{
  "id": "2106.09220",
  "version": "v1",
  "published": "2021-06-17T02:58:48.000Z",
  "updated": "2021-06-17T02:58:48.000Z",
  "title": "Infinite-time blowing-up solutions to small perturbations of the Yamabe flow",
  "authors": [
    "Seunghyeok Kim",
    "Monica Musso"
  ],
  "comment": "53 pages, Comments are welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension $N \\ge 3$ is known to exist for all time $t$ and converges to a solution to the Yamabe problem as $t \\to \\infty$. We prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the linear term of the Yamabe flow on any given Riemannian manifold $M$ of dimension $N \\ge 5$, the resulting flow may blow-up in the infinite time, forming singularities each of which looks like a solution of the Yamabe problem on the unit sphere $\\mathbb{S}^N$. This shows that the Yamabe flow is an equation at the borderline guaranteeing the global existence and uniformly boundness of solutions. In addition, we examine the stability of the blow-up phenomena, imposing some negativity condition on the Ricci curvature at blow-up points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-17T02:58:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "yamabe flow",
      "infinite-time blowing-up solutions",
      "small perturbations",
      "yamabe problem",
      "smooth compact riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}