{
  "id": "2106.04504",
  "version": "v1",
  "published": "2021-06-08T16:44:39.000Z",
  "updated": "2021-06-08T16:44:39.000Z",
  "title": "The axisymmetric $σ_k$-Nirenberg problem",
  "authors": [
    "Yanyan Li",
    "Luc Nguyen",
    "Bo Wang"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We study the problem of prescribing $\\sigma_k$-curvature for a conformal metric on the standard sphere $\\mathbb{S}^n$ with $2 \\leq k < n/2$ and $n \\geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $\\beta \\in [2,n)$. We prove among other things the following statements. (1) When $\\beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ \\beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $\\frac{2}{n-2k}\\le \\beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $\\beta < \\frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $\\beta$, there exists also some $K$ for which the solution set is empty.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T16:44:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "solution set",
      "nirenberg problem",
      "nonzero total degree counting",
      "south poles",
      "axisymmetric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}