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Bernstein problem of affine maximal type hypersurfaces on dimension N ≥ 3.
- Source :
-
Journal of Differential Equations . Oct2020, Vol. 269 Issue 9, p7429-7469. 41p. - Publication Year :
- 2020
-
Abstract
- Bernstein problem for affine maximal type equation (0.1) u i j D i j w = 0 , w ≡ det D 2 u − θ , ∀ x ∈ Ω ⊂ R N has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] 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 θ = 3 / 4 , and later extended by Jia-Li (2009) [12] to N = 2 , θ ∈ (3 / 4 , 1 (see also Zhou (2012) [16] for a different proof). On the past twenty years, many 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 compete for N ≥ 3 , θ ∈ (1 / 2 , (N − 1) / N). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 269
- Issue :
- 9
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 143779325
- Full Text :
- https://doi.org/10.1016/j.jde.2020.05.048