*Title*: **Constant scalar curvature Kahler metrics and properness theorem **

*Abstract*: Constant scalar curvature Kahler metric(cscK) is a natural higher dimensional generalization of the constant curvature metric on Riemann surfaces. One of the main difficulties in understanding the existence of cscK metric is the complexity of the underlying partial differential equation. We take up this challenge and show that the existence of cscK metric on compact Kahler manifold is equivalent to the properness of Mabuchi energy (whose critical point are cscK metrics).
I will also discuss future problems if time permits. This is based on joint work with Xiuxiong Chen.