2026-09-25T20:49:54Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/83982022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Gang Tian.
Cao, Xiaodong, 1972-
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2005-08-23T19:52:45Z
2005-08-23T19:52:45Z
2002
2002
http://hdl.handle.net/1721.1/8398
50594786
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.
Includes bibliographical references (p. 51-53).
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.
by Xiaodong Cao.
Ph.D.
53 p.
2968454 bytes
2968214 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
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. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Ricci flow on 3-manifolds with symmetry
Thesis
application/pdf
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="cb60b655-5307-4b4a-9678-77689dc06a32">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>eng</Language>
<Title>Ricci flow on 3-manifolds with symmetry</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2002</PublicationDate>
<Authors>
<Author>
<DisplayName>Cao, Xiaodong, 1972-</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://dspace.mit.edu/handle/1721.1/7582</License>
<Keyword>Mathematics.</Keyword>
<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>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>