Let $u$ be a global smooth pluri-subharmonic function such that $\sqrt{-1}\partial\bar{\partial}u$ defines a scalar flat metric on $\mathbb{C}^{n}$. If for some $p>3n(n-1)$, we have

$$\displaystyle\liminf_{r\rightarrow\infty}\;{1\over r^{2n}}\displaystyle\int_{B_{r}(0)\subset\mathbb{C}^{n}}\;(\Delta u)^{p}+\big(\sum_{k}\frac{1}{u_{k\bar{k}}}\big)^{p}<\infty,$$

then the Levi Hessian of $u$ is constant.