A Simple Proof of a Rigidity Theorem for an Affine Kähler–Ricci Flat Graph

Abstract

Li and Xu (Results Math 56:141–164, 2009) proved that any entire strictly convex ${C^\infty}$-solution of the Monge–Ampère equation

$${\rm det}\left(\frac{\partial^{2}u}{\partial \xi_{i} \partial \xi_{j}}\right) = \exp \left\{-\sum_{i=1}^n d_i \frac{{\partial}u}{\partial \xi_{i}} - d_0\right\}, \qquad \xi \in \mathbb{R}^n,$$

where ${d_0, d_1,\ldots,d_n}$ are constants, must be a quadratic polynomial. Their result extends a well-known theorem of Jörgens–Calabi–Pogorelov. In our paper we will give a relatively simple proof for this extension in any dimension.