{
  "id": "2109.06340",
  "version": "v1",
  "published": "2021-09-13T21:46:47.000Z",
  "updated": "2021-09-13T21:46:47.000Z",
  "title": "Harmonic flow of $\\mathrm{Spin}(7)$-structures",
  "authors": [
    "Shubham Dwivedi",
    "Eric Loubeau",
    "Henrique N. Sá Earp"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We formulate and study the isometric flow of $\\mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=\\mathrm{Spin}(7)\\subset\\mathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $\\varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$\\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $\\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T21:46:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C15",
      "53C43",
      "58J35",
      "58J60"
    ],
    "keywords": [
      "harmonic flow",
      "arbitrary isometric flows",
      "hamilton-type compactness theorems",
      "harmonic solitons",
      "self-similar solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}