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Ricci Yang-Mills Flow

by Streets, Jeffrey D.

Abstract (Summary)
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.

Bibliographical Information:

Advisor:Stern, Mark A.; Bray, Hubert L.; Bryant, Robert L.; Saper, Leslie D.

School:Duke University

School Location:USA - North Carolina

Source Type:Master's Thesis

Keywords:riemannian manifold global differential geometry ricci flow yang mills theory

ISBN:

Date of Publication:05/04/2007

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