New progress towards three open conjectures in geometric analysis
Author(s)Gallagher, Paul,Ph.D.Massachusetts Institute of Technology.
Massachusetts Institute of Technology. Department of Mathematics.
William P. Minicozzi.
MetadataShow full item record
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.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019Cataloged from PDF version of thesis.Includes bibliographical references (pages 68-70).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology