A Bernstein Property of Some Affine Kähler Scalar Flat Graph

Abstract

In this paper we consider the smooth, strictly convex solution u of the Abreu equation $$ \sum {U^{ij}} \left( {\left[ {\det (u_{kl} )} \right]^{ - 1} } \right)_{ij} = 0$$ (0.1) on \({\mathbb R}^n (2 \leq n \leq 4)\), where (Uij) denotes the cofactor matrix of the matrix (uij); we obtain a Bernstein property, that is, u must be a quadratic polynomial if the norm of affine Kahler–Ricci curvature is bounded from above.