A Survey on Kahler-Einstein Metric

Covering:
  1. Proposed Ricci curvature problem: Calabi-Yau theorem
  2. Existence of KE metric on negative Chern class manifold: Aubin-Yau theorem
  3. Obstructions of existence of KE metric on Fano manifold: Lie algebra of holomorphic fields- Matsushima Theorem, Futaki invariant
  4. Functional I,J,F; existence of KE metric equivalent to properness of F(Thm 1.2 in Tian 1997 Invent. paper)
  5. Alpha-invariant, existnece of KE metric on surface(Tian 1990 theorem)
  6. Generalized Futaki invariant of Ding and Tian, K-energy of Bando and Mabuchi
  7. Ultimate result: existence of KE metric equivalent to K-stability(sketch of proof).
One may refer to the original papers for the results mentioned in the note.