论文标题
$θ\ in（0，（n-1）/n] $
Non-quadratic Euclidean complete affine maximal type hypersurfaces for $θ\in(0,(N-1)/N]$
论文作者
论文摘要
Bernstein仿射最大类型方程的问题\ begin {equination} \ label {e0.1} u^{ij} d_ {ij} w = 0，\ \ w \ equiv [\ det d^2U] \ end {equation}一直是仿射几何形状的核心问题。首先，Chern提出了一个猜想（日本 - 日本国家的SEM。，东京，1977年，17-30），以延长Trudinger-Wang（发明的Math。在$ {\ mathbb {r}}}}^{n+1} $中必须是椭圆抛物面。同时，Trudinger-Wang完全解决了该猜想的尺寸$ n = 2 $和$θ= 3/4 $，然后由Jia-li（结果数学，{\ bf56} 2009，1009-139）扩展到$ n = 2，θ\ in（in（3/4,4,4,1] $），请参见zhou（bf）。 25-44）在过去的二十年中，在更高的维度问题上做了很多努力，但实际上并没有真正成功，即使在dimension $ n = 3 $的情况下，在\ cite {du2}中都发现了反对。 $θ\ in（1/2，（N-1）/N）$使用更复杂的参数，我们将明确构建各种新的Euclidean全新欧亲态最大型高空超过，这些高度呈椭圆形抛物面，以改善范围
Bernstein problem for affine maximal type equation \begin{equation}\label{e0.1} u^{ij}D_{ij}w=0, \ \ w\equiv[\det D^2u]^{-θ},\ \ \forall x\inΩ\subset{\mathbb{R}}^N \end{equation} 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 $C^4$-hypersurface in ${\mathbb{R}}^{N+1}$ 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 (Results Math., {\bf56} 2009, 109-139) to $N=2, θ\in(3/4,1]$ (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 \cite{Du2} (J. Differential Equations, {\bf269} (2020), 7429-7469) for $N\geq3$ and $θ\in(1/2,(N-1)/N)$ 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 $$N\geq2, \ \ θ\in(0,(N-1)/N].$$
