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New progress towards three open conjectures in geometric analysis

Author(s)
Gallagher, Paul,Ph.D.Massachusetts Institute of Technology.
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Massachusetts Institute of Technology. Department of Mathematics.
Advisor
William P. Minicozzi.
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MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582
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Abstract
This thesis, like all of Gaul, is divided into three parts. In Chapter One, I study minimal surfaces in R⁴ with quadratic area growth. I give the first partial result towards a conjecture of Meeks and Wolf on asymptotic behavior of such surfaces at infinity. In particular, I prove that under mild conditions, these surfaces must have unique tangent cones at infinity. In Chapter Two, I give new results towards a conjecture of Schoen on minimal hypersurfaces in R⁴. I prove that if a stable minimal hypersurface E with weight given by its Jacobi field has a stable minimal weighted subsurface, then E must be a hyperplane inside of R⁴. Finally, in Chapter Three, I do an in-depth analysis of the nodal set results of Logonov-Malinnikova. I give explicit bounds for the eigenvalue exponent in terms of dimension, and make a slight improvement on their methodology.
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
 
Cataloged from PDF version of thesis.
 
Includes bibliographical references (pages 68-70).
 
Date issued
2019
URI
https://hdl.handle.net/1721.1/122163
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Massachusetts Institute of Technology
Keywords
Mathematics.

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