Stability of Calabi flow near an extremal metric

Abstract

We prove that on a Kähler manifold admitting an extremal metric $\omega$ and for any Kähler 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.

Improved presentation

References in corpus (6)

• The Calabi flow on Kähler surface with bounded Sobolev constant
• The Calabi flow with small initial energy
• The Calabi flow on toric Fano surface
• Pseudo-Calabi Flow
• Liouville energy on a topological two sphere
• Local solution and extension to the Calabi flow

Cited by in corpus (7)

• Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow
• The global existence and convergence of the Calabi flow on ${\mathbb{C}}^n/{\mathbb{Z}}^n+i{\mathbb{Z}}^n$
• Pseudo-Calabi Flow
• Convergence of the calabi flow on toric varieties and related Kaehler manifolds
• Stability of Kähler-Ricci flow in the space of Kähler metrics
• Kähler non-collapsing, eigenvalues and the Calabi flow
• A splitting theorem on toric varieties