Stability of Calabi flow near an extremal metric

We prove that on a K"ahler manifold admitting an extremal metric $\omega$ and for any K"ahler potential ${\phi }_0$ close to $\omega$, the Calabi flow starting at ${\phi }_0$ exists for all time and the modified Calabi flow starting at ${\phi }_0$ 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.