Moduli space of extremal K\"ahler metrics and the Calabi flow

Xiuxiong Chen, University of Wisconsin,

This is a personal update of issues involves extremal K\"ahler metrics where the complex gradient vector field of the scalar curvature function is holomorphic. We will report the uniqueness result of extremal Kaehler metrics, as well as a sharp lower bound for the Calabi energy (L^2 norm of scalar curvature) in each K\"ahler class. Extremal Kahler metrics satisfy a 6th order equation and we treated as a triple 2nd order elliptic equation. We will discuss weak compactness of the moduli space and the possible application in the Calabi flow (gradient flow for the Calabi energy).