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 $C^{2,α}$ neighborhood of it, the pseudo-Calabi flow must converge exponentially to a nearby cscK metric.