The variational approach for canonical metrics in Kähler geometry

A central theme in Kähler geometry is the search for canonical Kähler metrics, such as Kähler-Einstein metrics, constant scalar curvature Kähler metrics, extremal metrics, etc…

The Mabuchi functional M was introduced by Mabuchi in the 80's in relation to the existence of such canonical metrics on a compact Kähler manifold. The critical points of M are indeed the special metrics we look for.

Since then, the properties of the Mabuchi functional (such as convexity, properness) have been intensively studied. In this talk I will give a panorama on the state of the art and on the recent breakthroughs in the field.