{
  "id": "1104.3794",
  "version": "v2",
  "published": "2011-04-19T16:28:45.000Z",
  "updated": "2012-10-07T19:41:30.000Z",
  "title": "The Cauchy Problem for Wave Maps on a Curved Background",
  "authors": [
    "Andrew Lawrie"
  ],
  "comment": "Fixed minor typos in previous version. To appear in Calculus of Variations and Partial Differential Equations",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for wave maps u: \\R times M \\to N for Riemannian manifolds, (M, g) and (N, h). We prove global existence and uniqueness for initial data that is small in the critical Sobolev norm in the case (M, g) = (\\R^4, g), where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe for proving global existence and uniqueness of small data wave maps u : \\R \\times \\R^d \\to N in the critical norm, for d at least 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-07T19:41:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cauchy problem",
      "curved background",
      "small data wave maps",
      "variable coefficient wave equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.3794L"
    }
  }
}