{
  "id": "0911.0375",
  "version": "v2",
  "published": "2009-11-02T18:15:12.000Z",
  "updated": "2009-11-24T16:22:58.000Z",
  "title": "On the prescribing $σ_2$ curvature equation on $\\mathbb S^4$",
  "authors": [
    "S. -Y. Alice Chang",
    "Zheng-Chao Han",
    "Paul Yang"
  ],
  "comment": "39 pages. Statement and proof of Corollary 1 have been revised. A remark added on p. 34, and a typo corrected two lines below equation (68) on p.32",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Prescribing $\\sigma_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $\\sigma_2$ curvature equation with the given $K$; and rule out the possibility of blowing up solutions when $K$ satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of $\\sigma_2$ curvature equations deforming $K$ to a positive constant under the same non-degeneracy condition on $K$, and prove the existence of a solution using degree argument to this deformation involving fully nonlinear elliptic operators under an additional, natural degree condition on a finite dimensional map associated with $K$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-24T16:22:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "53A30",
      "35J60",
      "35B45"
    ],
    "keywords": [
      "non-degeneracy condition",
      "prescribing",
      "scalar curvature equations",
      "fully nonlinear elliptic operators",
      "finite dimensional map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.0375C"
    }
  }
}