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Ricci-flat manifolds and a spinorial flow

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GTAW01 - General Relativity: from Geometry to Amplitudes

Joint work with Klaus Kröncke, Hartmut Weiß and Frederik Witt

We study the set of all Ricci-flat Riemannian metrics on a given compact manifold M.
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<sub>2</sub>-structure.

The set of unstructured Ricci-flat metrics is poorly understood.  Nobody knows whether unstructured 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, the set of structured Ricci-flat metrics on compact manifolds is now well-understood.

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.

These results build on previous work by J. Nordström, Goto, Koiso, Tian & Todorov, Joyce, McKenzie Wang and many others.
The important step is to pass from irreducible to reducible holonomy groups.

In the last part of the talk we summarize work on the L<sup>2</sup>-gradient flow of the functional $(g,\phi)\mapsto E(g,\phi):=\int_M|ablag\phi|2$.
This is a weakly parabolic flow on the space of metrics and spinors of constant unit length. The flow is supposed to flow against structured Ricci-flat metics. Its 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 stability questions.

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