Pseudo-Calabi Flow

Abstract

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial φ}\over {\partial t}}&= -f(φ), \triangle_varphi f(φ) &= S(φ) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continuous dependence on the initial data. Next, we point out that the $L^{\infty }$ bound on Ricci curvature is an obstruction to the extension of the pseudo-Calabi flow. Finally, we show that if there is a cscK metric in its Kähler class, then for any initial potential in a small $C^{2,\alpha }$ neighborhood of it, the pseudo-Calabi flow must converge exponentially to a nearby cscK metric.

46 pages

References in corpus (7)

• Interior estimates for solutions of Abreu's equation
• Space of Kähler metrics (IV)--On the lower bound of the K-energy
• The Calabi flow on Kähler surface with bounded Sobolev constant
• The Calabi flow with small initial energy
• Stability of Calabi flow near an extremal metric
• The Calabi flow on toric Fano surface
• Stability of Kähler-Ricci flow in the space of Kähler metrics

Cited by in corpus (3)

• Stability of Calabi flow near an extremal metric
• Stability of Kähler-Ricci flow in the space of Kähler metrics
• Kähler non-collapsing, eigenvalues and the Calabi flow