{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8398"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8398","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Ricci flow on 3-manifolds with symmetry","abstract":"In this thesis, we proved that for a 3-manifold S2 x S1 with warped product metric, the isoperimetric ratio on the base manifold S2 has a low positive bound away from zero, if the scalar curvature on the 3-manifold is positive. We also obtained a monotonicity result under the condition that the length of the optimal curve for isoperimetric ratio shrinks to zero under Ricci flow. This result excludes the product of a cigar soliton [Sigma]² with R¹ as the dilation limit of the Ricci flow equation. We also obtained an inequality of the curvature ratio Rmin/Rmax on the dilation limit for compact 3-manifold with positive scalar curvature.","abstract_html":"In this thesis, we proved that for a 3-manifold S2 x S1 with warped product metric, the isoperimetric ratio on the base manifold S2 has a low positive bound away from zero, if the scalar curvature on the 3-manifold is positive. We also obtained a monotonicity result under the condition that the length of the optimal curve for isoperimetric ratio shrinks to zero under Ricci flow. This result excludes the product of a cigar soliton [Sigma]² with R¹ as the dilation limit of the Ricci flow equation. We also obtained an inequality of the curvature ratio Rmin/Rmax on the dilation limit for compact 3-manifold with positive scalar curvature.","abstract_has_math":false,"creators":["Cao, Xiaodong, 1972-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Gang Tian."],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002","date_published":"2002","updated_at":"2026-07-22T22:22:10Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. 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