{
  "id": "0903.2326",
  "version": "v1",
  "published": "2009-03-13T07:36:39.000Z",
  "updated": "2009-03-13T07:36:39.000Z",
  "title": "Denjoy-Ahlfors Theorem for Harmonic Functions on Riemannian Manifolds and External Structure of Minimal Surfaces",
  "authors": [
    "Vladimir M. Miklyukov",
    "Vladimir G. Tkachev"
  ],
  "comment": "30 pages",
  "journal": "Comm. Anal. Geom., 4 (1996), no. 4, 547--587",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-13T07:36:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53A10",
      "30F15"
    ],
    "keywords": [
      "minimal surface",
      "external structure",
      "geodesic completeness requirement",
      "arbitrary riemannian manifolds",
      "well-known denjoy-ahlfors theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2326M"
    }
  }
}