Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics

Abstract

DMV Seminar, Band 8 Birkhäuser 1987 PREFACE These notes are based on the lectures I delivered at the German Mathe-matical Society Seminar in Schloss Michkeln in Düsseldorf in June, 1986 on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. The purpose of these notes is to present to the reader the state-of-the-art re-sults 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 prerequisites 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, refer-ences 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 presentations. For the problems of Hermitian-Einstein metrics for stable bundles and Kähler-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 priori estimates on, be it the continuity method or the heat equation method or even the method of minimizing sequences, is of rather minor importance when the required a priori estimates are available.