A Bernstein theorem for affine maximal-type hypersurfaces

Abstract

We obtain, in any dimension N and for a large range of values of θ, a Bernstein theorem for the fourth-order partial differential equation of affine maximal typeuijDijw=0,w=[det⁡D2u]−θ assuming the completeness of Calabi's metric. This contains the results of Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] for affine maximal equations and of Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] for Abreu's equation. In particular, we extend the result of Zhou from 2≤N≤4 to 2≤N≤5.