Stability of Calabi flow near an extremal metric

TL;DR

This paper proves the long-term stability and convergence of the Calabi flow near an extremal K"ahler metric on a manifold, demonstrating exponential convergence under symmetry conditions.

Contribution

It establishes the stability and exponential convergence of the Calabi flow near extremal metrics, extending understanding of geometric flow behavior in K"ahler geometry.

Findings

Calabi flow exists for all time near an extremal metric.

Modified Calabi flow remains close and converges exponentially under symmetry.

Flow stability is proven for initial potentials close to the extremal metric.

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 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.