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Construction and degeneration of Einstein 4-manifolds SemesterWiSe 2022 / 23 Lecturer Bernd Ammann Type of course (Veranstaltungsart) Seminar German title Konstruktion und Degeneration von Einstein-4-Mannigfaltigkeiten Contents A fundamental question in 4-dimensional Riemannian geometry is the following: What is the boundary of the space of Einstein metrics on a fixed compact 4-manifold?
By work of Anderson and Bando-Kasue-Nakajima it has been known since the late 80s that non-collapsed limits of sequences of compact Einstein 4-manifolds are Einstein 4-orbifolds with isolated singularities and ALE-spaces bubbling off. Conversely, the famous Kummer construction of the K3 surface shows that the 16 singular points of the orbifold T4/ℤ2 may be desingularised by gluing in Eguchi-Hanson spaces. The Eguchi-Hanson lives on the crepant resolution of the singularity ℂ2/ℤ2 which is biholomorphic to T*ℂℙ1. Hence one can ask the more refined question: Which Einstein 4-orbifolds do actually occur as boundary points, i.e. can be desingularised to Einstein 4-manifolds?
Recent progress towards this question has been made by Biquard, who discovered a new obstruction in the curvature tensor in the orbifold point preventing desingularisation. If the obstruction vanishes, the desingularistation can be constructed at least in the ALH (or conformally compact) setting. A good overview is provided by Biquard's ICM talk.
The aim of this seminar is to carefully review the relevant aspects of the geometry of Einstein metrics in 4 dimensions with the ultimate aim to study Biquard's first paper in detail. This includes topics such as Kähler and Hyperähler geometry, the Kummer construction, the Hitchin-Thrope inequality, the orbifold compactness theorem and ALH (or conformally compact) Einstein metrics. This covers the first two introductory sections of the program. The final section is devoted to Biquard's work.
MV, MSem ECTS 4,5 ECTS |