{
  "id": "0801.2729",
  "version": "v1",
  "published": "2008-01-17T17:35:53.000Z",
  "updated": "2008-01-17T17:35:53.000Z",
  "title": "Classification of solutions to the higher order Liouville's equation on R^{2m}",
  "authors": [
    "Luca Martinazzi"
  ],
  "journal": "Math. Z. (2009)",
  "doi": "10.1007/s00209-008-0419-1",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We classify the solutions to the equation (- \\Delta)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of \\Delta u(x) as |x|\\to \\infty. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and we observe that the pull-back of this metric to $S^{2m}$ via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-17T17:35:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher order liouvilles equation",
      "classification",
      "smooth riemannian metric",
      "finite total",
      "stereographic projection"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}