AI 中文总结
本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。
AI 中文摘要
仿射极大型方程的Bernstein问题是仿射几何中的核心问题，该方程为\\(\sum_{i,j=1}^n f^{ij} w_{ij}=0\\)，其中\\(w\equiv \left[\det\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)\right]^a\\)，\\(x\in\Omega\subset\mathbb R^n\\)。它源于1977年Chern针对\\(n=2\\)、\\(a=-\frac{3}{4}\\)情形下的整局部凸图猜想。该猜想于2000年由Trudinger和Wang完全解决，他们还在欧氏完备性假设下，将结论推广到任意维数\\(n\ge2\\)、\\(a=-\frac{n+1}{n+2}\\)的情况。随后，Li和Jia利用实仿射技术，通过建立\\(n=2\\)、\\(a\in(-\infty,-\frac{3}{4}]\\)情形下的Bernstein定理，给出了Chern猜想的全新纯解析证明。尽管过去二十年间研究者付出了大量努力，高维Chern猜想仍未解决。近期，Du构造了\\(a\in[-\frac{n-1}{n},0)\\)范围内的显式非二次欧氏完备解。本文从子流形理论与Calabi仿射几何的视角研究仿射极大型曲面，对Du构造的显式欧氏完备反例——包括二维情形下的Warren型、Trudinger-Wang型及其他解——给出了几何刻画。更重要的是，我们构造了一类新的非二次欧氏完备仿射极大型超曲面，对所有\\(n\ge 2\\)，将Du的参数范围拓展至\\(a\in[-\frac{n}{n+1},0)\\)。
英文摘要
The Bernstein problem for the affine maximal type equation \[ \sum_{i,j=1}^n f^{ij} w_{ij}=0,\qquad w\equiv \left[\det\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)\right]^a,\quad x\inΩ\subset\mathbb R^n, \] is a central problem in affine geometry. It originates from Chern's conjecture on entire locally convex graphs for the case $n=2$ and $a=-\frac{3}{4}$ in 1977. This conjecture was completely resolved by Trudinger and Wang in 2000, who moreover proposed a generalization to arbitrary dimension $n\ge2$ for $a=-\frac{n+1}{n+2}$ under the assumption of Euclidean completeness. Later, using real affine techniques, Li and Jia provided a new purely analytic proof of Chern's conjecture by establishing the Bernstein theorem for $n=2$ and $a\in(-\infty,-\frac{3}{4}]$. Despite considerable efforts over the past two decades, the higher-dimensional Chern's conjecture remains open. Recently, Du constructed explicit non-quadratic Euclidean complete solutions for $a\in[-\frac{n-1}{n},0)$. In this paper, from the perspective of submanifold theory and Calabi affine geometry, we investigate affine maximal type surfaces. It provides a geometric characterisation for Du's explicit Euclidean complete counterexamples---including Warren type, Trudinger-Wang type, and other solutions in dimension two. More importantly, we construct a new class of non-quadratic Euclidean complete affine maximal type hypersurfaces, which extends Du's parameter range, for all $n\ge 2$, to $a\in[-\frac{n}{n+1},\,0).$
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