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Title: Ricci flow in Milnor frames
Author: Johar, M. Syafiq
ISNI:       0000 0004 7971 6801
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2019
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In this work, we are going to find sufficient conditions on the initial metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as the work of Isenberg, Knopf, and Sesum, in which they introduced some ansatz for the problem setup. In the latter study, a global frame for the tangent bundle called the Milnor frame was used to set up the problem. We begin with some discussions on the symmetries of the manifold and its ansatz metric derived from Lie groups as well as the global tangent bundle frame developed by Milnor. The curvature quantities and the Ricci flow equation of this ansatz metric will then be explicitly computed. Numerical simulations of the Ricci flow on these manifolds are done, which provides insight and conjectures for the main problems. Some analytic results will be proven for the manifolds S1xS3 and S4 using maximum principles from parabolic PDE theory and sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on manifolds homeomorphic to CPM2\B4 and B4 will be discussed, which will provide a possible direction for future work.
Supervisor: Dancer, Andrew Sponsor: Huscher Family Scholarship ; Malaysian Government
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics