{
  "id": "1209.1285",
  "version": "v2",
  "published": "2012-09-06T14:10:57.000Z",
  "updated": "2016-06-03T11:14:25.000Z",
  "title": "$n$-harmonic coordinates and the regularity of conformal mappings",
  "authors": [
    "Tony Liimatainen",
    "Mikko Salo"
  ],
  "comment": "20 pages, updated reference",
  "journal": "Mathematical Research Letters 21, No. 2, (2014), 341-361",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \\infty$ on any Riemannian manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-06T14:10:57.000Z",
      "abstract": "This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal diffeomorphism. This result is due to Iwaniec \\cite{Iwaniec_thesis}, but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \\infty$ on any Riemannian manifold.",
      "comment": "20 pages",
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-06-03T11:14:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A30",
      "35J60",
      "35B65"
    ],
    "keywords": [
      "regularity",
      "riemannian manifold",
      "metric tensors",
      "standard harmonic coordinates",
      "harmonic coordinate system"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.1285L"
    }
  }
}