Bernd Ammann: The moduli space of Ricci-flat manifolds

We study the set of all Ricci-flat Riemannian metrics on a given compact manifold. These metrics are the Riemannian analogues to the solutions of the vaccuum Einstein equations, and therefore one would like to understand their structure. It is helpful to distinguish between ``good'' and ``bad'' Ricci-flat metrics: Good Ricci-flat Riemannian manifolds admit a parallel spinor on the universal covering, bad ones do not.

The set of bad Ricci-flat metrics is poorly understood. Nobody knows whether bad compact Ricci-flat Riemannian manifolds exist, and if they exist, there is no reason to expect that the set of such metrics on a fixed compact manifold should have the structure of a smooth manifold.

On the other hand, there was a lot of progress by many mathematicians on the set of good Ricci-flat metrics. It is an open and closed subset in the space of all Ricci-flat metrics. The holonomy group is constant along connected components. The dimension of the space of parallel spinors as well. The good Ricci-flat metrics form a smooth Banach submanifold in the space of all metrics. Furthermore the associated premoduli space is a finite-dimensional smooth manifold.

A recent joint preprint with Kröncke, Weiss and Witt adds a small final step to obtain the above statements by analyzing products of good Ricci-flat metrics and their deformations.