A Liouville Theorem on the PDE $\det (f_{i\bar{j}})=1$

Abstract

Let $f$ be a smooth plurisubharmonic function which solves $$ \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }Ω\subset \mathbb C^n.$$ Suppose that the metric ${\omega }_f=\sqrt{-1}{f}_{i\bar{j}}d{z}_i\wedge d{\bar{z}}_j$ is complete and $f$ satisfies the growth condition $$C^{-1}(1+|z{|}^2)\le f\le C(1+|z{|}^2),as|z|\to \infty .$$ for some $C>0,$ then $f$ is quadratic.