Donnerstag, 26.11.2020: Ricci-flat metrics, parallel spinors on Riemannian and Lorentzian manifolds and the dominant energy condition
Prof. Dr. Bernd Ammann (Universität Regensburg)
(based on joint work with Klaus Kröncke, Hartmut Weiß, Frederik Witt, Olaf
Müller and Jonathan Glöckle)
It is an interesting and challenging problem to construct Ricci-flat metrics.
There are several famous constructions to obtain compact Riemannian Ricciflat manifolds besides flat manifolds: e.g. Yau’s solution of the Calabi conjecture, or Joyce’s construction of “manifolds with special holonomy”. All these
constructions share the property that a finite covering of these spaces carry
a parallel spinor. We will call such metrics structured Ricci-flat metrics, as
the parallel spinor yields a structural reason for the Ricci-flatness. However,
it remains mysterious whether compact non-structured Ricci-flat Riemannian
manifolds exist.
In the talk we explain how the existence of a parallel spinor implies Ricciflatness in the Riemannian case, and to what extend this statement remains
true in the Lorentzian setup. I will discuss that the pre-moduli space of structured Ricci-flat Riemannian metrics on a compact manifold is a smooth, finitedimensional manifold, which carries a natural bundle, given by parallel spinors,
with a natural connection. Curves in this moduli space yield solutions of the
constraint equation for the Cauchy problem for Lorentzian manifolds with parallel spinors.
In the last part of the talk we discuss Lorentzian manifolds (strictly) satisfying the dominant energy condition, or more exactly initial data pairs (g, K)
for such manifolds. The space of such pairs has a rich topology, which can be
detected by a Lorentzian analogue of Hitchin’s ?-invariant. A related rigidity
discussion will lead us again to the same Cauchy problem.
| Uhrzeit: |
15:30 |
| Ort: |
online - wenn Sie Zugang haben wollen, schicken Sie bitte eine Nachricht an Angelika Spörer-Schmidle |
| Gruppe: |
Oberseminar |
| Einladender: |
Cederbaum, Huisken, Kröncke |
