{
  "id": "2302.13068",
  "version": "v1",
  "published": "2023-02-25T12:09:44.000Z",
  "updated": "2023-02-25T12:09:44.000Z",
  "title": "Classifying solutions of ${\\rm SU}(n+1)$ Toda system around a singular source via Fuchsian equations",
  "authors": [
    "Jingyu Mu",
    "Yiqian Shi",
    "Tianyang Sun",
    "Bin Xu"
  ],
  "categories": [
    "math.AP",
    "math.CV",
    "math.DG"
  ],
  "abstract": "Let $n$ be a positive integer, $\\gamma_1>-1,\\cdots,\\gamma_n>-1$, $D=\\{z\\in {\\Bbb C}:|z|<1\\}$, and $(a_{ij})_{n\\times n}$ be the Cartan matrix of $\\frak{su}(n+1)$. By using the Fuchsian equation of $(n+1)$th order around a singular source of ${\\rm SU}(n+1)$ Toda system discovered by Lin-Wei-Ye ($\\textit{Invent Math}$, $\\textbf{190}$(1):169-207, 2012), we describe precisely a solution $(u_1,\\cdots, u_n)$ to the ${\\rm SU}(n+1)$ Toda system \\begin{equation*} \\begin{cases} \\frac{\\partial^2 u_i}{\\partial z\\partial \\bar z}+\\sum_{j=1}^n a_{ij} e^{u_j}&=\\pi \\gamma _i\\delta _0\\,\\,{\\rm on}\\,\\, D\\\\ \\frac{\\sqrt{-1}}{2}\\,\\int_{D\\backslash \\{0\\}} e^{u_{i} }{\\rm d}z\\wedge {\\rm d}\\bar z &< \\infty \\end{cases} \\quad \\text{for all}\\quad i=1,\\cdots, n \\end{equation*} in terms of some $(n+1)$ holomorphic functions satisfying the normalized condition. Moreover, we show that for each $1\\leq i\\leq n$, $0$ is the cone singularity with angle $2\\pi(1+\\gamma_i)$ for metric $e^{u_i}|{\\rm d}z|^2$ on $D\\backslash\\{0\\}$, whose restriction near $0$ could be characterized by some $(n-1)$ holomorphic functions non-vanishing at $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-25T12:09:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "toda system",
      "singular source",
      "fuchsian equation",
      "classifying solutions",
      "th order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}