Gradient K\"ahler-Ricci solitons and a uniformization conjecture

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.