2006-03-23T16:00:00.000Z",
  "title": "Eternal Solutions to the Ricci Flow on $\\R^2$",
  "authors": [
    "Panagiota Daskalopoulos",
    "Natasa Sesum"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation $ \\frac{\\partial u}{\\partial t} = \\Delta \\log u $ on $ \\R^2 \\times \\R.$ We show that, under the necessary assumption that for every $t \\in \\R$, the solution $u(\\cdot, t)$ defines a complete metric of bounded curvature and bounded width, $u$ is a gradient soliton of the form $ U(x,t) = \\frac{2}{\\beta (|x-x_0|^2 + \\delta e^{2\\beta t})}$, for some $x_0 \\in \\R^2$ and some constants $\\beta >0$ and $\\delta >0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-03-23T16:00:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60"
    ],
    "keywords": [
      "eternal solutions",
      "fast diffusion equation",
      "two-dimensional ricci flow",
      "necessary assumption",
      "complete metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3525D"
    }
  }
}