{
  "id": "math/0502530",
  "version": "v2",
  "published": "2005-02-25T02:28:46.000Z",
  "updated": "2005-05-16T20:36:00.000Z",
  "title": "The rate of convergence of the mean curvature flow",
  "authors": [
    "Tom Ilmanen",
    "Natasa Sesum"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We study the flow $M_t$ of a smooth, strictly convex hypersurface by its mean curvature in $\\mathrm{R}^{n+1}$. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time $T$ and point $x^*$ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere ${\\bf S^n}$ of radius $\\sqrt{n}$. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us the rate of the exponential decay is at least $\\frac{2}{n}$. We can define the ''arrival time'' $u$ of a smooth, strictly convex $n$-dimensional hypersurface as it moves with normal velocity equal to its mean curvature as $u(x) = t$, if $x\\in M_t$ for $x\\in \\Int(M_0)$. Huisken proved that for $n\\ge 2$ $u(x)$ is $C^2$ near $x^*$. The case $n=1$ has been treated by Kohn and Serfaty, they proved $C^3$ regularity of $u$. As a consequence of obtained rate of convergence of the mean curvature flow we prove that $u$ is not $C^3$ near $x^*$ for $n\\ge 2$. We also show that the obtained rate of convergence $2/n$, that comes out from linearizing a mean curvature flow is the optimal one, at least for $n\\ge 2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-05-16T20:36:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44"
    ],
    "keywords": [
      "convergence",
      "rescaled mean curvature flow converges",
      "strictly convex",
      "surface remains smooth",
      "corresponding rescaled mean curvature flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......2530I"
    }
  }
}