Date of Award
3-2023
Document Type
Doctoral Thesis
Degree Name
Doctor of Philosophy
Department
School of Technology, Engineering and Mathematics
First Advisor
Professor Brendan Guilfoyle
Abstract
This work considers two related topics: Weingarten surfaces and classical curvature flows. The former being the stationary solutions of the latter. Firstly, the Weingarten relations satisfied by rotationally symmetric Weingarten surfaces in Euclidean 3-space E3 are considered from three perspectives: Restrictions on the slope of the relation at umbilic points, the action of SL2(R) as fractional-linear transformations on curvature space, and variational formulations for the relations. Concerning the first, bounds are obtained on the slope of a relation in terms of the fall-off of the astigmatism at an umbilic point. For the second, the action of SL2(R) on curvature space is shown to integrate to an action on surfaces in E3 and splits into three natural geometric actions. This is applied to semi-quadratic Weingarten surfaces on which the action is shown to be transitive and a classification result is given. Finally, a Lagrangian formulation is given for certain types of relations and stability established. Secondly, after restricting to rotationally symmetric surfaces, a curvature flow which is linear in the radii of curvature is considered. It’s known that the flow can be solved explicitly if the coefficients of the flow form certain integer ratios. In this work certain values of non-integer ratios are considered. Long time existence and convergence is established. If the focal points at the north and south poles on the initial surface coincide, the flow is shown to converge for large time to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the falloff of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow. The proof uses the spectral theory of singular SturmLiouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.
Recommended Citation
Robson, Morgan, "Weingarten Surfaces: Curvature Flows and Stationary Properties" (2023). Theses [online].
Available at: https://sword.cit.ie/allthe/828
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Access Level
info:eu-repo/semantics/openAccess
Project Identifier
info:eu-repo/grantAgreement/MTU///IE//
