{
  "id": "0902.3300",
  "version": "v1",
  "published": "2009-02-19T06:17:08.000Z",
  "updated": "2009-02-19T06:17:08.000Z",
  "title": "Lagrangian Mean Curvature flow for entire Lipschitz graphs",
  "authors": [
    "Albert Chau",
    "Jingyi Chen",
    "Weiyong He"
  ],
  "comment": "22 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\\R^{2n}$, we show that the parabolic equation \\eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\\to \\infty$. Our assumption on the Lipschitz norm is equivalent to the assumption that the underlying Lagrangian potential $u$ is uniformly convex with its Hessian bounded in $L^\\infty$. We apply this result to prove a Bernstein type theorem for translating solitons, namely that if such an entire Lagrangian graph is a smooth translating soliton, then it must be a flat plane. We also prove convergence of the evolving graphs under additional conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-19T06:17:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "53A10"
    ],
    "keywords": [
      "lagrangian mean curvature flow",
      "entire lipschitz graphs",
      "entire lagrangian graph",
      "second fundamental form decays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.3300C"
    }
  }
}