{
  "id": "2211.05197",
  "version": "v1",
  "published": "2022-11-09T21:08:03.000Z",
  "updated": "2022-11-09T21:08:03.000Z",
  "title": "Flows of geometric structures",
  "authors": [
    "Daniel Fadel",
    "Eric Loubeau",
    "Andrés J. Moreno",
    "Henrique N. Sá Earp"
  ],
  "comment": "63 pages, no figures. Comments are welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H \\subset \\mathrm{SO}(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\\mathrm{GL}(n,\\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen--Struwe \\cites{Struwe1988,Struwe1989} for the harmonic map heat flow. This yields an $\\epsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, with respect to the $L^\\infty$-norm of initial torsion, in the spirit of Chen--Ding \\cite{Chen-Ding1990}. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $\\pi_n(\\mathrm{SO}(n)/H)\\neq\\{1\\}$; e.g. when $n=7$ and $H=\\rm G_2$, or $n=8$ and $H=\\mathrm{Spin}(7)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-09T21:08:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E20",
      "53C43",
      "53C25",
      "53C15",
      "53C10"
    ],
    "keywords": [
      "geometric structures",
      "general first variation formula",
      "harmonic map heat flow",
      "initial homotopy class imposes",
      "dirichlet energy functional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}