A Rigidity Theorem for Affine K\"ahler-Ricci Flat Graph

Abstract

It is shown that any smooth strictly convex global solution of $$\det(\frac{\partial^{2}u}{\partial \xi_{i}\partial \xi_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial \xi_{i}} - d_0\right\},$$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of J\"{o}rgens-Calabi-Pogorelov.