Canonical Metrics in Kahler Geometry

There has been fundamental progress in complex differential geometry in the last two decades. For one, the uniformization theory of canonical Ka'hler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. The aim of this monograph is to give an essentially self-contained introduction to the theory of canonical Ka'hler metrics on complex manifolds. It also presents the reader with some advanced topics in complex differential geometry not easily found elsewhere. The topics include Calabi-Futaki invariants, extremal Ka'hler metrics, the Calabi-Yau theorem on existence of Ka'hler Ricci-flat metrics, and recent progress on Ka'hler-Einstein metrics with positive scalar curvature. Applications of Ka'hler-Einstein metrics to the uniformization theory are also discussed. Readers with a good general knowledge of differential geometry and partial differential equations should be able to grasp and appreciate the materials in this monograph.