New progress towards three open conjectures in geometric analysis

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.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2019

Author dc:creator

• Gallagher, Paul,Ph.D.Massachusetts Institute of Technology.

Advisor dc:contributor.advisor

• William P. Minicozzi.

• Mathematics.

Statement dc:rights

• 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.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/122163

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/122163