{
  "id": "1808.01685",
  "version": "v1",
  "published": "2018-08-05T21:07:06.000Z",
  "updated": "2018-08-05T21:07:06.000Z",
  "title": "Sharp estimate of global Coulomb gauge",
  "authors": [
    "Yu Wang"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $A$ be a $W^{1,2}$-connection on a principle $\\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\\|F_A\\|_{L^2(M)}\\le \\Lambda$. Our main result is the existence of a global section $\\sigma: M\\to P$ with finite singularities on $M$ such that the connection form $\\sigma^*A$ satisfies the Coulomb equation $d^*(\\sigma^*A)=0$ and admits a sharp estimate $\\|\\sigma^*A\\|_{\\mathcal{L}^{4,\\infty}(M)}\\le C(M,\\Lambda)$. Here $\\mathcal{L}^{4,\\infty}$ is a new function space we introduce in this paper that satisfies $L^4(M)\\subsetneq \\mathcal{L}^{4,\\infty}(M)\\subsetneq L^{4-\\epsilon}(M)$ for all $\\epsilon>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-05T21:07:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global coulomb gauge",
      "sharp estimate",
      "function space",
      "coulomb equation",
      "connection form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}