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

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