{
  "id": "2301.08309",
  "version": "v1",
  "published": "2023-01-19T20:49:05.000Z",
  "updated": "2023-01-19T20:49:05.000Z",
  "title": "A singular Kazdan-Warner problem on a compact Riemann surface",
  "authors": [
    "Xiaobao Zhu"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Let $(M,g)$ be a compact Riemann surface with unit area, $h\\in C^{\\infty}(M)$ a function which is positive somewhere, $\\rho>0$, $p_i\\in M$ and $\\alpha_i\\in(-1,+\\infty)$ for $i=1,\\cdots,\\ell$, we consider the mean field equation \\begin{align*} \\Delta v + 4\\pi\\sum_{i=1}^{\\ell}\\alpha_i\\left(1-\\delta_{p_i}\\right) = \\rho\\left(1-\\frac{he^v}{\\int_Mhe^vd\\mu}\\right), \\end{align*} on $M$, where $\\Delta$ and $d\\mu$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $\\rho=8\\pi(1+\\min\\{0,\\min_{1\\leq i\\leq\\ell}\\alpha_i\\})$. These results can be seen as partial generalizations of works of Chen-Li (J. Geom. Anal. 1: 359--372, 1991), Ding-Jost-Li-Wang (Asian J. Math. 1: 230--248, 1997), Mancini (J. Geom. Anal. 26: 1202--1230, 2016), Yang-Zhu (Proc. Amer. Math. Soc. 145: 3953--3959, 2017), Sun-Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and $h$ is positive, this is the most important contribution of our paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-19T20:49:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact riemann surface",
      "singular kazdan-warner problem",
      "mean field equation",
      "unit area",
      "laplace-beltrami operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}