Pseudo-Calabi Flow

Details Pseudo-Calabi Flow Pseudo-Calabi Flow

2010-04-15

arxiv.org/abs/1004.2663v3

Mathematics

Differential Geometry

46 pages

Scientific paper

We first define Pseudo-Calabi flow, as \begin{equation*} \left\{ \begin{aligned} {{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_\varphi f(\varphi) &= S(\varphi) - \ul S. \end{aligned} \right. \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.

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