{"id":{"repo_id":"duke","oai_identifier":"oai:dukespace.lib.duke.edu:10161/192"},"canonical_url":"https://search.dev.ndltd.org/etd/duke/oai:dukespace.lib.duke.edu:10161/192","repository":{"repo_id":"duke","name":"Duke University","base_url":"https://dukespace.lib.duke.edu/server/oai/request"},"display":{"title":"Ricci Yang-Mills Flow","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Streets, Jeffrey D."],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Stern, Mark A.","Bray, Hubert L.","Bryant, Robert L.","Saper, Leslie D."],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007-05-04T17:37:34Z","date_published":"2007-05-04T17:37:34Z","updated_at":"2026-07-24T02:06:56Z","subjects":["riemannian manifold","Global differential geometry","Ricci flow","Yang-Mills theory"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/192","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Stern, Mark A.","Bray, Hubert L.","Bryant, Robert L.","Saper, Leslie D."]},{"key":"dc:creator","label":"Author","values":["Streets, Jeffrey D."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2007-05-04T17:37:34Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2007-05-04T17:37:34Z"]},{"key":"dc:date.issued","label":"Date","values":["2007-05-04T17:37:34Z"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["riemannian manifold","Global differential geometry","Ricci flow","Yang-Mills theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/192"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Ricci Yang-Mills Flow"]}]}],"canonical_facts":{"dc:contributor.advisor":["Stern, Mark A.","Bray, Hubert L.","Bryant, Robert L.","Saper, Leslie D."],"dc:creator":["Streets, Jeffrey D."],"dc:date.accessioned":["2007-05-04T17:37:34Z"],"dc:date.available":["2007-05-04T17:37:34Z"],"dc:date.issued":["2007-05-04T17:37:34Z"],"dc:description.abstract":["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."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10161/192"],"dc:language.iso":["en_US"],"dc:subject":["riemannian manifold","Global differential geometry","Ricci flow","Yang-Mills theory"],"dc:title":["Ricci Yang-Mills Flow"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:06:56Z"}