{
  "id": "2403.19169",
  "version": "v1",
  "published": "2024-03-28T06:32:42.000Z",
  "updated": "2024-03-28T06:32:42.000Z",
  "title": "Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature",
  "authors": [
    "Hongyi Sheng"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T06:32:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scalar curvature",
      "mean curvature",
      "static manifolds",
      "arbitrary riemannian manifold",
      "classify simple non-generic domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}