Stability of Calabi flow near an extremal metric

Details Stability of Calabi flow near an extremal metric Stability of Calabi flow near an extremal metric

2010-07-26

arxiv.org/abs/1007.4571v2

Mathematics

Differential Geometry

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Scientific paper

We prove that on a K\"ahler manifold admitting an extremal metric $\omega$ and for any K\"ahler potential $\varphi_0$ close to $\omega$, the Calabi flow starting at $\varphi_0$ exists for all time and the modified Calabi flow starting at $\varphi_0$ will always be close to $\omega$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $\omega$ exponentially fast.

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