{
  "id": "2103.08921",
  "version": "v1",
  "published": "2021-03-16T09:02:49.000Z",
  "updated": "2021-03-16T09:02:49.000Z",
  "title": "Bernstein Problem of Affine Maximal Type Hypersurfaces on Dimension N>=3",
  "authors": [
    "Shi-Zhong Du"
  ],
  "comment": "Corrigendum to \"Bernstein problem of affine maximal type hypersurfaces on dimension N>= 3\", which has been published in Journal of Differential Equations 269 (2020), no. 9, 7429-7469",
  "journal": "Journal of Differential Equations 269 (2020) 7429-7469",
  "doi": "10.1016/j.jde.2020.05.048",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniofrmly convex C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and \\theta=3/4, and later extended by Jia-Li to N=2, \\theta\\in(3/4,1] (see also [Zhou]). On the past twenty years, much efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N=3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean complete for N>=3, \\theta\\in(1/2,(N-1)/N).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-16T09:02:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bernstein problem",
      "construct non-quadratic affine maximal type",
      "non-quadratic affine maximal type hypersurfaces",
      "affine maximal type equation",
      "euclidean complete"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}