Klaus Kröncke: The spectrum of the Einstein operator

Abstract.

A Riemannian metric g on a manifold M assigns to each point an inner product that depends smoothly on the point. The pair (M,g) is called a Riemannian manifold and admits natural notions of lengths, angles, volumes and curvature. A Riemannian metric is called Einstein if its Ricci curvature is proportional to the metric. Einstein metrics arise naturally as critical points of the Einstein-Hilbert action and are perhaps the most natural and interesting Riemannian metrics on a given manifold.

Linearization of the defining equation for Einstein metrics yields an elliptic operator, called the Einstein operator, which acts on perturbations of the metric. The spectrum of this operator determines the (linear) stability of the metric. Unfortunately, the spectrum of this operator is in general very hard to compute and only known for a few examples. In this talk, I will discuss some examples of Einstein manifolds, where this is actually possible. These examples are constructed from lower-dimensional Einstein manifolds and the spectrum can be computed in terms of the spectrum of the lower-dimensional manifold.