Non-quadratic Euclidean Complete Affine Maximal Type Hypersurfaces for θ∈(0,(N-1)/N

• Published in: The Journal of geometric analysis 2024-08, Vol.34 (8), Article 229
• Main Author: Du, Shi-Zhong
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Hyperspaces; Mathematics; Mathematics and Statistics; Parabolic bodies; Partial differential equations
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-024-01678-7

Bernstein problem for affine maximal type equation 0.1 u ij D ij w = 0 , w ≡ [ det D 2 u ] - θ , ∀ x ∈ Ω ⊂ R N has been a core problem in affine geometry. A conjecture (Version I in Section 1) initially proposed by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph with N = 2 , θ = 3 / 4 and then was strengthened by Trudinger-Wang (Invent. Math., 140 , 2000, 399-422) to its full generality (Version II), which asserts that any Euclidean complete, affine maximal, locally uniformly convex C 4 -hypersurface in R N + 1 must be an elliptic paraboloid. At the same time, the Chern’s conjecture was solved completely by Trudinger-Wang in dimension two. Soon after, the Affine Bernstein Conjecture (Version III) for affine complete affine maximal hypersurfaces was also shown by Trudinger-Wang in (Invent. Math., 150 , 2002, 45-60). Thereafter, the Bernstein problem has morphed into a broader conjectures for any dimension N ≥ 2 and any positive constant θ > 0 . The Bernstein theorem of Trudinger-Wang was then generalized by Li-Jia (Results Math., 56 2009, 109-139) to N = 2 , θ ∈ ( 3 / 4 , 1 ] (see also Zhou (Calc. Var. PDEs., 43 2012, 25-44) for a different proof). In the past twenty years, much effort was done toward higher dimensional issues but not really successful yet, even for the case of dimension N = 3 . Recently, counter examples were found in (J. Differential Equations, 269 (2020), 7429-7469), toward the Full Bernstein Problem IV for N ≥ 3 , θ ∈ ( 1 / 2 , ( N - 1 ) / N ) and using a much more complicated argument. In this paper, we will construct explicitly various new Euclidean complete affine maximal type hypersurfaces which are not elliptic paraboloid for the improved range N ≥ 2 , θ ∈ ( 0 , ( N - 1 ) / N ] .