{
  "id": "1907.06072",
  "version": "v1",
  "published": "2019-07-13T13:07:38.000Z",
  "updated": "2019-07-13T13:07:38.000Z",
  "title": "Harmonic flow of geometric structures",
  "authors": [
    "Eric Loubeau",
    "Henrique N. Sá Earp"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by C. M. Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As an application, we recover the divergence-free torsion equation for ${\\rm G}_2$-structures proposed by S. Grigorian (2017). We study the corresponding evolution problem, which runs among isometric ${\\rm G}_2$-structures, recovering some analytic results independently established by L. Bagaglini (2017), S. Grigorian (2019) and Dwiveti-Gianniotis-Karigiannis (2019) in that context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-13T13:07:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E20",
      "53C43",
      "53C10"
    ],
    "keywords": [
      "geometric structures",
      "harmonic flow",
      "natural dirichlet energy induces",
      "divergence-free torsion equation",
      "abstract harmonicity condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}