Stability of K\"ahler-Ricci flow in the space of K\"ahler metrics

Abstract

In this paper, we prove that on a Fano manifold $M$ which admits a K\"ahler-Ricci soliton (\om,X), if the initial K\"ahler metric \om_{\vphi_0} is close to \om in some weak sense, then the weak K\"ahler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if \vphi_0 is also $K_X$-invariant, then the weak modified K\"ahler-Ricci flow converges exponentially to a unique K\"ahler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the K\"ahler metrics are potentially applicable to other stability problem of geometric flow near a critical metric.Comment: 28 pages, 1 figure