{
  "id": "2006.03038",
  "version": "v1",
  "published": "2020-06-04T17:49:01.000Z",
  "updated": "2020-06-04T17:49:01.000Z",
  "title": "Generic scarring for minimal hypersurfaces along stable hypersurfaces",
  "authors": [
    "Antoine Song",
    "Xin Zhou"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $M^{n+1}$ be a closed manifold of dimension $3\\leq n+1\\leq 7$. We show that for a $C^\\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\\{\\Sigma_k\\}$ scarring along $S$, in the sense that the area and Morse index of $\\Sigma_k$ both diverge to infinity and, when properly renormalized, $\\Sigma_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-04T17:49:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimal hypersurface",
      "stable hypersurfaces",
      "generic scarring",
      "generic metric",
      "morse index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}