Convergence of a Kähler-Ricci flow

In this paper we prove that for a given Kähler-Ricci flow with uniformly bounded Ricci curvatures in an arbitrary dimension, for every sequence of times $t_i$ converging to infinity, there exists a subsequence such that $(M,g(t_i + t))\to (Y,\bar{g}(t))$ and the convergence is smooth outside a singular set (which is a set of codimension at least $4$) to a solution of a flow. We also prove that in the case of complex dimension $2$, we can find a subsequence of times such that we have a convergence to a Kähler-Ricci soliton, away from finitely many isolated singularities.

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Published 1 January 2005