  In this article we study the limiting behavior of the K\"ahler Ricci flow on
complete non-compact K\"ahler manifolds. We provide sufficient conditions under
which a complete non-compact gradient K\"ahler-Ricci soliton is biholomorphic
to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for
complete non-compact K\"ahler manifolds with positive holomorphic bisectional
curvature.
