An invariant for affine maximal type equations

Abstract

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. 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 Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.