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AuthorSiu, Yum-Tong. author
TitleLectures on Hermitian-Einstein Metrics for Stable Bundles and Kรคhler-Einstein Metrics [electronic resource] : Delivered at the German Mathematical Society Seminar in Dรผsseldorf in June, 1986 / by Yum-Tong Siu
ImprintBasel : Birkhรคuser Basel, 1987
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Descript 172 p. online resource


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 -- ยง1. Definition of Hermitian-Einstein metrics -- ยง2. Gradient flow and the evolution equation -- ยง3. Existence of solution of evolution equation for finite time -- ยง4. Secondary characteristics -- ยง5. Donaldsonโ{128}{153}s functional -- ยง6. 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. Kรคhler-Einstein metrics for the case of negative and zero anticanonical class -- ยง1. Monge-Ampรจre equation and uniqueness -- ยง2. Zeroth order estimates -- ยง3. Second order estimates -- ยง4. Hรถlder estimates for second derivatives -- ยง5. Derivation of Harnack inequality by Moserโ{128}{153}s iteration technique -- ยง6. Historical note -- 3. Uniqueness of Kรคhler-Einstein metrics up to biholomorphisms -- ยง1. The role of holomorphic vector fields -- ยง2. Proof of Uniqueness -- ยง3. Computation of the Differential. -- ยง4. Computation of the Hessian -- Appendix A. Lower bounds of the Greenโ{128}{153}s function of Laplacian -- 4. Obstructions to the Existence of Kรคhler-Einstein Metrics -- ยง1. Reductivity of automorphism group -- ยง2. The obstruction of Kazdan-Warner -- ยง3. The Futaki invariant -- 5. Manifolds with suitable finite symmetry -- ยง1. Motivation for the use of finite symmetry -- ยง2. Relation between supM? and infM? -- ยง3. Estimation of m+?? -- ยง4. The use of finite group of symmetry -- ยง5. Applications -- References

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