Stability of Calabi flow near an extremal metric

We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $φ_0$ close to $ω$, the Calabi flow starting at $φ_0$ exists for all time and the modified Calabi flow starting at $φ_0$ will always be close to $ω$. 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 $ω$ exponentially fast.

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