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Deformation of manifolds and continuity of eigenvalues Sun, Yucong
Abstract
In this thesis, we study the continuity of a family of operators Et acting on functions or sections. First we study two easier cases. 1. The operator is a matrix At; 2. The operator or a Laplacian −Δt which acts on a function. These cases can be solved using min-max techniques coming from linear algebra, which gives us the explicit expression of each eigenvalue. But generally, for elliptic operators Et which acts on a section of a vector bundle, min-max techniques does work. Compared to the Laplacian case, one can not use integration by parts to cancel the Laplacian. And therefore the explicit expression of each eigenvalue can not be obtained. We will introduce Kodaira-Spencer theory for general cases, which is much more powerful and complicated. We’ll introduce the continuity theorem and give detailed proofs for the main theorem.
Item Metadata
| Title |
Deformation of manifolds and continuity of eigenvalues
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| Creator | |
| Supervisor | |
| Publisher |
University of British Columbia
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| Date Issued |
2023
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| Description |
In this thesis, we study the continuity of a family of operators Et acting on functions or sections. First we study two easier cases. 1. The operator is a matrix At; 2. The operator or a Laplacian −Δt which acts on a function. These cases can be solved using min-max techniques coming from linear algebra, which gives us the explicit expression of each eigenvalue. But generally, for elliptic operators Et which acts on a section of a vector bundle, min-max techniques does work. Compared to the Laplacian case, one can not use integration by parts to cancel the Laplacian. And therefore the explicit expression of each eigenvalue can not be obtained. We will introduce Kodaira-Spencer theory for general cases, which is much more powerful and complicated. We’ll introduce the continuity theorem and give detailed proofs for the main theorem.
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| Genre | |
| Type | |
| Language |
eng
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| Date Available |
2023-04-21
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0431378
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
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| Graduation Date |
2023-05
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| Campus | |
| Scholarly Level |
Graduate
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International