{
  "id": "2505.02315",
  "version": "v1",
  "published": "2025-05-05T02:20:12.000Z",
  "updated": "2025-05-05T02:20:12.000Z",
  "title": "Four-dimensional shrinkers with nonnegative Ricci curvature",
  "authors": [
    "Guoqiang Wu",
    "Jia-yong Wu"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In this paper, we investigate classifications of $4$-dimensional simply connected complete noncompact nonflat shrinkers satisfying $Ric+\\mathrm{Hess}\\,f=\\tfrac 12g$ with nonnegative Ricci curvature. One one hand, we show that if the sectional curvature $K\\le 1/4$ or the sum of smallest two eigenvalues of Ricci curvature has a suitable lower bound, then the shrinker is isometric to $\\mathbb{R}\\times\\mathbb{S}^3$. We also show that if the scalar curvature $R\\le 3$ and the shrinker is asymptotic to $\\mathbb{R}\\times\\mathbb{S}^3$, then the Euler characteristic $\\chi(M)\\geq 0$ and equality holds if and only if the shrinker is isometric to $\\mathbb{R}\\times\\mathbb{S}^3$. On the other hand, we prove that if $K\\le 1/2$ (or the bi-Ricci curvature is nonnegative) and $R\\le\\tfrac{3}{2}-\\delta$ for some $\\delta\\in (0,\\tfrac{1}{2}]$, then the shrinker is isometric to $\\mathbb{R}^2\\times\\mathbb{S}^2$. The proof of these classifications mainly depends on the asymptotic analysis by the evolution of eigenvalues of Ricci curvature, the Gauss-Bonnet-Chern formula with boundary and the integration by parts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-05T02:20:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative ricci curvature",
      "four-dimensional shrinkers",
      "simply connected complete noncompact",
      "noncompact nonflat shrinkers satisfying",
      "connected complete noncompact nonflat shrinkers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}