{
  "id": "0801.3431",
  "version": "v2",
  "published": "2008-01-22T18:10:47.000Z",
  "updated": "2008-04-22T15:42:15.000Z",
  "title": "The modified Calabi-Yau problems for CR-manifolds and applications",
  "authors": [
    "JIanguo Cao",
    "Shu-Cheng Chang"
  ],
  "comment": "The new version is more accurate on citing other people's work",
  "categories": [
    "math.DG",
    "math.CV"
  ],
  "abstract": "In this paper, we derive a partial result related to a question of Yau: \"Does a simply-connected complete K\\\"ahler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?\" Main Theorem. Let $M^{2n}$ be a simply-connected complete K\\\"ahler manifold M with negative sectional curvature $ \\le -1 $ and $S_\\infty(M)$ be the sphere at infinity of $M$. Then there is an explicit {\\it bounded} contact form $\\beta$ defined on the entire manifold $M^{2n}$. Consequently, the sphere $S_\\infty(M)$ at infinity of M admits a {\\it bounded} contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster. We also discuss several open modified problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-22T15:42:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C23"
    ],
    "keywords": [
      "modified calabi-yau problems",
      "cr-manifolds",
      "applications",
      "simply-connected complete",
      "negative sectional curvature admit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}