{
  "id": "1210.4089",
  "version": "v2",
  "published": "2012-10-15T16:10:44.000Z",
  "updated": "2015-09-19T19:02:11.000Z",
  "title": "On type-II singularities in Ricci flow on $\\mathbb{R}^{N}$",
  "authors": [
    "Haotian Wu"
  ],
  "comment": "Revised version, typos corrected",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In each dimension $N\\geq 3$ and for each real number $\\lambda\\geq 1$, we construct a family of complete rotationally symmetric solutions to Ricci flow on $\\mathbb{R}^{N}$ which encounter a global singularity at a finite time $T$. The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate $(T-t)^{-(\\lambda+1)}$. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on $\\mathbb{R}^N$ whose blow-ups near the origin converge uniformly to the Bryant soliton.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-15T16:10:44.000Z",
      "title": "On type-$\\II$ singularities in Ricci flow on $\\RR^{n+1}$",
      "abstract": "For $n+1\\geq 3$, we construct complete solutions to Ricci flow on $\\RR^{n+1}$ which encounter global singularities at a finite time $T$. The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate $(T-t)^{-2\\lambda}$ for $\\lambda\\geq1$. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on $\\RR^{n+1}$ whose blow-ups near the origin converge uniformly to the Bryant soliton.",
      "comment": "26 pages, no figure. Comments are welcome!",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-19T19:02:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "35K59"
    ],
    "keywords": [
      "ricci flow",
      "bryant soliton",
      "solution converge",
      "construct complete solutions",
      "encounter global singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4089W"
    }
  }
}