{
  "id": "2308.02777",
  "version": "v1",
  "published": "2023-08-05T03:10:34.000Z",
  "updated": "2023-08-05T03:10:34.000Z",
  "title": "On some rigidity theorems of Q-curvature",
  "authors": [
    "Yiyan Xu",
    "Shihong Zhang"
  ],
  "comment": "23 pages, accepted by manuscripta mathematica",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\\geq6$) Riemannian manifold $(M,g)$ and prove the following results under the condition $\\int_{M} \\nabla R\\cdot\\nabla \\mathrm{Q}\\mathrm{d} V_g\\leq0$. (1) If $(M,g)$ is locally conformally flat with nonnegative Ricci curvature, then $(M,g)$ is isometric to a quotient of $\\mathbb{R}^n$, $\\mathbb{S}^n$, or $\\mathbb{R}\\times\\mathbb{S}^{n-1}$. (2) If $(M,g)$ has $\\delta^2 W=0$ with nonnegative sectional curvature, then $(M,g)$ is isometric to a quotient of the product of Einstein manifolds. Additionally, we investigate some rigidity theorems involving Q-curvature about hypersurfaces in simply-connected space forms. We also show the uniqueness of metrics with constant scalar curvature and constant Q-curvature in a fixed conformal class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-05T03:10:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C24",
      "53C18"
    ],
    "keywords": [
      "rigidity theorems",
      "constant scalar curvature",
      "einstein manifolds",
      "riemannian manifold",
      "nonnegative sectional curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}