{
  "id": "1807.02025",
  "version": "v1",
  "published": "2018-07-05T14:37:58.000Z",
  "updated": "2018-07-05T14:37:58.000Z",
  "title": "A note on singularities in finite time for the constrained Willmore flow",
  "authors": [
    "Simon Blatt"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional $\\mathcal W_{\\lambda_1, \\lambda_2}$, the sum of the Willmore energy, $\\lambda_1$ times the area, and $\\lambda_2$ times the signed volume of an immersed closed surface without boundary in $\\mathbb R^3$. We show that in the case that $\\lambda_1>1$ and $\\lambda_2=0$ any immersion develops singularities in finite time under this flow. If $\\lambda_1 >0$ and $\\lambda_2 > 0$, embedded closed surfaces with energy less than $$8\\pi+\\min\\{(16 \\pi \\lambda_1^3)/(3\\lambda_2^2), 8\\pi\\}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than $8 \\pi$, the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that $\\lambda_2$ is negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-05T14:37:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44"
    ],
    "keywords": [
      "finite time",
      "constrained willmore flow",
      "positive volume evolve singularities",
      "closed surface",
      "gradient flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}