Stability on K\"ahler-Ricci flow, I

TL;DR

This paper proves that the K"ahler-Ricci flow converges to a K"ahler-Einstein metric or soliton under initial conditions close to these metrics, enhancing the understanding of flow stability on certain compact K"ahler manifolds.

Contribution

It improves previous results by establishing stability of the K"ahler-Ricci flow near K"ahler-Einstein metrics and solitons on manifolds with positive first Chern class.

Findings

Flow converges to K"ahler-Einstein metric or soliton under initial closeness.

Enhances previous stability results for K"ahler-Ricci flow.

Applicable to manifolds with positive first Chern class.

Abstract

In this paper, we prove that K\"ahler-Ricci flow converges to a K\"ahler-Einstein metric (or a K\"ahler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial K\"ahler metric is very closed to $g_{KE}$ (or $g_{KS}$) if a compact K\"ahler manifold with $c_1(M)>0$ admits a K\"ahler Einstein metric $g_{KE}$ (or a K\"ahler-Ricci soliton $g_{KS}$). The result improves Main Theorem in [TZ3] in the sense of stability of K\"ahler-Ricci flow.