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 type u i j D i j w = 0 , w = [ det ⁡ D 2 u ] − 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. [ABSTRACT FROM AUTHOR]