Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg
Construction and degeneration of Einstein 4-manifolds
Semester
WiSe 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.

Literature
see program on homepage

Recommended previous knowledge
advanced knowledge in differential geometry

Time/Date
Blockseminar, Oct 3-8, 2022

Location
Sulzbürg, see webpage

Course homepage
https://ammann.app.uni-regensburg.de/conferences/2022Blockseminar/
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

  • by email
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 60 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Modules
MV, MSem

ECTS
4,5 ECTS