Ricci flow in Milnor frames

Thesis

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.

Authors: Johar, MS

• DOI: 10.5287/ora-g2a0m2qeg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: partial differential equations; geometric flows
• Subjects: Mathematics