{
  "id": "1308.4148",
  "version": "v2",
  "published": "2013-08-19T20:00:09.000Z",
  "updated": "2014-04-28T13:30:48.000Z",
  "title": "Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D",
  "authors": [
    "Paul M. Alsing",
    "Warner A. Miller",
    "Matthew Corne",
    "Xianfeng Gu",
    "Seth Lloyd",
    "Shannon Ray",
    "Shing-Tung Yau"
  ],
  "comment": "15 pages, 7 figures, submitted to Geometry, Imaging and Computation, minor revisions",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The 2-sphere cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice well-centered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-28T13:30:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "58B20"
    ],
    "keywords": [
      "neck pinch singularity",
      "simplicial ricci flow",
      "low-resolution discrete geometry yield",
      "axisymmetric piecewise flat geometry",
      "srf equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1249871,
      "adsabs": "2013arXiv1308.4148A"
    }
  }
}