{
  "id": "2309.15832",
  "version": "v1",
  "published": "2023-09-27T17:53:29.000Z",
  "updated": "2023-09-27T17:53:29.000Z",
  "title": "The relative $h$-principle for closed $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms",
  "authors": [
    "Laurence H. Mayther"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GT",
    "math.AT",
    "math.DG"
  ],
  "abstract": "This paper uses convex integration with avoidance and transversality arguments to prove the relative $h$-principle for closed $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms on oriented 6-manifolds. As corollaries, it is proven that if an oriented 6-manifold $\\mathrm{M}$ admits any $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-form, then every degree 3 cohomology class on $\\mathrm{M}$ can be represented by an $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-form and, moreover, that the corresponding Hitchin functional on $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms representing this class is necessarily unbounded above. Essential to the proof of the $h$-principle is a careful analysis of the rank 3 distributions induced by an $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-form and their interaction with generic pairs of hyperplanes. The proof also introduces a new property of sets in affine space, termed macilence, as a method of verifying ampleness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-27T17:53:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C10",
      "53C15",
      "15A69",
      "15A72",
      "57N75",
      "35R45",
      "58A30",
      "58A20"
    ],
    "keywords": [
      "transversality arguments",
      "cohomology class",
      "convex integration",
      "affine space",
      "generic pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}