Understanding the physical signatures of an M-Theory compactification on the product of four-dimensional Minkowski space and a seven-dimensional internal space requires a detailed mathematical understanding of possible geometries of these extra-dimensional models. Provided the effective four-dimensional theory should exhibit minimal supersymmetry, which is desirable to potentially resolve a variety of questions surrounding the standard model of particle physics, the internal space in such a setup should be a $\GG_2$-holonomy manifold. Such spaces also garner considerable interest in pure mathematics given their distinguished curvature properties and connections to spin geometry. The fundamental question for both mathematics and physics concerns which compact seven-manifolds (or singular generalizations thereof) admit these $\GG_2$-holonomy metrics. Although this question is foundational toward efforts to extract predictions from such compactifications, it requires the use of rigorous mathematics, in particular aspects related to differential geometry and geometric analysis. As such, the study of these $\GG_2$-holonomy manifolds provides an opportunity for physics and mathematics to grow by building off of each other, much as the two subjects have via their shared interest in Calabi-Yau threefolds over the past several decades, and more generally as they have throughout their entire history.
This thesis is squarely focused on understanding techniques which have potential to construct $\GG_2$-holonomy manifolds from a weaker class of structures called $\GG_2$-structures with torsion. The techniques under consideration fall under the class of geometric flows, which themselves constitute a fundamental tool in the arsenal of geometric analysis. The intuition for these geometric flows models itself largely off of the heat equation, but the objects evolving under these equations are usually tensors on differentiable manifolds and the flows are nonlinear. In this context, the goal is to flow certain types of $\GG_2$-structures with torsion to data underlying a $\GG_2$-holonomy metric. Two promising flows of $\GG_2$-structures are the Laplacian flow and the Laplacian coflow, both of which are investigated in this thesis.
After introducing the relevant physical and mathematical background in chapters 1 and 2, we begin studying the initial data for the Laplacian flow of $\GG_2$-structures. The initial data that is most relevant for this flow, and which is utilized in all known constructions of $\GG_2$-holonomy metrics on compact manifolds, is the class of closed $\GG_2$-structures. Much like the existence question for $\GG_2$-holonomy metrics themselves, the existence of closed $\GG_2$-structures on compact manifolds is not well-understood. Chapters 3 and 4 of this thesis aim to improve understanding of the closed $\GG_2$-structure condition in general while also characterizing additional conditions one may put on closed $\GG_2$-structures that affect their behavior under the Laplacian flow. Broadly-speaking, one can put conditions on a closed $\GG_2$-structure at the level of the structure itself, on its torsion, or on the curvatures of the associated metric.
Chapter 3 considers the most basic constraint one can put on a closed $\GG_2$-structure itself, namely that the differential three-form defining such a structure be exact. No known examples of exact $\GG_2$-structures exist on compact manifolds, so this question is closely-tied with the strength of the closed $\GG_2$-structure condition in the compact setting. Resolving this question has considerable implications for constructing $\GG_2$-holonomy metrics via general techniques, as well as for identifying promising initial data for the Laplacian flow. Exact $\GG_2$-structures also have considerable relevance for the Laplacian flow itself because certain types of these structures model self-similar solutions to this flow. To this end, we provide characterizations of exact $\GG_2$-structures on compact manifolds at the level of their defining two-form and in terms of their curvatures. We also provide specialized results for certain classes of self-similar solutions to the Laplacian flow.
Chapter 4 considers constraints that one can put on closed $\GG_2$-structures at the level of the curvatures of their associated metrics. In particular, this section studies the subclass of extremally Ricci-pinched $\GG_2$-structures, which in a sense constitute a threshold between closed $\GG_2$-structures and those underlying $\GG_2$-holonomy metrics. We provide a characterization of these structures within the broader class of closed $\GG_2$-structures obeying special properties, study natural perturbations of these structures, and then use the data of an extremally Ricci-pinched $\GG_2$-structure on a compact manifold to construct a family of closed $\GG_2$-structures. We then consider the behavior of the Laplacian flow starting at an arbitrary member of this family.
Chapter 5 considers the Laplacian coflow, which aims to perturb co-closed $\GG_2$-structures to those underlying $\GG_2$-holonomy metrics. Presently, the Laplacian coflow is not understood analytically as well as the Laplacian flow, and it is worth asking if the flow merits the kind of attention needed to develop it into a useful tool. We motivate developing the analytic theory of the Laplacian flow in greater detail by considering it in the context of what are called nearly-parallel $\GG_2$-structures, which themselves also have very special differential geometric properties. We reduce the study of the Laplacian flow, coflow, and Ricci flow in the context of co-closed $\GG_2$-structures originating from 3-Sasakian geometry to problems in dynamical systems theory. We then show that the coflow takes any initial data to the nearly-parallel $\GG_2$-structure within its ansatz. This is suggestive that the Laplacian coflow may behave well geometrically, and perhaps analytically, in certain contexts and is worth further developing as a tool for studying these nearly-parallel and $\GG_2$-holonomy geometries.