We study the set of all Ricci-flat Riemannian metrics on a given co mpact manifold M.

We say that a Ricci-flat met ric on M is structured if its pullback to the univ ersal cover admits a parallel spinor. The holonomy of these metrics is special as these manifolds ca rry some additional structure\, e.g. a Calabi-Yau structure or a G

The set of unstructured Ricci-flat metrics is poorly understood. \; Nobody knows whether unstructur ed 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 ma nifold should have the structure of a smooth manif old.

On the other hand\, the set of structu red Ricci-flat metrics on compact manifolds is now well-understood.

The set of structured Ri cci-flat metrics is an open and closed subset in t he space of all Ricci-flat metrics.

The holono my group is constant along connected components. < br>The dimension of the space of parallel spinors as well.

The structured Ricci-flat metrics for m a smooth Banach submanifold in the space of all metrics.

Furthermore the associated premoduli space is a finite-dimensional smooth manifold.

These results build on previous work by J. Nor dströ\;m\, Goto\, Koiso\, Tian & Todorov\, Joy ce\, McKenzie Wang and many others.

The import ant step is to pass from irreducible to reducible holonomy groups.

In the last part of the ta lk we summarize work on the L

This is a w eakly parabolic flow on the space of metrics and s pinors of constant unit length. The flow is suppos ed to flow against structured Ricci-flat metics. I ts geometric interpretation in dimension 2 is some kind of Willmore flow\, and in dimension 3 it is a frame flow.

We find that the functional E is a Morse-Bott functional. This fact is related to s tability questions.

Associated publications :

http://www.mathematik.uni-regensburg.de/ ammann/preprints/holrig

http://w ww.mathematik.uni-regensburg.de/ammann/preprints/s pinorflowI

http://www.mathemati k.uni-regensburg.de/ammann/preprints/spinorflowII< /a> \;

LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR