{
  "id": "2210.09127",
  "version": "v1",
  "published": "2022-10-17T14:20:18.000Z",
  "updated": "2022-10-17T14:20:18.000Z",
  "title": "On Bernstein Problem of Affine Maximal Hypersurfaces",
  "authors": [
    "Shi-Zhong Du"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Bernstein problem for affine maximal type hypersurfaces has been a core problem in affine geometry. A conjecture proposed firstly by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph and then extended by Trudinger-Wang (Invent. Math., 140, 2000, 399-422) to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly 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 (Results Math., 56, 2009, 109-139) to N=2, \\theta\\in(3/4,1] (see also Zhou (Calc. Var. PDEs., 43, 2012, 25-44) for a different proof). 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 firstly solved the Chern-Trudinger-Wang conjecture for affine maximal hypersurfaces on all dimensions N>=3, and then turn to construct non-quadratic affine maximal type hypersurfaces which are Euclidean complete for N>=2, \\theta\\in(0,(N-1)/N].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-17T14:20:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A15",
      "53A10",
      "35J60"
    ],
    "keywords": [
      "affine maximal hypersurfaces",
      "bernstein problem",
      "non-quadratic affine maximal type hypersurfaces",
      "construct non-quadratic affine maximal type",
      "euclidean complete"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}