Research

I research differential geometry, more specifically Riemannian geometry. My early work began the program of studying ambient obstruction solitons, which applies the study of geometric flows to tensors with conformal properties. In the past year, my work has expanded to include both the representations of Lie algebras and quantitative geometry.

Differential geometry is a diverse field that is based on the idea of applying principles from calculus to a more general set of objects. Broadly stated, research in differential geometry aims to answer the question ``what is the best metric?'' with the hope that the answer to this question will provide valuable insights to the universe we inhabit. The word “best” takes on different meanings in different contexts and thus we use different approaches. Riemannian geometry focuses on answering this question for Riemannian manifolds (M, g). A Riemannian manifold is a smooth manifold, M, paired with a Riemannian metric, g. Riemannian metrics are inner products defined on the tangent space of M that vary smoothly across M. Like the dot product we learn in calculus, Riemannian metrics are positive definite inner products defined on the tangent space of M at every point. Returning to the question of finding the best metric, limiting our scope to Riemannian metrics allows us to use curvature as a tool to help define what ``best'' might mean. (There are many ways to measure a manifold's curvature, but I will use ``curvature'' to mean these measures in general.)

The study of geometric flows 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. We are able to use tools from differential equations to analyze the relationship between metrics and curvature. One such tool 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. Heuristically: if the flow is supposed to make metrics ``nicer'' over time, solitons isolate metrics that are not improving as they are pushed through the flow.

Summarizing my main results:

I classified steady and shrinking homogeneous gradient Bach solitons. I also show that the study of these solitons is distinct from that of Ricci solitons by showing the existence of an expanding homogeneous gradient Bach soliton.
I proved that compact homogeneous ambient obstruction solitons are ambient obstruction flat. My approach using general q-solitons establishes a more refined system of results for general q.
Established that, with additional regularity or curvature conditions, non-compact $q$-solitons are also $q$-flat. Applied these results to Cotton solitons, homogeneous ambient obstruction solitons, and Bach solitons with harmonic Weyl curvature. (Joint with A.W. Cunha)
Proved that any closed ambient obstruction soliton is ambient obstruction flat. We defined extended ambient obstruction solitons to extend result to nonhomogeneous manifolds. Notably, solutions to the geometric flow that stay within their conformal class are extended solitons. (Joint work with R. Poddar, R. Sharma, W. Wylie)
We are establishing the existence of Einstein metrics on homogeneous manifolds with contractible nerve complex and 3 or 4 irreducible isotropy summands. (Joint with I. Beach, H. Contreras Peruyero, M. Kerr, and C. Searle)
We establish natural geometric conditions that yield length estimates for the shortest possible orthogonal geodesic chords. (Joint with I. Beach, H. Contreras-Peruyero, R. Rottmann, and C. Searle)

Papers

Extended Solitons of the Ambient Obstruction Flow with R. Poddar, R. Sharma, and W. Wylie (May 2024)

Abstract. In this paper we expand on the work of the first author on ambient obstruction solitons, which are self-similar solutions to the ambient obstruction flow. Our main result is to show that any closed ambient obstruction soliton is ambient obstruction flat and has constant scalar curvature. We show, in fact, that the first part of this result is true for a more general extended soliton equation where we allow an arbitrary conformal factor to be added to the equation. We discuss how this implies that, on a compact manifold, the ambient obstruction flow has no fixed points up to conformal diffeomorphism other than ambient obstruction flat metrics. These results are the consequence of a general integral inequality that can be applied to the solitons to any geometric flow. Additionally, we use these results to obtain a generalization of the Bourguignon-Ezin identity on a closed Riemannian manifold and study the converse problem on a closed extended q-soliton. We also study this extended equation further in the case of homogeneous and product metrics.
Citation. Griffin, E.; Poddar, R.; Ramesh, R.: Wylie, W. Extended solitons of the ambient obstruction flow. Journal of Geometric Analysis. Submitted.
Link. https://arxiv.org/abs/2405.15870

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

Gradient Ambient Obstruction Solitons on Homogeneous Manifolds (August 2020)

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

Dissertation: Gradient Ambient Obstruction Solitons on Homogeneous Manifolds (August 2020)

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/

In Preparation

Lengths of the Orthogonal Geodesic Chords on Riemannian Manifolds (Anticipated Fall 2024)

Joint with: Isabel Beach, Haydee Contreras Peruyero, Reigina Rotman, Catherine Searle

Einstein Metrics on Homogeneous Manifolds (Anticipated Winter 2025)

Joint with: Isabel Beach, Haydee Contreras Peruyero, Megan Kerr, Catherine Searle

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griffine@northwestern.edu