{
  "abstract": "We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection $\\pm1$. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is $C^{14}$-open. It is nonempty if and only if the manifold contains a sphere of self-intersection $\\pm1$. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct $C^{14}$-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard $k\\mathbb{CP}^2\\#\\ell\\overline{\\mathbb{CP}}{}^2$, through finitely many blowdown singularities. (2) On the standard $2\\mathbb{CP}^{2}\\#9\\overline{\\mathbb{CP}}{}^{2}$, there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00139",
  "author_total": 2,
  "authors": [
    {
      "name": "Dain Kim"
    },
    {
      "name": "Tristan Ozuch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/95gsaqr6dk9j8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00139",
  "fulltext_url": null,
  "id": "95gsaqr6dk9j8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00139",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00139",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/95gsaqr6dk9j8.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00139",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00139"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Blowdowns along Ricci flow in dimension four",
  "title_text": "Blowdowns along Ricci flow in dimension four",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/95gsaqr6dk9j8",
  "venue": "arXiv",
  "versions": []
}