{
  "id": "2101.12681",
  "version": "v1",
  "published": "2021-01-29T16:51:32.000Z",
  "updated": "2021-01-29T16:51:32.000Z",
  "title": "Gradient Steady Ricci Solitons with Harmonic Weyl Curvature",
  "authors": [
    "Fengjiang Li"
  ],
  "comment": "27 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Our main aim in this paper is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\\geq 5$) complete noncompact gradient steady Ricci soliton with harmonic Weyl tensor is either Ricci flat or isometric to the Bryant soliton up to scaling. We also derive a classification result for complete noncompact gradient expanding Ricci solitons with harmonic Weyl tensor. Meanwhile, for $n\\ge 5$, we provide a local structure theorem for $n$-dimensional connected (not necessarily complete) gradient Ricci solitons with harmonic Weyl curvature, thus extending the work of Kim [31] for $n=4$. Furthermore, a similar method can be applied to treat vacuum static spaces and CPE metrics with harmonic curvature [32, 11], as well as quasi-Einstein manifolds with harmonic Weyl curvature [12].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-29T16:51:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic weyl curvature",
      "complete noncompact gradient steady ricci",
      "noncompact gradient steady ricci soliton",
      "harmonic weyl tensor",
      "gradient expanding ricci"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}