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

TL;DR

This paper proves the global existence and convergence of the weak K"ahler-Ricci flow on Fano manifolds near a K"ahler-Ricci soliton, with exponential convergence under symmetry assumptions, using metric geometry methods.

Contribution

It establishes stability results for the K"ahler-Ricci flow near solitons on Fano manifolds, including exponential convergence and removing symmetry assumptions when the Futaki invariant vanishes.

Findings

Weak K"ahler-Ricci flow exists globally near a soliton.

Flow converges in Cheeger-Gromov sense.

Exponential convergence under symmetry conditions.

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.