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 [3]. 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.

arXiv comment27 pages; condition (c2) modified in section 3; correction to statement (part (b)) and proof of Proposition 3.1; reference [11] added; typos corrected

References in corpus (3)

• Poincaré-Lelong equation via the Hodge Laplace heat equation
• Deforming complete Hermitian metrics with unbounded curvature
• Kähler-Ricci flow with unbounded curvature

Cited by in corpus (2)

• Kähler-Ricci flow with unbounded curvature
• An existence time estimate for Kähler-Ricci flow