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Permanent link (DOI): https://doi.org/10.7939/R3T01P

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Numerical simulation of Ricci flow on a class of manifolds with non-essential minimal surfaces Open Access

Descriptions

Other title
Subject/Keyword
Ricci flow
Numerical Simulation
Riemannian Geometry
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Wilkes, Jason
Supervisor and department
Woolgar, Eric (Mathematical and Statistical Sciences)
Examining committee member and department
Belhamadia, Youssef (Mathematical and Statistical Sciences)
van Roessel, Henry (Mathematical and Statistical Sciences)
Moore, Roger (Physics)
Department
Department of Mathematical and Statistical Sciences
Specialization

Date accepted
2011-10-03T20:22:03Z
Graduation date
2011-11
Degree
Master of Science
Degree level
Master's
Abstract
In the last three decades, the Ricci flow has proved to be an extremely useful tool in mathematics and physics. We explore numerically the long time existence of the Ricci-DeTurck flow and the List flow for a one-parameter family of Riemannian manifolds with non-essential minimal surfaces. This class of metrics is constructed to be an intermediate case between the corseted spheres examined by Garfinkle and Isenberg, and the RP3 geon explored by Balehowsky and Woolgar. We find that the Ricci-DeTurck flow of these manifolds depends on the value of a geometric parameter, with immortal flow below a critical parameter value, and singularity formation above it. We also examine the List flow of this family of manifolds with and without a stable minimal surface, we compare the long time existence properties to those observed in the case of the Ricci flow, and we use these results to gain insights into both the results obtained by Gulcev, Oliynyk, and Woolgar, and the general phenomena of singularity formation and critical behavior in Ricci flow.
Language
English
DOI
doi:10.7939/R3T01P
Rights
License granted by Jason Wilkes (jtwilkes@ualberta.ca) on 2011-10-02 (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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