# Stability of Calabi flow near an extremal metric - Mathematics > Differential Geometry

Abstract: 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 flowstarting at $\varphi 0$ exists for all time and the modified Calabi flowstarting at $\varphi 0$ will always be close to $\omega$. Furthermore, when theinitial data is invariant under the maximal compact subgroup of the identitycomponent of the reduced automorphism group, the modified Calabi flow convergesto an extremal metric near $\omega$ exponentially fast.

Author: Hongnian Huang, Kai Zheng

Source: https://arxiv.org/