ABSTRACT
This chapter discusses the recent progress on Kahler-Einstein manifolds. It explores a brand new non-linear inequality on compact Kahler-Einstein manifolds. This new inequality involves the complex Hessian of functions, and generalizes the classical Moser-Trudinger inequality on S2. The chapter establishes the stability of the underlying manifold if there is a Kahler-Einstein metric using previous result on the connection between Kahler-Einstein metrics and stability. More than forty years ago, E. Calabi asked if a compact Kahler manifold M admits any Kahler-Einstein metrics. A metric is Kahler-Einstein if it is Kahler and its Ricci curvature form is a constant multiple of its Kahler form. Such a metric provides a special solution of the Einstein equation on Riemannian manifolds. The only known Kahler-Einstein metrics were either homogeneous or of cohomogeneity one. A long-standing conjecture claims that there is a Kahler-Einstein metric on any compact Kahler manifold.
