Convergence of a K\"ahler-Ricci flow

TL;DR

This paper proves convergence results for the K"ahler-Ricci flow under bounded Ricci curvature, showing subsequential limits converge to solutions or solitons outside singular sets, with special results in complex dimension two.

Contribution

It establishes subsequential convergence of the K"ahler-Ricci flow to solutions or solitons, with detailed analysis of singularities and without curvature assumptions in complex dimension two.

Findings

Convergence to a flow solution outside a codimension at least 4 singular set.

In complex dimension 2, convergence to a K"ahler-Ricci soliton with finitely many singularities.

Existence of subsequences with smooth convergence outside singularities.

Abstract

In this paper we prove that for a given K\"ahler-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, without any curvature assumptions we can find a subsequence of times such that we have a convergence to a K\"ahler-Ricci soliton, away from finitely many isolated singularities.