Index estimates and compactness for constant mean curvature surfaces and Steklov eigenvalues
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Hong, Han
issued: 2021-06-02T13:32:09Z
issued: 2021
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/78506
note: Index estimates and compactness for constant meancurvature surfaces and Steklov eigenvaluesbyHan HongA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)May 2021© Han Hong, 2021The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Index estimates and compactness for constant mean curvature surfacesand Steklov eigenvaluessubmitted by Han Hong in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mathematics.Examining Committee:Jingyi Chen, Professor, Mathematics, UBCSupervisorAilana Fraser, Professor, Mathematics, UBCCo-supervisorFei Zhou, Professor, Physics & Astronomy, UBCUniversity ExaminerRichard Froese, Professor, Mathematics, UBCUniversity ExaminerMartin Man-chun Li, Associate Professor, Mathematics, CUHKExternal ExaminerAdditional Supervisory Committee Members:Albert Chau, Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractIn this work, we focus on three problems. First, we give a relationship betweenthe number of eigenvalues of the Jacobi operator below a certain threshold and thetopology of closed constant mean curvature (CMC) surfaces in three-dimensionalRiemannian manifolds. We then obtain that the (weak) Morse index of CMC sur-faces in an arbitrary 3-manifold is bounded below by a linear function of the genuswhen the constant mean curvature is greater than a certain nonnegative value. Inparticular, this implies that stable CMC surfaces are topological spheres. Corre-sponding results for CMC surfaces with free boundary in 3-manifolds with bound-ary are obtained as well. Second, we consider the space of embedded free bound-ary CMC surfaces with bounded topology, bounded area, and bounded boundarylength in a 3-manifold N with boundary. We show that this space is…
has type: Thesis/Dissertation
date issued: 2021-11
is shown at: 10.14288/1.0398225
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Chen, Jingyi
contributor: Fraser, Ailana
type: Text
identifier URI: http://hdl.handle.net/2429/78506