Stability of Kähler-Ricci flow in the space of Kähler metrics

Abstract

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified Kähler-Ricci flow converges exponentially to a unique Kähler-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 Kähler metrics are potentially applicable to other stability problem of geometric flow near a critical metric.

arXiv comment28 pages, 1 figures

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Cited by in corpus (3)

• Kähler Ricci flow on Fano manifolds with vanished Futaki invariants
• Pseudo-Calabi Flow
• Kähler Ricci flow with vanished Futaki invariant