Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics Delivered at the German Mathematical Society Seminar in Düsseldorf in June, 1986

These notes are based on the lectures I delivered at the German Mathematical Society Seminar in Schloss Michkeln in DUsseldorf in June. 1986 on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics. The purpose of these notes is to present to the reader the state-of-the-art results in the simplest and the most comprehensible form using (at least from my own subjective viewpoint) the most natural approach. The presentation in these notes is reasonably self-contained and prerequisi tes are kept to a minimum. Most steps in the estimates are reduced as much as possible to the most basic procedures such as integration by parts and the maximum principle. When less basic procedures are used such as the Sobolev and Calderon-Zygmund inequalities and the interior Schauder estimates. references are given for the reader to look them up. A considerable amount of heuristic and intuitive discussions are included to explain why certain steps are used or certain notions introduced. The inclusion of such discussions makes the style of the presentation at some places more conversational than what is usually expected of rigorous mathemtical prese"ntations. For the problems of Hermi tian-Einstein metrics for stable bundles and Kahler-Einstein metrics one can use either the continuity method or the heat equation method. These two methods are so very intimately related that in many cases the relationship betwen them borders on equivalence. What counts most is the a. priori estimates. The kind of scaffolding one hangs the a.

1. The heat equation approach to Hermitian-Einstein metrics on stable bundles.- §1. Definition of Hermitian-Einstein metrics.- §2. Gradient flow and the evolution equation.- §3. Existence of solution of evolution equation for finite time.- §4. Secondary characteristics.- §5. Donaldson’s functional.- §6. The convergence of the solution at infinite time.- Appendix A. Hermitian-Einstein metrics of stable bundles over curves.- Appendix B. Restriction of stable bundles.- 2. Kähler-Einstein metrics for the case of negative and zero anticanonical class.- §1. Monge-Ampère equation and uniqueness.- §2. Zeroth order estimates.- §3. Second order estimates.- §4. Hölder estimates for second derivatives.- §5. Derivation of Harnack inequality by Moser’s iteration technique.- §6. Historical note.- 3. Uniqueness of Kähler-Einstein metrics up to biholomorphisms.- §1. The role of holomorphic vector fields.- §2. Proof of Uniqueness.- §3. Computation of the Differential..- §4. Computation of the Hessian.- Appendix A. Lower bounds of the Green’s function of Laplacian.- 4. Obstructions to the Existence of Kähler-Einstein Metrics.- §1. Reductivity of automorphism group.- §2. The obstruction of Kazdan-Warner.- §3. The Futaki invariant.- 5. Manifolds with suitable finite symmetry.- §1. Motivation for the use of finite symmetry.- §2. Relation between supM? and infM?.- §3. Estimation of m+??.- §4. The use of finite group of symmetry.- §5. Applications.- References.