{
  "id": "1905.01664",
  "version": "v1",
  "published": "2019-05-05T11:45:03.000Z",
  "updated": "2019-05-05T11:45:03.000Z",
  "title": "Recognizing shape via 1st eigenvalue, mean curvature and upper curvature bound",
  "authors": [
    "Yingxiang Hu",
    "Shicheng Xu"
  ],
  "comment": "35 pages,1 figure",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "abstract": "Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close to a geodesic sphere $S(p_0,R_0)$ in $N$, but also the ``enclosed'' ball $B(p_0,R_0)$ is close to be of constant curvature, provided with a uniform control on the volume and mean curvature of $M$. We raise a conjecture for $M$ to be a diffeomorphic sphere, and give positive partial answers for several special cases, one of which is as follows: If in addition, the renormalized $L^q$ norm ($q>n$) of $M$'s 2nd fundamental form admits a uniform upper bound, then $M$ is an embedded diffeomorphic sphere, almost isometric to $S(p_0, R_0)$, and intrinsically $C^{\\alpha}$-close to a round sphere of constant curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-05T11:45:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C21",
      "53C24"
    ],
    "keywords": [
      "mean curvature",
      "upper curvature bound",
      "1st eigenvalue",
      "recognizing shape",
      "constant curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}