{
  "id": "0804.0783",
  "version": "v5",
  "published": "2008-04-04T17:18:40.000Z",
  "updated": "2010-08-11T18:18:44.000Z",
  "title": "Mean Curvature Flow of Spacelike Graphs",
  "authors": [
    "Guanghan Li",
    "Isabel M. C. Salavessa"
  ],
  "comment": "version 5: Math.Z (online first 30 July 2010). version 4: 30 pages: we replace the condition $K_1\\geq 0$ by the the weaker one $Ricci_1\\geq 0$. The proofs are essentially the same. We change the title to a shorter one. We add an application",
  "doi": "10.1007/s00209-010-0768-4",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We prove the mean curvature flow of a spacelike graph in $(\\Sigma_1\\times \\Sigma_2, g_1-g_2)$ of a map $f:\\Sigma_1\\to \\Sigma_2$ from a closed Riemannian manifold $(\\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\\leq -c$, $c>0$ constant, any map $f:\\Sigma_1\\to \\Sigma_2$ is trivially homotopic provided $f^*g_2<\\rho g_1$ where $\\rho=\\min_{\\Sigma_1}K_1/\\sup_{\\Sigma_2}K_2^+\\geq 0$, in case $K_1>0$, and $\\rho=+\\infty$ in case $K_2\\leq 0$. This largely extends some known results for $K_i$ constant and $\\Sigma_2$ compact, obtained using the Riemannian structure of $\\Sigma_1\\times \\Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2010-08-11T18:18:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C40",
      "58D25",
      "35K55"
    ],
    "keywords": [
      "mean curvature flow",
      "spacelike graph",
      "complete riemannian manifold",
      "sectional curvatures",
      "riemannian structure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.0783L"
    }
  }
}