An invariant for affine maximal type equations

Abstract

Let $y:M\to R^{n+1}$ be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space $R^{n+1}$, given as the graph of a smooth, strictly convex function $x_{n+1}=f(x_1,...,x_n)$ defined on a domain $\Omega \subset R^n$. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete $T^n$-invariant Kähler metric on complex torus ${\left(C^{⁎}\right)}^n$ with vanishing scalar curvature for $n\le 5$.