Let (Mn, g) be a Riemannian manifold. Say K ! E ! M is a principal K-bundle
with connection A. We define a natural evolution equation for the pair (g,A) combining
the Ricci flow for g and the Yang-Mills flow for A which we dub Ricci Yang-Mills
flow. We show that these equations are, up to di eomorphism equivalence, the gradient
flow equations for a Riemannian functional on M. Associated to this energy
functional is an entropy functional which is monotonically increasing in areas close
to a developing singularity. This entropy functional is used to prove a non-collapsing
theorem for certain solutions to Ricci Yang-Mills flow.
We show that these equations, after an appropriate change of gauge, are equivalent
to a strictly parabolic system, and hence prove general unique short-time existence
of solutions. Furthermore we prove derivative estimates of Bernstein-Shi type.
These can be used to find a complete obstruction to long-time existence, as well as
to prove a compactness theorem for Ricci Yang Mills flow solutions.
Our main result is a fairly general long-time existence and convergence theorem
for volume-normalized solutions to Ricci Yang-Mills flow. The limiting pair (g,A)
satisfies equations coupling the Einstein and Yang-Mills conditions on g and A respectively.
Roughly these conditions are that the associated curvature FA must be
large, and satisfy a certain “stability” condition determined by a quadratic action of
FA on symmetric two-tensors.
| Identifer | oai:union.ndltd.org:DUKE/oai:dukespace.lib.duke.edu:10161/192 |
| Date | 04 May 2007 |
| Creators | Streets, Jeffrey D. |
| Contributors | Stern, Mark A., Bray, Hubert L., Bryant, Robert L., Saper, Leslie D. |
| Source Sets | Duke University |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
| Format | 511873 bytes, application/pdf |
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