TITLE: When a Kähler manifold is isometric to $\mathbb C^n$
QUESTION [7 upvotes]: Let $(X,\omega)$ be an $n$-dimensional complete Kähler manifold. Then when it  is isometric to complex Euclidean space $\mathbb C^n$

REPLY [12 votes]: 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.