{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(N, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1$, let $M$ be a closed manifold of dimension $n$, and let $T \u003e 0$.\nA family of immersions $F \\colon M \\times [0, T) \\to N$ is a solution of *inverse mean curvature flow* if for every $t \\in [0, T)$:\n\n1. the hypersurface $M_t = F(M, t)$ has an outward unit normal $\\nu(\\cdot, t)$;\n2. the [[mean curvature|PA78NFEPCF]] $H(\\cdot, t)$ of $M_t$ with respect to $\\nu$, the sum of the principal curvatures, is positive everywhere;\n3. for all $x \\in M$,\n$$\\frac{\\partial F}{\\partial t}(x, t) = \\frac{1}{H(x, t)}\\, \\nu(x, t).$$\n\n## Remarks\n\nExample: in $N = \\R^{n+1}$ take $M_t$ the sphere of radius $r(t) = r_0 e^{t/n}$ about the origin.\nIts mean curvature is $H = n/r$, so the speed required by clause 3 is $r/n$, and indeed $r'(t) = r(t)/n$.\nIts area is $e^t$ times the area of $M_0$.\nNon-example: a hypersurface with a point where $H \\le 0$ fails clause 2, and the flow cannot start from it.\nA torus of revolution in $\\R^3$ with tube radius $r$ and center circle of radius $R \u003c 2r$ has such points on its inner circle.\nUnder the flow the area element is multiplied by $e^t$ in general, since its time derivative is $H \\cdot (1/H) = 1$ times itself.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "5XVEF5DKAG",
    "kind": "definition",
    "msc": [
      "53E10",
      "53C21"
    ],
    "name": "Inverse mean curvature flow",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Inverse_mean_curvature_flow"
      }
    ],
    "tag": "ZZFTPP8JTG",
    "uses": [
      "PA78NFEPCF",
      "7CBT4SM83Y"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/ZZFTPP8JTG/",
  "links": [
    {
      "field": "defines",
      "target": "5XVEF5DKAG"
    },
    {
      "field": "uses",
      "target": "PA78NFEPCF"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZZFTPP8JTG.md",
  "msc": [
    "53E10",
    "53C21"
  ],
  "name": "Inverse mean curvature flow",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZZF/ZZFTPP8JTG.md",
  "status": null,
  "tag": "ZZFTPP8JTG",
  "uri": "m2://ZZFTPP8JTG",
  "url": "https://math2.tamnd.com/wiki/inverse-mean-curvature-flow-definition/",
  "writing": {
    "tag": "ZZFTPP8JTG",
    "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
      }
    ]
  }
}