{
  "abstract": "Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\\geqslant 3$. We prove short--time existence and uniqueness for the \"critical\" {\\em Ricci--Bourguignon flow} \\begin{equation} \\partial_t g=-2\\operatorname{Ric}_g+\\frac{R_g}{n-1}g\\,. \\end{equation} Up to a constant rescaling of time, this is the {\\em Schouten flow}, since the right--hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual \"DeTurck modification\" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an \"extended\" system consisting of a strictly parabolic equation for the metric coupled to a transport--reaction equation for $r$ with a curvature--square forcing. The extended system belongs to the class of nonlinear composite parabolic--hyperbolic systems studied by Vol'pert and Khudyaev~\\cite{VolpertKhudyaev1972}. Then, a standard compact--manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar--curvature constraint \"propagates\" and the original critical Ricci--Bourguignon flow is recovered. This settles the short--time existence and uniqueness problem at the critical ``Schouten'' value $ρ=1/(2(n-1))$, which was left open by the previous Ricci--Bourguignon theory in~\\cite{CatinoEtAl2017}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06503",
  "author_total": 2,
  "authors": [
    {
      "name": "Giovanni Catino"
    },
    {
      "name": "Carlo Mantegazza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23qtx4xe1gpjt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06503",
  "fulltext_url": null,
  "id": "23qtx4xe1gpjt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06503",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7211996052",
  "pdf_url": "https://arxiv.org/pdf/2609.06503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23qtx4xe1gpjt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Short-time existence for the Schouten flow",
  "title_text": "Short-time existence for the Schouten flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/23qtx4xe1gpjt",
  "venue": "arXiv",
  "versions": []
}