{
  "id": "2005.10688",
  "version": "v1",
  "published": "2020-05-21T14:30:21.000Z",
  "updated": "2020-05-21T14:30:21.000Z",
  "title": "Self-similar solutions to the mean curvature flow in $\\mathbb{R}^{3}$",
  "authors": [
    "Benedito Leandro",
    "Rafael Novais",
    "Hiuri F. S. dos Reis"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper we made an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in $\\mathbb{R}^{3}$ are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under an helicoidal motion in $\\mathbb{R}^{3}$ in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under an helicoidal motion in $\\mathbb{R}^3$. Explicit families of exact solutions for the MCF by cylindrical surfaces in $\\mathbb{R}^{3}$ are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-21T14:30:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar solutions",
      "mean curvature flow",
      "helicoidal motion",
      "cylindrical surfaces",
      "revolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}