Ricci Yang-Mills Flow
Stern, Mark A.
Bray, Hubert L.
Bryant, Robert L.
Saper, Leslie D.
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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.
CitationStreets, Jeffrey D. (2007). Ricci Yang-Mills Flow. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/192.
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