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

We produce longtime solutions to the K"ahler-Ricci flow for complete K"ahler metrics on ${\mathbb{C}}^n$ 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 $t\to \mathrm{\infty }$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general K"ahler metrics on ${\mathbb{C}}^n$ which are not necessarily $U(n)$-invariant.