Let $(X,\omega)$ be an $n$-dimensional complete Kähler manifold. Then when it is isometric to complex Euclidean space $\mathbb C^n$



When a Kähler manifold is isometric to $\mathbb C^n$
The story of characterization of the isometry class of $\mathbb C^n$ equipped with the flat metric comes back to around 80 of the nice work of Burns in Annals of Mathematics and also others


I know the following theorem, it may help in the foliation language.


**Theorem**: An $n$-dimensional Kähler manifold $(X,\omega)$ is isometric to $\mathbb C^n$ if and only if the Kähler metric equals $\omega= \partial\bar\partial\tau$, where 
$ \tau:M\to [0,∞)$ is an $\mathbb C^\infty$ strictly plurisubharmonic (psh) exhaustion function which satisfies the following Monge-Ampère foliation $$(\partial\bar\partial\log\tau)^n=0$$


See 


[Burns, Dan, 
Curvatures of Monge-Ampère foliations and parabolic manifolds. 
Annals of Mathematics. (2) 115 (1982), no. 2, 349–373.](https://www.jstor.org/stable/1971395?seq=1#page_scan_tab_contents) 


