1 Introduction to Kรคhler manifolds -- 1.1 Kรคhler metrics -- 1.2 Curvature of Kรคhler metrics -- 2 Extremal Kรคhler metrics -- 2.1 The space of Kรคhler metrics -- 2.2 A brief review of Chern classes -- 2.3 Uniformization of Kรคhler-Einstein manifolds -- 3 Calabi-Futaki invariants -- 3.1 Definition of Calabi-Futaki invariants -- 3.2 Localization formula for Calabi-Futaki invariants -- 4 Scalar curvature as a moment map -- 5 Kรคhler-Einstein metrics with non-positive scalar curvature -- 5.1 The Calabi-Yau Theorem -- 5.2 Kรคhler-Einstein metrics for manifolds with c1(M) < 0 -- 6 Kรคhler-Einstein metrics with positive scalar curvature -- 6.1 A variational approach -- 6.2 Existence of Kรคhler-Einstein metrics -- 6.3 Examples -- 7 Applications and generalizations -- 7.1 A manifold without Kรคhler-Einstein metric -- 7.2 K-energy and metrics of constant scalar curvature -- 7.3 Relation to stability

1. Mathematics
2. Global analysis (Mathematics)
3. Manifolds (Mathematics)
4. Differential geometry
5. Physics
6. Mathematics
7. Differential Geometry
8. Global Analysis and Analysis on Manifolds
9. Mathematical Methods in Physics