An invariant for affine maximal type equations (Q6648746)

A graph defined by a function \(f: \Omega\subset{\mathbb{R}}^n\to{\mathbb{R}}\) is called an affine maximal type hypersurface of \({\mathbb{R}}^{n+1}\) if it satisfies the equation \N\[\N\sum_{i,j}F^{ij}w_{ij}=0, \ \ w:=\begin{cases} (\det(f_{kl}))^a, & a\not=0,\\\N\log\det(f_{kl}), & a=0 \end{cases} \N\]\Nfor some parameter \(a\not=0\) and cofactor matrix \(F^{ij}\) of the Hessian matrix \((f_{ij})\). For convex graph function \(f\), \(G:=\sum f_{ij}dx_idx_j\) is called to be Calabi metric, and \(G^{(a)}:=\rho^\alpha G, \rho:=(\det(f_{ij}))^{-\frac{1}{n+2}}\) is called the \(\alpha\)-relative metric of the hypersurface.\N\NVarious Bernstein problems for affine maximal hypersurfaces concern the question whether a convex affine maximal hypersurface must be an elliptic paraboloid. The validity of the statement is usually called the Bernstein theorem for hypersurfaces.\N\NIn this paper, for some relation between \(n, a\) and \(\alpha\), the author shows that the Bernstein property of the equation holds true under the completeness of the \(\alpha\)-relative metric. As an application, the author derives a rigidity result for the complete \(T^n\)-invariant Kähler metric on complex torus \((C^*)^n\) with vanishing scalar curvature for \(n\geq5\).