{
  "id": "2304.04663",
  "version": "v1",
  "published": "2023-04-10T15:40:01.000Z",
  "updated": "2023-04-10T15:40:01.000Z",
  "title": "Kazdan-Warner Problem on Compact Riemann Surfaces with Smooth Boundary",
  "authors": [
    "Jie Xu"
  ],
  "comment": "15 Pages, all comments are welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary $ (M, \\partial M, g) $ can be realized as a Gaussian curvature function; (ii) any smooth function on $ \\partial M $ can be realized as a geodesic curvature function for some metric $ \\tilde{g} \\in [g] $. The essential steps are the existence results of Brezis-Merle type equations $ -\\Delta_{g} u + Au = K e^{2u} \\; {\\rm in} \\; M $ and $ \\frac{\\partial u}{\\partial \\nu} + \\kappa u = \\sigma e^{u} \\; {\\rm on} \\; \\partial M $ with given functions $ K, \\sigma $ and some constants $ A, \\kappa $. In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-10T15:40:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "35J60",
      "53C18"
    ],
    "keywords": [
      "compact riemann surface",
      "kazdan-warner problem",
      "smooth function",
      "gaussian curvature function",
      "brezis-merle type equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}