New progress towards three open conjectures in geometric analysis
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2019
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| Online Access: | https://hdl.handle.net/1721.1/122163 |
| _version_ | 1811069375542722560 |
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| author | Gallagher, Paul,Ph.D.Massachusetts Institute of Technology. |
| author2 | William P. Minicozzi. |
| author_facet | William P. Minicozzi. Gallagher, Paul,Ph.D.Massachusetts Institute of Technology. |
| author_sort | Gallagher, Paul,Ph.D.Massachusetts Institute of Technology. |
| collection | MIT |
| description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019 |
| first_indexed | 2024-09-23T08:10:01Z |
| format | Thesis |
| id | mit-1721.1/122163 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T08:10:01Z |
| publishDate | 2019 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1221632019-11-22T03:47:52Z New progress towards three open conjectures in geometric analysis Gallagher, Paul,Ph.D.Massachusetts Institute of Technology. William P. Minicozzi. Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics Mathematics. Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019 Cataloged from PDF version of thesis. Includes bibliographical references (pages 68-70). 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. by Paul Gallagher. Ph. D. Ph.D. Massachusetts Institute of Technology, Department of Mathematics 2019-09-16T22:33:41Z 2019-09-16T22:33:41Z 2019 2019 Thesis https://hdl.handle.net/1721.1/122163 1117775036 eng 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 70 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Gallagher, Paul,Ph.D.Massachusetts Institute of Technology. New progress towards three open conjectures in geometric analysis |
| title | New progress towards three open conjectures in geometric analysis |
| title_full | New progress towards three open conjectures in geometric analysis |
| title_fullStr | New progress towards three open conjectures in geometric analysis |
| title_full_unstemmed | New progress towards three open conjectures in geometric analysis |
| title_short | New progress towards three open conjectures in geometric analysis |
| title_sort | new progress towards three open conjectures in geometric analysis |
| topic | Mathematics. |
| url | https://hdl.handle.net/1721.1/122163 |
| work_keys_str_mv | AT gallagherpaulphdmassachusettsinstituteoftechnology newprogresstowardsthreeopenconjecturesingeometricanalysis |