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SUMMARY:Ricci-flat manifolds and a spinorial flow - Bernd Ammann (Universi
tät Regensburg)
DTSTART:20160627T090000Z
DTEND:20160627T100000Z
UID:TALK66594@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Joint work with Klaus Krö\;ncke\, 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 i
s structured if its pullback to the universal cover admits a parallel spin
or. The holonomy of these metrics is special as these manifolds carry some
additional structure\, e.g. a Calabi-Yau structure or a G2-str
ucture.
The set of unstructured Ricci-flat metrics is poorly unders
tood. \; Nobody knows whether unstructured compact Ricci-flat Riemanni
an 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 struc
ture of a smooth manifold.
On the other hand\, the set of structure
d Ricci-flat metrics on compact manifolds is now well-understood.
The set of structured Ricci-flat metrics is an open and closed subset in t
he space of all Ricci-flat metrics.
The holonomy group is constant alo
ng connected components.
The dimension of the space of parallel spinor
s as well.
The structured Ricci-flat metrics form a smooth Banach subm
anifold in the space of all metrics.
Furthermore the associated premod
uli space is a finite-dimensional smooth manifold.
These results bu
ild on previous work by J. Nordströ\;m\, Goto\, Koiso\, Tian & Todorov
\, Joyce\, McKenzie Wang and many others.
The important step is to pas
s from irreducible to reducible holonomy groups.
In the last part o
f the talk we summarize work on the L2-gradient flow of the fun
ctional $(g\,\\phi)\\mapsto E(g\,\\phi):=\\int_M|\\nabla^g\\phi|^2$.
T
his is a weakly parabolic flow on the space of metrics and spinors of cons
tant unit length. The flow is supposed to flow against structured Ricci-fl
at metics. Its geometric interpretation in dimension 2 is some kind of Wil
lmore flow\, and in dimension 3 it is a frame flow.
We find that the fu
nctional E is a Morse-Bott functional. This fact is related to stability q
uestions.
Associated publications:
http://www.mathematik.uni-regensburg.de/ammann/preprints/holrig
http://www.mathematik.uni-regensburg.d
e/ammann/preprints/spinorflowI
http://www.mathematik.uni-regensburg.de/ammann/preprints/spinorflowII \;
LOCATION:Seminar Room 1\, Newton Institute
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