{
  "id": "0903.1018",
  "version": "v2",
  "published": "2009-03-05T15:07:32.000Z",
  "updated": "2009-07-06T04:50:03.000Z",
  "title": "Boundaries of Graphs of Harmonic Functions",
  "authors": [
    "Daniel Fox"
  ],
  "journal": "SIGMA 5 (2009), 068, 8 pages",
  "doi": "10.3842/SIGMA.2009.068",
  "categories": [
    "math.DG",
    "math.CA"
  ],
  "abstract": "Harmonic functions $u:{\\mathbb R}^n \\to {\\mathbb R}^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,{\\mathcal I},\\omega)$. To this system one associates the space of conservation laws ${\\mathcal C}$. They provide necessary conditions for $g:{\\mathbb S}^{n-1} \\to M$ to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary $g({\\mathbb S}^{n-1})$. The proof uses standard linear elliptic theory to produce an integral manifold $G:D^n \\to M$ and the completeness of the space of conservation laws to show that this candidate has $g({\\mathbb S}^{n-1})$ as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in ${\\mathbb C}^m$ in the local case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-07-06T04:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "35J25",
      "53B25"
    ],
    "keywords": [
      "harmonic functions",
      "integral manifold",
      "conservation laws",
      "standard linear elliptic theory",
      "exterior differential system"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "SIGMA",
      "year": 2009,
      "month": "Jul",
      "volume": 5,
      "pages": "068"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009SIGMA...5..068F"
    }
  }
}