{
  "id": "0806.2080",
  "version": "v1",
  "published": "2008-06-12T13:54:27.000Z",
  "updated": "2008-06-12T13:54:27.000Z",
  "title": "$C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\\R^n$",
  "authors": [
    "Guy David"
  ],
  "comment": "115 pages, 4 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\\R^3$ are locally $C^{1+\\alpha}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-12T13:54:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K99",
      "49Q20"
    ],
    "keywords": [
      "two-dimensional almost-minimal sets",
      "local separation result",
      "minimal cones",
      "reifenbergs parameterization theorem",
      "almgren almost-minimal sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 115,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2080D"
    }
  }
}