Ricci-flat manifolds and a spinorial flow
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| Description: |
Ammann, B (Universität Regensburg)
Monday 27th June 2016 - 10:00 to 11:00 |
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| Created: | 2016-07-06 10:25 |
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| Collection: | Gravity, Twistors and Amplitudes |
| Publisher: | Isaac Newton Institute |
| Copyright: | Ammann, B |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | 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,ϕ)↦E(g,ϕ):=∫ M |∇ g ϕ| 2 (g,ϕ)↦E(g,ϕ):=∫M|∇gϕ|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. Associated publications: http://www.mathematik.uni-regensburg.de/ammann/preprints/holrig http://www.mathematik.uni-regensburg.de/ammann/preprints/spinorflowI http://www.mathematik.uni-regensburg.de/ammann/preprints/spinorflowII |
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