Research

My research is in differential geometry, specifically Riemannian geometry. I focus on examining geometric flows, their solitons, and the relation of both to conformal geometry. Recently, my work has focused on the ambient obstruction flow and the Bach flow.

The study of geometric flow evolved as a way to use curvature to identify best metrics. A geometric flow is a differential equation in which the metric is considered as a function of time, g(t), and is changed over time in accordance with the curvature of the manifold. Looking at the geometric flow associated with a tensor, we are able to use tools from differential equations to analyze the relationship between metrics and curvature. Researchers use this shift in perspective to examine the behavior of the flow itself, to better understand how the curvature behaves, and, consequently, to refine the idea of what "best" might mean for a specific measure of curvature. 

One of the major ideas from differential equations is locating and classifying fixed points. In the study of geometric flows, this manifests as the examination and classification of solitons. Solitons are solutions to the flow that, over time, change only by diffeomorphism and/or rescaling. Studying the solitons of a geometric flow provides insight into the nature of the flow while narrowing down the number of metrics that one is considering. 

One can limit the number of solitons they're looking at by choosing a particular vector field. The gradient of a function is a natural choice of vector field because it provides a more familiar setting, which is helpful when beginning to investigate solitons. 

A first step in understanding a collection objects is understanding those with the most symmetry. In the case of Riemannian manifolds, these objects are homogeneous manifolds. A Riemannian manifold is said to be homogeneous if for each pair of points p, q on the manifold there is an isometry (a map preserving angle and distance) mapping p to q. Generally speaking, this means that the manifold looks the same at every point. Some familiar examples of homogeneous manifolds are spheres and R^n. 

Papers

On non-compact gradient solitons with A.W. Cunha (July 2021)

Abstract. In this paper we extend existing results for generalized solitons, called q-solitons, to the complete case by considering non-compact solitons. By placing regularity conditions on the vector field X and curvature conditions on M, we are able to use the chosen properties of the tensor q to see that such non-compact q-solitons are stationary and q-flat.We conclude by applying our results to the examples of ambient obstruction solitons, Cotton solitons, and Bach solitons to demonstrate the utility of these general theorems for various flows.
Citation. Cunha, A.W.; Griffin, E.. On non-compact gradient solitons. Annals of Global Analysis and Geometry 63, 27 (2023). https://doi.org/10.1007/s10455-023-09904-1
Link.  https://link.springer.com/article/10.1007/s10455-023-09904-1
Abstract. We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 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. We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor q. 
Citation. Griffin, E. Gradient ambient obstruction solitons on homogeneous manifolds. Ann Glob Anal Geom (2021). https://doi.org/10.1007/s10455-021-09784-3
Link.  https://link.springer.com/article/10.1007/s10455-021-09784-3 
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.
Citation. Griffin, E. Ambient Obstruction Solitons And Homogeneous Gradient Bach Solitons (2021). Dissertations - ALL. 1309. https://surface.syr.edu/etd/1309
Link. https://surface.syr.edu/etd/1309/ 

Citations