Pseudo-Calabi Flow

TL;DR

This paper introduces the Pseudo-Calabi flow, proves its well-posedness, discusses obstructions to its extension, and shows convergence to constant scalar curvature Kähler metrics under certain conditions.

Contribution

It defines a new geometric flow, establishes its mathematical properties, and demonstrates its convergence behavior near cscK metrics.

Findings

Well-posedness of the Pseudo-Calabi flow

Obstruction due to Ricci curvature bounds

Exponential convergence to cscK metrics

Abstract

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_varphi f(\varphi) &= S(\varphi) - \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 K\"ahler 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.