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Index bounds and existence results for minimal surfaces and harmonic maps Sargent, Pam
Abstract
In this work, we focus on three problems. First, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in ℝ³ tends to infinity as its genus or the number of boundary components tends to infinity. Second, we consider the relationship between the kth normalized eigenvalue of the Dirichlet-to-Neumann map (the kth Steklov eigenvalue) and the geometry of rotationally symmetric Möbius bands. More specifically, we look at the problem of finding a metric that maximizes the kth Steklov eigenvalue among all rotationally symmetric metrics on the Möbius band. We show that such a metric can always be found and that it is realized by the induced metric on a free boundary minimal Möbius band in B⁴. Third, we consider the existence problem for harmonic maps into CAT(1) spaces. If Σ is a compact Riemann surface, X is a compact locally CAT(1) space and φ : Σ → X is a continuous finite energy map, we use the technique of harmonic replacement to prove that either there exists a harmonic map u : Σ → X homotopic to φ or there exists a conformal harmonic map v : S² → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.
Item Metadata
Title |
Index bounds and existence results for minimal surfaces and harmonic maps
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2018
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Description |
In this work, we focus on three problems. First, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in ℝ³ tends to infinity as its genus or the number of boundary components tends to infinity.
Second, we consider the relationship between the kth normalized eigenvalue of the Dirichlet-to-Neumann map (the kth Steklov eigenvalue) and the geometry of rotationally symmetric Möbius bands. More specifically, we look at the problem of finding a metric that maximizes the kth Steklov eigenvalue among all rotationally symmetric metrics on the Möbius band. We show that such a metric can always be found and that it is realized by the induced metric on a free boundary minimal Möbius band in B⁴. Third, we consider the existence problem for harmonic maps into CAT(1) spaces. If Σ is a compact Riemann surface, X is a compact locally CAT(1) space and φ : Σ → X is a continuous finite energy map, we use the technique of harmonic replacement to prove that either there exists a harmonic map u : Σ → X homotopic to φ or there exists a conformal harmonic map v : S² → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.
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Genre | |
Type | |
Language |
eng
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Date Available |
2018-06-05
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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 |
10.14288/1.0368560
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2018-09
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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