{
  "id": "2309.04706",
  "version": "v1",
  "published": "2023-09-09T07:15:17.000Z",
  "updated": "2023-09-09T07:15:17.000Z",
  "title": "Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation",
  "authors": [
    "Xuezhang Chen",
    "Shihong Zhang"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from $0$, which serves as an intermediate one between Chang-Hang's inequalities under first and second order moments constraints. A threshold for the deviation is a uniqueness criterion for the mean field equation $$-a\\Delta_{\\mathbb{S}^2}u+1=e^{2u} \\quad \\mathrm{~~on~~} \\quad \\mathbb{S}^2$$ when the constant $a$ is close to $\\frac{1}{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-09T07:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean field equation",
      "constrained moser-trudinger-onofri inequality",
      "uniqueness criterion",
      "second order moments constraints",
      "establish moser-trudinger-onofri inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}