Non-quadratic Euclidean Complete Affine Maximal Type Hypersurfaces for $\theta \in (0,(N-1)/N]$

Abstract

Bernstein problem for affine maximal type equation

$$\begin{aligned} u^{ij}D_{ij}w=0, \ \ w\equiv [\det D^2u]^{-\theta },\ \ \forall x\in \Omega \subset {\mathbb {R}}^N \end{aligned}$$

(0.1)

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 $N=2, \theta =3/4$ 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 ${\mathbb {R}}^{N+1}$ 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 $N\ge 2$ and any positive constant $\theta >0$. The Bernstein theorem of Trudinger-Wang was then generalized by Li-Jia (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). 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 $N\ge 3,\theta \in (1/2,(N-1)/N)$ 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

$$\begin{aligned} N\ge 2, \ \ \theta \in (0,(N-1)/N]. \end{aligned}$$