Liouville Property

Suppose $\det(\partial_{i}\partial_{\overline{j}}u)=1$ on $\mathbb{C}^{n}$ and

for all $z\in\mathbb{C}^{n}$. Is it true that $u$

It known if, in addition, $\sqrt{-1}\partial\overline{\partial}u$ is a complete Kahler metric on $\mathbb{C}^{n}$.

It is known if $u(z)=\langle Az,\overline{z}\rangle+o(|z|)$ as $|z|\to\infty$ (Yu Wang 2012).

Is there a way to use the Kahler-Ricci flow to solve this problem? Is the Kahler-Ricci flow starting at $\sqrt{-1}\partial\overline{\partial}u$ instantaneously complete?

Suppose $u$ is such that $\sqrt{-1}\partial\overline{\partial}u$ is complete, close to the Taub-Nut (maybe in the sense that $u(z)-P(z)=o(|z|)$, where $P$ is the Kahler potential for the Taub NUT metric) and is Ricci flat.