{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $M_0$ be a compact, connected, smooth hypersurface without boundary embedded in $\\R^{n+1}$.\nAssume that $M_0$ is uniformly convex, that is, all [[principal curvatures|FA8QE1SGP4]] of $M_0$ are positive at every point.\nThen:\n\n1. The mean curvature flow starting from $M_0$ has a unique smooth solution $(M_t)_{0 \\le t \u003c T}$ on a maximal time interval with $T \u003c \\infty$.\n2. Each $M_t$ is uniformly convex.\n3. As $t \\to T$, the hypersurfaces $M_t$ shrink to a single point $p \\in \\R^{n+1}$.\n4. The rescaled hypersurfaces $(2n(T - t))^{-1/2}(M_t - p)$ converge smoothly to the unit sphere $S^n$ as $t \\to T$.\n\n## Remarks\n\nThe proof first shows that convexity is preserved by the maximum principle applied to the evolution of the second fundamental form.\nThe key estimate is a pinching estimate: the ratio of the smallest to the largest principal curvature tends to $1$ where the [[mean curvature|PA78NFEPCF]] blows up.\nTogether with bounds on the derivatives of the curvature, this forces the rescaled surfaces to become round.\nFor a round sphere of radius $r_0$, the flow gives spheres of radius $r(t) = \\sqrt{r_0^2 - 2nt}$, so $T = r_0^2/(2n)$ and the rescaled surfaces are all the unit sphere.\nConvexity is needed: a dumbbell, made of two large spheres joined by a thin neck, can pinch off at the neck before it shrinks to a point.\nThe case $n = 1$ of convex curves in the plane is the theorem of Gage and Hamilton, part of the [[Gage–Hamilton–Grayson theorem|FVGW1381TZ]].\n\n## Explanations\n\n### L1 human\n\nMean curvature flow moves each point of a surface inward at a speed given by how curved the surface is there.\nHuisken's theorem says that a convex closed surface, such as an egg shape, stays convex under this flow.\nIt shrinks to a point in finite time.\nJust before it disappears, it looks more and more like a round sphere.\n\n### L2 human\n\nHuisken proved this in 1984, for hypersurfaces of dimension at least two.\nIt is the model result for geometric flows that make a shape round.\nIt parallels Hamilton's theorem that the Ricci flow deforms a closed $3$-manifold with positive Ricci curvature into a round one.\nThe method combines the parabolic maximum principle with curvature pinching estimates.\nLater work of Huisken and others studies what happens without convexity, where singularities such as neck pinches appear.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "statement",
    "msc": [
      "53E10",
      "53C44"
    ],
    "name": "Huisken's theorem on convex hypersurfaces",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mean_curvature_flow"
      }
    ],
    "tag": "14RHHMGF2D",
    "uses": [
      "PA78NFEPCF",
      "FA8QE1SGP4"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/14RHHMGF2D/",
  "links": [
    {
      "field": "uses",
      "target": "PA78NFEPCF"
    },
    {
      "field": "uses",
      "target": "FA8QE1SGP4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/14RHHMGF2D.md",
  "msc": [
    "53E10",
    "53C44"
  ],
  "name": "Huisken's theorem on convex hypersurfaces",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/14R/14RHHMGF2D.md",
  "status": null,
  "tag": "14RHHMGF2D",
  "uri": "m2://14RHHMGF2D",
  "url": "https://math2.tamnd.com/wiki/huiskens-theorem-on-convex-hypersurfaces/",
  "writing": {
    "tag": "14RHHMGF2D",
    "kind": "statement",
    "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
      }
    ]
  }
}