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

• Author / Creator

  Wilkes, Jason

• 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.

• Subjects / Keywords

  • Riemannian Geometry
  • Ricci flow
  • Numerical Simulation

• Graduation date

  Fall 2011

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R3T01P

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.