{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Sigma$ be a closed oriented surface and let $T \u003e 0$.\nLet $f \\colon \\Sigma \\times [0, T) \\to \\R^3$ be a smooth family of immersions, and write $f_t = f(\\cdot, t)$.\nFor each $t$, let $\\nu$ be the unit normal of $f_t$.\nLet $H = \\tfrac{1}{2}(\\kappa_1 + \\kappa_2)$ be the [[mean curvature|PA78NFEPCF]] of $f_t$ with respect to $\\nu$, so that the mean curvature vector is $2H\\nu$.\nLet $K$ be the [[Gaussian curvature|SQVQ5DQT6W]] of $f_t$.\nLet $\\Delta$ be the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the metric induced by $f_t$.\nThe family is a solution of the *Willmore flow* if\n$$\\frac{\\partial f}{\\partial t} = -\\left(\\Delta H + 2H\\left(H^2 - K\\right)\\right) \\nu.$$\n\n## Remarks\n\nThe right side is minus the $L^2$ gradient of the [[Willmore energy|R7WFJGJBV4]] $W(f_t) = \\int_\\Sigma H^2 \\, dA$, so $W(f_t)$ does not increase.\nExample: the constant family of a round sphere of radius $r$ is a solution.\nIndeed $H = \\pm 1/r$ is constant, so $\\Delta H = 0$, and $H^2 - K = 1/r^2 - 1/r^2 = 0$, so the right side vanishes.\nNon-example: the spheres of radius $1 + t$ do not form a solution, since the right side vanishes for each of them but the surfaces move.\nA surface for which $\\Delta H + 2H(H^2 - K) = 0$ is a Willmore surface; these are exactly the stationary solutions.\nThe equation is of fourth order and parabolic, so the maximum principle fails and convexity need not be preserved.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "MSRAD0X7H4",
    "kind": "definition",
    "msc": [
      "53E40",
      "53A05"
    ],
    "name": "Willmore flow",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Willmore_energy"
      }
    ],
    "tag": "CMK92XDD0W",
    "uses": [
      "R7WFJGJBV4",
      "PA78NFEPCF",
      "SQVQ5DQT6W",
      "F4NWZNTRMD"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CMK92XDD0W/",
  "links": [
    {
      "field": "defines",
      "target": "MSRAD0X7H4"
    },
    {
      "field": "uses",
      "target": "R7WFJGJBV4"
    },
    {
      "field": "uses",
      "target": "PA78NFEPCF"
    },
    {
      "field": "uses",
      "target": "SQVQ5DQT6W"
    },
    {
      "field": "uses",
      "target": "F4NWZNTRMD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CMK92XDD0W.md",
  "msc": [
    "53E40",
    "53A05"
  ],
  "name": "Willmore flow",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CMK/CMK92XDD0W.md",
  "status": null,
  "tag": "CMK92XDD0W",
  "uri": "m2://CMK92XDD0W",
  "url": "https://math2.tamnd.com/wiki/willmore-flow-definition/",
  "writing": {
    "tag": "CMK92XDD0W",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}