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

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

2010-04-15

arxiv.org/abs/1004.2695v2

Pacific J. Math. 251, no.2, (2011), 469--497

Mathematics

Differential Geometry

28 pages, 1 figures

Scientific paper

10.2140/pjm.2011.251.469

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.

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