{
  "id": "1111.0390",
  "version": "v1",
  "published": "2011-11-02T05:36:02.000Z",
  "updated": "2011-11-02T05:36:02.000Z",
  "title": "Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources",
  "authors": [
    "Chang-Shou Lin",
    "Dong Ye",
    "Juncheng Wei"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider the following Toda system \\Delta u_i + \\D \\sum_{j = 1}^n a_{ij}e^{u_j} = 4\\pi\\gamma_{i}\\delta_{0} \\text{in}\\mathbb R^2, \\int_{\\mathbb R^2}e^{u_i} dx < \\infty, \\forall 1\\leq i \\leq n, where $\\gamma_{i} > -1$, $\\delta_0$ is Dirac measure at 0, and the coefficients $a_{ij}$ form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: $$\\sum_{j=1}^n a_{ij}\\int_{\\R^2}e^{u_j} dx = 4\\pi (2+\\gamma_i+\\gamma_{n+1-i}), \\;\\;\\forall\\; 1\\leq i \\leq n.$$ This generalizes the classification result by Jost and Wang for $\\gamma_i=0$, $\\forall \\;1\\leq i\\leq n$. (ii) We prove that if $\\gamma_i+\\gamma_{i+1}+...+\\gamma_j \\notin \\mathbb Z$ for all $1\\leq i\\leq j\\leq n$, then any solution $u_i$ is \\textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \\textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-02T05:36:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B20"
    ],
    "keywords": [
      "toda system",
      "singular sources",
      "nondegeneracy",
      "standard tri-diagonal cartan matrix",
      "dirac measure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00222-012-0378-3",
      "journal": "Inventiones Mathematicae",
      "year": 2012,
      "month": "Oct",
      "volume": 190,
      "number": 1,
      "pages": 169
    },
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012InMat.190..169L"
    }
  }
}