{
  "id": "2207.05587",
  "version": "v1",
  "published": "2022-07-12T15:10:00.000Z",
  "updated": "2022-07-12T15:10:00.000Z",
  "title": "Rigidity results on Liouville equation",
  "authors": [
    "Alexandre Eremenko",
    "Changfeng Gui",
    "Qinfeng Li",
    "Lu Xu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We give a complete classification of solutions bounded from above of the Liouville equation $$-\\Delta u=e^{2u}\\quad\\mbox{in}\\quad \\R^2.$$ More generally, solutions in the class $$N:=\\{ u:\\limsup_{z\\to\\infty} u(z)/\\log|z|:=k(u)<\\infty\\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\\to-\\infty$ as $z\\to\\infty$, if and only if $u$ is radial about some point. Third, if $u$ is strictly decreasing along any radial direction, then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of $\\R^2$ with the metric $e^{2u}\\delta$ is $\\pi$, where $\\delta$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition we also extend the concavity rigidity result on Liouville equation in higher dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-12T15:10:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville equation",
      "concavity rigidity result",
      "higher dimensions",
      "complete classification",
      "euclidean metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}