{
  "id": "2307.13942",
  "version": "v1",
  "published": "2023-07-26T03:47:54.000Z",
  "updated": "2023-07-26T03:47:54.000Z",
  "title": "The $σ_{2}$-curvature equation on a compact manifold with boundary",
  "authors": [
    "Xuezhang Chen",
    "Wei Wei"
  ],
  "comment": "102 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We first establish local $C^2$ estimates of solutions to the $\\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the background metric has nonnegative mean curvature on totally non-umbilic boundary, for dimensions three and four we prove the existence of a conformal metric with a prescribed positive $\\sigma_2$-curvature function and a prescribed nonnegative boundary mean curvature function. The local estimates play an important role in the blow up analysis in the latter existence result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-26T03:47:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "curvature equation",
      "compact manifold",
      "nonnegative boundary mean curvature function",
      "nonlinear neumann boundary condition",
      "prescribed nonnegative boundary mean curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 102,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}