2022/10/17 by Shi-Zhong Du, Du, Shi-Zhong
Mathematics · #35J60 #53A10 #53A15 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2210.09127
openalex publication_date 2022/10/17 · openalex created_date 2022/10/20 · openalex updated_date 2026/07/28
Bernstein problem for affine maximal type equation uijDijw=0, w≡[det D2u]-θ, ∀ x∈Ω⊂ℝN 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., \bf140, 2000, 399-422) to its full generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C4-hypersurface in ℝN+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 (Results Math., \bf56 2009, 109-139) to N=2, θ∈(3/4,1] (see also Zhou (Calc. Var. PDEs., \bf43 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. Recently, counter examples were found in \citeDu2 (J. Differential Equations, \bf269 (2020), 7429-7469) for N≥3 and θ∈(1/2,(N-1)/N) 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].