Date of Award

Spring 5-23-2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Wylie, William

Keywords

ambient obstruction tensor, Bach flow, geometric flow, gradient soliton, homogeneous manifold

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

Differential geometry is a diverse field which applies principles from calculus to a more general set of objects. Endowing a smooth manifold with a Riemannian metric allows us to measure length and angle in a way such that length is positive. This enables us to examine measures of curvature on a manifold. The study of manifolds with such metrics is called Riemannian geometry. Using geometric flows associated with tensors, we are able to analyze the relationship between metrics and curvature. Examining solitons, specifically gradient solitons, is one way we investigate this relationship.

This thesis focuses on the geometric flows associated with the Bach tensor and the ambient obstruction tensor. The Bach tensor is realized as the gradient of the Weyl energy functional. Consequently, the minimizers of the Weyl energy are the metrics where the Bach tensor vanishes. There are a number of metrics that are widely considered interesting that are known to be Bach flat. Studying the Bach flow and broadening our understanding of Bach flat metrics could produce other such metrics. At the crux of our investigation is the fact that the Bach tensor is divergence-free (in dimension 4) and trace-free. To generalize this to higher dimensions and maintain these properties, we consider the ambient obstruction tensor, $\calO$. For $n=4$ the ambient obstruction tensor is the Bach tensor.

In this thesis we begin a new program of studying ambient obstruction solitons and homogeneous gradient Bach solitons. Examining higher dimensions, we establish a number of results for solitons to the geometric flow for a general tensor $q$ and apply these result to the ambient obstruction flow. This method enables us to prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. For $n=4$, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat, shrinking gradient solitons are product metrics on $\R^2\times S^2$ and $\R^2 \times H^2$. Moreover, we construct a non-Bach-flat expanding homogeneous gradient Bach soliton.

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