New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry

Abstract

The Bernstein problem for the affine maximal type equation \[ \sum_{i,j=1}^n f^{ij} w_{ij}=0,\qquad w\equiv \left[\det\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)\right]^a,\quad x\in\Omega\subset\mathbb R^n, \] 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 $a\in[-\frac{n}{n+1},\,0).$