Longtime existence of the K\"ahler-Ricci flow on $\Bbb C ^n$

We produce longtime solutions to the K\"ahler-Ricci flow for complete K\"ahler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant K\"ahler metric with non-negative holomorphic bisectional curvature, and that the solution converges as $t\to \infty$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general K\"ahler metrics on $\Bbb C ^n$ which are not necessarily $U(n)$-invariant.