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

Abstract:

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\mathbb {C}^n$ without assuming the initial metric has bounded curvature, thus extending results in an earlier work of the authors. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler 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 Kähler metrics on $\mathbb {C}^n$ which are not necessarily $U(n)$-invariant.