The main part of the talk is a collaboration with Klaus Kröncke, Hartmut Weiss, and Frederik

Witt, with ongoing further work Olaf Müller, Mattias Dahl, Thomas Leistner, and Andree Lischewski.

We say that a Ricci-flat metric on $M$ is structured if its pullback to the universal cover admits a parallel spinor.

The holonomy of these metrics is special as these manifolds carry some additional structure, e.g.

a Calabi-Yau structure or a $G_2$-structure. All known compact Ricci-flat manifolds are structured.

The set of structured Ricci-flat metrics on compact manifolds is now well-understood, and we will

explain this in the first part of the talk.

The set of structured Ricci-flat metrics 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 structured 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, and the parallel spinors form a natural bundle with metric an connection over this premoduli space.

The contribution of the collaboration with Kröncke, Weiss and Witt was to extend previous results

for simply connected manifold with irreducible holonomy to arbitrary manifolds.

The results were recently used by D. Wraith to detect topological information about the

space of metrics with non-negative scalar curvature.

If time admits, we will sketch how this knowledge prvovides initial data sets

for the Cauchy problem for Lorentzian manifolds with parallel spinors.