Abstract

In this paper, we study the global Kahler–Ricci flow on a complete non-compact Kahler manifold. We prove the following result. Assume that \((M,g_0)\) is a complete non-compact Kahler manifold such that there is a potential function f of the Ricci tensor, i.e., $$\begin{aligned} R_{i{\bar{j}}}(g_0)=f_{i{\bar{j}}}. \end{aligned}$$ Assume that the quantity \(|f|_{C^0}+|\nabla _{g_0}f|_{C^0}\) is finite and the \(L^2\) Sobolev inequality holds true on \((M,g_0)\). Then the Kahler–Ricci flow with the initial metric \(g_0\) either blows up at finite time or infinite time to Ricci-flat metric or exists globally with Ricci-flat limit at infinite time. Related results are also discussed.

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