{
  "id": "math/0010036",
  "version": "v2",
  "published": "2000-10-03T16:07:01.000Z",
  "updated": "2001-08-01T15:18:21.000Z",
  "title": "Evolution equations for special Lagrangian 3-folds in C^3",
  "authors": [
    "Dominic Joyce"
  ],
  "comment": "54 pages, LaTeX. (v2) references updated, minor corrections",
  "journal": "Annals of Global Analysis and Geometry 20 (2001), 345-403.",
  "categories": [
    "math.DG"
  ],
  "abstract": "This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P in R^n, and constructed a family of special Lagrangian m-folds N in C^m, which are swept out by the image of P under a 1-parameter family of linear or affine maps phi_t : R^n -> C^m, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C^3. We find a 1-1 correspondence between sets of evolution data with m=3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C^3. Our main results are a number of new families of special Lagrangian 3-folds in C^3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R^3 or S^1 x R^2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind. We hope these 3-folds will be helpful in understanding singularities of compact special Lagrangian 3-folds in Calabi-Yau 3-folds. This will be important in resolving the SYZ conjecture in Mirror Symmetry.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-08-01T15:18:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "evolution equations",
      "evolution data",
      "compact special lagrangian",
      "papers constructing explicit examples",
      "special lagrangian m-folds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....10036J"
    }
  }
}